Science and the Horizon of Reason on The Horizon of Reason

The Bidirectional Relationship Between Magic and Science

Peter Prevos — Thu, 12 Apr 2018 10:00:00 +0000

Scientists often describe the relationship between magic and science as opposed to each other. In this view, magicians break the laws of physics and change the world in ways ordinary people cannot. Our mind is conditioned to view the world as a chain of cause and effect. Reading somebody's mind, mutilating pretty girls without harming them or causing coins to disappear without a chain of causality between the actions and the results is impossible in our everyday experience. This view diametrically opposes magic and science as two incompatible human endeavours.

The relationship between magic and science is much more complicated than this simple view, as Arthur C. Clarke famously expressed in the last of his three laws:

"Any sufficiently advanced technology is indistinguishable from magic."

Arthur C. Clarke, Profiles of the Future: An Inquiry Into the Limits of the Possible.

The closest we can come in our everyday life to supernatural magic is the theatrical magic or conjuring. Magicians are actors who play the role of a supernatural magician. This form of art has a very close and bidirectional relationship with the various sciences. Magicians use science to create the illusion of magic, and scientists study magicians and their audiences. This essay discusses the bidirectional relationship between magic and science.

The bi-directional relationship between magic and science

Magicians present theatrical illusions that seemingly breach the laws of the physical sciences. Still, they often deploy the principles of these and other sciences to create these illusions. Scientists are interested in magic because they seek to understand this unique performance art better.

The relationship between science and magic is thus bi-directional. Magicians use science to create the illusion of supernatural magic, while scientists study magicians and their craft to learn more about the world around us.

The diagram below visualises the relationship between magic and science. The outer circle show the sciences that are involved with magic. Performers use some of these sciences as methods to create the illusion of magic—scientists in most of these fields research magicians and their performances.

This model divides the science in formal, physical, social and applied sciences. The formal sciences, such as mathematics, help us to understand the world, but they are not empirical. The physical sciences, such as physics and chemistry, mathematically describe the material world. The social sciences study human beings in all its facets. The social sciences study human individuals through psychology, how they work together in a group through sociology. Lastly, the social sciences study the artefacts of human culture, one of which is theatrical magic.

The bi-directional relationship between magic and science.

Each of these sciences has a different relationship with theatrical magic. Some sciences are used by magicians to perform their tricks, while other sciences are used by scholars as a perspective of magic. Some sciences perform both roles.

Imagine you are in a small magic theatre, the conjurer introduces two cups and two little balls. Using a magic wand, he commands the balls to dematerialise from his bare hands and materialise under the cups. The balls seem possessed by supernatural powers as they disappear, reappear and penetrate solid matter at the mere whim of the magician. This supernatural choreography ends with a giant pompom appearing under the cups, leaving the spectators mystified and amused. The video below shows a masterful performance of this ancient magic trick by the Dutch master, the late Tommy Wonder.

Tommy Wonder performs the Cups and Balls.

Although the magician entertains the audience, a nagging thought lingers in their minds:

How did he do it?

The Science of Magic

We know that magic is not real, but what Tommy Wonder just showed seems like a real miracle. Among the spectators is a group of university colleagues who reflect intellectually on what they just experienced. They enjoyed the show as much as everyone else, but their questions are different from the rest of the audience. These issues are frankly a lot more interesting than knowing how the magician did the magic trick.

The psychologist wonders how it is possible that the performer so easily deceived his mind. How is it possible that people can be tricked to see something that contradicts our common-sense view of the world? The psychologist's friend, a professor of humanities, also enjoyed the show. She wonders how the pompom appeared under the cups like everybody else. Still, she also asks questions about the cultural significance of magic why it has remained popular for millennia, across different cultures. Her husband, who works as an occupational therapist at the local hospital, is also an amateur magician. He contemplates the incredible hand-eye coordination and muscle control required to perform the trick he just saw, and he wonders whether he could apply magic skills to his profession.

Magic is a unique performance art that can teach us a lot about how humans relate to reality. Although magic tricks are always a game of wits between the spectator and the performer, a great magic show conveys a more profound message about our relationship with the world. Studying the relationship between magic and science helps us to better understand how we experience the world.

Perspectives on Magic

This book explores some of the answers to the questions that scholars from different fields of science have asked about the performances of magicians.

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The Magic of Science

Art is defined by limitations. Musicians play a mathematically defined range of tones, actors express the spectrum of human emotions, and painters have a palette of colours. With these building blocks, these artists can create great works of art.

Magicians also have a palette of possibilities to create their art. Firstly, the number of magic effects they can perform is limited to about seventeen categories, such as disappearances, levitation, mind-reading and other apparent miracles. Magicians perform these tricks using a large number of techniques that all fit into four categories.

Secret actions (sleight of hand)
Hidden mechanics (gimmicks)
Psychology (misdirection)
Laws of nature (physical and formal sciences)

Although a magic performance portrays an event that contravenes our intuitions about nature, the laws of nature can be used in such a way that the outcome is counter-intuitive. Magicians have used almost every science in their quest to perform the art of deception.

Applied Science

The applied sciences are areas where professionals, such as engineers, use the findings of science for the benefit of humanity. Magicians often build elaborate apparatus that requires a reasonable level of knowledge of engineering to make sure they are safe to use. Information technology is a new method in magic with many new magic tricks that use an iPhone or hidden electronics to perform magic.

Magic tricks can also have a purpose beyond frivolous entertainment. Professionals from many areas use magic tricks in their day job.

Although the formal and physical sciences do not study magicians, teachers in these subjects use magic methods to explain the abstract principles of their science to students. Magic tricks are especially effective to teach abstract mathematics because they can visualise many principles.

Magic tricks are also popular with people that work in healthcare. Clown doctors help to reduce anxiety in little patients, and occupational therapists use magic to help people recover from injuries.

Formal Sciences

The formal sciences mathematics and information theory are the only areas of research that are totally abstract as they do not study anything physical. Mathematics and magic are perfect partners. Many tricks use principles of mathematics by using a process that seems random but is, in fact, controlled. Most mathematical magic tricks are self-working in they achieve the desired result even as long as you follow the process.

This video of an 1991 David Copperfield television show is an example of beautiful magic created by an algorithm. Mathematics teacher Sydney Kolpas wrote a paper about the principles behind this trick. The beauty of mathematical magic tricks is that because they are algorithmic and always work, the trick is purely a theatrical performance.

The Magic of David Copperfield XIII: Mystery On The Orient Express (April 3, 1991).

Magicians can also use topology (the science of shapes) and geometry (the science of size) to create magical theatre.

The Möbius Strip in Magic: A Treatise on the Afghan Bands

Magicians use the Möbius strip to perform magic under the name Afghan Bands. This ebook describes the principles, history and performance this topological trick.

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Paradoxes of Size: A Treatise on Geometric Vanishes

This book discusses the history and principles of the three types of geometric vanishes: The Vanishing Leprechaun, the Curry Paradox and the Tangram Paradox.

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Physical Sciences

The physical science chemistry and physics are the magician's best friend. A famous example is a Light and Heavy Chest, which uses electromagnets to create the illusion that its weight has increased. This method was used by nineteenth-century magician Jean-Eugène Robert Houdin to convince Algerian rebels that the French were more powerful than them.

Magic and science are most closely related to the ancient craft of alchemy. Many magic tricks use chemistry to create illusions, some hidden, some more visible, such as the pyrotechnics to create dramatic effects. Most tricks with liquids use the principles of hydraulics to perform miracles.

Social Sciences

Scholars in the social sciences and the humanities have extensively studied magic from a wide variety of topics. This section only mentions three main areas of interest.

Social scientists have researched gender issues in magic. The vast majority of magicians are male, and women mostly perform a passive role as they are the ones being cut in half. Why is this the case, is magic a reflection of society or is it even more male-oriented?

Magicians are fascinated by the history of their craft, and recently professional historians started writing about magic performances of the past. One such topic is the role magicians played in the development of film as a form of entertainment. The earliest film makers, such a Georges Méliès, were magicians who invented many of the techniques in special effects.

Another example of the social science of magic are anthropologists who study the subculture of magicians. Because magicians surround their work in secrecy, they form strong ties with each other to discuss their craft in a safe surrounding. These studies are exciting as the Internet has revolutionised how information is shared between people.

Psychology is a principle in almost every magic trick as the performer needs to manage the spectator's attention. The science of perception is a particular area of psychology that studies how we perceive the world around us and how we interpret the information we receive to our senses. Our mind is fallible in how it reconstructs the world, which gives rise to misdirection and optical illusions, which are nature's own magic tricks.

The Jastrow Illusion in Magic: A Treatise on the Boomerang Illusion

This ebook describes the psychology of this stunning optical illusion and how magicians use it to perform magic.

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Perspectives on Magic

If you are interested in the relationship between science and magic then you should consider reading Perspectives on Magic by Peter Prevos. This book discusses the many perspectives on magic of scientists from various fields of endeavour.

Perspectives on Magic

This book explores some of the answers to the questions that scholars from different fields of science have asked about the performances of magicians.

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Magic Research Resources: Studying theatrical magic

Peter Prevos — Thu, 12 Apr 2018 10:00:00 +0000

Magic is a secretive art form, or so it seems. Anyone interested in magic research has access to a large number of physical and online resources, which I used to write Perspectives on Magic.

Perspectives on Magic

This book explores some of the answers to the questions that scholars from different fields of science have asked about the performances of magicians.

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Many magicians are avid collectors of magic tricks, DVDs, books and anything else related to their passion. Some are quite fanatical and amass thousands of volumes on the art of deception, like the Conjuring Arts Research Centre in New York. Some have bequeathed their collections to academic institutions, who make these accessible to researchers.

Cool Hunting Video: The Conjuring Arts Research Center.

Parallel to the literature about magic, written by magicians, a sub-genre of scientific writings from many fields of endeavour exists. These books and journal articles are mainly written for the colleagues of the professionals and scientists that created this work. Historians, social scientists, psychologists, occupational therapists, neuroscientists, film researchers and so on have deliberated on the role, workings and practical purpose of conjuring. Most of this work is, however, locked behind pay walls and publishers charge exorbitant prices for a single ten-page journal article. Over the past years, I have extensively researched this literature and created an online annotated bibliography on the science of conjuring to help unlock this vast resource.

List of magic research resources

Internet libraries

Ask Alexander: the world's largest online magic library (Conjuring Arts Research Center).
People's Magic Library: Eight centuries of open access books about magic (Mariano Tomatis).
Magic Knowledge Base: Search a large collection of magic ebooks (Lybrary.com).
Conjuring Archive: Annotated bibliography of magic tricks (Denis Behr).
MagicPedia: Digital magic encyclopedia.
Conjuring Credits: Traces the history and origins of magical sleights, plots and concepts back to their known origins.

Academic Libraries

University of Texas: McManus-Young Collection and Houdini Collection
Brown University: H. Adrian Smith Collection of Conjuring and Magicana
State Library of New South Wales: Robbins Stage Magic Collection.
WG Alma Conjuring Collection: National Library of Victoria, Australia. Including some digitised books.
H. Adrian Smith Collection of Conjuring and Magicana: Brown University, Rhode Island.
Robbins Stage Magic Collection: State Library of New South Wales.

Dominique Dunstan on the WG Alma Conjuring Collection.

Journals

Journal of Performance Magic: Open access peer-reviewed journal publishing articles that explore the relationship between mindfulness and performance (theatre, dance, music, live art).
Gibecière: Dedicated to the history of magic published by the Conjuring Arts Research Center.

Perform Optical Illusions as Magic Tricks

Peter Prevos — Fri, 28 Dec 2018 11:00:00 +0000

The art of magic creates a world where expectations of the way the world functions are temporarily violated. Magicians use the fact that our mind does not perceive the world as it is to create the illusion of magic. Optical and visual illusions are nature's own magic trick as they expose how our limited perception of the world around us. Science and theatrical magic have always gone hand-in-hand. This article describes how magicians can use optical illusions as magic tricks. A previous version of this article was published in the Austrian magazine Aladin.

Railway track illusion (Source: pyzata).

Optical Illusions

Imagine that you are sitting on a train. You look out the window and believe that the train is moving. You soon realise that in reality, the neighbouring wagons are moving. After your train finally leaves, you look through the back window, and it seems like the tracks physically meet in the distance, although you know with certainty that they are perfectly parallel to each other. This anecdote is examples of illusions as we experience them in everyday life.

Psychologists have studied the nature of optical illusions to understand how our brain is cheating itself. The study of these illusions also has practical implications. For example, factory control rooms contain many complicated dials and displays that help operators make correct decisions. Engineers design these instruments, so there is no misunderstanding how to interpret the information they display. Another application can be found in traffic engineering. Roads must be designed so that visual information is correctly understood by the driver to reduce the likelihood of accidents.

Optical illusions are also deliberately introduced to avoid perceptual errors. Ancient Greek architects formed stone pillars in a gentle curve with their base slightly wider than the top, so that they seem perfectly straight. Optical refinements to simulate geometric perfection are common in Doric Greek temple architecture. The Parthenon on the Acropolis of Athens is admired for its architectural perfection. To realise this perfect geometry, the architects Ictinus and Callicrates have curved almost every line in the design and created the illusion of straight lines and geometric perfection.

Parthenon on the Acropolis in Athens, Greece (Source: Pakhnyushchyy).

Scientists have systematically studied optical illusions since 1855, the year when the German psychologist Joseph Oppel published an article on the subject. Since that first publication, thousands of books and journal articles have been written on the topic.

One of the simplest illusions is the Müller-Lyer illusion. The horizontal lines have the same length, but the arrows inward or outward confuse the brain because the horizontal lines do not seem to be the same size. The German psychologist Franz Carl Müller-Lyer first published this illusion in work on fifteen geometric illusions.

Müller-Lyer Illusion.

The Ponzo Illusion is similar to the first example. The converging oblique lines distort our perception of identical horizontal lines. The angled lines cause our brain to recognise the upper line as slightly smaller than the lower one. This illusion is named after the Italian psychologist Mario Ponzo.

Ponzo illusion.

If two circles of the same diameter are surrounded by larger and smaller circles, they appear to be different in size. This deception was discovered by the German psychologist Hermann Ebbinghaus. The presence of smaller or larger objects affects our perception of size. This illusion can be used in real life to make portions of food appear smaller than they are. When a chef serves a meal on a giant plate, it seems smaller than when placed on a standard plate.

Ebbinghaus Illusion.

The last example here is the Poggendorff illusion, first described in 1860. In this illusion, a rectangle covers an oblique line. Although the diagonal line is straight, it seems as if the line is unequal. This illusion was designed by Johann Zöllner, but it was named after Johann Poggendorff, the publisher of the journal in which it appeared.

Poggendorf Illusion.

Although these phenomena have been studied for more than a century, the actual cause of these deceptions is still not fully certain. There are several competing explanations of how the brain causes them. However, none of these theories is universally accepted by psychologists.

Visual and cognitive illusions

Visual illusions occur when the physical circumstances outside our mind deceive us. Distortions of light in our eyes, such as the mirrors in an amusement park, cause these illusions. If you look in a curved mirror, you appear shorter, longer, thicker or thinner. Optical illusions also form the basis of the art of photography. Photographers combine focal length, aperture and shutter speed to make an artful photo. Photography is not a perfect replica of reality, but an interpretation of reality by the photographer through the way she manipulates light.

While visual illusions have a physical cause, cognitive illusions arise within our brain. The way our mind interprets the visual stimuli delivered to the brain by the optic nerve causes cognitive illusions. Psychologists and artists have discovered and developed hundreds of cognitive optical illusions, such as the one shown above. Some illusions cause us to perceive movements where there are none, cause us to assign size or colour wrongly, or to show us forms where there is the only space. The psyche interprets the information it receives, and our perception thus systematically deviates from reality.

Optical Illusions as Magic Tricks

These physical and neurological phenomena make visual and cognitive delusions a perfect tool for magicians to create the illusion of magic.

Magicians use a variety of techniques to perform magic. Both visual and cognitive delusions form an integral part of the show. Visual illusions are widespread in stage magic, or as they say in English, "with smoke and mirrors". The nineteenth-century trick called the Pepper's Ghost is one of the best examples of visual illusions in magic theatre. In this illusion, the magician conjures-up ghostlike figures on stage using an elaborate setup of smoke and mirrors.

Designers of stage illusions often rely on the principle of cognitive illusions. The magician's assistant hides in carefully designed boxes that seem smaller than they really are. This cognitive illusion creates the idea that a hemicorporectomy is performed, while the assistant remains uninjured. The craft of the illusionist requires a good understanding of the physics of visual illusions and the psychology of cognitive illusions.

In addition to large boxes used to mutilate and restore women, there are also demonstrations of optical illusions in close-up magic. In the Marlo Tilt, the magician seems to place a card in the middle of the deck, while in actual fact it is much closer to the top.

In these examples, optical illusions form the secret method and are not part of the magical effect itself. In the case of stage illusions, the discrepancy in size is not revealed because the cognitive illusion is not the effect but part of the method. Also in 'Marlo Tilt' the cognitive illusion is part of the secret and not the actual magic trick.

Magic tricks where the effect itself is an optical illusion are rare as they are usually easily reconstructed by audiences. The audience is aware of optical illusions, and the method quickly becomes clear. Alleviation of this problem requires additional layers of deception or an appealing presentation to promote the magical appeal.

The following sections describe some magic tricks that fully exploit our tendency to be fooled by our senses. The Jastrow or Boomerang Illusion is a classic in magic that uses a strong optical illusion. Say Cheese is a cute trick invented by Robert Hill that uses the ambiguous depth illusion.

Jastrow Illusion

The best-known cognitive illusion is the classic boomerang illusion. This magic trick seems to be a cognitive deception that is only useful for entertaining children. A recent Twitter post by BBC journalist Mark Blank-Settle shows that this illusion can also capture the imagination of adults. His video went viral, and news agencies around the world drew attention to his magical illusion.

My toddler's train track is freaking me out right now. What is going on here?! pic.twitter.com/9o8bVWF5KO
— Marc Blank-Settle marcsettle.bsky.social (@MarcSettle) April 6, 2016

The first psychologist to describe this illusion as we now know it was not Joseph Jastrow but the German psychologist Franz Müller-Lyer. His article from 1889 presents an extensive collection of geometrical illusions of size. The illusion first published in the English literature by Polish psychologist Joseph Jastrow, whose name is still linked to this principle.

Shogu Imai from Japan examined the Jastrow illusion in detail. He experimented with different designs and determined the optimal radius and angle to maximise the illusion. The results of his research can be used to produce the perfect prop to perform the Boomerang Illusion.

Ideal Jastrow / Boomerang illusion based on Imai (1960).

The magical literature contains many ideas on how to perform this routine, and there are many versions available in magic shops. The boomerang illusion is a perfect little secret that provides an ideal way to express your creativity. If you want to know more about this venerable trick, then read my book The Jastrow Illusion in Magic. This book describes the history of this illusion both from a scientific perspective and its performance history as a magic trick.

The Jastrow Illusion in Magic: A Treatise on the Boomerang Illusion

This ebook describes the psychology of this stunning optical illusion and how magicians use it to perform magic.

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Tenyo Sakkaku Scale (Ruler) Puzzle.

Ambiguous Depth Illusion

When interpreting lines as three-dimensional objects, the mind is sometimes confused. The brain interprets some drawings in multiple ways. Famous examples such as the Necker Cube and the Schroeder Stairs, shown below, demonstrate this illusion. As you stare at these two images, you notice your mind constantly switching its interpretation between the two possibilities.

For the Necker cube, either the left or right square appears to be at the front. The Schroeder stairs can be viewed as either sitting on the ground or hanging from the ceiling.

Necker cube and Penrose Stairs.

Say Cheese is a magic trick that uses this principle. Other magic tricks based on the Ambiguous Depth Illusion are Parabox by Jerry Andrus and Escheresque by Daryl.

Say Cheese!

What is the difference between Dutch and Swiss cheese? This quirky magic trick uses an optical illusion to explain the difference.

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Jerry Andrus demonstrate his Paradox Box illusion.

Topological Magic Tricks: Shapes as a Method for Deception

Peter Prevos — Fri, 28 Dec 2018 11:00:00 +0000

Magicians often use the principles of science as a method to create the illusion that real magic has occurred. Most tricks that use mathematics, such as the many card routines, rely on number theory through counting and sorting patterns. Some magic tricks, such as the enigmatic vanishing square puzzles, use principles of geometry. A small subgroup of scientific magic is topological magic tricks both as a method and as a plot.

What is topology?

The word topology was first introduced by the 19th-century German mathematician Johann Listing. He is also known for simultaneously discovering what is now known as the Möbius strip. Topology is a branch of mathematics concerned with the (logos) of space (topos).

Mathematicians involved with topology study transform into other shapes. To a topologist, a doughnut made from silly putty is equivalent to a coffee mug because one can be transformed into the other without drilling holes or adding material. Another favourite past time of topologists is the study of knots. One of the significant problems in knot theory is determining whether a knot is actually a knot or just a tangled mess.

Transforming a coffee mug into a doughnut.

Topology is an artful and speculative branch of mathematics, but it also has many practical applications. Topology is an essential input into many technologies we rely on, such as the internet. This form of mathematics also plays a role in physics and biology. Lastly, topology can be used to create the illusion of magic.

Topology is the silly putty of mathematics.

Clifford Pickover

Topological magic tricks

Topology appears in magic as an effect as a method. Basically, all rope tricks are topological effects. Most tricks with rings involve linking and unlinking them by melting metal through metal. Linking ring tricks feign to violate a principle of topology which states that two closed curves cannot link or unlink without breaking them apart. Magic books also describe a lot of fake knots, within the vocabulary of topology are unknots.

Topology as a Method

The most famous of all topological magic tricks is the Afghan Bands. This trick uses the Möbius Strip as a technique to create magic. The primary effect is that a loop is cut or ripped in half with unexpected consequences.

The Möbius Strip is a two-dimensional loop that paradoxically only has one side. This principle has been studied by magicians, used by inventors and has inspired artists since it was first discovered. Magicians have performed this trick for over a century with paper, cloth and with zippers.

The video below shows Harry Blackstone performing his version with cloth bands which he called the Red Rags.

Harry Blackstone performs the Afghan Bands.

The Möbius Strip in Magic: A Treatise on the Afghan Bands

Magicians use the Möbius strip to perform magic under the name Afghan Bands. This ebook describes the principles, history and performance this topological trick.

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Topology as a Plot

All rope tricks are essentially topological magic tricks. Magicians cut and restore ropes and magically tie or untie knots. In knot theory, a branch of topology, knots can be deformed in any way, except by cutting the cord. Magic tricks thus don't use topology but appear to be topological while

In knot theory, all knots are shows as loops. One of the essential questions in knot theory is a loop is a knot or knot. Mathematicians call a loop that has no knot an unknot. Some of these fake knots are quite complex and look like real knots. Magicians know various ways to apparently knot a rope and magically dissolve it into an unknot.

A knot theorists way to untangle a knot.

Daryl's rope routine.

Unknot Diagrams: The Art and Magic of the Trivial Knot

Peter Prevos — Wed, 07 Jul 2021 00:00:00 +0000

One of my biggest frustrations with watering the garden is that the hose always seems to be knotted. No matter how careful I am coiling it when done, knots seem to be everywhere where you don’t want them — garden hoses, headphone wires, extension cords or even your muscles. However, these knots are just an illusion most of the time, and my hose is merely a complex’ unknot’, as mathematicians would say.

Mathematicians study knots in great detail. Not to prevent their garden hoses from becoming a tangled mess but to better understand reality. For example, this branch of mathematics provides insight into the structure of DNA and entanglement in quantum mechanics and the behaviour of fluids. But knots can also be beautiful and a fascinating object of study by itself.

The unknot, also known as the trivial knot, is the most simple knot. In its simplest form, it is just a loop but with hidden complexity. The unknot is deceptive because what looks like a knot is actually not a knot. Mathematicians have designed complex unknots to study their properties. The unknot has also inspired painters as works of art. Lastly, magicians have developed methods to create unknots that look like real knots.

This article is an ode to the trivial knot and contains a gallery of mathematical, artistic and magical unknot diagrams.

The Unknotting Problem

Knot theory is a young branch of mathematics that started about a century ago. A mathematical knot is not the same as a knot in the real world. Instead, mathematical knots are closed loops in three-dimensional space. The loops need to be closed to ensure that their properties stay the same when we deform them. Mathematicians have defined thousands of knots and analysed their properties. The image below shows the first four basic knots.

Projections of the first four basic knots.

The unknot is a closed loop that is not knotted. This is a convoluted way of saying that an unknot is a circle. Although this description is as simple as possible, it is not always that clear whether a loop is knotted or just a ring. The diagram below shows two examples that are indisputably not knotted.

Two simple projections of the unknot.

One of the fundamental and most challenging problems in knot theory is the unknotting problem. While in the physical world, you would simply pull the string to test its knottedness, we require a more rigorous and reproducible process in the abstract world of mathematics. The example below looks like a proper knot, but you’ll see that it is just a loop when you create one from a piece of string.

Projection of a nasty trivial knot with 7 crossings.¹

The unknotting problem is deceptive because complex projections of the unknot look like genuine knots. Mathematicians have developed a plethora of systematic methods to test whether a tangled loop is an actual knot or merely a trivial one.

Reidemeister Moves

The Reidemeister moves are the oldest test of whether a knot is trivial. This is a set of three allowed changes to a knot to transfer it from one state to the next. A sequence of these moves is proof that a knot is really an unknot.

The diagram below shows how to untangle the Culprit knot with Reidemeister moves.² Note that the first step increases the number of crossings before it can be untangled.

Reidemeister moves to untie the Culprit knot (10 crossings).

The main problem with this method is that we don’t know how long we need to continue before we simply give up. How do you know when to stop trying? There is no way to look at a knot and calculate the number of moves you need to transfer it from one to another state. All we know is that the theoretical maximum number of moves is unimaginably large and estimated to be $10^{2n10^{11}}$, where $n$ is the number of crossings.

In other words, if you take a complex unknot with lots of crossings and perform one random Reidemeister move per second, it could take much longer than a multitude of ages of the universe to get to a solution. Note that this unfathomably large number is the maximum number of random moves. With a bit of human intelligence, most unknots can be resolved much faster.

Unknotting algorithms

Reidemeister moves are a graphical and practical way to untangle mathematical knots. But the process requires some human insight and creativity. Mathematicians ask themselves how we can automatically solve the unknotting problem.

Wolfgang Haken devised an algorithm to tell if a knot projection is an unknot. His algorithm is so complicated, however, that it has never been implemented. Nevertheless, other mathematicians have worked on this problem and have developed different methods to automatically unknot a knot.

Animation of untying the Culprit knot by Rob Scharein using the KnotPlot software.

Knot theory is an exciting area of mathematics with new insights. For example, Oxford Mathematician Marc Lackenby has created an algorithm that determines whether a knot is the unknot in $n^{C \log{n}}$ steps, which is not quite as much as several ages of the universe.

In 1954 Alan Turing wrote "No systematic method is yet known by which one can tell whether two knots are the same."

In 2021, in a remarkable Gordian tour-de-force, Marc Lackenby reveals a new unknot recognition algorithm that runs in quasi-polynomial timehttps://t.co/0jGImTa9Uc pic.twitter.com/Y0n15Bu8DT
— Oxford Mathematics (@OxUniMaths) February 2, 2021

If you like to know about knot theory, then I highly recommend the lecture on the Mathematics of Knots by Jessica Purcell from the School of Mathematics at Monash University.

Unknot Diagram Hall of Fame

Knot theorists working on the unknotting problem have developed some fiendishly complex trivial knots to test their conjectures about the unknotting problem and their software.³

While these designs certainly have mathematical relevance and play a role in untangling the mysteries of the universe, the unknot can also be a thing of beauty. Complex unknots are inherently deceptive because they are not what they seem to be. Unknots are knots in disguise, and trying to untangle one in your mind can be challenging.

Goeritz

Lebrecht Goeritz was a German mathematician who designed some trivial knots almost a century ago. His most famous unknot has eleven crossings. The beauty of this knot is that you can extend the number of crossings by adding tangles on the left and right parts in the middle.

Goeritz unknot (11 crossings).⁴

Thistlethwaite

Morwen Thistlethwaite is a knot theorist who made essential contributions to the field. He co-developed the Dowker–Thistlethwaite notation, which is a tool to encode knots for computers. He also defined a trivial knot with fifteen crossings that is often cited in the literature.

Thistlethwaite (15 crossings).⁵

Ochiai

Japanese researcher Mitsuyuki Ochiai developed a computer program to decide the unknotting problem. He also constructed four complex trivial knots with an increasing number of crossings to test his theorems and software. The image below shows the first two of his creations.

Ochiai’s first two trivial knots (16 and 45 crossings).⁶

Haken

Wolfgang Haken developed the most complex trivial knots available in the literature. Haken is famous for co-solving the four-colour theorem, which states that any map can be filled with only colours so that no two adjacent regions have the same colour. You could also apply that theorem to his famous unknot shown below.

Haken’s Gordian Knot (141 crossings).⁷

The name of this knot relates to a classic legend about the Gordian knot. Alexander the Great was challenged to untie the knot. The local legend said that whoever was able to undo it would rule Asia. Instead of using Reidemeister moves or carefully untangling the rope, he boldly cut it with his sword.

Constructing Haken's Gordian Knot by Madi S., inspired by the work of Mick Burton.

Unknots in Art

Knots have been part of human culture for tens of thousands of years. Humans of the early stone age used knots to create fishing nets and tie things together. Knots are not only practical; they also feature in the arts. We find images of knots in the art of cultures around the globe and through the ages.⁸

The traditional sand drawings of Vanuatu, which I discussed in a previous article, look like geometric projections of knots. Most designs are drawn with a finger in the sand in one continuous line. Because the finger does not leave the sand, these designs are trivial knots.

Outline of a Ni-Vanuatu Sand Drawing (Depiction of a turtle and a yam).⁹

The unknot has inspired not only mathematicians but also visual artists. For example, abstract painter James Sienna painted some works inspired by the unknot. Whether these designs are actually trivial knots remains to be seen.

Mick Burton is a continuous line artist, which means he draws pictures in one continuous line. Mick found inspiration in knot theory and Haken’s Gordian Knot in particular, which produced this wonderful painting. He also devised a method to construct Haken's Gordian Knot, which was the basis of the above animation.

Mick Burton, Twisting, overlapping colouring of Haken’s Gordian Knot.

Magical Unknotting

Magicians love to use ropes and knots to create the illusion of magic. The Conjuring Archive lists hundreds of effects that involve knots. Most of these either cause knots to appear or dissolve.

A dissolving knot is a trick where a magician appears to tie a genuine knot, but it melts away. These types of tricks are unknots in motion. Magicians have developed a range of techniques to tie an unknot.

This video by magician Doug Conn teaches a short knot routine that shows how to tie the two most common dissolving knots.

Doug Conn, Rope Magic Knots Tutorial.

Dissolving knots are deceptive because, in our experience, every tangled mess of rope is knotted. Unfortunately, we cannot determine whether a piece of rope is knotted by merely looking at it. The complexity of the unknotting problem in mathematics supports this intuition.

Just like mathematical unknots, dissolving knots are named after the person who composed them. The two most famous versions are the Chefalo knot and the Hunter Bow Knot, which Doug teaches in his video.

Magic and topology often go hand-in-hand as a myriad of tricks uses topology either as a method or as a plot. Famous knot theorist Louis Kaufman loves to perform knot magic tricks in his lectures.

So let’s treat these two dissolving knots the same way as the mathematical projections we seen above. First, I created the knots, connected the ends and laid them out on the table before dissolving them. I then photographed the ropes and drew the projection.

The resulting diagrams show the projections of the Chefalo Knot (11 crossings) and Hunter Bowknot (9 crossings). You can also combine these knots by tying the Chefalo Knot on top of the bowknot, which gives an unknot with 20 crossings.

Perhaps a mathematician reading this page can work out the proof that these are indeed trivial knots without pulling the ends.

Mathematical projection of the Chefalo Knot and Hunter Bowknot.

An Ode to the Unknot

This ode to the unknot shows that it is by no means trivial. On the contrary, the unknot hides a deep mathematical complexity and has inspired both artists and magicians.

Complex trivial knots are deceptive because they look more elaborate than they are. So next time you are stuck with a tangled garden hose, don’t panic but find the unknot that hides within. Perhaps the more important lesson is that no matter how tangled up your life is, there is a way to simplify the situation.

Feel free to contact me or leave a comment below if you have any additions to my collection of unknots.

Magical unknot knot board.

Notes

Adams, C. C., The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (1994), New York: W.H. Freeman.

Henrich, A., & Kauffman, L. H., Unknotting Unknots, The American Mathematical Monthly, 121(5), 379 (2014) DOI 10.4169/amer.math.monthly.121.05.379.

Burton, B. A. et al.Hard diagrams of the unknot, (2021). arXiv:2104.14076.

⁴

Kauffman, L. H., & Lambropoulou, S., Hard unknots and collapsing tangles, Series on Knots and Everything, 187–247 (2011) DOI 10.1142/9789814313001_0009.

⁵

Petronio, C., & Zanellati, A., Algorithmic simplification of knot diagrams: New moves and experiments (2016). arXiv:1508.03226v3.

⁶

Ochiai, M., Nontrivial projections of the trivial knot, Astérisque, 192, 7–10 (1990).

⁷

Stewart, I., Professor Stewart’s Cabinet of Mathematical Curiosities (2008), London: Profile Books.

⁸

Jablan, S., Radović, L., Sazdanović, R., & Zeković, A., Knots in art, Symmetry, 4(2), 302–328 (2012) DOI 10.3390/sym4020302.

⁹

Deacon, A. B., & Wedgwood, C. H., Geometrical Drawings from Malekula and Other Islands of the New Hebrides, The Journal of the Royal Anthropological Institute of Great Britain and Ireland, 64, 129–175 (1934) DOI 10.2307/2843952.

Optical Illusion Architecture in Melbourne

Peter Prevos — Thu, 21 Mar 2019 00:00:00 +0000

Last weekend, I visited the Escher X Nendo exhibition at the National Gallery of Victoria. Dutch graphic artist M.C. Escher is famous for designing speculative architecture. He created paradoxical buildings where water flows uphill, and people effortlessly walk upside down. Escher's drawings are optical illusions and impossible in the real world. This impossibility does not mean that optical illusions only exist on paper. Optical illusion architecture exploits our perception to create real-life illusions in the urban space.

This article discusses three buildings in Melbourne designed by ARM Architecture that are fine examples of optical illusion architecture. The Barak Building in Carlton, the customs building in Docklands and the Southbank Theatre are fascinating reminders that the world we perceive might not be the way we think it is.

Optical Illusion Architecture in Melbourne. Barak Building, 1010 La Trobe Street and the MTC Southbank theatre.

Barak Building, Carlton. Seeing Patterns in Chaos

Our brains are wired to recognise patterns in the chaos of impressions we receive from the world around us. As a result of this tendency, our interpretation of these patterns does not always correspond with reality. We look at the clouds and see animals and a rock reminds us of a face. The software in our mind that causes us to recognise these patterns where there are none is called pareidolia. Our brains are particularly prone to see faces in random patterns.

Pareidolia causes people to see Jesus in a piece of buttered toast, or a human face on the surface of Mars. This phenomenon is not a hallucination, but a necessary part of our psychology. It enables us to recognise our friends and family in a see of people or perhaps a predator hiding in the bushes.

Essentially, all images of faces are optical illusions. Even the most realistic portraits of seventeenth-century masters such as Rembrandt are not even close to a real face. When looking at these works carefully, we just see blobs of paint. Only when we view the painting from a distance, do we recognise a face.

The facade of the Barak Building looks at first sight like a random set of white wavy parallel lines. Looking at this building from a distance, however, reveals a bearded face. The face that appears in these lines is an abstracted portrait of William Barak (1824–1903), who was one of the last traditional elders of the Wurundjeri people. Barak was an influential leader who fought for justice of the first inhabitants of the land that we now call Melbourne.

The location of the building is not without significance as Barak faces the Shrine of Remembrance. The shrine is a secular sacred space to commemorate the Australians that have lost their lives in war and a symbol of Australian identity. The Barak building complements this axis of remembrance by including Aboriginal culture in the identity of this country.

The face we see in this building is technically not pareidolia because it was intentionally designed to be this way. When we see the face of Jesus in a piece of buttered toast, we assume that there was no intent. The psychological principle that causes us to recognise the face of Barak is, however, the same as is the case with random patterns.

The Barak Building, Carlton by ARM Architecture.

1010 La Trobe Street, Docklands: Geometrical Optical Illusions

When psychology first developed as a science in the nineteenth century, many scholars researched geometrical optical illusions. These illusions are simple drawings that deceive the senses. Lines seem bent, objects look smaller or larger than they are and shapes can be distorted.

The Café Wall illusion is a famous example of a geometrical optical illusion. In this illusion, a chequered pattern makes a straight line seem bent. The Café Wall Illusion was popularised by psychologist Richard Gregory after a member of his laboratory noticed it on the wall of in Bristol.

Docklands in Melbourne is filled with contemporary architecture in a mostly lifeless jungle of concrete, steel and glass. One building by ARM Architecture that stands out is 1010 La Trobe Street. This building was completed in 2007 and is occupied by the Australian Customs Services.

The external face of the building is decorated with black and white squares, separated by orange lines that give the illusion that the horizontal lines are wavy. In reality, these lines are perfectly parallel. This illusion is convincing and can only be broken by placing a ruler along the line.

The way the brain processes the light on the retina causes this distortion of reality. The difference in contrast between the squares and the horizontal lines causes your neurons to interpret the image as small wedges, which makes the horizontal line appear to be wavy. The Cafe Wall Illusion is only one of many examples of geometrical illusions. The existence of these deceptions shows that we can never trust our eyes when perceiving the world.

1010 La Trobe Street, ARM Architecture.

MTC Southbank Theatre, Southbank. The Ambiguity of Depth Perception

We live in what some psychologists call the ‘carpented world'. Architecture most often uses straight lines and angles to form rectangular buildings. The shapes that surround us only exist in the cultural landscape and not in nature. Because we live in this artificial world of straight lines, our brains interpret drawings in a certain way. When we see lines on a piece of paper, we might recognise these as a three-dimensional carpented space.

This perception mechanism is the ultimate form of self-deception. The image on our retina is only a two-dimensional projection of the physical world, but our mind interprets it as a three-dimensional space. Smaller objects are further away, parallel lines vanish in the horizon, are some of the interpretive schemes that exist in our mind.

Early research in psychology suggested that tribal people who don't live in the creations of Western architecture are less susceptible to geometrical optical illusions. The carpented world hypothesis states that people who reside in a manufactured environment are quicker to interpret lines as three-dimensional objects. Because our eyes are used to see carpented shapes, geometrical drawings can easily deceive our mind.

One of the great triumphs of art is the development of perspective to create the illusion of a third dimension. Artists have reverse-engineered our mind to creates the illusion of depth from a two-dimensional image. All art is therefore an optical illusion.

M.C. Escher exploited the ambiguity between two-dimensional drawings and three-dimensional objects in many of his famous works. He basically breaks the rules of perspective drawing to create his impossible worlds.

The third optical illusion building is the Southbank Theatre. The exterior of this building is decorated with a white tubular ornamental structure that contrasts with the dark building.

This building exploits our mind's propensity to interpret certain images as solid figures. The stark white lines on the facade of the theatre look like three-dimensional objects because our mind is programmed to interpret lines in this way.

This last optical illusion building blurs the boundary between two a three-dimensional perception.

MTC Theatre Southbank, ARM Architecture.

Optical Illusion Architecture

These three buildings are not unique in their application of optical illusions. Trompe-l'œil, French for deceiving the eye, is a form of art that uses embraces illusions. Ancient Greek architects used optical illusions to create the impression of perfection. While columns of a temple seem to be spaced equally, the outer ones are actually a little closer than the inner columns. Architects and interior designers still use optical illusions. Forced perspective is a technique to make a room, a theatre set or whatever else seem larger or smaller than it is in reality.

In conclusion, optical illusions are not a human weakness, but a necessary part of our psychology. Our mind continually deceives itself by interpreting the light on our retina in a certain way. This deception a mental shortcut that helps us navigate the world. These three buildings are thus artistic reminders of the fickle relationship between our perception and reality.

The Reason for Daylight Savings: Blessing or a Nuisance?

Peter Prevos — Thu, 05 Oct 2017 00:00:00 +0000

Moving the clock an hour forward is an excellent day of the year because we can enjoy more daylight. Instead of sleeping through the first hours of morning light, I wake up close to when the sun rises, allowing me to get most out of my day. This blissful state unfortunately only lasts for about six months as the clocks eventually are moved back again. This article discusses the reason for daylight savings time.

I am writing this article from Coolangatta, on the border between New South Wales and Queensland in Australia. The border runs through the middle of the built-up area. There is no real indication which part of the street is in which state. Queensland does not observe daylight-saving. This means that there is a one-hour time difference between the two sides of the road.

Crossing the street is like travelling backwards and forwards in time, without having to worry about time travel paradoxes. You can safely kill your grandfather without vanishing into thin air. In Coolangatta and Tweed Heads you can even be in two time zones at the same time, without being ripped apart by temporal tidal forces.

Daylights saving time is a controversial subject. This article discusses some of the reasons that people love daylight saving and some of the arguments against the bi-annual time shift.

Tweed Heads and Coolangatta from the air.

The Reason for Savings

Daylight saving time makes perfect rational sense for an urbanised population at a fair distance from the equator. Why would anyone want to sleep through the first hours of daylight during summer?

The original argument for daylight saving was to save energy because people would spend less time in the dark and require less illumination. If there is more sunlight at the end of the day, there will be less time we are awake when it is dark. Although that might be logically correct, there are so many other factors that influence energy consumption that the effect of daylight saving is minimal. Also, the cost of illumination is now so small that it hardly matters anymore.

The best reason for daylight saving is that sunlight benefits your health, even when you are indoors. Natural light positively impacts our mood and has been shown to improve performance. Spending the first hours of daylight in bed thus seems to be an awful waste. In an industrial society where most people spend their time indoors, it makes perfect sense to maximise the amount of daylight they can use.

The Unreason for Daylight Savings

Some people struggle with the concept of changing time zones during the warmer months. There are reports of people that report suffering from jetlag due to the time change.

There are, however, also individuals who struggle intellectually with the concept of time changes. Some people believe that the very act of moving the clock an hour forward implies that we are adding an hour of daylight to the day. They appear to believe that changing the reference point of our clocks somehow influences the rotation of the earth.

The letter shown below was published in a regional newspaper during the height of the Millennium Drought in southern Australia. The author blames the dire situation on daylight saving. He claims that the extra hour of sunlight is "slowly evaporating the moisture". Evaporation indeed increases in summer, but changing the clocks is not most certainly not causing the drought.

Blaming the drought on daylight saving time.

Daylight saving time does have negative effects because many people sleep an hour less on the day of change. This reduced sleep can lead to increased accidents, lowering productivity. Perhaps to break the ongoing discussion by changing the way we measure time.

Measuring Time

Coolangatta and Tweed Heads illustrate that time is relative. Not only in the physical sense of Einstein's special theory of relativity, but also in the social sense. The time of day is not the result of scientific research but is agreed by social convention. Time is whatever we decide it to be so that we can have a functioning society where people can do things simultaneously.

Our clock is defined by 24 hours in a day, divided by noon. Each hour has sixty minutes, each of which has sixty seconds. This system might seem strange as it would make more sense to have two times ten hour days with 100 minutes per hour and 100 seconds per minute.

The clock with two periods of twelve hours has its roots in the sexagesimal number system used by the Sumerians five thousand years ago. This system uses sixty digits instead of our system based on ten digits. The Sumerians used this method because sixty is divisible by many numbers, which helps with accurate calculations without decimals. We have inherited this number system in the way we measure trigonometry and time. It is incredible to realise that the way we measure time has not changed much in five thousand years!

Solar time

In the Roman Empire, time was based on [sunset and sunrise][6], measured with a sundial. The day was divided into twelve hours, which meant that the length of an hour varied over the course of a year, depending on your location within the empire. At the Forum Romanum, an hour was about 45 minutes in winter and 75 minutes during summer. Further north, near Hadrian's wall, an hour was between 35 and 85 minutes.

With the invention of mechanical clocks, the day became divided into 24 hours of equal length. Hours of equal length are easier to comprehend, but the downside is that the times for sunrise and sunset are always changing.

Before official time zones existed, each village kept its own time with minor differences between them. The official time was whatever was displayed on the local church clock, with noon defined by the highest position of the sun. This system worked fine for centuries, but with the invention of instant communication, it became necessary to synchronise the clocks around the country and formal time zones came into existence.

Church with clock in my home town of Hoensbroek, the Netherlands.

Local unofficial time zones still exist in Australia. The hamlets of Cocklebiddy, Madura, Eucla and Border Village in the east of the state of West Australia use their own time zone.

They keep their own unofficial time to stay closer to the timezone of their South Australian neighbours. The sun in that part of the state rises almost a full hour earlier than in their capital Perth.

A similar situation occurs in Coolangatta. When New South Wales changes to daylight savings, the town of Tweed Heads sticks to winter time to avoid cross-border confusion.

A new way to measure time?

Perhaps we can use technology to solve the daylight saving time problem. The way we measure time during the day has changed once before; maybe we should change it again.

We could develop electronic clocks that synchronise sunrise to always occur at six or seven in the morning. The time between successive sunrises then defines the 24-hour day. This new system provides daylight saving every day of the year, with only a gradual daily change each day.

Implementing this method would be a gigantic task which dwarfs the issues we had with the Millennium Bug. Perhaps it is time to modify the way we measure time for a third time in the past five thousand years to maximise our daylight, aided by technology.

Pi Approximation Day: Esoteric Secrets of the Number Pi

Peter Prevos — Thu, 22 Jun 2017 00:00:00 +0000

The number Pi ($\pi $) is without a doubt the only number in the world with celebrity status. It is so popular that it has a holiday. Many people around the globe observe Pi Day on March the 14th because writing this date the American way we get 3.14. This date does not make sense for most of the rest of the world as 14 March is 14.03. Because of this inconsistency how we write dates, some people prefer to celebrate Pi Approximation Day on 22 July. This day is a good alternative because 22 divided by 7 is approximately 3.14.

Why does a number deserve a holiday? Even though Pi is an irrational and transcendental number with an infinite number of decimals, it is also a practical number as it appears in many equations that describe the physical world. As an engineer, I used to hit the π button on my calculator many times a day. When I have no calculator nearby, I divide 22 by 7 to get a reasonable approximation.

As a practical mathematician, I am interested in the science and history of this significant number. In the photo below, I am smoking a sheesha pipe in Egypt and wearing my favourite geek T-shirt. The number Pi on the shirt consists of the first 4,493 digits of the number Pi in the shape of the Greek letter itself.

The relationship between Pi and Egypt is not coincidental. Many people believe that the design of the Khufu pyramid at Giza reveals that Egyptians of the fourth Dynasty knew about Pi, long before any other culture did. The perimeter of the pyramid, divided by its height, is approximately two times pi. Whether the designers of the Khufu pyramid knew about Pi is doubtful as there is no evidence of the time the tombs were built.

The article discusses methods for Pi Approximation and what its properties teach us about the relationship between mathematics and reality.

Enjoying a Sheeha in my favourite T-Shirt in Luxor, Egypt.

The Digits of Pi

Pi is an irrational and transcendental number with an infinite array of randomly distributed digits. Pi is not irrational because it behaves like a crazy person. The number Pi is irrational because it cannot be expressed as a fraction.

Countless mathematicians have tried to 'square the circle', which means constructing a square equal with the same area as a given circle. Squaring the circle is only possible if Pi is a rational number, for example, 22 / 7, 333 / 106 or 8958937768937 / 2851718461558. These fractions become more accurate as the numbers grow larger, without ever achieving full accuracy. Mathematicians realised in 1761, after thinking about the problem for more than two thousand years, that squaring the circle is a dead end and that Pi must be an irrational number.

To make matters worse, Pi is not only irrational; it is also transcendental, which adds a glow of mystique to the number. These are numbers that cannot be determined with a finite equation. The transcendentality of Pi does not mean that the digits of Pi are random, but the equations to describe this number are an infinite series. This series produces an infinite number of digits. I will need to find an infinitely large t-shirt (∞ XL) to fit them all on.

$$\frac{\pi}{2} =\frac{2}{1} \cdot \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdots$$

Although the never-ending procession of digits in the number pi might seem random because there is no recognisable pattern, computers can calculate them with ridiculously high levels of precision. The current record for calculating the digits of Pi stands at $22 \pi^e$ trillion digits. Printed at 11 point font, the numbers would stretch to the moon and back and beyond. The only practical reason for doing so is to test the limits of supercomputers. In most engineering applications, a handful of digits is sufficient. In my work as a civil engineer, I have often used 22/7 to work out calculations in my head.

Even after calculating trillions of digits, we cannot seem to find any pattern in the order of the digits. The digits can be calculated, but there is no structure to the sequence of numbers. But within the chaos, we can find beauty. The image below shows the first 400 colour-coded digits of Pi, which is based on art by Martin Krzywinski.

The digits of Pi based on work by Martin Krzywinski.

Pi and the Horizon of Reason

The number Pi might be irrational and transcendental; it is nevertheless one of the most rational tools to understand the universe. We need Pi to understand the curves of nature, and Pi appears in statistics. Without an accurate understanding of Pi, science would be impossible.

If the universe is a rational and ordered place, then why do we need an irrational number to describe it? Does this apparent contradiction mean that Pi hides an inner meaning and that it encompasses an internal logic that we have yet to discover?

In the novel Contact by Carl Sagan, the heroine Ellie discovers a hidden message in the base-11 representation of Pi. This discovery is not as miraculous as it sounds like Pi consist of infinite decimals. The complete works of Shakespeare and the Bible could eventually be found inside the digits of Pi. My birthday appears at the 107,070,083 rd decimal digit of Pi and my PIN appears many times in the first million decimals of Pi. The infinity of Pi assures that every pattern we want to find in it will exist. The chances of finding anything meaningful are, however, infinitesimal.

The idea that an irrational number helps us to understand the universe can be a worrying thought. Fully understanding Pi requires us to deal with infinity. What does the fact that we cannot grasp Pi in simple terms tell us about the structure of the universe? Is Pi a property of the universe, or is it just a construct of our mind?

Flying back from Egypt, I watched The Oxford Murders. In this movie, the question of whether mathematics is the underlying truth of the world is discussed by the two main characters. Martin, a student, played by Elijah Wood, believes that the universe is mathematical:

Things are organised following a model, a scheme, a logical series. Even the tiny snowflake includes a numerical basis in its structure. Therefore, if we manage to discover the secret meaning of numbers, we will know the secret meaning of reality.

But is Martin correct? Can all philosophical questions and truths be expressed in mathematics? Will we eventually calculate our way out of ethical dilemmas? Can we improve our understanding of Shakespeare by expressing his prose in mathematics? Should we say $2b \lor \neg 2b$ instead of "To be or not to be"?

I tend to agree with Professor Martin Seldon, played by John Hurt in the same movie, who disagrees with his student.

Since man is incapable of reconciling mind and matter he turns to confer some sort of entity on ideas because he cannot bear the notion that the purely abstract only exists in our brain.

Pi exists on the Horizon of Reason. It is the product of a rational series of thoughts that eventually leads to the incomprehensible concept of infinity.

Does the infinite sequence of random numbers mean that Pi is an artificial construct of the human mind and not something that exists in reality? After all, how can something that is infinite exist in reality? The idea that we need a transcendental number to describe reality does not seem to match our perception of the world as a discrete set of things. Our common sense view of the world consists or integers and fractions, not of irrational numbers.

Nature doesn't care about perimeters and diameters because they are merely constructs of the human mind. The ubiquitous nature of Pi in physics and statistics seems to suggest differently. The existence of transcendental numbers reveals a divide between our common sense view of the world and our scientific description of the world. Common sense is most likely wrong, just like almost all our perceptions of the world are a construction of reality. Perhaps reality is not as discrete and countable as we believe it to be and we are projecting our rationality upon the world?

Pi Approximation Day

After these metaphysical thoughts back to more mundane matters. Pi Approximation day celebrates the fact that we can only ever approximate the number Pi. This limitation might be frustrating to mathematicians; it is great for engineers and anyone else who wants to use numbers to describe the world.

The Bystander Effect in Helping Behaviour: An Experiment

Peter Prevos — Tue, 03 Jan 2006 11:00:00 +0000

In 1964, Kitty Genovese was murdered outside her home in New York, while 38 witnesses did nothing to save her. This incident sparked a public outcry and was the catalyst for a considerable amount of research into what motivates people to help others in obvious need or what prevents them from helping.¹ The common-sense explanation for this seeming lack of compassion are vague concepts such as ‘‘alienation’’ and ‘‘apathy’’. These explanations stem from the idea that our moral actions are determined by character traits. This explanation of morality has, however, been contradicted by results from contemporary research in social psychology.²

Bystander Effect | Kitty Genovese | Social Psychology.

Most research on helping behaviour has used experimental methodologies to study situations in which someone has a sudden need for help. Factors such as clarity, the urgency of the need and skin colour, gender, age or handicap of the ‘‘victim’’, how many potential helpers are present and the relationship between victim and subject have been manipulated.³

Researchers comparing helping behaviour in rural and urban areas consistently find that helping strangers is more likely in less densely populated areas. North, Tarrant & Hargreaves found that participants are more likely to help when they are in a positive mood and stimulated by music.⁴ Wegner & Crano found that that contrasting skin colour of the victim and helper can also be a determinant for helping behaviour.⁵

Several studies have demonstrated that the presence of other observers reduces the likelihood that any one person will display a helping response.⁶ Contrary to common sense, there does not seem to be safety in numbers as the victim appears to have a greater likelihood of receiving help when there is a single witness rather than a group. Two possible psychological explanations proposed to explain the bystander effect are diffusion of responsibility among bystanders and a social norms explanation.

Diffusion of Responsibility

Latané & Darley developed a model that bystanders follow to decide if they will provide help or not. According to this model, a bystander goes through a five-step decision tree before assistance is provided. Helping responses can, however, be inhibited at any stage of the process, and no support is provided:

The bystander needs to notice that an event is taking place, but may fail to do so and not provide help.
The bystander needs to identify the event as some form of emergency. The situation may be ambiguous, preventing from help being given.
The bystander needs to take responsibility for helping but might avoid taking responsibility by assuming that somebody else will (diffusion of responsibility).
The bystander needs to decide on the appropriate helping response, but may not believe themselves to be competent to do so.
The bystander needs to implement that response, but this may be against their interest to do so, especially in dangerous situations.

In the diffusion of responsibility in stage three, each bystander notices the event and recognises that help is required, but fails to act because they assume that somebody else will take responsibility. This can be viewed as a means of reducing the psychological cost of not helping. The cost (e.g. embarrassment and guilt) are shared among the group, reducing the likelihood of intervention.

Social Norms and Helping Behaviour

Bryan & Test have shown that the bystander effect does not seem to appear if a helping response is first modelled by another observer, which seems to contradict the diffusion of responsibility concept.⁷ They suggest that this behaviour can be explained by the process of conformity to social norms. The social norms explanation holds that people use actions from others as cues to decide what an appropriate response to specific situations should be, as demonstrated by Asch’sAsch’s conformity experiments.⁸ Cialdini, Reno & Kallgren conducted five experiments to determine how social norms influence littering in public places and concluded that norms have a considerable impact on behaviour.⁹

The Ash Experiment.

The methodology employed by Bryan & Test is, however, not fully comparable with the traditional helping model as described by Latané & Nida. The study by Bryan & Test involved two separate events—the driver first sees a driver in need being helped by somebody and a while later sees another driver in need that is not being helped. Separating these two moments eliminates the possibility of diffusion of responsibility as there are no bystanders in the second situation and the subject is alone in his or her car.

The objective of this study is to test whether the diffusion of responsibility or the social norms explanation applies to helping behaviour in a non-emergency situation. If the diffusion of responsibility explanation is correct, then the number of people providing help will be less when non-helping bystanders are present than when no bystanders are present. The social norms explanation predicts that helping behaviour is increased when a bystander offers help as compared to when no bystanders are present.

Method

Participants

The study consisted of a task where a naive subject had an opportunity to help the experimenter in a non-emergency situation. All subjects were selected randomly when the circumstances were suitable for undertaking the experiment. A confederate was used to act as a helping or non-helping bystander in the investigation. The experiment consisted of 135 trials in total. The data was obtained from 75 trials on four Monash University campuses, and 47 responses were obtained by distance education students working in the general community. The data was appended with thirteen observations by the author obtained in a municipal park in central Victoria.

Materials & Procedure

The experimenter looked for a person standing alone in a public place, with no other person present within ten metres. The subject was not participating in any specific activity to ensure they would notice the event. The experimenter then ‘‘accidentally’’ dropped a pile of loose pages from a manilla folder close to the subject. The subject was defined as helping if he or she picked up one or more pages within thirty seconds from the drop. In cases where a third person started helping, or the subject was not able to help, the trial was not included in the results.

In the control condition, only the subject and the experimenter were present. In the test conditions, a confederate was standing nearby, and the papers were dropped equidistant between the subject and the confederate. In one condition, the associate did not help, while in the other condition, the confederate started to pick up the papers, providing a model for the appropriate behaviour. The helping behaviour of the confederate bystander was the independent variable and the percentage of subjects helping to pick up the papers the dependent variable.

The raw data shows an increase in helping behaviour in those scenarios where a confederate is present, as summarised in figure 1. In the control situation, 41% (n=44) of the subjects provided help. With a non-helping bystander present, the helping behaviour of subjects increased to 46% (n=48), and for a helping bystander, the percentage of helping subjects was increased to 56% (n=43).

Figure 1. Results of helping behaviour experiment.

A $\chi^2$ test for goodness of fit at a 5% confidence level was undertaken to compare the results with the control situation. The presence of a non-helping confederate resulted in an increase of helping compared to the control situation (41% v.s. 46%), albeit not significant: χ²(1,n=48)=0.48, p>0.05. The presence of a helping confederate resulted in a significant increase over the control situation (41% v.s. 56%), $\chi^2 (1, n=43)=3.95, p<0.05)$.

Discussion

The results show an increase in helping behaviour when a bystander is present, failing to support the diffusion explanation, which predicts a decrease in helping behaviour. The results do, however, not provide a firm ground to reject the diffusion explanation, as the increase is not statistically significant. The social norms explanation predicts that helping behaviour is increased when a bystander offers help as compared to when no bystanders are present. The results support the social norms explanation as there is a statistically significant increase in helping behaviour when first modelled by another bystander.

Although Latané & Nida have shown that the bystander effect has been replicated in many studies in many different circumstances, it has not occurred in 100% of the cases. It is unlikely that all these studies suffer from the same internal validity problems as this study. There could thus also be theoretical reasons for the abnormal results. Both the diffusion of responsibility explanation and the social norms explanation can be true simultaneously as the diffusion of responsibility is extinguished by a bystander who models the appropriate behaviour. Further research is required to untangle the relationship between the diffusion of responsibility mechanism and social norms as determinants for helping behaviour.

Methodology

The study suffers from some methodological problems, weakening its internal validity. Subject variables, such as gender and age, were not controlled, nor where they noted in the results. The data can thus not be tested for any significant effects of subject variables. There is also some doubt whether the methodology has been consistent because the experiment consists of groups of trials by different experimenters. There are also situational nuisance variables, such as weather conditions, location and time of day the investigation was held, which were not controlled because of the fragmented execution of the experiment. On a windy day, for example, the need to help to pick up the papers is much more apparent to any bystander. Situational variables can also influence mood, which in turn can influence helping behaviour. The increase in helping behaviour in the non-helping bystander condition has most likely been confounded by any of these uncontrolled variables.

Practical Application

Latané & Nida are pessimistic about the possibility of generating practical outcomes of the helping behaviour experiments. The significance of these experiments is of a more philosophical than practical nature. A critical aspect of the helping behaviour research is that it shows that our moral behaviour is not governed by moral virtues or character traits but by much more mundane social mechanisms. When things go wrong, it is usually the bystander who is being blamed for failing to act morally. We attribute these failures, like in the Genovese case, to expressions of bad character traits. Experiments in helping behaviour are valuable in that they can provide a greater understanding of why people fail to do what is morally expected and thus lead to greater tolerance and understanding of others.

Notes

Brehm, S. S., & Kassin, S. M. (1996). Social psychology (3rd ed.). Boston: Houghton Mifflin.

Harman, G. (1999). Moral philosophy meets social psychology: Virtue ethics and the fundamental attribution error. In Proceedings of the Aristotelian Society/ (Vol. CXIX, pp. 316–331).

Piliavin, J. A. (2001). Sociology of altruism and prosocial behavior. In N. J. Smelser & P. B. Baltes (Eds.), International encyclopedia of the social and behavioral sciences (pp. 411–415). Elsevier.

⁴

North, A. C., Tarrant, M., & Hargreaves, D. J. (2004). The effects of music on helping behavior: A field study. Environment and Behavior, 36(2), 266–275.

⁵

Wegner, D. M., & Crano, W. D.(1975). Racial factors in helping behavior: An unobtrusive field experiment. Journal of Personality and Social Psychology, 32(5), 901–905.

⁶

Latané, B., & Nida, S. (1981). Ten years of research on group size and helping. Psychological Bulletin, 89, 308–324.

⁷

Bryan, J. H., & Test, M. A. (1967). Models and helping: Naturalistic studies in aiding behavior. Journal of Personality and Social Psychology, 6, 400–407.

⁸

Asch, S.(1995). Opinions and social pressure. In E. Aronson (Ed.), Readings about the social animal (7 ed., pp. 17–26). New York: Freeman.

⁹

Cialdini, R. B., Reno, R. R., & Kallgren, C. A. (1990). A focus theory of normative conduct: Recycling the concept of norms to reduce littering in public places. Journal of Personality and Social Psychology, 58(6), 1015–1026.

Difference Between Male and Female Body Image Statistics

Peter Prevos — Sat, 09 Jul 2005 00:00:00 +0000

This analysis of body image statistics investigates the difference between male and female body image for men and women of different ages and reviews two hypotheses. The first hypothesis is that the ideal body image for women is that their ideal body shape is thinner than their perceived actual shape.

The most important indication of beauty for women, and to a lesser extent for men, is the prevailing ideal of thinness. The models on catwalks and in magazines portray unrealistic images that redefine the expectations that young men and women have of themselves. The result of this onslaught of idealised body shapes is that many women and some desire an unattainable and even unhealthy thin body (Lamb et al. 1993).

The power of marketing is that we don't buy things for what they do, but because of the kind of person we think it makes us. The first law of consumer behaviour states is that your real self, plus a product equals your perceived self.

This study measures the current and ideal body shape and the body shape of the most attractive other sex. The results confirm previous research which found that body dissatisfaction for females is significantly higher than for men. The research also found a mild positive correlation between age and ideal body shape for women. Older women are less concerned about being skinny than younger ones. For men, their age and the female body shape they found most attractive also correlated positively.

Perceptions of Beauty

This preoccupation with thinness is a recent development. The perception of women's body shapes has significantly changed over the past decades. In the early 1940s, thin people with ectomorph bodies were perceived as nervous, submissive and socially withdrawn. By the late 1980s, this perception had changed, and thin people were the most sexually appealing (Turner et al., 1997).

Several researchers have found that the female body depicted in the media has become increasingly thin. Bust and hip measurements of centrefold models show that between 1960 and 1979 there was a trend towards non-curvaceousness of women. This trend was, however, reversed in the early 1990s (Garner et al., 1980; Turner et al., 1997; Sypeck et al., 2006; Wiseman et al. 1992).

The Body-Mass-Index (BMI) of Playmates shows a steep decline from the 1950s to the 1990s and has slowly crawled back. The current index of around 19 kg/m² is at the lower edge of healthy weight. The waist-to-hip ratio has gradually increased from the first publications until recently (Figure 1). The higher this ratio, the less curvaceous a woman is. You can download the data and the analytical code from GitHub.

Figure 1: Playboy playmate BMI and Bust-Hip Ratio statistics.

Body Dissatisfaction

Parallel to the decrease of the ideal body shape for women, the dissatisfaction that women have with their body shape increased (Cash et al., 2004. In recent years, some researchers have found that females are more likely to judge themselves as overweight than males. This tendency was most active in adolescent and young adult women (Fallon and Rozin, 1985 Tiggeman and Penningto 1990; Tiggeman, 1992).

Concerns with body image have been linked to a decrease in self-esteem and an increase in dieting among young women (Hill and Rogers, 1992). This increase in dieting among young women indicates the onset of eating disorders such as anorexia nervosa and bulimia nervosa (Barker and Galambos, 2003; Fear et al., 1996; Lamb et al., 1993).

Numerous researchers have studied body dissatisfaction in recent years (Abel & Richards, 1996; Byrne & Hills, 1996; Cash et al., 2004; Fallon & Rozin, 1985; Fear et al., 1996; Lamb et al., 1993; Tiggeman & Pennington, 1990; Tiggeman, 1992). In this study replicates the findings of Fallon and Rozin. They concluded that 33% of men and 70% of women rate their current figure as larger than ideal and that body dissatisfaction among women is much larger than for men. Fallon and Rozin also found that men judge the female figure they found most attractive as heavier than women's ratings of the ideal body shape.

Hypotheses

The first hypothesis tested in this study is that the ideal body shape for women ($ideal$) is thinner than their current self-assessed body image ($current$) and the perfect body shape for men is heavier than their current body shape.

$$H_{1w} : ideal - current < 0$$

$$H_{1m} : ideal - current > 0$$

Some research has been undertaken to determine generational differences in body shape preferences (Lamb et al., 1993). Tiggeman and Pennington (1990) researched body size dissatisfaction for children, adolescents and adults and found significant differences between the age groups.

It should be noted that they used different types of body image scales for each age group. Lamb et al. (1993) compared two generations and found significance in gender and cohort differences. These cohort differences confirm the recent increase in body dissatisfaction and eating disorders among mainly young women. In this study, the ideal image for females of different ages and the most attractive female body shape, as judged by men, will be determined for various age groups. The second hypothesis of this study is that there is a positive correlation between age and the ideal figure for females and between age and the female body image that men find most attractive.

$$H_2:R_{ideal,age} >0$$

Body Image Experiment Method

Participants

The participants of the survey are unsolicited visitors to the Monash University website. Most participants were students undertaking the psychology course. No socioeconomic information is available. The survey data contains 166 surveys completed between 3 January 2003 and 6 March 2004. Of the respondents, 59 are male and 107 female. Fourteen surveys were only partially completed and were not considered in the analysis. Table 1 shows the age distribution over the full data set. The cohorts taken into account for this survey are men and women between 16 and 30 years of age.

	<16	31–50	16–30	>50	sum
Female	1	46	56	4	107
Male	0	29	24	6	59
Sum	1	85	70	10	166

Table 1: Age profile of survey participants.

Materials

This body image experiment consisted of questionnaire with seven questions regarding body shape, age, gender and possible concerns regarding body shape and dieting. Question 1 (current body shape), 2 (ideal body shape) and question 6 (body shape of the other gender found most attractive) and the questions about gender and age of the online questionnaire were used for analysis. The remaining survey results have not been considered.

Participants compared a set of nine drawings, showing an increasing body size (Figure 2). Subjects scored the first seven questions between one and nine. This type of survey has been widely used in similar research regarding body dissatisfaction (Abel and Richards, 1996; Byrne and Hills, 1996; Fallon and Rozin, 1985; Fear et al., 1996; Hill and Rogers, 1992; Lamb et al., 1993; Tiggeman and Pennington, 1990; Tiggeman, 1992).

Figure 2: Body shape measurement scale.

The independent variables for this experiment are the gender and age of the participants. The dependent variables under consideration are the current body shape (current) the ideal body shape ($ideal$) and the gender found most attractive ($other$).

Of the 16–30 cohort, the results consist of 29 men and 56 women. A random sample of 29 women and all responses submitted by men ensured symmetry in the data. The complete data set determine the correlations between age and ideal female figures for both men and women.

Body Image Statistics

Body Image

The arithmetic mean and standard deviation of the three questions under consideration are summarised in Table 2. The results have not been tested for statistical significance. The results show that for women, the average current figure is larger than the average ideal. The perceived current body shape for men is much closer to the ideal. The percentage of women that considered their current body shape larger than the ideal ($current − ideal > 0$) is 75.9%. Only 37.9% of men thought that their current body shape was larger than their ideal.

	$n$	$current$	$ideal$	difference
Female	29	3.93(1.19)	3.03(0.94)	0.9(0.86)
Male	29	4.14(1.55)	4.03(0.73)	0.1(1.37)

Table 2: Mean and standard deviation of body image.

Attractiveness

The results of this body image experiment also show that the ideal body shape for women increases as the age of the participant’s increases. The data shows a mild positive correlation between ideal body shape and age ($r = 0.3$). The female body shape that men find most attractive also changes slightly as age increases ($r = 0.33$). The ideal female body shape found attractive by men is slightly larger than the female ideal for the cohorts between 16 and 50 years of age, but significantly lower for the group older than 51 (Figure 3).

Figure 3: Attractiveness of the other gender for females and males.

Discussion

The body dissatisfaction value for women found in this survey confirms previous research conducted in this area and is very close to the figure found by Fallon (1985). There is thus no indication that the high body dissatisfaction among young women has been decreasing over the past twenty years. One of the reasons most often cited for this continuing body dissatisfaction among young women is the influence of the media.

The media often reply that they are merely reflecting the ideals of the current generation. Previous research has, however, shown that the press indeed plays a significant role in shaping, rather than reflecting, perceptions of the female body (Turner, 1997).

There seems to be a circularity that needs to be broken to decrease body dissatisfaction among young women and reduce the occurrence of eating disorders. The only group that can take the first step is the media and the fashion industry. It is, however, doubtful that this will happen, given the commercial interests at stake.

Research results

The results of this body image experiment indicate that men are also slightly dissatisfied with their body shape. The ideal body image of men is slightly larger than their current shape (Fallon, 1985; Tiggeman et al., 1992). There are, however, differences in age cohorts for men. Younger men were shown to display positive body dissatisfaction older men a negative body dissatisfaction.

If the outcomes of this survey regarding the body dissatisfaction of men are statistically significant, then there are two possible reasons for the difference in the results. The ideal body image for men could have decreased in the twelve years between this study and the most recent reference cited above. Another reason could be an increase in actual body size. The real body shape for men in this study is indeed slightly larger, and the ideal body shape for men is slightly slimmer than previously reported Lamb et al. (1993).

Different body shape scales should be used to measure body dissatisfaction for the various age groups (Byrne & Hills 1996). Results can change significantly, depending on the type of body scale used (Tiggeman 1992). To test the sensitivity of the results of this study, the age group of 16–30 were divided into 16–21 and 22–30 (Table 2). When looking at the date for these two sub-groups, the results change only slightly. The age groups used in this study are broad, and further refinement could be achieved by using different body image scales.

Only the first part of the first hypothesis for this study has thus been confirmed. Further research into body dissatisfaction among young men needs to be conducted to verify the increase in body dissatisfaction measured in this study.

Conclusions

Fallon (1985) theorised that the difference between ideal body shape for women and the female body shape found desirable by men exists because women are misinformed about the magnitude of thinness that men desire.

This misinformation is, according to Fallon, caused by the prevalence of thin women in the media. They seem to assume that a woman’s primary motivation for preferring thinner bodies is that they want to be attractive to men. This motivation is not necessarily the case, as the desire to be thinner could also be caused by peer pressure from other females. No conclusion can be drawn about the personal motives for wanting to be thinner from the results of this study, nor any of the other studies used for this study.

The results of this body image experiment show that the ideal body shape increases as women get older. The female body shape found ideal by men also increases with age. This result could support the hypothesis proposed by Fallon (1985) which suggests that being attractive to the other gender plays a lesser role in the lives of older women.

Another reason could be that images in the media are mainly of thin young women. The jump in ideal body shape for women over 51 years of age is significant. The body shape found ideal by men of the same age does, however, only increase slightly. One could theorise that, as women reach menopause, they relax their quest for the ideal thin body, while men only marginally relax their preferences.

This study has confirmed most of the findings of earlier research. Further research into male body dissatisfaction is required to verify the results of this study. Also, study into the motivation for young men and women to be thinner is needed to determine how this trend of increasing body dissatisfaction can be turned around.

Hidden Personality according to Sigmund Freud and Carl Rogers

Peter Prevos — Sat, 29 Jan 2005 11:00:00 +0000

Personality is a complex phenomenon in psychology. Sigmund Freud (1856–1939) and Carl Rogers (1905–1987) are widely recognised as two of the most influential psychotherapists of the twentieth century. Both theorised that people have a hidden personality of which they are not aware of. Although both theories are developed through years of clinical experience, they are based on very different assumptions. In this essay, these assumptions on hidden personality will be discussed, and it is argued that Rogers’ theory is to be preferred over the Freudian model because it is more in tune with findings of contemporary scientific research.

Sigmund Freud

The basic assumption of Freud’s psychoanalytic view of the person is an energy system in which all mental processes are considered to be energy flows, which can flow freely or can get sidetracked or dammed up. Freud argues that the goal of all behaviour is the reduction of tension through the release of energy, which produces pleasure. People function by hedonistic principles, seeking unbridled gratification of all desires. The endless pursuit of happiness is, however, in conflict with society and civilisation, as the uncontrolled satisfaction of pleasure is not accepted. In Freud’s view, humans are primarily driven by sexual and aggressive instincts. Sexual and aggressive energy prevented from expression more directly is converted to cultural activities such as art and science. Energy used for cultural purposes is, however, no longer available to sexual purposes and Freud concludes that the price of civilisation is misery, the forfeit of happiness and a sense of guilt (Pervin 1997).

Sigmund Freud.

Freud’s theory of personality is based on the idea that much of human behaviour is determined by forces outside awareness. The relation between the person and society is controlled by primitive urges buried deep within ourselves, forming the basis of the hidden self. Freud argues that much of our psychic energy is devoted either to finding acceptable expressions of unconscious ideas or to keeping them unconscious. Freud constructed his concept of the unconscious from analysis of slips of the tongue, dreams, neuroses, psychoses, works of art and rituals (Pervin 1997). In psychoanalytic theory, mental life is divided into three levels of awareness. The most considerable portion of the mind is formed by the unconscious-system and only a tiny part by the conscious. The preconscious-system stands like a partition screen between the unconscious-system and consciousness. (Ekstrom, 2004). The conscious mind is like the tip of an iceberg, with its most significant part—the unconscious—submerged. Psychoanalytic theory is fundamentally a motivational theory of human behaviour. Freud claimed that “psychoanalysis aims at and achieves nothing more than the discovery of the unconscious in mental life” (Cited in Pervin 1997: 71).

Carl Rogers

Humanist psychologist Carl Rogers opposed psychoanalytic personality theory. He was dissatisfied with the ‘dehumanising nature’ of this school of thought (Pervin 1997). The central tenet of humanistic psychology is that people have drives that lead them to engage in activities resulting in personal satisfaction and a contribution to society: the actualising tendency. This tendency is present in all organisms and can be defined as the motivation present in every life form to develop its potentials to the fullest extent. Humanistic psychology is based on an optimistic view of human nature, and the direction of people’s movement is basically towards self-actualisation. Some might criticise Rogers as being a naive optimist and point out the violent history of humanity. Rogers defends his view by referring to the fact that his theory is based on more than twenty-five years of experience in psychotherapy (Pervin 1997).

Carl Rogers.

A person’s identity is formed through a series of personal experiences. These experiences reflect how the individual is perceived by both him or herself and the outside world—the phenomenological field. Individuals also have experiences of which they are unaware, and the phenomenological field contains both conscious and unconscious perceptions. The concept of the self is, according to Rogers, however, primarily conscious. The most important determinants of behaviour are the one’s that are conscious or are capable of becoming aware. Roger argues that a definition of the self that includes a reference to the unconscious (as with Freud) can not be objectively studied as it can not be directly known (Pervin 1997).

Rogerian personality theory distinguishes between two personalities. The real self is created through the actualising tendency, it is the self that one can become. The demands of society, however, do not always support the actualising tendency, and we are forced to live under conditions that are out of step with our preferences. The ideal self is the ideal created through the demands of society. Rogers does not see it as something to strive form (that is the real self) but a standard imposed on us we can never fully reach (Boeree 1998). Rogers’ view of ‘hidden’ personality relates to the person one could be given the right circumstances within society and for an individual to be truly happy (and for self-actualisation to be realised) their public and private selves must be as similar as possible. For an individual to be truly happy and for self-actualisation to be realised, the public and hidden selves must be as identical as possible. Rogers believed that when all aspects of a person’s life, surroundings and thoughts are in harmony and thus, the ideal state of congruence is reached (Pervin 1997).

Foundations of Personality

Freud and Roger’s theories of personality are based on some very different assumptions. Their concept of human nature and the role of rationality in human motivation are diametrically opposed. Although both theories include a hidden personality, both concepts are very different in that for Freud, it is our natural state. At the same time, for Rogers, it is the self, created by the demands of society.

Human Nature

Freud theorised that people have an unconscious mind that would, if permitted, manifest itself in incest, murder and other activities which are considered crimes in contemporary society. Freud believes that neuroticism is a result of tensions caused by suppression of our unconscious drives, which are fundamentally aggressive towards others (Pervin 1997).

Rogers agrees that we may behave aggressive and violent at times, but at such times we are neurotic and are not functioning as fully developed human beings (Pervin 1997). Rogers reverses Freud’s concept of neuroticism and thinks that what Freud has construed as our natural state of being is actually unnatural and unhealthy behaviour. For Rogers, the core of our nature is essentially positive and aligned towards self-actualisation, while for Freud, we solely are driven by sexual and aggressive instincts. Recent research support Rogers’ point of view as it has shown that people with an optimistic style of thought tend to cope more effectively with stress than do people who have a pessimistic tone (Gray 2002).

Reason in Human Behaviour

Revolutions in the history of science have one common feature: they deconstruct our convictions about our own self-importance. Copernicus moved our home from centre of the universe to its periphery, Darwin relegated us to descent from an animal world, and Freud discovered the unconscious and deconstructed the myth of an entirely rational mind (Ekstrom 2004). In Freud’s view, human beings are basically irrational, and the unconscious mind is a-logical. We are forever driven by irrational, practically uncontrollable unconscious instincts that are the ultimate cause of all activity (Pervin 1997).

Rogers sees human beings as basically rational, and behaviour is controlled through reason. Rationality and the actualising tendency are inseparably connected in Rogers’ image of personality (Ziegler 2002). Human behaviour is, according to Rogers: “exquisitely rational, moving with subtle and ordered complexity toward the goals the organism is endeavouring to achieve” (Cited in Ziegler 2002: 82). The natural course of the actualising tendency is, however, often blocked by psychosocial conditions. When this happens, people become estranged from their true nature and may behave irrationally through anti-social and destructive behaviour.

Hidden Personality

The Freudian concept of the unconscious mind was never experimentally verified by him and remained a theoretical construct. Critical questions about what is available to immediate observation and what occurs unconsciously could never be fully answered by Freud as he did not possess any of the current day technological possibilities. Through contemporary cognitive science, it has been discovered that most of our thought actually is unconscious, not in the Freudian sense of being repressed, but in the sense that it operates beneath the level of cognitive awareness, inaccessible to consciousness and operating too fast to focus on (Ekstrom 2004).

Unconscious processing goes on in the mind of humans, not because we have to filter out threatening stimuli and impulses, but because many cognitive operations go on without conscious participation. The brain operates in this way in order not to flood the conscious part of the mind with impressions. The unconscious is a type of process, a form of constructing perception, memories and other kinds of cognition, not a portion of the mind (Ekstrom 2004). This view agrees with Roger’s concept of the unconscious, who theorised that the unconscious is only a part of the phenomenological field and does not control our personality.

Conclusion

Both Freud’s and Rogers’ theories of personality are based on some fundamental assumptions and occupy opposite ends of the spectrum of views on human motivation. When comparing both approaches with contemporary research in cognitive psychology, Rogers’ personality is to be preferred over the Freudian model.

References

Boeree, C. G. (1998). Personality theories: Carl Rogers.
Ekstrom, S. R. (2004). The mind beyond our immediate awareness: Freudian, Jungian and cognitive models of the unconscious. Journal of Analytical Psychology, 49, 657–682.
Gray, P. (2002). Psychology (4 ed.). New York: Worth Publishers.
Pervin, L., & Oliver, O., 1997). Personality: Theory and research (7th ed.). New York: John Wiley.
Ziegler, D. J. (2002). Freud, Rogers and Ellis: A comparative theoretical analysis. Journal of Rational-Emotive & Cognitive Behaviour Therapy, 20(2), 75–91.