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Alexina Campbell
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Click here to download a Printable pdf version of this page. We live in a space with three different degrees of freedom for movement. We can go to the left or to the right.

Any point in our space can be reached by combining the three possible types of motion. Up / down motions are hard for humans. We are tied to the surface of the Earth by gravity.

Space is more 3-D for a bird or a fish than it is for us. Two dimensions requires only two numbers to specify the location of any point. There are two degrees of freedom in 2-D.

1 Click here to download a Printable pdf version of this page. We live in a space with three different degrees of freedom for movement. We can go to the left or to the right. We can go forward or backward. We can go up or we can go down. We are allowed no more options. Any movement we make must be some combination of these degrees of freedom. Any point in our space can be reached by combining the three possible types of motion. Up / down motions are hard for humans. We are tied to the surface of the Earth by gravity. Hence it is not hard for us to walk along the surface anywhere not obstructed by objects, but we find it difficult to soar upwards and then downwards. Space is more 3-D for a bird or a fish than it is for us. Two dimensions requires only two numbers to specify the location of any point. There are two degrees of freedom in 2-D. Three dimensions requires three numbers to specify the location of any point. There are three degrees of freedom in 3-D. Note that the direction outwards is represented by a slanted line. (1 of 14) [04/10/2002 5:27:50 PM]

2 Brief history of ideas on dimensionality. Euclid: In the formulation of Euclidean geometry a fourth dimension was not even considered. Aristotle: He was the first person to state categorically that the fourth dimension is impossible. In his work 'On Heaven' he wrote, "The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all." Ptolemy:(A.D.150) In his book 'On Distance' Ptolemy gave a 'proof' that the fourth dimension is impossible. Draw three mutually perpendicular lines he suggested. Try to draw another line perpendicular to all of these lines. It is impossible. The fourth perpendicular line is "entirely without measure and without definition." The fourth dimension is impossible. This is really not a legitimate proof of the 4th dimension. It is merely a proof that we cannot visualize the 4th dimension. Riemann: On June 10, 1854 a new way of looking at geometry was put forward in a famous lecture by the mathematician Bernhard Riemann. He generalized Euclidean geometry to a non-euclidean geometry allowing for curved surfaces and any number of higher dimensions. Riemann's ideas started many people thinking about higher dimensions. People found that the ramifications of the existence of higher dimensions were astounding. If you could manipulate the fourth or higher dimensions you would have god-like powers. You could walk in such a way that no wall could stop you. (2 of 14) [04/10/2002 5:27:50 PM]

3 You would appear to others to be passing through walls or doors. When hungry you could simply reach into the refrigerator, without opening the door. You could extract a section of an orange without peeling it. You could do surgery without cutting skin. You could disappear and reappear at will. You would be able to see people who had been buried by an avalanche. The standard method by which people try to understand higher dimensions is to try to see how a lower-dimensional creature would see a three-dimensional world. For this purpose 2-D worlds are often used. Let us consider a 2-D world. To jail a criminal in such a world a circular boundary would have to be placed around the criminal. To extricate the criminal, all a 3-D creature has to do is to peel him off the 2-D world, and redeposit him elsewhere on his world. This feat, which is quite ordinary in 3-D, appears fantastic in 2-D. No one in the 2-D world understands what the up direction means. The internal organs of a 2-D creature would be visible to us. It would be trivial to reach inside a 2-D creature and perform surgery without cutting the skin. Viewing this 2-D flatland, notice that we are omnipotent. The 2-D creature cannot hide from us. He would see us as having magical powers. In the later part of the 1800's the idea of a fourth dimension became very popular. In 1877, a scandalous trial in London gave the idea of extra dimensions international notoriety. A magician and psychic by the name of Henry Slade was arrested for fraudulently using palmistry, etc, to deceive his clients. Prominent physicists of the time came to Slade's defense claiming that his psychic feats actually proved that he could summon spirits from the fourth dimension. Detractors said that scientists, because they are trained to trust their senses, are the worst possible people to evaluate a magician. To objectively test a magician/psychic you need another magician. They will know when any tricks are being made. Flatland In 1884 a headmaster in London, named Edwin Abbott ( ) published a satirical novel called 'Flatland: A Romance of Many dimensions.' This book works on several levels outlined below. (3 of 14) [04/10/2002 5:27:50 PM]

Flatland is a story that can interpreted several ways. (1) It is a satire on the staid and heartless Victorian society, a place filled with bigotry and suffocating prejudice.

4 Flatland is a story that can interpreted several ways. (1) It is a satire on the staid and heartless Victorian society, a place filled with bigotry and suffocating prejudice. "Irregulars" (cripples) are put to death, women have no rights at all, and when the protagonist in the story Mr. A. Square tries to teach his fellows about the third dimension, he is imprisoned. (2) It is a scientific work. By thinking about A. Square's difficulties in understanding the third dimension, we become better able to deal with our own problems with the fourth (3) At the deepest level, we can perhaps view Flatland as Abbott's roundabout way to talk about some intense spiritual experiences Flatland is a plane inhabited by creatures that slide about. The lower classes in Flatland are triangles with only two sides equal. The upper classes are objects with all equal sides. The more sides one has in Flatland, the greater one's social standing. The highest caste of all are objects which have so many small equal sides that they are indistinguishable from perfect circles. The pure circles are the high priests. Women in Flatland are not even skinny triangles, they are but lines, infinitely less respected than the priestly circles. Discussion of the third dimension in Flatland is strictly forbidden. Mr. Square's life is disrupted one day by a 3-D creature called Lord Sphere. Lord Sphere manifests himself as a (4 of 14) [04/10/2002 5:27:50 PM]

5 circle in the 2-D world, that can magically change size. Lord Sphere peels Mr. Square off the 2-D Flatland and hurls him into 'spaceland.' Mr. Sphere only sees the cross sections of 3-D objects. Things appear and disappear and change shape rapidly. When Mr. Square is returned to Flatland he tells others of his experience. He is jailed and put into solitary confinement as punishment for telling others that the third dimension exists. The years 1890 to 1910 may be thought of as the golden years of the 4th dimension. Ideas about higher dimensions permeated literary (5 of 14) [04/10/2002 5:27:50 PM]

In particular this took the form of an artistic revolt against the laws of perspective. In the Middle Ages religious art was distinctive for its deliberate lack of perspective.

6 circles, the avant garde, and the thoughts of the general public, affecting trends in art, literature, and philosophy. Some art historians have argued that the fourth dimension crucially influenced the development of Cubism and Expressionism in the art world. In particular this took the form of an artistic revolt against the laws of perspective. In the Middle Ages religious art was distinctive for its deliberate lack of perspective. These pictures were full of flat people and flat surroundings. This art reflected the church's view that God was omnipotent and could therefore see all parts of our world equally. There was no need for perspective in God's view of things. Hence according to the church art had to reflect God's point of view. All paintings had to be two dimensional. Art of the Middle Ages: The Bayeux Tapestry Renaissance art was a revolt against this restricted form of art. Perspective in art began to be much more popular. Renaissance Art: Leonardo Da Vinci's painting 'The Last Supper' (6 of 14) [04/10/2002 5:27:50 PM]

Cubist art revolted against the restrictions that perspective imposed. Picasso's art shows a clear rejection of the perspective, with women's faces viewed simultaneously from several angles.

7 Cubist art revolted against the restrictions that perspective imposed. Picasso's art shows a clear rejection of the perspective, with women's faces viewed simultaneously from several angles. Picasso's paintings show multiple perspectives, as though they were painted by someone from the 4th dimension, able to see all perspectives simultaneously. Cubist Art: Picasso's painting 'Portrait of Dora Maar' In Europe at the end of the 1800's talking about the 4th dimension was the in thing at parties and other social gatherings. Eventually the ideas of the fourth dimension crossed the Atlantic and came to the United States. The main proponent of all things 4-dimensional (7 of 14) [04/10/2002 5:27:50 PM]

was a colorful English mathematician named Charles Hinton. Hinton spent his entire adult life obsessed with the notion of popularizing and visualizing the fourth dimension.

8 was a colorful English mathematician named Charles Hinton. Hinton spent his entire adult life obsessed with the notion of popularizing and visualizing the fourth dimension. In 1885, he was arrested for bigamy in England and put on trial. He was imprisoned for three days. Shortly after that he left for Japan, eventually ending up in the U.S. in Hinton wanted there to be a name for going one way or the other in the fourth dimension just like : forward and backward left and right up and down His corresponding made-up words for the fourth dimension was ANA and KATA. If you were going in the ana direction in the fourth dimension then you were going in the opposite direction from someone going in the kata direction. Hinton was also the person who thought up the name for the 4-dimensional hypercube. He called it the TESSERACT. This hypercube is the generalization of the three-dimensional cube. All of its sides must be of the same length. There are three ways of visualizing the Tesseract. (1) The unfolding analogy method. (2) The method of looking at the shadow in a lower dimension. (3) The slicing method (i.e., use cross-sections). Method 1 Suppose you have to make a model so that a Flatlander can visualize what a 3-D cube looks like. One way of doing this is to unfold the cube and stretch it out flat. The 2-D creature now will at least be able to observe the sides of the full object and the 2-D creature will be able to get some feeling of what it looks like when it is folded back into it's usual 3-D shape. Note that when the 2-D creature watches as the cube is folded up once again, she will see in the end only one face still existing in her 2-D world What would a unfolded hypercube look like stretched out in 3-D? Instead of stretched out areas as there are in the diagram above there should be stretched out volumes. Just as the areas above are 2-D squares, the volumes that would emerge with an unfolded hypercube should be 3-D cubes. The unfolded 3-D representation of a hypercube should look like the following object. (8 of 14) [04/10/2002 5:27:50 PM]

The unfolded hypercube stretches out to an arrangement of eight 3-D cubes.

This form of the hypercube projection has been made into kites as shown below.

(Click to make bigger) Method 2 Another way to show what the 3-D cube actually looks like to a 2-D creature would be to shine a light on the object to

9 The unfolded hypercube stretches out to an arrangement of eight 3-D cubes. Note that when folded back again the hypercube leaves only one cube left in our 3-D world. This form of the hypercube projection has been made into kites as shown below. (Click to make bigger) It has also shown up in art as in the painting by Salvador Dali as shown below. (Click to make bigger) Method 2 Another way to show what the 3-D cube actually looks like to a 2-D creature would be to shine a light on the object to see how its shadow projects on a 2-D surface. The 2-D creature would then try to infer from the shadow pattern what the 3-D object might look like. Note that there are six bounded areas in the 2-D creatures space coming from the 3-D object. This corresponds to the six areas that were present in the unfolding case discussed above. (9 of 14) [04/10/2002 5:27:50 PM]

Method 3 The last way to show what the 3-D cube actually looks like to a 2-D creature would be to slice the 3-D cube up into areas and give each slice to the 2-D

10 What is the analogous situation for the hypercube? Above there were 8 volumes. We expect that this should still be true this time. There should be 8 bounded volumes in the 'shadow' projection of the 4-D object to 3-D space. Method 3 The last way to show what the 3-D cube actually looks like to a 2-D creature would be to slice the 3-D cube up into areas and give each slice to the 2-D creature to analyze. A slice of a 3-D cube is just a square. A cube is many squares all compressed together. Hence any cube can be decomposed into a sequence of square slices. (10 of 14) [04/10/2002 5:27:50 PM]

A slice of an hypercube is going to be a generalization of the slice of the 3-D cube. The slices should be volumes instead of areas.

This design for the hypercube was first put forward by Claude Bragdon in his 1913 book "A Primer of Higher Space.

The Chamber of Commerce Building in Rochester, New York, is one of his buildings based on 4-D ideas.

11 A slice of an hypercube is going to be a generalization of the slice of the 3-D cube. The slices should be volumes instead of areas. To draw the sliced version of the hypercube you first draw two side by side cubes. Then connect each corresponding point on each cube. This design for the hypercube was first put forward by Claude Bragdon in his 1913 book "A Primer of Higher Space." Bragdon was an architect who incorporated this and other 4-D designs into some of his buildings. The Chamber of Commerce Building in Rochester, New York, is one of his buildings based on 4-D ideas. For a animated hypercube click here: ROTATING HYPERCUBE. A good link for the discussion of 4-D structures is (11 of 14) [04/10/2002 5:27:50 PM]

12 Criticism of 4-D ideas In retrospect, the ideas of Riemann put forward in 1854, were popularized to a wide audience in the late 1800's and the early 1900's by mystics, philosophers and artists. Our understanding of nature was not measurably increased by these early attempts at analyzing higher dimensions as Riemann had hoped. There was no attempt to use hyperspace to simplify the laws of nature. No attempt was made to use Riemann's mathematics in an applied way. It became a branch of pure mathematics. There was no experimental confirmation of the 4th dimension. There was no good physical justification of the 4th dimension; ghost stories weren't enough. Within a few decades of the peak of interest in the extra 4th space dimension, Einstein related time to the 4th dimension. This changed the way that people viewed the 4th dimension. Modern Resurgence of Interest in Higher Dimensions In physics the extra space dimension was to come back. In 1919, Einstein received a letter from Theodor Kaluza. The letter astounded him because Kaluza had successfully generalized Einstein's theory of gravity to five dimensions ( one time dimension plus 4 space dimension). What amazed Einstein was that by adding one extra space dimension Kaluza obtained, not only a theory of gravity, but a theory of electromagnetism as well. These ideas never received widespread attention and fell by the wayside. In the 1980's, spurred on by a desire to construct a theory of nature that could incorporate all of the fundamental forces, physicists resurrected the higher-dimensional ideas of Kaluza, Einstein and others. Since then the theory of higher dimensions has become a well respected tool for the attempt to describe nature in a unified way. Today one of the most respected attempts to describe all of the fundamental forces in a unified theory is called the Theory of Superstrings. This theory assumes that the most fundamental entities are string-like bundles of energy that all fundamental particles can be constructed from. The standard superstring theory assumes that (12 of 14) [04/10/2002 5:27:50 PM]

13 space has 9 spatial dimensions and that there is one extra dimension for time. Six of the 9 extra dimensions are curved up into a very small radius. This supposedly happened very early in the evolution of the universe. Hence, we now live in a 3-D universe. Dimensions and Self-aware Subsystems Arguments that the only life is 3-D life are common. When the dimensionality of space is greater than three, people have shown that if similar forms of electromagnetism and gravity exist in the higher dimensional world then atoms will not hold together. The reasoning then goes that if atoms can't form, how can living things that must be made from atoms exist? Obviously they can't, so therefore there are no higher dimensional lifeforms. When the dimensionality of space is less than three dimensions it has been argued that neurological systems for any 2-D creatures would not be complex enough. Nerves would cross in 2-D creatures, making thought transmission very difficult. The computer scientist A.K. Dewdney from the University of Western Ontario has proposed that this objection can be gotten around by the presence of switching nodes in the 2-D brain. When a thought has to be transmitted it gets switched through the crossed grid of neural paths just as car traffic gets switched through intersections by traffic lights. Another problem with 2-D creatures is that digestion might be a problem. A digestive tract based on a digestive tract that 3-D creatures have would result in the 2-D creature being cut into two pieces. (13 of 14) [04/10/2002 5:27:50 PM]

This objection to 2-D life, however, can easily be countered. We can allow the 2-D creature to be constructed like some flatworms, which have only one opening in their digestive tracts.

14 This objection to 2-D life, however, can easily be countered. We can allow the 2-D creature to be constructed like some flatworms, which have only one opening in their digestive tracts. These creatures eat food and expel wastes from the same opening. Alternatively, the 2-D creatures can have a 'self-gripping gut.' Each side of the digestive tube would have interlocking projections. This zipper-like structure is open at the mouth end when the creature eats. As the food passes through the body, the zipper closes behind the food and opens ahead of it. The body stays together this way. It is not clear whether any arguments we put forward about different dimensional life are correct since they are somewhat anthropocentric. Why should energy structures in disjoint dimensionally different universes be the same? If they are not the same, then any analysis that people have done to point out the uniqueness of 3-D is flawed. (14 of 14) [04/10/2002 5:27:50 PM]

We are grateful to Hester Lewellen and Pam McKee (Baldwin Wallace College) whose own Study Guide provided many of these questions.

æ Study Questions. We are grateful to Hester Lewellen and Pam McKee (Baldwin Wallace College) whose own Study Guide provided many of these questions. 1. Of the Nature of Flatland a. What appearance do