From Abracadabra to Zeno's Paradoxes

Sample Entries

Alhambra | Banach-Tarski paradox | Boole, Alicia | bridges of Königsberg | Brouwer fixed-point theorem | Császár polyhedron | Eddington number | fly-between-trains problem | fourth dimension | hairy ball theorem | ham sandwich theorem | Hamilton, William Rowan | Hinton, Charles Howard | Ishango bone | Kepler-Poinsot solids | loculus of Archimedes | Menger sponge | Möbius band | Peano curve | Senet | sphericon


The former palace and citadel of the Moorish kings of Granada, and perhaps the greatest monument to Islamic mathematical art on Earth. Because the Qur'an considers the depiction of living beings in religious settings blasphemous, Islamic artists created intricate patterns to symbolize the wonders of creation: the repetitive nature of these complex geometric designs suggested the limitless power of God. The sprawling citadel, looming high above the Andalusian plain, boasts a remarkable array of mosaics with tiles arranged in intricate patterns. The Alhambra tilings are periodic; in other words, they consist of some basic unit that is repeated in all directions to fill up the available space. All 17 different groups of isometries – the possible ways of repeatedly tiling the plane – are used at the palace. The designs left a deep impression on Maurits Escher, who came here in 1936. Subsequently, Escher's art took on a much more mathematical nature and over the next six years he produced 43 colored drawings of periodic tilings with a wide variety of symmetry types.

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Banach-Tarski paradox

There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.
-William Shakespeare

A seemingly bizarre and outrageous claim that it is possible to take a ball, break into a number of pieces and then reassemble those pieces to make two identical copies of the ball. The claim can be made even stronger: it is possible to decompose a ball the size of a marble and then reassemble the pieces to make another ball the size of the Earth, or, indeed, the size of the known universe!

Before writing off Messieurs Banach and Tarski as being either very bad mathematicians or very good practical jokers, it's important to understand that this is not a claim about what can actually be done with a real ball, a sharp knife, and some dabs of glue. Nor is there any chance of some entrepreneur being able to slice up a gold ingot and assemble in its place two new ones like the original. The Banach-Tarski Paradox tells us nothing new about the physics of the world around us but a great deal about how "volume," "space," and other familiar-sounding things can assume unfamiliar guises in the strange abstract world of mathematics.

Stefan Banach and Alfred Tarski announced their startling conclusion in 1924, having built on earlier work by Felix Hausdorff who proved that it's possible to chop up the unit interval (the line segment from 0 to 1) into countably many pieces, slide these bits around, and fit them together to make an interval of length 2. The Banach-Tarski paradox, which mathematicians often refer to as the Banach-Tarski decomposition because it's really a proof not a paradox, highlights the fact that among the infinite set of points that make up a mathematical ball, the concept of volume and of measure can't be defined for all possible subsets. What this boils down to is that quantities that can be measured in any familiar sense are not necessarily preserved when a ball is broken down into subsets and then those subsets reassembled in a different way using just translations (slides) and rotations (turns). These unmeasurable subsets are extremely complex, lacking reasonable boundaries and volume in the ordinary sense, and thus are not attainable in the real world of matter and energy. In any case, the Banach-Tarski paradox doesn't give a prescription for how to produce the subsets: it only proves their existence and that there must be at least five of them to produce a second copy of the original ball. The fact that the Banach-Tarski paradox depends on the axiom of choice (AC), yet is so strongly counterintuitive, has been used by some mathematics to suggest that AC must be wrong; however, the benefits of adopting AC are so great that such black sheep of the mathematical family as the paradox are generally tolerated.

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Boole (Stott), Alicia (1860-1940)

The third daughter of George Boole and an important mathematician in her own right. At the age of 18, she was introduced to a set of wooden cubes devised by her brother-in-law Charles Hinton as an aid to visualization of the fourth dimension. Despite having had no formal education, she surprised everyone by becoming adept with the cubes and developing an amazing feel for four-dimensional geometry. She introduced the word "polytope" to describe a four-dimensional convex solid, and went on to explore the properties of the six regular polytopes and to make 12 beautiful card models of their three-dimensional central cross-sections. She sent photographs of these models to the Dutch mathematician Pieter Schoute (1846-1923), who had done similar work and with whom she subsequently published two papers. The models themselves are now housed in the Department of Pure Mathematics and Mathematical Statistics at Cambridge University.

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bridges of Königsberg

A famous routing problem that was analyzed and solved by Leonhard Euler in 1736, and that helped spur the development of graph theory. The old city of Königsberg, once the capital of East Prussia, is now called Kaliningrad, and falls within a tiny part of Russia known as the Western Russian Enclave, between Poland and Lithuania, which (to the surprise even of many modern Russians) is not connected with the rest of the country! Königsberg lay some four miles from the Baltic Sea on rising ground on both sides of the river Pregel (now the Pregolya), which flowed through the town in two branches before uniting below the Grune Brocke (Green Bridge). Seven bridges (numbered in the diagram) crossed the Pregel and connected various parts of the city (letters A to D), including Kneiphof Island (B), the site of Königsberg University and the grave of its most famous son, the great philosopher Emmanuel Kant (1724-1804).

A question arose among the town's curious citizens: Was it possible to make a journey across all seven bridges without having to cross any bridge more than once? No one had been able to do it, but was there a solution? Euler, who was in St. Petersburg, Russia, at the time, heard about this puzzle and looked into it. In 1736 he published a paper called "Solutio problematis ad geometriam situs pertinentis" (The solution of a problem relating to the geometry of position) in which he gave his answer. Euler reasoned that, for such a journey to be possible, each land mass would need to have an even number of bridges connected to it, or, if the journey began at one land mass and ended at another then those two land masses alone could have an odd number of connecting bridges while all the other land masses would have to have an even number of connecting bridges. Since the Königsberg bridges violated this layout, a grand tour that involved only one crossing per bridge was impossible. Euler's paper was important because it solved not just the Königsberg conundrum but the much more general case of any network of points, or vertices, that are connected by lines, or arcs. What is more, the words "geometry of position" in the title shows that Euler realized that he was dealing with a different type of geometry where distance is irrelevant; so this work can be seen as a prelude to the subject of topology. See also Euler path.

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Brouwer fixed-point theorem

An amazing result in topology and one of the most useful theorems in mathematics. Suppose there are two sheets of paper, one lying directly on top of the other. Take the top sheet, crumple it up, and put it back on top of the other sheet. Brouwer's theorem says that there must be at least one point on the top sheet that is in exactly the same position relative the bottom sheet as it was originally. The same idea works in three dimensions. Take a cup of coffee and stir it as much as you like. Brower's theorem insists that there must be some point in the coffee that is in exactly the same spot as it was before you started stirring (though it might have moved around in between). Moreover, if you stir again to move that point out of its original position, you can't help but move another point back into its original position! Not surprisingly, the formal definition of Brouwer's theorem makes no mention of sheets of paper or cups of coffee. It states that a continuous function from an n-ball into an n-ball (that is, any way of mapping points in one object that is topologically the same as the filling of an n-dimensional sphere to another such object) must have a fixed point. Continuity of the function is essential: for example, if you rip the paper in the example above then there may not be a fixed point.

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Császár polyhedron

A polyhedron, first described in 1949 by the Hungarian mathematician Ákos Császár, that is a solution to an interesting problem, namely: How many polyhedra exist such that every pair of vertices is joined by an edge? The first clear example is the well known tetrahedron (triangular pyramid). Some simple combinatorics specify how many vertices, edges, faces, and holes such polyhedra must have. It turns out that, other than the tetrahedron, any such polyhedron must have at least one hole. The first possible polyhedron beyond the tetrahedron has exactly one hole; this is the Császár polyhedron, which is thus topologically equivalent to a torus (donut). The Császár polyhedron has 7 vertices, 14 faces, and 21 edges, and is the dual of the Szilassi polyhedron. It isn't known if there are any other polyhedra in which every pair of vertices is joined by an edge. The next possible figure would have 12 faces, 66 edges, 44 vertices, and 6 holes, but this seems an unlikely configuration—as, indeed, to any even greater extent, does any more complex member of this curious family.

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Eddington number

"I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231, 425,076,185,631,031,296 protons in the universe and the same number of electrons." So wrote the English astrophysicist Sir Arthur Eddington (1882-1944) in his book Mathematical Theory of Relativity (1923). Eddington arrived at this outrageous conclusion after a series of convoluted (and wrong!) calculations in which he first "proved" that the value of the so-called fine-structure constant was exactly 1/136. This value appears as a factor in his prescription for the number of particles (protons + electrons; neutrons were not discovered until 1930) in the universe: 2×136×2256 = 17×2260 ~ 3.149544...×1079 (double the number written out in full in the quote above). This is the Eddington number, notable for being the largest specific integer (as opposed to an estimate or approximation) ever thought to have a unique and tangible relationship to the physical world. Unfortunately, experimental data gave a slightly lower value for the fine-structure constant, closer to 1/137. Unfazed, Eddington simply amended his "proof" to show that the value had to be exactly 1/137, prompting the satirical magazine Punch to dub him "Sir Arthur Adding-One." See also large numbers.

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fly-between-trains problem

Two trains are approaching each another and a fly is buzzing back and forth between the two trains. Given the (constant) speed of the trains and their initial separation distance, and the (constant) speed of the fly, calculate how far the fly will travel before the trains collide. This problem appears to have been first posed by Charles Ange Laisant (1840-1921) in his Initiation Mathématique. There is a long-winded method of getting the answer and a much shorter way. Suppose the trains start out 200 miles apart and are each traveling at 50 miles per hour, and the fly-a speedster of its kind-is moving at 75 miles per hour. The long method involves considering the length of the back-and-forth path that the fly takes and evaluating this as the sum of an infinite series. The quick solution is to notice that the trains will collide in 2 hours and that in this time the fly will travel 2 x 75 = 150 miles! When this problem was put to John von Neumann, he immediately gave the correct answer. The poser, assuming he had spotted the shortcut, said: "It is very strange, but nearly everyone tries to sum the infinite series." Von Neumann replied: "What do you mean, strange? That's how I did it!"

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fourth dimension

"Do you think that there are things which you cannot understand, and yet which are; that some people see things that others cannot?" said Dr. Van Helsing in Bram Stoker's Dracula. Instead of vampires, he may just as easily have been talking about the fourth dimension-an extension at right-angles to the three familiar directions of up-down, forward-backward, and side-to-side. In physics, especially relativity theory, time is often regarded as the fourth dimension of the spacetime continuum in which we live. But what meaning can be attached to a fourth spatial dimension? The mathematics of the fourth dimension can be approached through a simple extension of either the algebra or the geometry of one, two, and three dimensions.

Algebraically, each point in a multidimensional space can be represented by a unique sequence of real numbers. One-dimensional space is just the number line of real numbers. Two-dimensional space, the plane, corresponds to the set of all ordered pairs (x, y) of real numbers, and three-dimensional space to the set of all ordered triplets (x, y, z). By extrapolation, four-dimensional space corresponds to the set of all ordered quadruplets (x, y, z, w). Linked to this concept is that of quaternions, which can also be viewed as points in the fourth dimension.

Geometric facts about the fourth dimension are just as easy to state. The fourth dimension can be thought of as a direction perpendicular to every direction in three-dimensional space; in other words, it stretches out along an axis, say the w-axis, that is mutually perpendicular to the familiar x-, y-, and z-axes. Analogous to the cube is a hypercube or tesseract, and to the sphere is a 4-d hypersphere. Just as there are five regular polygons, known as the Platonic solids, so there are six four-dimensional regular polytopes. They are: the 4-simplex (constructed from five tetrahedra, with three tetrahedra meeting at an edge); the tesseract (made from eight cubes, meeting three per edge); the 16-cell (made from 16 tetrahedra, meeting four per edge); the 24-cell (made from 24 octahedra, meeting three per edge); the 120-cell (made from 120 dodecahedra, meeting three per edge); and the monstrous 600-cell (made from 600 tetrahedra, meeting five per edge). Geometers have no difficulty in analyzing, describing, and cataloging the properties of all sorts of 4-d figures. The problem starts when we try to visualize the fourth dimension. This is a bit like trying to form a mental picture of a color different from any of those in the known rainbow from red to violet, or a "lost chord," different from any that has ever been played. The best that most of us can hope for is to understand by analogy. For example, just as a sketch of a cube is a 2-d perspective of a real cube, so a real cube can be thought of as a perspective of a tesseract. At a movie, a 2-d picture represents a 3-d world, whereas if you were to watch the action live, in three-dimensions, this would be like a screen projection in four dimensions.

Many books have been written and schemes devised to nudge our imaginations into thinking four-dimensionally. One of the oldest and best is Edwin Abbott’s Flatland written more than a century ago, around the time that mathematical discussion of higher dimensions was becoming popular. H. G. Wells also dabbled in the fourth dimension, most notably in The Time Machine (1895), but also in The Invisible Man (1897), in which the central character drinks a potion "involving four dimensions," and in "The Plattner Story" (1876), in which the hero of the tale, Gottfried Plattner, is hurled into a four spatial dimension by a school chemistry experiment that goes wrong and comes back with all his internal organs switched around from right to left. The most extraordinary and protracted attack on the problem, however, came from Charles Hinton, who believed that, through appropriate mental practice involving a complicated set of colored blocks, a higher reality would reveal itself, "bring[ing] forward a complete system of four-dimensional thought [in] mechanics, science, and art."

Victorian-age spiritualists and mystics also latched on to the idea of the fourth dimension as a home for the spirits of the departed. This would explain, they argued, how ghosts could pass through walls, disappear and reappear at will, and see what was invisible to mere three-dimensional mortals. Some distinguished scientists lent their weight to these spiritualist claims, often after being duped by clever conjuring tricks. One such unfortunate was the astronomer Karl Friedrich Zöllner who wrote about the four-dimensional spirit world in his Transcendental Physics (1881) after attending séances by Henry Slade, the fraudulent American medium.

Art, too, became enraptured with the fourth dimension in the early twentieth century. When the Cubist painter and theorist, Albert Gleizes said, "Beyond the three dimensions of Euclid we have added another, the fourth dimension, which is to say, the figuration of space, the measure of the infinite," he united math and art and brought together two major characteristics of the fourth dimension in early modern art theory – the geometric orientation as a higher spatial dimension and the metaphorical association with infinity. See also Klein bottle.

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hairy ball theorem

If a sphere is covered with hair or fur, like a tennis ball, the hair cannot be brushed so that it lies flat at every point. In mathematical terms: any continuous tangent vector field on the sphere must have a point where the vector is zero. This theorem also means that somewhere on the Earth's surface there has to be a point where the horizontal wind speed is zero, even if it's windy everywhere else. Does the same apply to a torus? Is there a hairy donut theorem? No! The number of "problem points," where the hair would stick up on a surface, is related to a quantity called the Euler characteristic of that surface. Basically, every point on a surface has an index that describes how many times the vector field rotates in a neighborhood of the problem point. The sum of the indices of all the vector fields is the Euler characteristic. Since the torus has Euler number 0, it is possible to have a covering of hair – a vector field – on it that lies flat at every point.

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ham sandwich theorem

Given a sandwich in which bread, ham, and cheese (three finite volumes) are mixed up, in any way at all, there is always a flat slice of a knife (a plane) that bisects each of the ham, bread, and cheese. In other words, however messed up the sandwich — even if it's been in a blender – you can always slice through it in such a way that the two halves have exactly equal amounts, by volume, of the three ingredients. This theorem generalizes to higher-dimensional ham sandwiches, when it essentially becomes the Borsuk-Ulam theorem: in n-dimensional space in which there are n globs of positive volume, there is always a hyperplane that cuts all the globs exactly in half.

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Hamilton, William Rowan (1805-1865)

An Irish mathematician who, among other things, invented quaternions and a new theory of dynamics. Having excelled in Greek and mathematical physics at Trinity College, Cambridge, Hamilton was appointed Astronomer Royal of Ireland; in this position he served from 1827 to his death and, during all that time, lived in Dunsink Observatory, Dunsink Lane, to the northwest of Dublin. However, he quickly lost interest in staying up at nights to make observations – he hired three of his sisters to help run the place – and preferred instead to write poetry (badly). He was friends with Samuel Coleridge, who introduced him to the philosophy of Kant, which had a great influence on him, and with William Wordsworth, who advised him against writing any more poems.

Hamilton did early work on caustic curves and was led from this to his discovery of the law of least action, which enabled many physical problems to be expressed more elegantly. One of his greatest triumphs was his treatment of complex numbers as pairs of real numbers, an approach that finally exorcised long-standing suspicions about the reality of imaginary numbers, and helped clear the way for other algebras. From this he was led to consider ordered quartets of numbers, which he called quaternions. The idea for quaternions came to Hamilton suddenly on October 16, 1843, while he was standing on Brougham ("Broom") Bridge, where Broombridge Street crosses the Royal Canal, Dublin; a plaque under the bridge, on the towpath, was unveiled by the Taoiseach (head of the Irish parliament), Eamon De Valera, on November 13, 1958. Of his invention, Hamilton wrote:

The quaternion was born, as a curious offspring of a quaternion of parents,
say of geometry, algebra, metaphysics, and poetry. . . I have never been able to
give a clearer statement of their nature and their aim than I have done in two
lines of a sonnet addressed to Sir John Herschel:
"And how the One of Time, of Space the Three
Might in the Chain of Symbols girdled be."

Hamilton's interest in complex numbers was stimulated by his friend and compatriot John Graves, who pointed Hamilton in the direction of John Warren's A Treatise on the Geometrical Representation of the Square Root of Negative Quantities. This book explained the concept of the complex plane, which Hamilton turned from geometry into algebra. One of Hamilton's last inventions was a curiosity called the icosian calculus, which was another outcome of his friendship with Graves. After a visit to the latter's house, Hamilton wrote: "Conceive me shut up and revelling for a fortnight in John Graves' Paradise of Books! of which he has really an astonishingly extensive collection, especially in the curious and mathematical kinds. Such new works from the Continent he has picked up! and such rare old ones too!" Graves posed some puzzles to Hamilton, and either Graves or his books got Hamilton to thinking about regular polyhedra. When Hamilton returned to Dublin he thought about the symmetry group of the icosahedron, and used it to invent an algebra he called the "icosians" and also a game called the Icosian Game. The only complete example of this game, inscribed to Graves, is now in the keeping of the Royal Irish Academy, of which Hamilton was the president from 1837 to 1847. (In early 1996, a second example of the Icosian Game came to light but only included the board.)

In some ways, Hamilton was too far ahead of his time. The operator now referred to as the Hamiltonian and the so-called Hamilton-Jacobi equation that relates waves and particles only became important when quantum mechanics came along, and Felix Klein introduced Erwin Schrödinger, the father of wave mechanics, to Hamilton's work.

Hamilton's personal life was not always happy. He fell deeply in love with a woman named Catherine Disney, who was forced by her parents to marry a wealthy man 15 years older than her. Hamilton remained hopelessly in love with her the rest of his life, though he eventually married someone else. He became an alcoholic, then foreswore drink, then relapsed. Many years after their early romance, Catherine began a secret correspondence with Hamilton. Her husband became suspicious and she attempted suicide by taking laudanum. Five years later, she became seriously ill. Hamilton visited her and gave her a copy of his Lectures on Quaternions; they kissed at last, and she died two weeks later. He carried her picture with him ever afterward and talked about her to anyone who would listen.

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Hinton, Charles Howard (1853-1907)

An English-born mathematician best known for his writings and inventions aimed at helping to visualize the fourth dimension; he may also have coined the name tesseract for the four-dimensional analogue of a cube. Hinton matriculated at Oxford and continued to study there, earning a B.A. (1877) and an M.A. (1886), while he also taught, first at Cheltenham Ladies' School and then, from 1880 to 1886, at Uppingham School. At this time, another teacher at Uppingham was Howard Candler, who was a friend of Edwin Abbott and thus provides a possible link between these two explorers of other dimensions. In the early 1880s Hinton published a series of pamphlets starting with "What is the Fourth Dimension?" and "A Plane World" (a contemporary of Abbott's Flatland), which were reprinted in the two-volume Scientific Romances (1884). Hinton's descriptions owed much to the mathematical models of William Clifford, whose theories about 4-d spaces were then in vogue. But Hinton went much further in his attempts to break free of three-dimensional thought. He devised an elaborate set of small colored cubes to represent the various cross sections of a tesseract and then memorized the cubes and their many possible orientations in order to gain a window on the fourth dimension.

At the time he was teaching in England, Hinton married Mary Everest Boole, the eldest daughter of George Boole, the founder of mathematical logic. Regrettably, he also married a Maud Wheldon and was tried at the Old Bailey in London for bigamy. After serving a day in prison for the offence, he fled with his (first) family to Japan, where he taught for some years, before taking up a post at Princeton University. There, in 1897, he designed a species of baseball gun which, with the help of gunpowder charges, would shoot out balls at speeds of 40 to 70 miles per hour. It was used by the Princeton Nine for several seasons before being abandoned by the players in fear of their lives.

After a brief spell at the University of Minnesota, Hinton joined the Naval Observatory in Washington, D.C. At the same time, he developed more rigorously his ideas on the fourth dimension and presented his results before the Washington Philosophical Society in 1902. What would prove, Hinton asked, the existence of a real fourth spatial dimension? He offered three possibilities, two of which, involving a specific molecular structure and a particular case of electrical induction, have since been explained by science in more mundane ways. However, Hinton’s other case, to do with right and left handedness remains open because there are instances of right and left-handedness in nature, such as the spin of elementary particles, to which his example could be applied. In any event, Hinton’s final assessment that we can only regard a four-dimensional space as possible if three-dimensional mechanics fails to explain known physical phenomena still rings true today. See also Boole (Stott), Alicia.

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Ishango bone

A bone tool handle discovered around 1960 in the African area of Ishango, near Lake Edward. It has been dated to about 9,000 B.C. and was at first thought to have been a tally stick. At one end of the bone is a piece of quartz for writing, and the bone has a series of notches carved in groups on three rows running the length of the bone. The markings on two of these rows each add to 60. The first row is consistent with a number system based on 10, since the notches are grouped as 20 + 1, 20 - 1, 10 + 1, and 10 - 1, while the second row contains the prime numbers between 10 and 20! A third seems to show a method for multiplying by 2 that was used in later times by the Egyptians. Additional markings suggest that the bone was also used a lunar phase counter. The Ishango Bone is kept at the Royal Institute for Natural Sciences of Belgium in Brussels. See also Lebombo bone.

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Kepler-Poinsot solids

The four regular non-convex polyhedra that exist in addition to the five regular convex polyhedra known as the Platonic solids. As with the Platonic solids, the Kepler-Poinsot solids have identical regular polygons for all their faces, and the same number of faces meet at each vertex. What is new is that we allow for a notion of "going around twice," which results in faces that intersect each other. In the great stellated dodecahedron and the small stellated dodecahedron, the faces are pentagrams (five-pointed stars). The center of each pentagram is hidden inside the polyhedron. These two polyhedra were described by Johannes Kepler in 1619, and he deserves credit for first understanding them mathematically, though a sixteenth century drawing by the Nuremberg goldsmith Wentzel Jamnitzer (1508-1585) is very similar to the former and a fifteenth century mosaic attributed to the Florentine artist Paolo Uccello (1397-1475) illustrates the latter. The great icosahedron and great dodecahedron were described by Louis Poinsot in 1809, though Jamnitzer made a picture of the great dodecahedron in 1568. In these the faces (20 triangles and 12 pentagons, respectively) which meet at each vertex "go around twice" and intersect each other, in a manner that is the three-dimensional analog to what happens in two-dimensions with a pentagram. Together, the Platonic solids and these Kepler-Poinsot polyhedra form the set of nine regular polyhedra. Augustin Cauchy first proved that no other polyhedra can exist with identical regular faces and identical regular vertices.

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loculus of Archimedes

A dissection game, similar to tangrams, which consists of 14 polygonal shapes that fit together to make a square. These pieces can be rearranged to make pictures of people, animals, and objects, or reassembled into their original form. There are many references to the game in ancient literature, including a description by the Roman poet and statesman Magnus Ausonius (A.D. 310-395). Only two fragmentary manuscripts, one an Arabic translation and the other a Greek manuscript dating from the 10th century discovered in Constantinople in 1899, connect the puzzle to Archimedes by calling it loculus Archimedius ("Archimedes's box"). More generally, but for unclear reasons, it is known as the ostomachion (Greek for stomach), or, in Latin texts, as the syntemachion.

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Menger sponge

A famous fractal solid that is the three-dimensional equivalent of the Sierpinski carpet (which, in turn, is the one-dimensional equivalent of Cantor dust). To make a Menger sponge, take a cube, divide it into 27 (= 3 x 3 x 3) smaller cubes of the same size and remove the cube in the center and the six cubes that share faces with it. What's left are the eight small corner cubes and twelve small edge cubes holding them together. Now, imagine repeating this process on each of the remaining 20 cubes. Repeat it again. And again ... ad infinitum. The Menger sponge was invented in 1926 by the Austrian mathematician Karl Menger (1902-1985).

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Möbius band

A simple and wonderfully entertaining two-dimensional object, also known as the Möbius strip, that has only one surface and one edge. It is named after the German mathematician and theoretical astronomer August Ferdinand Möbius (1790-1868), who discovered it in September 1858, although his compatriot and fellow mathematician Johann Benedict Listing (1808-1882) independently devised the same object in July 1858. Making a Möbius Band is simple: take an A4 sheet of paper, cut an 11" x 1" rectangle, bring the two long ends together, twist one of the ends 180º, and tape the two ends together. To prove that the band is single-sided, take a pen and start drawing a line around the Band's circumference. When drawing the line, never take the pen off the paper; just keep drawing the line until the starting point is reached. Once you are finished look at both sides, there should be a line on both sides, thus proving that it is all the same side because you never took the pen off the paper.

The Möbius band has a lot of curious properties. If you cut down the middle of the band, instead of getting two separate strips, it becomes one long strip with two half-twists in it. If you cut this one down the middle, you get two strips wound around each other. Alternatively, if you cut along the band, about a third of the way in from the edge, you will get two strips; one is a thinner Möbius band, the other is a long strip with two half-twists in it. Other interesting combinations of strips can be obtained by making Möbius bands with two or more flips in them instead of one. Cutting a Möbius band, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings.

The Möbius band has provided inspiration both for sculptures and for graphical art. M. C. Escher who was especially fond of it and based many of his lithographs on it. It is also a recurrent feature in science fiction stories, such as Arthur C. Clarke's "The Wall of Darkness." A common fictional theme is that our universe might be some kind of generalized Möbius band. There have been technical applications; giant Möbius bands have been used as conveyor belts (to make them last longer, since "each side" gets the same amount of wear) and as continuous-loop recording tapes (to double the playing time).

A closely related strange geometrical object is the Klein bottle, which can be produced by gluing two Möbius bands together along their edges; however, this can't be done in ordinary three-dimensional Euclidean space without creating self-intersections.

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Peano curve

The first known example of a space-filling curve. Discovered by Guiseppe Peano in 1890, its effect was like that of an earthquake on the traditional structure of mathematics. Commenting in 1965 on the impact of the curve in Peano's day, N. Ya Vilenkin said: "Everything has come unstrung! It's difficult to put into words the effect that Peano's result had on the mathematical world. It seemed that everything was in ruins, that all the basic mathematical concepts had lost their meaning." Today, the Peano Curve is recognized as just one of an infinite class of familiar objects known as fractals. But at the end of the nineteenth century it was an extravagant, completely counterintuitive thing; indeed, it was something that had been believed impossible. Writing of Peano's result in Grundzüge der Mengenlehre in 1914, Felix Hausdorff said: "This is one of the most remarkable facts of set theory." Originally, the Peano Curve was derived purely analytically, without any kind of drawing or attempt at visualization. But the first few steps in drawing it, as shown in the diagrams, are easy enough, even though the finished product is unattainable in this way-and totally unimaginable. To fill the unit square, as the Peano Curve does, without leaving any holes, the curve has to be both continuous and self-intersecting.

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A popular two-player board game in ancient Egypt, enjoyed by both commoners and nobility, that may be an ancestor of modern backgammon. The rules are not known, though about 40 sets have been found in tombs, some in very good condition, together with paintings of games on tomb walls, dating back to the reign of Hesy (c. 2686-2613 B.C.). Senet, or the "game of passing," was played on a rectangular board consisting of three rows of 10 squares called "houses" that represented good or bad fortune The board could be a grid drawn on a smooth surface, or an elaborate box of wood and other precious materials. A perfectly preserved traveling version of Senet was found in Tutankhamen's tomb. The pieces, called ibau ("dancers" in Egyptian), varied in number from five to ten per player-five and seven being commonest. Cone-shaped pieces were pitted against reel-shaped pieces. The object was to get one's pieces on the board, then around the board in an S-shaped pattern, and finally off again at the far end. Strategy was mixed with chance (as it is in backgammon), introduced by the throw of four, two-sided sticks (as depicted in the Hesy painting) or, in later times, of knucklebones. Later depictions of the game, in the New Kingdom period, often showed just one player in competition-the opponent being a spirit from the afterlife. This has been interpreted as a change in the significance of Senet, from a simple amusement to a symbolic representation of the deceased's journey through the underworld. See also nine mens' morris.

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A curious and mathematically delightful three-dimensional object made from a right double-cone – two identical, 90-degree cones joined base to base – and an added twist. To create a sphericon, a right double-cone is sliced along a plane that includes both vertices. The resulting cross-section is a square, which enables one of the halves to be rotated through a right angle and the two halves to be glued back together without any overlap. This final twist enables the sphericon to roll-but in an unusual way. An ordinary cone placed on a flat surface rolls around in circles. A double-cone can roll in a clockwise circle or a counterclockwise one. A sphericon, in contrast, performs a controlled wiggle, with first one conical sector in contact with the flat surface, then the other. Two sphericons placed next to each other can roll on each other's surfaces. Four sphericons arranged in a square block can all roll around one another simultaneously. And eight sphericons can fit on the surface of one sphericon so that any one of the outer solids can roll on the surface of the central one. The sphericon was first found by the Englishman Colin Roberts in 1969, while he was still in school. In 1999 he brought his discovery to the attention of Ian Stewart who subsequently wrote about the new object in his "Mathematical Recreations" column in Scientific American.

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