THE UNIVERSAL BOOK OF MATHEMATICS
From Abracadabra to Zeno's Paradoxes
Sample Entries
Alhambra  BanachTarski paradox
 Boole, Alicia  bridges of Königsberg
 Brouwer fixedpoint theorem  Császár
polyhedron  Eddington number  flybetweentrains
problem  fourth dimension  hairy
ball theorem  ham sandwich theorem  Hamilton,
William Rowan  Hinton, Charles Howard  Ishango
bone  KeplerPoinsot solids  loculus
of Archimedes  Menger sponge  Möbius
band  Peano curve  Senet  sphericon
Alhambra
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.
~ ~ ~
BanachTarski 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 BanachTarski Paradox tells us nothing new about the
physics of the world around us but a great deal about how "volume," "space," and other familiarsounding 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 BanachTarski paradox,
which mathematicians often refer to as the BanachTarski 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 BanachTarski
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 BanachTarski
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.
~ ~ ~
Boole (Stott), Alicia
(18601940)
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 brotherinlaw 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 fourdimensional geometry. She introduced
the word "polytope" to describe a fourdimensional convex solid, and
went on to explore the properties of the six regular polytopes and to make
12 beautiful card models of their threedimensional central crosssections.
She sent photographs of these models to the Dutch mathematician Pieter Schoute
(18461923), 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.
~ ~ ~
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 (17241804).
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.
~ ~ ~
Brouwer fixedpoint
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 nball into an nball (that is,
any way of mapping points in one object that is topologically the same as
the filling of an ndimensional 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.
~ ~ ~
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.
~ ~ ~
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 (18821944) 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 socalled finestructure
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: 2×136×2^{256} = 17×2^{260} ~ 3.149544...×10^{79} (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 finestructure 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 AddingOne." See also large numbers.
~ ~ ~
flybetweentrains 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 (18401921) in his Initiation Mathématique. There is a longwinded 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 flya speedster
of its kindis moving at 75 miles per hour. The long method involves considering
the length of the backandforth 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!"
~ ~ ~
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 dimensionan extension at
rightangles to the three familiar directions of updown, forwardbackward,
and sidetoside. 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. Onedimensional
space is just the number line of real numbers. Twodimensional space,
the plane, corresponds to the set of all ordered pairs (x, y)
of real numbers, and threedimensional space to the set of all ordered triplets
(x, y, z). By extrapolation, fourdimensional 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 threedimensional space; in other words, it stretches
out along an axis, say the waxis, that is mutually perpendicular to the familiar
x, y, and zaxes. Analogous to the cube is a hypercube
or tesseract, and to the sphere is a 4d hypersphere. Just as
there are five regular polygons, known as the Platonic solids, so there
are six fourdimensional regular polytopes. They are: the 4simplex
(constructed from five tetrahedra, with three tetrahedra meeting at an edge);
the tesseract (made from eight cubes, meeting three per edge); the 16cell
(made from 16 tetrahedra, meeting four per edge); the 24cell (made from 24
octahedra, meeting three per edge); the 120cell (made from 120 dodecahedra,
meeting three per edge); and the monstrous 600cell (made from 600 tetrahedra,
meeting five per edge). Geometers have no difficulty in analyzing, describing,
and cataloging the properties of all sorts of 4d 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 2d perspective of
a real cube, so a real cube can be thought of as a perspective of a tesseract.
At a movie, a 2d picture represents a 3d world, whereas if you were to watch
the action live, in threedimensions, this would be like a screen projection
in four dimensions.
Many books have been written and schemes devised to nudge
our imaginations into thinking fourdimensionally. 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 fourdimensional thought [in] mechanics, science,
and art."
Victorianage 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 threedimensional
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 fourdimensional
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.
~ ~ ~
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.
~ ~ ~
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 higherdimensional ham sandwiches, when it essentially becomes the BorsukUlam
theorem: in ndimensional space in which there are n globs of positive
volume, there is always a hyperplane that cuts all the globs exactly in half.
~ ~ ~
Hamilton, William Rowan
(18051865)
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 longstanding 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 socalled HamiltonJacobi
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.
~ ~ ~
Hinton, Charles Howard
(18531907)
An Englishborn 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 fourdimensional
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 twovolume Scientific Romances (1884). Hinton's descriptions
owed much to the mathematical models of William Clifford, whose theories
about 4d spaces were then in vogue. But Hinton went much further in his attempts
to break free of threedimensional 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
lefthandedness 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 fourdimensional space as possible if threedimensional
mechanics fails to explain known physical phenomena still rings true today.
See also Boole (Stott), Alicia.
~ ~ ~
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.
~ ~ ~
KeplerPoinsot solids
The four regular nonconvex polyhedra that exist in addition
to the five regular convex polyhedra known as the Platonic solids.
As with the Platonic solids, the KeplerPoinsot 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 (fivepointed 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
(15081585) is very similar to the former and a fifteenth century mosaic attributed
to the Florentine artist Paolo Uccello (13971475) 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 threedimensional analog to what happens in twodimensions
with a pentagram. Together, the Platonic solids and these KeplerPoinsot
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.
~ ~ ~
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. 310395). 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.
~ ~ ~
Menger sponge
A famous fractal solid that is the threedimensional equivalent
of the Sierpinski carpet (which, in turn, is the onedimensional 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 (19021985).
~ ~ ~
Möbius band
A simple and wonderfully entertaining twodimensional 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 (17901868), who discovered it in September 1858, although
his compatriot and fellow mathematician Johann Benedict Listing (18081882)
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 singlesided, 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 halftwists 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 halftwists 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 continuousloop 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 threedimensional
Euclidean space without creating selfintersections.
~ ~ ~
Peano curve
The first known example of a spacefilling 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
wayand totally unimaginable. To fill the unit square, as the Peano Curve
does, without leaving any holes, the curve has to be both continuous and selfintersecting.
~ ~ ~
Senet
A popular twoplayer 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. 26862613 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 playerfive and seven
being commonest. Coneshaped pieces were pitted against reelshaped pieces.
The object was to get one's pieces on the board, then around the board in
an Sshaped 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, twosided
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 competitionthe 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.
~ ~ ~
sphericon
A curious and mathematically delightful threedimensional
object made from a right doublecone – two identical, 90degree
cones joined base to base – and an added twist. To create a sphericon,
a right doublecone is sliced along a plane that includes both vertices. The
resulting crosssection 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 rollbut in
an unusual way. An ordinary cone placed on a flat surface rolls around in
circles. A doublecone 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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