The Math Book: Milestones in the History of Math

I love math (though it's debatable whether math loves me back, I suspect not) so it's a pleasure to read Cliff Pickover's newest creation, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics.

Don't let the title fool you - The Math Book is a thoroughly enjoyable "walk" through the history of mathematics with each milestone narrated by Pickover in a short and sweet fashion (and surprisingly, with very little equations) that even non-mathemagicians like myself can enjoy. If you've ever heard the terms Bessel functions, Transcendental numbers, and Riemann hypothesis, and want to know more, then this is the book for you.

Below is an excerpt from the book (selecting which ones to show was a hard thing to do - there were just so many interesting articles!):

Cicada-Generated Prime Numbers

Cicadas are winged insects that evolved around 1.8 million years ago during the Pleistocene epoch, when glaciers advanced and retreated across North America. Cicadas of the genus Magicicada spend most of their lives below the ground, feeding on the juices of plant roots, and then emerge, mate, and die quickly. These creatures display a startling behavior: Their emergence is synchronized with periods of years that are usually the prime numbers 13 and 17. (A prime number is an integer such as 11, 13, and 17 that has only two integer divisors: 1 and itself.) During the spring of their 13th or 17th year, these periodical cicadas construct an exit tunnel. Sometimes more than 1.5 million individuals emerge in a single acre; this abundance of bodies may have survival value as they overwhelm predators such as birds that cannot possibly eat them all at once. (Photo: Joelmills [Wikipedia])

Some researchers have speculated that the evolution of prime-number life cycles occurred so that the creatures increased their chances of evading shorter-lived predators and parasites. For example, if these cicadas had 12-year life cycles, all predators with life cycles of 2, 3, 4, or 6 years might more easily find the insects. Mario Markus of the Max Planck Institute for Molecular Physiology in Dortmund, Germany, and his coworkers discovered that these kinds of prime-number cycles arise naturally from evolutionary mathematical models of interactions between predator and prey. In order to experiment, they first assigned random life-cycle durations to their computer-simulated populations. After some time, a sequence of mutations always locked the synthetic cicadas into a stable prime-number cycle.

Of course, this research is still in its infancy and many questions remain. What is special about 13 and 17? What predators or parasites have actually existed to drive the cicadas to these periods? Also, a mystery remain as to why, of the 1,500 cicada species worldwide, only a small number of the genus Magicicada are known to be periodical.

Borromean Rings

(L) Borromean Rings; (M) Valknut, or three interlocked triangles, on the Stora Hammar Stone; (R) Molecular Borromean Rings by J. Fraser SToddart

Peter Guthrie Tait (1831 - 1901) - A simple yet intriguing set of interlocking objects of interest to mathematicians and chemists is formed by Borromean rings - three mutually interlocked rings named after the Italian Renaissance family who used them on its coat of arms in the fifteenth century. (Image: Theon [Wikipedia])

Notice that Borromean rings have no two rings that are linked, so if we cut any one of the rings, all three rings come apart. Some historians speculate that the ancient ring configurations once represented the three families of Visconti, Sforza, and Borromeo, who formed a tenuous union through intermarriages. The rings also appear in 1467 in the Church of San Pancrazio in Florence. Even older, triangular versions were used by the Vikings, one famous example of which was found on a bedpost of a prominent woman who died in 834.

The rings appear in mathematical context in the 1876 paper on knots by Scottish mathematical physicist Peter Tait. Because two choices (over or under) are possible for each ring crossing, 26 = 64 possible interlaced patterns exist. If we take symmetry into account, only 10 of these patterns are geometrically distinct.

Mathematicians now know that we cannot actually construct a true set of Borromean rings with flat circles, and in fact, you can see this for yourself if you try to create the interlocked rings out of wire, which requires some deformation or kinks in the wires. In 1987, Michael Freedman and Richard Skora proved the theorem stating that Borromean rings are impossible to construct with flat circles.

In 2004, UCLA chemists created a molecular Borromean ring compound that was 2.5 nanometers across and that included six metal ions. Researchers are currently contemplating ways in which they may use molecular Borromean rings in such diverse fields as spintronics (a technology that exploits electron spin and charge) and medical imaging.

Golden Ratio

Fra Luca Bartolomeo de Pacioli (1445 - 1517) - In 1509, Italian mathematician Luca Pacioli, a close friend of Leonardo da Vinci, published Divina Proportione, a treatise on a number that is now widely known as the "Golden Ratio." This ratio, symbolized by , appears with amazing frequency in mathematics and nature. We can understand the proportion most easily by dividing a line into two segments so that the ratio of the whole segment to the longest part is the same as the ratio of the longer part to the shorter part, or (a+b)/b = b/a = 1.61803 ...

If the lengths of the sides of a rectangle are in the golden ratio, then the rectangle is a "golden rectangle." It's possible to divide a golden rectangle into a square and a golden rectangle. Next, we can cut the smaller golden rectangle into a smaller square and golden rectangle. We may continue this process indefinitely, producing smaller and smaller golden rectangles.

If we draw a diagonal from the top right of the original rectangle to the bottom left, then from the bottom right of the baby (that is, the next smaller) golden rectangle to the top left, the intersection point shows the point to which all the baby golden rectangles converge. Moreover, the lengths of the diagonals are in golden ratio to each other. The point to which all the golden rectangles converge is sometimes called the "Eye of God."

The golden rectangle is the only rectangle from which a square can be cut so that the remaining rectangle will always be similar to the original rectangle. If we connect the vertices in the diagram, we approximate a logarithmic spiral that "envelops" the Eye of God. Logarithmic spirals are everywhere - seashells, animal horns, the cochlea of the ear - anywhere that nature needs to fill space economically and regularly. A spiral is strong and uses a minimum of materials. While expanding, it alters its size but never its shape.

Benford's Law

Simon Newcomb (1835 - 1909), Frank Benford (1883 - 1948) - Benford's Law, also called the first-digit law or leading-digit phenomenon, asserts that in various number lists, the digit 1 tends to occur in the leftmost position with probability of roughly 30 percent, much greater than the expected 11.1 percent that would result if each digit occurred with a 1 to 9 probability. Benford's law can be observed, for instance, in tables that list populations, death rates, stock prices, baseball statistics, and the area of rivers and lakes. Explanations for this phenomenon are very recent. (Photo from Mark J. Nigrini)

Benford's law is named after Dr. Frank Benford, a physicist at the General Electric Company who publicized his work in 1938, although it had been previously discovered by mathematician and astronomer Simon Newcomb in 1881. Pages of logarithms, with numbers starting with the numerals 1 are said to be dirtier and more worn by other pages, because the number 1 occurs as the first digit about 30 percent more often than any other. In numerous kinds of data, Benford determined that the probability of any number n from 1 through 0 being the first digit is log10 (1 + 1/n). Even the Fibonacci sequence - 1, 1, 2, 3, 5, 8, 13 - follows Benford's law. Fibonacci numbers are far more likely to start with "1" than any other digit. It appears that Benford's law applies to any data that follows a "power law." For example, large lakes are rare, medium-size lakes are more common, and small lakes are even more common. Similarly, 11 Fibonacci numbers exist in the range 1 - 100, but only one in the next three ranges of 100 (101 - 200, 201- 300, 301- 400)

Benford's law has often been used to detect fraud. For example, accounting consultants can sometimes use the law to detect fraudulent tax returns in which the occurrence of digits does not follow what would be expected according to Benford's law.

Menger Sponge

Menger Sponge by Jeannine Mosely, at the Institute for Figuring. Photo: Ravi Apte

Karl Menger (1902 - 1985) - The Menger sponge is a fractal object with an infinite number of cavities - a nightmarish object for any dentist to contemplate. The object was first described by Austrian mathematician Karl Menger in 1926. To construct the sponge, we begin with a "mother cube" and subdivide it into 27 identical smaller cubes. Next, we remove the cube in the center and the six cubes that share faces with it. This leaves behind 20 cubes. We continue to repeat the process forever. The number of cubes increases by 20n, where n is the number of iterations performed on the mother cube. The second iteration gives us 400 cubes, and by the time we get to the sixth iteration, we have 64,000,000 cubes.

Each face of the Menger sponge is called a Sierpinski carpet. Fractal antennae based on the Sierpinski carpet are sometimes used as efficient receivers of electromagnetic signals. Both the carpets and the entire cube have fascinating geometrical properties. For example, the sponge has an infinite surface area while enclosing zero volume.

According to the Institute for Figuring, with each iteration, the Sierpinski carpet face "dissolves into a foam whose final structure has no area whatever yet possesses a perimeter that is infinitely long. Like the skeleton of a beast whose flesh has vanished, the concluding form is without substance - it occupies a planar surface, but no longer fills it." This porous remnant hovers between a line and a plane. Whereas a line is one-dimensional and a plane two-dimensional, the Sierpinski carpet has a "fractional" dimension of 1.89. The Menger sponge has a fractional dimension (technically referred to as the Hausdorff Dimension) between a plane and a solid, approximately 2.73, and it has been used to visualize certain models of a foam-like space-time. Dr. Jeannine Mosely has constructed a Menger sponge model from more than 65,000 business cards that weights about 150 pounds (70 kilograms).

The Quest for Lie Group E8

E8 graph as a 2-dimensional projection, by Peter McMullen
(image by Claudio Rocchini [wikipedia])

Marius Sophus Lie (1842 - 1899), Wilhelm Karl Joseph Killing (1847 - 1923) - For more than a century, mathematicians have sought to understand a vast, 248-dimensional entity, known to them only as E8. Finally, in 2007, an international team of mathematicians and computer scientists made use of a supercomputer to tame the intricate beast.

As background, consider the Mysterium Cosmographicum (The Sacred Mystery of the Cosmos) of Johannes Kepler (1571 - 1630), who was so enthralled with symmetry that he suggested the entire solar system and planetary orbits could be modeled by Platonic Solids, such as the cube and dodecahedron, nestled in each other forming layers as if in a gigantic crystalline onion. These kinds of Keplerian symmetries were limited in scope and number; however, symmetries that Kepler could have hardly imagined may indeed rule the universe.

In the late nineteenth century, the Norwegian mathematician Sophus Lie (pronounced "Lee") studied objects with smooth rotational symmetries, like the sphere or doughnut in our ordinary three-dimensional space. In three and higher dimensions, these kinds of symmetries are expressed by Lie groups. The German mathematician Wilhelm Killing suggested the existence of the E8 group in 1887. Simpler Lie groups control the shape of electron orbital and symmetries of subatomic quarks. Larger groups, like E8, may someday hold the key to a unified theory of physics and help scientist understand string theory and gravity.

Fokko du Cloux, a Dutch mathematician and computer scientist who was one of the E8 team members, wrote the software for the supercomputer and pondered the ramifications of E8 while he was dying of amyotrophic lateral sclerosis and breathing with a respirator. He died in November 2006, never living to see the end of the quest for E8.

On January 8, 2007, a supercomputer computed the last entry in the table for E8, which describes the symmetries of a 57-dimensional object that can be imagined as rotating in 248 ways without changing its appearance. The work is significant as an advance in mathematical knowledge and in the use of large-scale computing to solve profound mathematical problems.

Mathematical Universe Hypothesis

Max Tegmark (b. 1967) - In this book, we have encountered various geometries that have been thought to hold the keys to the universe. Johannes Kepler modeled the solar system with Platonic Solids such as the dodecahedron. Large Lie groups, like E8, may someday help us create a unified theory of physics. Even Galileo in the seventeenth century suggested that "nature's great book is written in mathematical symbols." In the 1960s, physicist Eugene Wigner was impressed with the "unreasonable effectiveness of mathematics in the natural sciences." (Photo: MIT Physics Faculty website)

In 2007, Swedish-American cosmologist Max Tegmark published scientific and popular articles on the Mathematical Universe Hypothesis (MUH) that states that our physical reality is a mathematical structure and that our universe is not just described by mathematics - it is mathematics. Tegmark is a professor of physics at the Massachusetts Institute of Technology and scientific director of the Foundational Questions Institute. He notes that when we consider equations like 1 + 1 = 2, the notations for the numbers are relatively unimportant when compared to the relationship that are being described. He believes that "we don't invent mathematical structures - we discover them, and invent only the notation for describing them."

Tegmark's hypothesis implies that "we all live in a gigantic mathematical object - one that is more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names like Calabi-Yau manifolds, tensor bundles, and Hilbert spaces, which appear in today's most advanced theories. Everything in our world is purely mathematical - including you." If this idea seems counterintuitive, this shouldn't be surprising, because many modern theories, like quantum theory and relativity, can defy intuition. As mathematician Ronald Graham once said, "Our brain have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions."


Cliff Pickover is a prolific author, having published more than 40 books, translated into over a dozen languages, on topics ranging from science and mathematics to religion, art, history, computers and creativity, human intelligence, higher dimensions, time travel, and science fiction. He received his Ph.D. from Yale University's Department of Molecular Biophysics and Biochemistry, holds over 50 U.S. patents, and is an associate editor for several scientific journals. His computer graphics have appeared on the cover of numerous magazines, and his research has received considerable attention from media outlets ranging from CNN and WIRED to The New York Times. His website,, receives millions of visits.

Links: The Math Book website | The Math Book on Amazon | Cliff Pickover's Reality Carnival


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I am puzzled about the cicada prime number life cycle idea. Just because other animals have even year life cycles, isn't their always an abundent amount of preditors any given year? Just because their life cycle is an even numbers of years, doesn't mean each animal is the same age.

I am not saying this is wrong. I am just puzzled by the idea.
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Can someone explain how flat circles cannot be made into Borromean Rings? Perhaps I do not know what "flat circles" are?

Assume that the upper most circle in the picture above is "A". The lower most circle is "B" and the right most circle is "C".

In the picture above we have the following:

1) A is over B
2) A is under C
3) B is over C
4) B is under A
5) C is over A
6) C is under B

(Once can see that 1 and 4, 2 and 5 & 3 and 6 are merely saying the same thing in the opposite manner.)

I cut three identical circles out of a piece of paper and the cut a slit in each one to allow for interlocking. (One must assume a slit in the circle because one cannot (if not an illusionist, a magician with trick rings or a physicist with quantum mechanical rings) make solid objects pass through each other.) After doing so I was able to construct a BR immediately. No twisting, kinks or deformations were necessary - the feat was accomplished by merely passing the slits through at the appropriate points.

Now perhaps by flat circles it is meant that the circles are always on the same plane such that they would bump into each other and therefore interlocking could not occur. However, if that were the case you would have to say that two flat circles could not interlock. The implication would then be that BRs aren't that special; therefore, I have to assume that flat circles mean something else (and cause the impossibility of creating BRs) or that I have missed something that is blatantly obvious.

I fear in my naiveté that a "flat circle" in the "physical" world; i.e., that I can cut out of a flat sheet of paper, is not the same as a flat circle in the scientific world.

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