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.
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.
"We Are All Connected" was made from sampling Carl Sagan's Cosmos, The History Channel's Universe series, Richard Feynman's 1983 interviews, Neil deGrasse Tyson's cosmic sermon, and Bill Nye's Eyes of Nye Series, plus added visuals from The Elegant Universe (NOVA), Stephen Hawking's Universe, Cosmos, the Powers of 10, and more. It is a tribute to great minds of science, intended to spread scientific knowledge and philosophy through the medium of music.
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).
(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.
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.
