Quantumaniac

Symmetree

Fullerenes and Buckyballs

Despite the seemingly complex name, a fullerene is nothing more than a molecule composed entirely of carbon. That’s all! Fullerenes can come in the shape of a hollow sphere, ellipsoid or tube. When a fullerene is spherical, they are known as buckyballs - and when cylindrical they are called carbon nanotubes or can be affectionately called buckytubes. Fullerenes are similar in structure to graphite, which is composed of stacked graphene sheets of linked hexagonal rings; but they may also contain pentagonal (or sometimes heptagonal) rings.

The first fullerene to be discovered, buckminsterfullerene (C₆₀), was prepared in 1985 by Richard Smalley, Robert Curl, James Heath, Sean O’Brien, and Harold Kroto at Rice University. The name was an homage to Buckminster Fuller, whose geodesic domes it resembles. The structure was also identified some five years earlier by Sumio Iijima, from an electron microscope image, where it formed the core of a “bucky onion.” Fullerenes have since been found to occur in nature. More recently, fullerenes have been detected in outer space. According to astronomer Letizia Stanghellini, “It’s possible that buckyballs from outer space provided seeds for life on Earth.”

The discovery of fullerenes greatly expanded the number of known carbon allotropes, which until recently were limited to graphite, diamond, andamorphous carbon such as soot and charcoal. Buckyballs and buckytubes have been the subject of intense research, both for their unique chemistry and for their technological applications, especially in materials science, electronics, and nanotechnology.

Fractal Tetrahedron

Calculus Board

The Book that Started It All

Klein Quartic

Happy Face Math - Charlie Smith

Zeno’s Dichotomy Paradox

Zeno (ca. 490 B.C. - ca. 430 B.C.) created a number of famous paradoxes, some which baffle students (and cause great amusement for teachers) to this day. One of this most famous, the dichotomy paradox - seems to destroy the basic concept of motion and destination.

Zeno’s dichotomy paradox simply states that in order for an object to move to a certain destination, it must reach certain ‘goals’ along the way. Let’s use the example of a train. In order for a train to get from New York to Washington D.C., it must travel half of the distance. Once that is covered, it will reach halfway to Washington D.C. again (now 1/4th of the full distance away), and will then reach another halfway - the train will inevitably reach an infinite amount of halfway points, without ever reaching Washington D.C. Another way to think of it is in order for the train to even reach the halfway point, it must travel 1/4th of the full distance, then 1/8th, then 1/16th - and it will never actually arrive. 

Although intuitively this cannot be true, mathematically it is not easily disproven at first.  However, one can set up a sum to simplify this nicely. Let’s assume that each ‘halfway point,’ is its own unique interval, and there are an infinite amount of them. So, we can set up a sum by setting 1/2 to the nth power: 

We find that the sum is equal to 1 - or the full distance. For our less mathematically inclined readers, think of it this way: bring your finger about halfway from your face to the computer screen. Now take that distance and move it halfway to the screen again. If you continued this process, although you would technically be moving halfway each time, and there would always be another halfway point to get to - the distance becomes trivial. In this same way, the sum will eventually reach a point where it is equal to 0.99999999999999999999 etc., and 1 will suffice. Nice try Zeno - better luck next time. 

The Poincaré Conjecture

Imaginestretching a rubber band around the surface of an apple, then shrinking it down slowly. This shrinking could occur without tearing the rubber band or breaking the apple - and the band would never have to leave the surface. However, if this rubber band were to be stretched across, say, a tire - there is no way to shrink to a point without breaking one or the other. The surface of such an apple is “simply connected,” but the tire is not. Henri Poincaré (shown below), during the early twentieth century - knew that two dimensional spheres had this ‘connected’ property - and he asked if the same applied for three dimensional spheres. 

The conjecture turned out to be immensely difficult to prove. After more than a century, Grigori Perelman finally devised a solution. In 2006, Perelman was awarded the Fields Medal for this contribution, but he decided to turn it down, stating that:

“I’m not interested in money or fame, I don’t want to be on display like an animal in a zoo.”