by Preston MacDougall
June 17, 2005
The constellation Crux is prominent in the night skies of the southern hemisphere, as well as on the flags of Australia and New Zealand. But long before Captain Cook sailed the seven seas, Aborigines used the stars for guidance on their long nomadic treks across their expansive island home.
Much like ancient European civilizations, such as the Celts and Romans, created different mythologies in association with the same northern skies, different Aborigine peoples had unique associations for the same stellar patterns, but always relating to their creation myth, called the Dreamtime.
For instance, what to European explorers was the Southern Cross, was a stingray to the Ngarindjeri people of early Australia. Also, for people who worry too much, such as what to do when our sun burns out many billion years from now, the nearby stars Alpha and Beta Centauri are among the closest solar systems to our own. For the Ngarindjeri people, however, they are two sharks chasing the stingray.
When I was down under recently, I saw a kite. I also saw an arrowhead to the north of the kite. Come to think of it, Sagittarius was nearby to the west.
The arrowhead constellation that I saw was made up by three stars from Libra, the balance, as well as the brightest star, Spica, from Virgo. Long before John Ashcroft's blue burqa was hung, Virgo was the name given to the constellation representing the Roman goddess Astraea, who is holding the scales of justice in the original night court.
I suppose that the Ngarindjeri would see a boomerang instead of the angled arms of a balance, and that most Westerners would connect Spica to Zuben El Genubi, the pivotal star of the balance, forming a perfect Y shape. Instead, in my mind I connected Spica to the ends of the balance, forming an arrowhead shape.
The wings of a simple paper airplane, or a "paper dart", have an arrowhead shape when viewed top-down, and I suspect that this is what triggered my abnormal night-thoughts. You see, kites and darts are the names of the shapes first used by Oxford mathematician Roger Penrose to crush the central dogma of pattern formation: You cannot have five-fold symmetry in an extended pattern that has no vacancies. This applies whether you are tiling floors or growing crystals.
For instance, in classic Roman architecture, a floor of terracotta tiles has four-fold symmetry. This means that, for an ant anywhere on the floor, the floor looks the same in four directions, with angles of ninety degrees between them. Ninety degrees because that is what you get when you divide the 360 degrees of a circle into four equal parts.
Likewise, the honeycomb of a beehive has six-fold symmetry. For a honeybee situated anywhere on a large honeycomb, there are six directions, separated by sixty degrees, that appear to be equivalent. Mathematicians were sure that it was impossible to tile a surface with five-fold symmetry, and the insect world seemed to concur.
That was until Roger Penrose created one in 1974. He originally used kites and darts, but later demonstrated five-fold tiling using just rhombus, or diamond, shapes. One is "skinny" and one is "fat", with their smaller, or acute, angles being thirty-six and seventy-two degrees, respectively.
One of the most amazing things about this tiling pattern is that, although there are five equivalent directions from the central point, the pattern does not repeat itself. In this regard, it is sort of like the number pi.
The architect of the brand new, five-storey Biomolecular and Chemical Sciences Building at the University of Western Australia, in Perth, must have also been impressed by the genius of Professor Penrose's pattern.
Having had enough trouble with square linoleum tiles myself, I can only imagine the difficulty of designing the non-repeating pattern and laying the thousands of individually numbered rhombi to form the expansive tile floor of the building's stunningly beautiful atrium, with perfect five-fold symmetry!
A cube is the three-dimensional analogue of a square, and even a child has no trouble building a big cube out smaller cubic building blocks. It is one, of only five, so-called Platonic solids - objects with identical faces. A cube has six identical faces, and can also be called a hexahedron.
If you have ever played Dungeons and Dragons, and during the course of the game the odds of getting pole-axed by a Celtic marauder were one in twelve, then you have used a dodecahedron. It is also a Platonic solid, having twelve identical pentagonal faces. It has five-fold symmetry.
Since crystals are the nano-scale versions of periodic building block arrangements, but with extended and repeating patterns of atoms instead of wooden blocks, it should be impossible to grow a crystal in the shape of a dodecahedron.
Don't tell that to Dan Schechtman and his colleagues at the National Institute of Standards and Technology. They did just that in 1984! However, since atoms are too small to see, and nobody has yet figured out how the atoms could possibly be arranged, the objects that are grown are called quasicrystals.
Perhaps the answer to this
chemical puzzle is hermetically sealed.