Objects can be classified by the type of symmetry they have. For example, starfish (and other echnioderms) have pentagonal symmetry, or order-10 dihedral symmetry. Most vertabrates, by comparison, only have bilateral symmetry. But both of these pale in comparison to the microscopic Radiolaria and Adenoviruses (responsible for common colds), which have the spectacular icosahedral/dodecahedral symmetry, with an order of 120.
How can such symmetries be classified? It turns out that classifying symmetries with reflections, such as the three examples above, is easier than classifying purely rotational symmetries, such as the order-3 cyclic symmetry of the Isle of Man's triskelion. In two dimensions, we have the symmetries of each of the regular polygons. In three dimensions, we also have the symmetries of the tetrahedron, cube/octahedron and icosahedron/dodecahedron, with order 24, 48 and 120, respectively. These five symmetrical objects were known to the Greeks, and are collectively referred to as Platonic solids.
In four dimensions, the situation gets even more interesting. Instead of five Platonic solids, there are six regular polychora. The pentachoron is the 4-dimensional analogue of the tetrahedron; the tesseract and 16-cell are 4-dimensional versions of the cube and octahedron, respectively; the 120-cell and 600-cell are the 4-dimensional counterparts of the dodecahedron and icosahedron, respectively. The sixth polychoron has no 3-dimensional equivalent, and is called the 24-cell.
In five and higher dimensions, the only symmetrical objects are the simplex, hypercube and cross-polytope (analogues of the tetrahedron, cube and octahedron, respectively). However, there are three other symmetry groups, E6, E7 and E8. The subscript denotes the number of dimensions that they inhabit. These are even more symmetrical than any of the objects I mentioned earlier.
In conclusion, the different symmetries are those of the regular polygons (I2[p]), simplices (An), dodecahedron/icosahedron (H3), 24-cell (F4), 120-cell/600-cell (H4) and the higher-dimensional objects, E6, E7 and E8. Most of these symmetries can be found in lattices; the exceptions are H3, H4, and I2[p], where p is not 2, 3, 4 or 6. It was long believed that these could not appear in crystals, as all known crystals has a repetitive, periodic structure.
Previously, the only known macroscopic covalent structures fell into two categories: crystals and glass. Crystals have a periodic structure, similar to the mathematical tilings and lattices mentioned above. Glass, on the other hand, has a chaotic, non-uniform structure. Nevertheless, an intermediate structure was discovered in magnesium-aluminium alloys.
In 1961, Hao Wang realised that Turing machines and tiling problems are intimately linked. More specifically, he noticed that the problem of whether a generic Turing machine halts can be modelled as a problem of tiling squares, known as Wang tiles. He thought that if a set of tiles admits a non-periodic tiling, then it must, necessarily, also admit a periodic tiling of the plane. For example, certain triangles can be made to tile non-periodically, but can also be arranged into a periodic tiling. Wang concluded that all decision problems could be solved, eventually, by determining whether the tiles formed a repeatable unit cell, or whether they failed to tile a finite section of the plane.
In reality, Wang's assumption was incorrect. Sets of tiles can be engineered that only admit non-periodic tilings of the plane. These are known as aperiodic tilings. In 1966, Robert Berger found an aperiodic set of 20 426 Wang tiles. He later reduced this figure to 104 tiles, and there is even an aperiodic set of 13 Wang tiles. Robinson found a set of just six square tiles, which admit reflections and rotations (unlike Wang tiles).
More interestingly, Sir Roger Penrose and Robert Ammann independently discovered tilings with pentagonal symmetry. This is easy to prove impossible for periodic tilings, which offers an easy proof that the Penrose tiling is indeed aperiodic. Penrose's original set comprised six tiles, five of which had pentagonal symmetry, but there are two other sets of Penrose tiles, each of which have only two members. Arguably, the most interesting one is the tiling by thin and thick rhombi.
The Penrose tiling has many interesting properties. Firstly, any finite section of the Penrose tiling appears infinitely often throughout the tiling. This means that no matter how large an area you scan, it is impossible to determine where you are located in a Penrose tiling. This contrasts with, for example, Lincolnshire, where nearby signs indicate your location. This property alone does not seem so remarkable. The decimal expansions of e, pi and phi are also believed to possess this property. However, in a Penrose tiling, one only has to travel twice the diameter of a given section to find an identical copy of this section. This is awe-demanding -- in random patterns, the distance between sections is exponential in the size of the section!
As I mentioned earlier, these patterns have been discovered in alloys of metals, such as aluminium-magnesium alloys, which are known as quasicrystals. Some of these are periodic in one direction, and aperiodic in the other two, and are equivalent to the Cartesian product of a Penrose tiling and a straight line. Others have icosahedral/dodecahedral symmetry, same as the Radiolaria, and are aperiodic in all three dimensions.
Quasicrystals can be constructed by projecting irrational slices of higher-dimensional lattices. For example, the rhombic Penrose tiling is a projection of the five-dimensional hypercubic lattice. This construction generalises to any polygonal symmetry, allowing quasicrystals with heptagonal symmetry, octagonal symmetry, and so on, ad infinitum.
So far we have encountered aperiodic tilings with I2(p) symmetry, for all values of p, and a three-dimensional aperiodic tiling with H3 (icosahedral/dodecahedral) symmetry. This just leaves one finite reflection group not covered by periodic or aperiodic tilings: the symmetry of the 120-cell and 600-cell, H4.
As a minor detour, consider the infinite line of real numbers. You probably already knew that this can be extended to a two-dimensional plane of numbers, the complex numbers. This was later extended by Sir William Rowan Hamilton to a four-dimensional space of numbers, known as quaternions. Quaternions, unlike real and complex numbers, are not commutative; the product of a and b is not the same as the product of b and a. The set of quaternions is equivalent to the set of 2*2 matrices, which are also non-commutative.
Points in four-dimensional space can be expressed as quaternions. For example, the vertices of a 16-cell correspond to the basic unit quaternions and their inverses, 1, -1, i, -i, j, -j, k and -k. Henceforth, I will represent a quaternion ai + bj + ck + d as a vector, (a,b,c,d). The vertices of the 16-cell then become:
Interestingly, these form a group under multiplication, i.e. multiplying any two quaternions in this set results in another quaternion in this set. It happens that the 24-cell and 600-cell also have this property, when appropriately scaled and orientated. The vertices of the 24-cell contain the 8 vertices of the 16-cell, together with the 16 quaternions of the form (±½,±½,±½,±½). The 120 vertices of the 600-cell contain the 24 aforementioned vertices, together with the 96 quaternions of the form:
φ and ψ represent the irrational numbers ½(1+√5) and ½(1-√5), respectively. φ is the golden number, or the ratio between successive Fibonacci numbers. The golden number is precisely the ratio of thin and thick rhombi in the Penrose tiling -- another proof that it is aperiodic. Finally, the Fibonacci numbers and golden number appear in plant phyllotaxis; the number of petals on a sunflower is a Fibonacci number, and the numbers of spirals in the seed arrangement are also Fibonacci numbers. Seeds (or primordiaM) are offset by an angle of 137.51° from each other, which is also related to the golden number.
(Picture by L. Shyamal)
The 120 unit quaternions that form the vertices of the 600-cell are referred to as the icosian group. This should not be confused with the icosian ring, which form the (infinite) set of numbers obtained by adding a finite number of group icosians together. The ring icosians are closed under both addition and multiplication, whereas the group icosians are only closed under multiplication.
The 600-cell and 120-cell have H4 symmetry, of order 14400. Does an aperiodic tiling exist with this same group of symmetries? Such a tiling, if proven to exist, would be a natural four-dimensional extension to the three-dimensional quasicrystals found in nature. A possible construction would be to take an irrational four-dimensional slice through the eight-dimensional E8 lattice.