Mathematics of Symmetry and Art: Week 2

As I dove deeper into my research into the mathematics of symmetry, the math definitely became more advanced, but the things that were discussed also became more interesting. The main thing that I focussed on this week was color symmetry. When it comes to symmetric patterns there can be symmetry between fundamental regions and also symmetry in the way colors are interchanged throughout the pattern.  The Symmetries of Things, defines an n-fold coloring as “one in which there are regions of n distinct ‘colors’ that have symmetries that take any one of the n ‘colors’ to another one” and focusses on first 2-fold colorings, then 3-fold, and finally other prime-fold colorings (Conway 153). As we are interested in the way colors are interchanged, each color symmetry “achieves a permutation” of colors and can thus be described by a homomorphism from the full group G to the symmetric group Sym(n) where is the number of colors. These color symmetries are denoted G/H/K where H is the stabilizer group which fixes a chosen color and is the kernel which fixes all colors. Often this signature is shortened to G//K as in the type 3*3//333 where the plane repeating pattern has a 3-fold rotation and a 3-fold kaleidoscope and the colors are fixed by three 3-fold rotations as shown:

Image of symmetry type: 3*3//333 from The Symmetry of Things

The way to discover the possible color symmetries of each group was simple for 2-fold colorings where the group presentation was used to describe the ways to interchange two colors and then the descriptions gradually became more difficult to uncover.  For 3-fold colorings, the color type was “determined by a homomorphism from G onto some subgroup of the group S[3] of all permutations of the three colors” (Conway 154). Finally, p-fold colorings were described by subgroups characterized by there slope in  Cp × Cp the lattice formed by the direct product of the cyclic groups of order and fixed by there quotient groups Q.

This week I also learned the justification for each group presentation. The presentations are formed by cutting the “orbifold” of each symmetry group following paths that represent “elements of the fundamental group of this orbifold that correspond to our chosen generators” (Conway 176). When the orbifold is unfolded the resulting graph can be used to determine the relations between each generator.

For the rest of this week and into next week I will be learning about different “tiles” for each symmetry group and then delve into the enumeration of the different abstract groups.

textbook: Conway, John H, et al. The Symmetries of Things. A K Peters/CRC Press, 2008.

image: “ERRATA for The Symmetries of Things.” Errata for The Symmetries of Things, www.mit.edu/~hlb/Symmetries_of_Things/SoTerrors.html.

1. teramage says:

Hello!

What an amazing project! While a lot of the math is beyond me, I can appreciate the merits of having a G/H/K or G//K notation established for describing these phenomena. It seems quite elegant and beautiful that symmetry, which can be instantly recognized (or perhaps “perceived” is a better word) by the human eye, can also be represented so precisely in mathematical terms.

I was especially interested by the value K, which you say is “the kernel which fixes all colors”. In OS design, the kernel is the piece of the system which interfaces with peripherals, controls interrupts, and performs many other essential functions. What is the meaning of “kernel” here? And, how does this kernel differ from H? Once again, most of the math you use is beyond me, so I’m sure my questions are silly, but all the same, I’d be interested to hear how you’d define/describe this term.

Thank you so much!

2. arvanessendelf says:

Hi!

Thank you for your interest in my project. In group theory, groups are collections of elements that are held together by certain properties. The elements must be associative, have inverses, have an identity, and be closed under multiplication. A kernel is a subgroup, K, of a larger group G such that the elements in K are elements of the domain that map to the identity of the codomain. For instance the integers are a group under addition by the properties above. If you take the function f from integers to integers where f(x) = x, the identity of the codomain is 0 as x+0=x and 0+x=x for all x in integers. Thus the kernel of the group is all x in integers such that f(x) = 0. Under the definition of our function the kernel is the element 0 in the domain as this is the only x that maps to 0. These functions can not be any function however they must be a special function called a homomorphism. These are functions that preserve structure between groups. Ex: Let G and H be two groups. Then f: G to H is a homomorphism if for all x,y in G, f(x)f(y) = f(xy). Within the framework of color symmetry, there exists a homomorphism from the symmetric group G which is all the symmetries of a design into the permutation of the colors of the group. The kernel of that homomorphism is permutations that map elements of each color to elements of that same color. This is what I meant by the kernel fixes the colors of the symmetry group. For a symmetry type, the kernel is another type of symmetry that is the symmetry of between elements of the same color.

I don’t know if this will help or make you more confused, it probably isn’t the best explanation, but I hope that answered your question!