Symmetry: art, nature, and math.
Symmetry follows us around everywhere. We seek it out as much as we seek to see friendly faces. We equate it with beauty, order, and sometimes spirituality. With our symmetric bodies we walk the earth seeing, creating, and categrorising symmetries. There is a particular way to do the latter.
another section
BEWARE! This is very much under construction. I suck at computer stuff and i am a slow learner. Do not expect anyhting but half finished ramblings.
Defining types of symmetry
How do we decide the way in which symmetries... symmetry? In what way they look like themselves? Through rotation, mirroring, or something else entirely? First, we'll need some group therory.
Defining groups:
A group consists of a Set of elements, such as G={A, B, C, D, E} and an operation (○). In order for any couple of G, circly to be a group, the following requirements must be fulfilled:
- the operation of two of the elements of the set G is also an element of G. Such as A ○ B = D (which is an element of G). This has to be true for all elements of G
- Association has to be valid. Namely, A ○ (B ○ C) = (A ○ B) ○ C.
- There exists a neutral element E, so that, when applied to any element of set G, it doesn't change it. Such as A ○ E = A, and B ○ E = B.
- For each element of the set G, there exists an inverse element, where A ○ A^(-1) = E.
furthermore:
Groups where the commutative law applies, namely where A ○ B = B ○ A for every element of the group, are called "Abelian groups".
The order g of a Group is the total number of elements in the group. For our chosen G, that would be g = 5.
A Group table is what you get when you write all elements of G in a row, then all the elements of G in a column, and execute the operation between each of them. This way one can obtain a Table containing all possible combinations (through ○) of two elements of G. These are not unlike multiplication tables you might have used to learn basic multiplication.
Ähnlichkeitstransformation: Seien 𝐴, 𝐵 ∈ 𝐺; 𝐴, 𝐵 heißen konjugiert zueinander, wenn eine Ähnlichkeitstransformation existiert, d.h. es gibt ein Element 𝑈 ∈ 𝐺, so dass A = 𝑈 ⊙ 𝐵 ⊙ 𝑈−1 (und damit auch B = 𝑈−1 ⊙ 𝐴 ⊙ 𝑈) (Bem.: Wenn zwei Elemente B und C konjugiert zu A sind, dann sind sie auch zueinander konjugiert) Def: Alle Elemente einer Gruppe, die zueinander konjugiert sind, bilden eine konjugierte Klasse. • Jedes Element gehört genau zu einer Klasse • Die Anzahl der Elemente in einer Klasse muss ein Teiler der Ordnung der Gruppe sein • Elemente, die miteinander vertauschen 𝐴 ⊙ 𝐵 = 𝐵 ⊙ 𝐴 sind immer in verschiedenen Klassen => 𝐸 ist immer eine eigene Klasse, da 𝐸 mit allen Elementen vertauscht => In Abelschen Gruppen bilden alle Elemente eine eigene Klasse