A convoluted and bad way to notate symmetry.

Higher-dimensional geometry (previously "Polyshapes").

A convoluted and bad way to notate symmetry.

Postby ubersketch » Fri Jan 11, 2019 10:44 pm

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Alright, time for some handwaving. (cue track of little kids yelling "yay")
[1] by itself represents an asymmetrical figure.
[2] represents a figure with 2-fold rotational symmetry.
To absolutely nobody's surprise [code][n][/code] has n-fold rotational symmetry.
[*n] has n axis of mirror symmetry.
Now on to the cool stuff.
[*6-*3-*2] is the symmetry of the euclidean hexagonal tiling.
A - represents a sort of mutual symmetry, i.e how Conway's orbifold notation works.
Introducing the ->
An arrow represents translational symmetry. ->1 just translates it and doesn't do anything. ->n translates and rotates the number 1/nth of a circle. ->* translates it and reflects it.
On to nested brackets. Let's say you have a symbol [1->1]. What if we replaced "1", the asymmetrical figure with a symmetrical figure. [[[6*-3-*2]-2]->1] would represent the symmetry of a euclidean uniform honeycomb made entirely of hexagonal prisms.
This is fairly informal, mostly because the concepts I want to share are kinda hard to represent with words. And this cannot obviously enumerate all of the symmetry groups, due to the lack of hyperparellel symmetries, and possibly ambiguity. I'll leave off with a few things.
[1->*] = glide reflection frieze pattern.
[1->6] = Hexagonal helical thingmabob.
[[1->1]->1] = wonder-ring.
I would enjoy criticism and improvements considering how badly written this is.
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