A transformation T of a structure D is structure-preserving if T(D) has the same symmetry group as the original one: Aut(T(D)) ~= Aut(D).https://twitter.com/literalbanana/status/982816860566532096 …
Anti-authoritarian, bleeding-heart Stirnerite,
. Banned in Sweden. SubGenius, Zhuangist, white-hat troll. Defrocked mathematician. Brain problems.
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St. Rev ☯️ 🏴 😻 Retweeted Simply Knitted Banana
A transformation T of a structure D is structure-preserving if T(D) has the same symmetry group as the original one: Aut(T(D)) ~= Aut(D).https://twitter.com/literalbanana/status/982816860566532096 …
St. Rev ☯️ 🏴 😻 added,
This naive first take on a definition actually runs into some subtle problems: Consider a square divided into four smaller squares. The large square has the same symmetry group as the smaller ones, but automorphisms of the large square are not automorphisms of the small ones!
Nevertheless, this approach holds out the prospect of an algebraically precise language for discussing aspects of the problem. Consider that in @literalbanana's poll, the original rectangle has symmetry group Z_2 x Z_2, while the transformed versions are all Z_2.
That is, they all break the structure in the same way: they lose symmetry across the horizontal axis, while keeping it across the vertical axis.
Why did Alexander believe there was a correct choice among the three? Perhaps as an architect, symmetry across the horizontal axis is less important, since gravity always breaks it. His preferred elaboration resembles an object standing on legs! cc: @literalbanana
his original degree was in math I think - but “structure-preserving transformation” isn’t formalized in the math way I don’t think
Yeah, what I'm arguing I think is that group actions are one language to (very rigorously) describe some of these notions, not all of them. Certainly when you look at something like https://en.wikipedia.org/wiki/Islamic_geometric_patterns … it's almost necessary.
But it tells you nothing about why he prefers the third rectangle, except by omission.
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