, 7 tweets, 3 min read Read on Twitter
A fractal impossible object, packed with impossibility at multiple scales!

Nidhal Selmi drew this. It's a blend of the Sierpinski triangle, a famous fractal, and the Penrose tribar, a famous "impossible object".

Let me explain how it's connected to sheaf cohomology.

First, the Penrose tribar! Penrose has pointed out that this is "locally possible" - there's nothing wrong with any *small piece*. But if you follow it all the way around, you can't consistently interpret it.

This is precisely what sheaf cohomology can detect.
Second, the Sierpinski triangle: a triangle with triangular holes poked out of it, in an iterated way.

While the Penrose tribar only has one hole, this has a countable infinity of them... each giving the opportunity for an impossible twist!
A "sheaf" on a space X assigns a set of "sections" F(U) to any open subset U of that space, with the ability to restrict sections to smaller subsets, and to glue sections that agree on overlaps to get sections on bigger subsets!

Sheaves let us study local vs. global issues.
The "first cohomology" of a sheaf describes the set of things that *locally* look like sections but perhaps do not *globally* come from sections.

In this paper, Penrose showed that his tribar gives a nontrivial element of first cohomology:

So, you should be able to work out the sheaf cohomology of the Sierpinski triangle and show Nidhal Selmi's impossible object gives an interesting element of the first cohomology!

I would do this here, but this tweet is getting a bit technical and I need to get some work done.
If you want to learn more about sheaves and impossible objects, besides Penrose's article I recommend this:


Have fun!
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