The Math of Hunting Lions | Hacker News Reader

The Math of Hunting Lions

gwern.net

36 pointsfinite_jest5 comments

> We observe that a lion has the connectivity of the torus

Dear topologists of HN, how do I visualize the deformation of a lion made of clay into a donut, if we model that a biologically accurate lion has one entrance orifice for solids and liquids and not one, but two, exit orifices which form a connected cave system inside the lion? (We may ignore nostrils, lungs, skin pores, and all other orifice systems.)

I was looking at the Wikipedia genus article:

https://en.wikipedia.org/wiki/Genus_%28mathematics%29

If my layman's understanding is correct, this means if we start with a clay sphere of genus 0 and bore a hole through the center of it we get genus 1, a torus. If we continue to bore additional holes through the center of the sphere that connect to our existing cave system we never get to genus 2, because with genus 2 you have two separate cave systems, a lump with two holes A and B such that entering hole A to explore it means that you can't explore hole B until you first exit hole A, so I conclude we must remain at genus 1.

But I don't see how you do a continuous deformation of a clay lump with three or more entrances into tunnels that meet in the middle into a donut with only one hole.

You can't; the sentence is an intentional simplification for the purposes of a joke. A torus has two non-trivial elements in its 1-dimensional homology group: there are two types of loops that I can't contract to a point on a torus, around the handle and orthogonal to the handle. A lion with 1 entrance and 2 exits has at least 3 such loops one around each entrance, and 3 more (I think) passing through each of the 3 choose 2 aforementioned holes.

Extremely helpful, thank you!

This vsauce video is about humans, but I imagine pretty much every mammal will be just about topologically equivalent. As it turns out we humans have quite a few more holes than fleshy donuts would.

https://www.youtube.com/watch?v=egEraZP9yXQ

Great vid! (Was rather cringe though when he eats the toilet dunked donut. Some years ago I recall the Exploratorium science museum in SF had a psychology exhibit consisting of a toilet drinking fountain made from a toilet that had never been used. The question posed was, if you aren't willing to drink from the toilet fountain, why not? It's hard to articulate why so many of us find it gross even if we are logically perfectly aware there isn't a sanitary risk.)

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