Puzzle Time - Potato Curves

[jon-nyc]

Given two potatoes, can you draw a closed curve on the surface of each so that the two curves are identical as curves in three-dimensional space?

[Klaus]

I assume this is possible.

I assume this follows from Brouwer's fixed point theorem.

Are there any assumptions about the shape of the potatoes? I assume the assumption is that they are topologically equivalent, e.g., the case where one of the potatoes is shaped like a donut or Moebius ring isn't allowed, right?

I assume the potatoes are not necessarily convex.

[Horace]

I assume this is possible.

And I assume that, in practice, the largest possible identical closed curves would be so small as to render the real-world framing of the question absurd.

[jon-nyc]

@Horace said in Puzzle Time - Potato Curves:

I assume this is possible.

And I assume that, in practice, the largest possible identical closed curves would be so small as to render the real-world framing of the question absurd.

Actually there are an infinite number of such curves and some are roughly as large as the smaller potato

[jon-nyc]

@Klaus said in Puzzle Time - Potato Curves:

I assume this is possible.

I assume this follows from Brouwer's fixed point theorem.

Are there any assumptions about the shape of the potatoes? I assume the assumption is that they are topologically equivalent, e.g., the case where one of the potatoes is shaped like a donut or Moebius ring isn't allowed, right?

I assume the potatoes are not necessarily convex.

Assume they’re regular potatoes. In actual fact this works with any two shapes, you could use a spiky polyhedron and a donut.

[jon-nyc]

Here’s the hint that gives it away:

click to show

Imagine solving it with holograms of the potatoes.

[Horace]

oh right. Overlap them. Clever.

[jon-nyc]

The answer comes out Saturday. I’m curious if they’ll share a rigorous theorem or just say “imagine the holograms”