What's the nicest way to draw a shape with many holes?
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- Опубликовано: 2 окт 2024
- For millennia human beings have been drawn to the question: what's the nicest possible shape I can make? "Nicest" could mean the most symmetric-for instance, the ancient Greeks discovered there were five so-called Platonic solids where every face and every vertex looks the same (tetrahedron, cube, octahedron, dodecahedron, icosahedron). Or, "nicest" could mean smoothest. For instance, the smoothest shape with no holes or handles is a round sphere; the smoothest shape with a single hole is (by some definitions) a perfectly round donut, also known as a torus of revolution.
What's the nicest way to draw a shape with many holes? As shown in this video, the smoothest possible shapes often want to have a high degree of symmetry-just like the Platonic solids. In each case we start with an initial guess of several donuts glued together along a straight line. If we make this surface smoother and smoother and smoother, we end up with nice smooth shapes that look like the tetrahedron, the cube, and the icosahedron.
More precisely, these surfaces are (conjectured!) minimizers of something called the tangent-point energy. This energy tries to keep all pairs of points on the surface far away from each other-just like two electrons pushing away from one another. At the same time, the area of the surface must remain the same, so that it doesn't just blow up to infinity.
These motions were computed using the algorithm described in
Chris Yu, Caleb Brakensiek, Henrik Schumacher, Keenan Crane
"Repulsive Surfaces" (2021)
For more information, see www.cs.cmu.edu/...
What does it look like with even more holes? How high can you go?
Presumably icosahedron with 19 holes, though it would be interesting to see what happens with non-platonic solids.
infinitely high. Also agree with Mc. - Presumably the holes would end up being dual to *some* sort of repulsive potential for points on a sphere, but I know that those don't always give platonic solids as an answer (for the appropriate number of points) so I wonder which potentials end up giving these results.
We talk a bit about this in the paper linked above. Basically as the number of holes continue to increase you start to get "volumetric" shapes, rather than something that looks like a sphere. More like a grid in 3D space. This makes a certain amount of sense, because if you had a big, hollow sphere then you could reduce energy by moving some of the material to the (empty) sphere center. *Exactly* what these shapes look like is an interesting open question…
okay but why does the cube have 5 holes and 6 faces? where does the 6th hole come from??? is it a kind of ficticious hole based on the boundary? Try higher numbers maybe we can find some like not quite platonic solids.
Think of drilling holes into a cube. The first hole can go through two faces (say top and bottom). Then you only need to drill half way through each of the side faces until you get a hole, since it meets up with the original hole. So it's just 5 holes total, even though it seems like 6.
@@adelelopez1246 Great explanation! Yes, exactly. This perspective is similar in spirit to thinking about the exterior of a planar graph being a single "infinite" face. A funny thing people often do in mathematics is think of the plane as actually just a close-up view of the sphere, much like when we stand on the Earth it looks like a plane-even though we know it's a big ball. Another closely related idea is that you can decompose the 3-sphere into two solid tori: math.stackexchange.com/questions/4014174/s3-as-union-of-2-solid-tori. Again you can think of 3-dimensional space as just a really huge sphere, rather than as being infinite in all directions.
Your talk (and this video) did mention shapes where the holes happen to be one less than the sides of a platonic solid (for three examples)
Will going for seven holes give you an octahedron and nineteen an icosahedron?
And what about any other number of holes?
@@duyaa9526 thanks I'll check it out
It doesn't *quite* work out this way, because as you add more and more holes the shapes start becoming more "volumetric," sort of like a regular grid in 3D space. The intuition is that if you had something like a big empty sphere, then you could reduce the energy by moving some of the material to the empty interior of the sphere.
@@keenancrane ah that makes sense. So you might have, say, a bunch of "pillars" that go to the center, adding several more holes there. I *think* if you did that to a tetrahedron you'd add like four more holes maybe?
Trying to imagine this in my head right now, might be off a bit. But that'd put you at 3 holes for the tetrahedron, and then 4 more, so like seven holes in total. - Is that the shape you get in that case? A tetrahedron with extra edges connecting to a fifth point in its center?
@@Kram1032 I think in that case it's still better to try to distribute the holes around a sphere, rather than in a solid volume. But yes, as you keep adding more and more "pillars" things eventually get more volumetric…
@@keenancrane Makes sense, thanks!
By adding more holes can you generalize the platonic solids? What do they look like?
Also, small note, a sphere technically has one hole (the hollow inside), and a torus has two :) a filled sphere (which has no holes) is usually called a ball in topology, and a filled torus I think is usually called "donut".
We talk a bit about this in the paper, but basically as you keep adding holes you end up with more "volumetric" objects: the surface ends up looking like a regular grid of tunnels. So you don't recover all the platonic solids: as the space on the "interior" gets bigger and bigger, it eventually becomes better to move some of the material to the big empty space on the inside.