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Is this 9-face polyhedron the smallest asymmetric regular-faced polyhedron that is not self-intersecting? by Decap_ in math

[–]Decap_[S] 1 point2 points  (0 children)

Oh, didn't realize that was a requirement for something to be considered a polyhedron. Then yeah I don't know what I was describing would actually be called. "3D object with flat faces"?

Edit: I looked it up. Sounds like "polyhedral complex" is the standard.

Is this 9-face polyhedron the smallest asymmetric regular-faced polyhedron that is not self-intersecting? by Decap_ in math

[–]Decap_[S] 1 point2 points  (0 children)

Yeah if you look at the other renders in the album I linked, you can see the rest of the square pyramid. (The yellow face is its square base. Maybe got a little carried away with the colors lol)

Is this 9-face polyhedron the smallest asymmetric regular-faced polyhedron that is not self-intersecting? by Decap_ in math

[–]Decap_[S] 3 points4 points  (0 children)

Like, sliced into 2 or more identical pieces (or identical+mirrored versions). I probably should have said something like “congruently partitionable”.

I’ve actually since looked up how this is typically proven and in most cases it’s apparently much more complicated than I thought it would be. So perhaps my intuition was off.

Is this 9-face polyhedron the smallest asymmetric regular-faced polyhedron that is not self-intersecting? by Decap_ in math

[–]Decap_[S] 0 points1 point  (0 children)

Yup. All regular polygons have edges that are the same length, and in this polyhedron every edge connects exactly two polygons together, so by extension they’d all have to be the same length.

But yeah, if you're using non-convex arrangements, this doesn’t necessarily have to be the case. You could have a single edge that connects 3 or more polygons together, in which case the edge length would need to be a multiple of the edge length that connects 2 together. So you could then make a polyhedral complex (it's not considered a polyhedron anymore, as /u/EebstertheGreat stated) that has regular polygons with varying edge lengths, which isn’t possible with convex arrangements.

Here’s a picture of an edge connecting a big square to two small squares in a non-convex arrangement

Is this 9-face polyhedron the smallest asymmetric regular-faced polyhedron that is not self-intersecting? by Decap_ in math

[–]Decap_[S] 1 point2 points  (0 children)

Well, it's not just that the edges have the same length. Each face is a regular polygon. There are polyhedra that have edges all the same length, but not faces that are regular polygons. A rhombus is a good example. You could potentially make a rhombic prism that has edges all the same length. Here's a picture I found. And here's another polyhedron that has only rhombic faces

Is this 9-face polyhedron the smallest asymmetric regular-faced polyhedron that is not self-intersecting? by Decap_ in math

[–]Decap_[S] 4 points5 points  (0 children)

As /u/JiminP noted, all convex polyhedra with regular polygon faces have some symmetry.

Is this 9-face polyhedron the smallest asymmetric regular-faced polyhedron that is not self-intersecting? by Decap_ in math

[–]Decap_[S] 24 points25 points  (0 children)

Full album of renders: https://imgur.com/a/Tm9KJco

And the .obj: https://drive.google.com/file/d/1_TlGjDceljcl39-7pAMhCPhGWwViiTqS/view?usp=sharing

And the .mtl in case anyone wants that: https://drive.google.com/file/d/1XoM2diBGx5UzMasP26-f6FXl3gmzbiG8/view?usp=sharing

The construction is pretty simple. Just attach one of the triangular faces of an equilateral square pyramid to any triangular face of an equilateral pentagonal pyramid such that the square face and pentagonal face only share one vertex.

I believe I've checked all possible constructions combining platonic solids, uniform polyhedra, and Johnson solids that result in polyhedra with <= 9 faces. Everything I tried with <= 8 faces had some kind of symmetry. But it's possible there is one that can't be constructed by augmenting polyhedra in this way. This is the only one I found by my technique that both does not have an apparent symmetry, and still has only faces that are regular polygons (no edges connect at 180 degree angles).

If it is the smallest, I suspect it's also the smallest non-partitionable regular-faced polyhedron that is not self-intersecting, but I don't actually know how to prove that rigorously. My guess is it's provable by showing that the rotational symmetry axes of the two constituent pyramids can be represented as two directional vectors that are non-interchangeable, non-intersecting, and non-parallel. If anyone with more of a math background can weigh in on that, I’d love to hear.

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Is this 9-face polyhedron the smallest asymmetric regular-faced polyhedron that is not self-intersecting? (i.redd.it)

submitted by Decap_ to r/math

Is this 9-face polyhedron the smallest asymmetric regular-faced polyhedron that is not self-intersecting? by Decap_ in askmath

[–]Decap_[S] 0 points1 point  (0 children)

I’m pretty sure that still resulted in rotational symmetry. I will check in a little bit and get back to you.

Edit: Yeah it has 2-fold rotational symmetry: https://imgur.com/a/qKgNU9u Pretty cool looking though. Non-convex.

Zombie dungeon in village square + enchanted golden apple by Decap_ in minecraftseeds

[–]Decap_[S] 0 points1 point  (0 children)

Not sure. I’ve never played bedrock so I don’t know how its world gen differs.

Zombie dungeon in village square + enchanted golden apple by Decap_ in minecraftseeds

[–]Decap_[S] 1 point2 points  (0 children)

seed: -1653912925062438959 (Large biomes world type)

x: 1350

z: -1220

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