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…and the answer is 
!
Each one is a *hectohedron* (a solid with 100 planar faces)
The first correct answer was from @0x686b, followed by @inaba_darkfox then @lanky_yankee. Congratulations! https://twitter.com/KangarooPhysics/status/1322189144953704458
!Each one is a *hectohedron* (a solid with 100 planar faces)
The first correct answer was from @0x686b, followed by @inaba_darkfox then @lanky_yankee. Congratulations! https://twitter.com/KangarooPhysics/status/1322189144953704458
When I found these shapes I thought they were surprising on a number of levels:
-The only hectohedra I've seen documented before are some completely irregular D100 dice, or have only trivial symmetry such as faceted cones. So to find 3 new symmetrical ones at once was neat.
-The only hectohedra I've seen documented before are some completely irregular D100 dice, or have only trivial symmetry such as faceted cones. So to find 3 new symmetrical ones at once was neat.
-At first glance they appear v similar to more familiar icosahedrally symmetric polyhedra. Like these they have 12 pentagons and the rest hexagons(12 valence 5 vertices and the rest v 6 for the triangular one), but look carefully and you see the distance between 5gons varies
-Despite having the same no. of faces and the same symmetry group, the 2 solids on the left really are different, and not just as mirrors of each other.
-The triangular one is not just a dual of either of the others (since that would swap the number of faces and verts)
-The triangular one is not just a dual of either of the others (since that would swap the number of faces and verts)
-I found it slightly counter-intuitive that rotational tetrahedral symmetry can give something this uniform. From my experience with meshes and subdivision, I was used to thinking that the only Platonic solids that gave remotely even spacing were the dodecahedron and icosahedron.
I first thought about similar sized circle packings with tetrahedral symmetry when learning about golf ball dimples. Several popular balls use a tetrahedral arrangement, though most have >300 dimples, and all the ones I've seen have more than 12 irregular faces.
