foldworksDo you know a polyhedron that has 300 faces? Ideally nice and non-trivial.
Or a polyhedron with 300 edges or 300 vertices?
Obot 50549535@foldworks I guess "non-trivial" means you don't want a prism or an antiprism?
@obot50549535 Yeah, I'm trying to avoid obvious shapes. It's harder than I thought it would be.
I tried subdividing other shapes but still couldn't get 300 faces, or even 300 edges š¤
@obot50549535
Apparently the compound of ten tetrahedra has 300 edges (when counted as non-intersecting).
Also, some octahedral geodesic polyhedra and Goldberg polyhedra have 300 edges: u5O and c5C at https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra#Octahedral
Nice, but still looking for 300 faces (or vertices)...
#askfedi #mathematics #geometry #iTeachMath #polyhedron #geodesic
Rahul Narain@foldworks If you have a polyhedron with N edges, you can get a polyhedron with N vertices by rectification: https://en.wikipedia.org/wiki/Rectification_(geometry)
@foldworks Since rectification gives you N vertices, the dual of rectification will give you N faces. This is the "join" operation in Conway's notation (https://en.wikipedia.org/wiki/Conway_polyhedron_notation#Original_operations), and turns, for example, a cube or an octahedron (12 edges) into a rhombic dodecahedron (12 faces).
Steven Dollins@foldworks 152 vertices makes for 300 triangles. This arrangement's dual gives 24 pentagons, 116 hexagons, and 12 heptagons sharing 300 vertices in tetrahedral symmetry.
@foldworks Or 152 points can arrange in icosahedral symmetry with 72 pentagons, 20 hexagons, and 60 heptagons sharing 300 vertices. Note that few of these polygons are regular or even planar.
@foldworks And here is a tetrahedral arrangement of 300 points that gives 596 triangles. The dual has 48 pentagons, 216 hexagons, and 36 heptagons for a total of 300 (non-planar) faces.
@scdollins Excellent, Iām now trying to visualise a construction starting from the icosahedron...
A straightforward solution from an icosahedron: add two octahedra and a tetrahedron on each face of the icosahedron to get 20 * 15 faces.
Yes, the faces of the tetrahedra and octahedra merge as rhombic face, but Hedron App @hedron counts distinct faces (the tetrahedra could be distorted to make 3 distinct faces.)
Hedron App@foldworks @scdollins If you actually want have the rhombic faces, then use the shape, I called: Heptaeder
@hedron @scdollins Thanks for the tip.
In this case, I did want the triangles counted separately. But other times I would want each rhombic face to be counted as one š
morph@foldworks
Ehm ... is this stuff still foldable? š³
Ehm ... is this stuff still foldable? š³