begvend wrote:[...]Envelopes into 3D of 16 cell are:
- Cell-first projection is a cube.
- Face-first projection is a hexagonal dipyramid.
- Edge-first projection is a ?
A flattened square dipyramid (flattened octahedron):
The nearest edge is the vertical one in the center of the image.
- Vertex-first projection is an octahedron.
Envelopes into 3D of 24 cell are:
- Cell-first projection is a cuboctahedron.
- Face-first projection is a ?
I don't know what to call this shape, it's something like two irregular hexagonal antiprisms with enlarged bases, joined to each other at the enlarged bases.
The nearest face is highlighted in red for ease of identification.
- Edge-first projection is a ?
Hexakis hexagonal prism (hexagonal prism with two hexagonal pyramids on its hexagonal faces).
Nearest edge highlighted in red for easy spotting. Note that the hexagonal prism is
not uniform; its tetragonal faces are rectangular, not square, so this shape is not the Johnson solid.
- Vertex-first projection is a rhombic dodecahedron.
Envelopes into 3D of 120 cell are:
- Cell-first projection is a ?
Truncated rhombic triacontahedron (almost like a buckyball/truncated icosahedron, except with a different number and arrangement of hexagons, in this case non-uniform elongated hexagons).
- Face-first projection is a ?
It's an interesting shape, but I've no idea what to call it. It has a fascinating arrangement of pentagonal and decagonal faces. For clarity, I show only the faces/edges/vertices that lie on the envelope plus the closest face.
- Edge-first projection is a ?
No idea what to call this shape either. All I can say is, there are dodecahedra, dodecahedra everywhere!
- Vertex-first projection is a ?
Something with tetrahedral symmetry, of course. Still no idea what to call it.
Like I said, dodecahedra, dodecahedra everywhere!
Envelopes into 3D of 600 cell are:
- Cell-first projection is a ?
An interesting shape, no doubt. I don't think there's a name for it:
There are lots of triangles and kite-shaped faces. For clarity, I'm rendering only the envelope here.
- Face-first projection is a ?
Who knows what this is called? I don't. At any rate, it's something with triangular symmetry.
- Edge-first projection is a ?
An interesting shape with two dodecagonal pyramids.
Let's see. It consists of two identical halves, each of which looks like some kind of modified pentakis icosidodecahedron with dodecagonal pyramids at the top. Good luck naming this thing. I suggest "envelope of the edge-first projection of the 600-cell into 3D".
- Vertex-first projection is a ?
A pentakis icosidodecahedron (an icosidodecahedron with pentagonal pyramids on its pentagonal faces).
Many thanks in advance for your help.
You're welcome. Sorry for replying so late; I've been quite busy recently.
P.S. I doubt you'll gain very much insight into these shapes just by looking at these projection envelopes. Much more insightful is the internal structure of these projections (which I've omitted in many of these images 'cos they're kinda complex to convey in a single 2D image), as described in:
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http://eusebeia.dyndns.org/4d/regular.html</shameless plug>