Simplicial polytope
In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a simplicial polyhedron in three dimensions contains only triangular faces[1] and corresponds via Steinitz's theorem to a maximal planar graph.
They are topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons.
Examples
Simplicial polyhedra include:
- Bipyramids
- Gyroelongated dipyramids
- Deltahedra (equilateral triangles)
- Platonic
- Johnson solids:
- triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, gyroelongated square dipyramid
- Catalan solids:
Simplicial tilings:
- Regular:
- Laves tilings:
- tetrakis square tiling, triakis triangular tiling, kisrhombille tiling
Simplicial 4-polytopes include:
- convex regular 4-polytope
- Dual convex uniform honeycombs:
- Disphenoid tetrahedral honeycomb
- Dual of cantitruncated cubic honeycomb
- Dual of omnitruncated cubic honeycomb
- Dual of cantitruncated alternated cubic honeycomb
Simplicial higher polytope families:
- simplex
- cross-polytope (Orthoplex)
gollark: This is why ALL are to utilize zig.
gollark: ```c#include <sys/socket.h>#include <sys/types.h> #include <netinet/in.h>#include <stdio.h>#include <string.h>#include <sys/select.h>#include <fcntl.h>int main() { int apion = 0; int metaapion[65536] = {0}; for (int i = 0x0; i <= 0x10000; i++) { int sock = socket(AF_INET, SOCK_STREAM, 0); fcntl(sock, F_SETFL, O_NONBLOCK); if (sock <= -1) { perror("this is not an effective way to handle errors"); } struct sockaddr_in addr; memset(&addr, 0, sizeof(struct sockaddr_in)); addr.sin_family = AF_INET; addr.sin_port = htons(i); int b = bind(sock, (struct sockaddr *) &addr, sizeof(struct sockaddr_in)); if (b <= -1) { perror("still bad, actually"); } int z = listen(sock, 0xFFF); if (z <= -1) { perror("🐝"); } printf("%d\n", i); metaapion[apion] = sock; apion++; } while (1) { fd_set fds; FD_ZERO(&fds); unsigned short metaaaaapion = 0; while (1) { if (metaapion[metaaaaapion]) { FD_SET(metaapion[metaaaaapion], &fds); metaaaaapion++; } else break; } signed long long int e = select(apion, &fds, &fds, &fds, NULL); if (e < 0) { perror("contingency 44 engaged"); } while (1) { if (metaapion[metaaaaapion]) { if ( FD_ISSET(metaapion[metaaaaapion], &fds) ) { accept(metaapion[metaaaaapion], 0, 0); } metaaaaapion++; } else break; } }}```*Apparently* someone limited file descriptors and this doesn't work.
gollark: Actually, \🐝.
gollark: "Done".
gollark: Possibly.
See also
Notes
- Polyhedra, Peter R. Cromwell, 1997. (p.341)
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