4D Euclidean space
In February 2018, we introduced the tetrahedral ursachoron, a polychoron that consists of 4 tridiminished icosahedra (J63) folded in tetrahedral formation around a tetrahedral cell. This month, we introduce its big cousin, a polytope that consists of 4 tridiminished rhombicosidodecahedra (J83) folded in tetrahedral formation around a cuboctahedral cell:
This is the tetrahedral magnaursachoron, an interesting CRF polychoron that consists of 4 J83 cells, 8 J63 cells, 6 metabidiminished icosahedra (J62), 34 triangular prisms, 4 triangular cupolae (J3), 5 cuboctahedra, and one truncated tetrahedron.
It bears many parallels with J83 itself: just as J83 has 3 decagons folded around a framework of alternating triangles and squares, so the magnaursachoron has 4 J83 cells folded around a framework of J63 cells and cuboctahedra alternating with triangular prisms.
22 Mar 2019:
19 Mar 2019:
18 Mar 2019:
17 Mar 2019:
13 Mar 2019:
12 Mar 2019:
Added more Johnson solids:
11 Mar 2019:
8 Mar 2019:
Full coordinates are provided for both of these rare solids in algebraic form, from which arbitrarily precise coordinate values may be computed.
Added the gyrate bidiminished rhombicosidodecahedron (J82), yet another Johnson solid.
7 Mar 2019:
1 Mar 2019:
The Polytope of the Month for March is up!
This particular polychoron is notable because 6 of the triangular prisms of the cantitruncated 5-cell lie on the same hyperplane as the 6 triangular prisms of the augment in gyrated orientation, and therefore merge into gyrobifastigium (J26) cells.
25 Feb 2019:
1 Feb 2019:
The Polytope of the Month for February is up!
In generalizing 3D polyhedra to 4D, a question that frequently crops up is how to generalize the 3D antiprisms. The common understanding of a 3D antiprism—two polygons in parallel planes rotated relative to each other and connected by triangles—is difficult to generalize to higher dimensions in a consistent way that is also aesthetically-pleasing.
A different way of understanding a 3D antiprism is that the top and bottom
polygons are duals of each other. Since the dual operation
applies across all dimensions, and furthermore preserves the symmetries of the
starting polytope, this provides a nice basis on which to construct a
higher-dimensional definition of an antiprism. Under this analysis, the
triangles connecting the polygons may be thought of as
which become full-dimensioned pyramids in the higher-dimensional
In the case of 4D, we can form a polyhedron antiprism by placing the polyhedron and its dual in parallel hyperplanes, and connecting the vertices of one to the faces of the other with polygonal pyramids, and vice versa. Furthermore, the edges of one can be connected to the other by tetrahedra. The image above shows one such example: the cube antiprism (K4.15), formed by placing a cube and an octahedron in two parallel hyperplanes and connecting them with 6 square pyramids and 20 tetrahedra, 8 of which connect the faces of the octahedron to the vertices of the cube, and 12 of which connect their respective edges.
Note: the above definition of a 4D antiprism is not compatible with the structure of the so-called Grand Antiprism, one of the uniform polychora. The grand antiprism has an interesting structure based on the Hopf fibration, which is peculiar to 4D and thus difficult to generalize to other dimensions. Furthermore, the grand antiprism stands alone as the only uniform (and indeed, the only CRF) member of its family. Nevertheless, its rings of antiprisms connected by tetrahedra do seem like a fitting analogue of the 3D antiprisms, so it possibly represents a different way to generalize the family of 3D antiprisms to higher dimensions.
25 Jan 2019:
An important internal rendering system upgrade took place today. You should not notice anything different about the site, but if you notice any images that have glitches, rendering artifacts, or that otherwise look wrong, please let us know. Thanks, and enjoy the site!
23 Jan 2019:
Added more Johnson solids:
21 Jan 2019:
16 Jan 2019:
15 Jan 2019:
10 Jan 2019:
9 Jan 2019:
The Polytope of the Month for January is up!