Tetrahedral prism

In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices.

Tetrahedral prism

Schlegel diagram
TypePrismatic uniform 4-polytope
Uniform index48
Schläfli symbol{3,3}×{} = h{4,3}×{}
s{2,4}×{}
sr{2,2}×{}
Coxeter diagram =

Cells2 (3.3.3)
4 (3.4.4)
Faces8 {3}
6 {4}
Edges16
Vertices8
Vertex configuration
Equilateral-triangular pyramid
Symmetry group[3,3,2], order 48
[4,2+,2], order 16
[(2,2)+,2], order 8
Propertiesconvex

Net

It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.

Images

An orthographic projection showing the pair of parallel tetrahedra projected as a quadrilateral divided into yellow and blue triangular faces. Each tetrahedra also have two other uncolored triangles across the opposite diagonal.
Transparent Schlegel diagram seen as one tetrahedron nested inside another, with 4 triangular prisms between pairs of triangular faces.
Rotating on 2 different planes

Alternative names
  1. Tetrahedral dyadic prism (Norman W. Johnson)
  2. Tepe (Jonathan Bowers: for tetrahedral prism)
  3. Tetrahedral hyperprism
  4. Digonal antiprismatic prism
  5. Digonal antiprismatic hyperprism

Structure

The tetrahedral prism is bounded by two tetrahedra and four triangular prisms. The triangular prisms are joined to each other via their square faces, and are joined to the two tetrahedra via their triangular faces.

Projections

The tetrahedron-first orthographic projection of the tetrahedral prism into 3D space has a tetrahedral projection envelope. Both tetrahedral cells project onto this tetrahedron, while the triangular prisms project to its faces.

The triangular-prism-first orthographic projection of the tetrahedral prism into 3D space has a projection envelope in the shape of a triangular prism. The two tetrahedral cells are projected onto the triangular ends of the prism, each with a vertex that projects to the center of the respective triangular face. An edge connects these two vertices through the center of the projection. The prism may be divided into three non-uniform triangular prisms that meet at this edge; these 3 volumes correspond with the images of three of the four triangular prismic cells. The last triangular prismic cell projects onto the entire projection envelope.

The edge-first orthographic projection of the tetrahedral prism into 3D space is identical to its triangular-prism-first parallel projection.

The square-face-first orthographic projection of the tetrahedral prism into 3D space has a cuboidal envelope (see diagram). Each triangular prismic cell projects onto half of the cuboidal volume, forming two pairs of overlapping images. The tetrahedral cells project onto the top and bottom square faces of the cuboid.

It is the first in an infinite series of uniform antiprismatic prisms.

Convex p-gonal antiprismatic prisms
Name s{2,2}×{} s{2,3}×{} s{2,4}×{} s{2,5}×{} s{2,6}×{} s{2,7}×{} s{2,8}×{} s{2,p}×{}
Coxeter
diagram







Image
Vertex
figure
Cells 2 s{2,2}
(2) {2}×{}={4}
4 {3}×{}		 2 s{2,3}
2 {3}×{}
6 {3}×{}		 2 s{2,4}
2 {4}×{}
8 {3}×{}		 2 s{2,5}
2 {5}×{}
10 {3}×{}		 2 s{2,6}
2 {6}×{}
12 {3}×{}		 2 s{2,7}
2 {7}×{}
14 {3}×{}		 2 s{2,8}
2 {8}×{}
16 {3}×{}		 2 s{2,p}
2 {p}×{}
2p {3}×{}
Net

The tetrahedral prism, -131, is first in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The tetrahedral prism is the vertex figure for the second, the rectified 5-simplex. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each uniform polytope in the sequence is the vertex figure of the next.

k31 dimensional figures
n 4 5 6 7 8 9
Coxeter
group		 A3A1 A5 D6 E7 = E7+ =E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [33,3,1] [34,3,1]
Order 48 720 23,040 2,903,040 ∞
Graph - -
Name −131 031 131 231 331 431

References
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
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