The Tetrahedral Canticupola

The Tetrahedral Canticupola, also known as Truncated Tetrahedron Atop Truncated Octahedron (truncated octahedron || truncated octahedron), K4.76, or its Bowers Acronym tutatoe, is one of Richard Klitzing's convex segmentochora, 4D CRF polytopes whose vertices lie on two parallel hyperplanes and which are orbiform (can be inscribed into a 4D sphere).

Its surface consists of a truncated tetrahedron and a truncated octahedron lying on parallel hyperplanes, with 6 triangular prisms, 4 triangular cupolae, and 4 hexagonal prisms connecting them to each other. It has a total of 16 cells, 58 polygons (16 triangles, 30 squares, 12 hexagons), 78 edges, and 36 vertices.

The tetrahedral
canticupola

It is notable for combining with a few other polychora to produce CRF polytopes that contain gyrobifastigium (J26) cells.

Structure

We shall explore the structure of the tetrahedral canticupola using its parallel projections into 3D, centered on its truncated tetrahedral cell.

Parallel projection of
K4.76, showing truncated tetrahedron

This image shows the truncated tetrahedron, which is the nearest cell to the 4D viewpoint. For clarity, we render all the other cells in a light transparent color.

The triangular faces of this cell are attached to 4 triangular cupolae, as shown in the next image:

Parallel projection of
K4.76, showing 4 triangular cupolae

The hexagonal faces of the truncated tetrahedron are attached to 4 hexagonal prisms, shown next:

Parallel projection of
K4.76, showing 4 hexagonal prisms

The remaining edges of the truncated tetrahedron are attached to 6 triangular prisms:

Parallel projection of
K4.76, showing 6 triangular prisms

Here are all these cells together:

Parallel projection of
K4.76, showing all preceding cells

The last cell is the truncated octahedron, that closes up the polytope:

Parallel projection of
K4.76, showing truncated octahedron

The following table summarizes the cell counts of the tetrahedral canticupola:

Layer
Near side	1	4	6	4	0
Far side	0	0	0	0	1
Grand total	1	4	6	4	1
Grand total	16 cells

Coordinates

The Cartesian coordinates of the tetrahedral canticupola, with edge length 2, are:

( 1/√2, 1/√2, 3/√2, −√(5/2))
( 1/√2, −1/√2, −3/√2, −√(5/2))
(−1/√2, 1/√2, −3/√2, −√(5/2))
(−1/√2, −1/√2, 3/√2, −√(5/2))
( 1/√2, 3/√2, 1/√2, −√(5/2))
( 1/√2, −3/√2, −1/√2, −√(5/2))
(−1/√2, 3/√2, −1/√2, −√(5/2))
(−1/√2, −3/√2, 1/√2, −√(5/2))
( 3/√2, 1/√2, 1/√2, −√(5/2))
( 3/√2, −1/√2, −1/√2, −√(5/2))
(−3/√2, 1/√2, −1/√2, −√(5/2))
(−3/√2, −1/√2, 1/√2, −√(5/2))
(0, ±√2, ±2√2, 0)
(±√2, 0, ±2√2, 0)
(±√2, ±2√2, 0, 0)
(0, ±2√2, ±√2, 0)
(±2√2, 0, ±√2, 0)
(±2√2, ±√2, 0, 0)

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