# 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.

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.

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:

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

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

Here are all these cells together:

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

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 |

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)