# The Deca-augmented 5,10-duoprism

The deca-augmented 5,10-duoprism is a CRF polychoron bounded by 5 decagonal prisms, 50 square pyramids, and 10 pentagonal bipyramids (J13) for a total of 65 cells, 205 polygons (5 decagons, 50 squares, and 150 triangles), 200 edges, and 60 vertices.

It is constructed by augmenting the 5,10-duoprism with 10 pentagonal prism pyramids. Due to the dichoral angle between the pentagonal prisms in the 5,10-duoprism being exactly 144°, and the dichoral angle between the pentagonal pyramids and pentagonal prism of the pentagonal prism pyramid being exactly 18°, the pentagonal pyramid cells of adjacent augments lie on the same hyperplanes, thus merging into 10 pentagonal bipyramids (J13).

It can be partially Stott-expanded along the symmetries of the decagons to yield the deca-augmented 5,20-duoprism, which has the same structure except with elongated pentagonal bipyramids (J16) instead of the J13 cells, and icosagonal prisms instead of the decagonal prisms.

## Structure

### Centered on Ring of Decagonal Prisms

The deca-augmented 5,10-duoprism consists of a ring of 5 decagonal prisms:

This ring of cells intertwines with an orthogonal ring of pentagonal bipyramids, strung to each other by their apical vertices:

These pentagonal bipyramids are straddled by square pyramids, 5 surrounding each shared apex of the pentagonal bipyramids, for 50 square pyramids in total:

These are all the cells of the deca-augmented 5,10-duoprism.

### Centered on a Pentagonal Bipyramid

#### The Near Side

Now we look at the deca-augmented 5,10-duoprism from a different 4D viewpoint, one that allows us to see the J13 cells more clearly.

The above image shows 5 of the pentagonal bipyramid (J13) cells that face this new 4D viewpoint.

These J13 cells are straddled by 20 square pyramids:

These are all the cells that lie on the near side of the polychoron.

#### The Far Side

Next, we come to the far side of the polytope. Here are the other 5 pentagonal bipyramids:

There are 30 square pyramids straddling these cells:

Finally, the square faces of all the square pyramids we've seen so far, including those on the near side of the polychoron, are where the 5 decagonal prisms are fitted:

These decagonal prisms lie at a steep angle from the 4D viewpoint, and therefore appear quite squished by foreshortening. They are, of course, perfectly uniform in 4D.

#### Summary

The following table summarizes the cell counts:

Region | |||
---|---|---|---|

Near side | 5 | 20 | - |

Far side | 5 | 30 | 5 |

Grand total | 10 | 50 | 5 |

65 cells |

## Coordinates

The Cartesian coordinates of the deca-augmented 5,10-duoprism, centered on the origin and with edge length 2, are:

- (A, 0, ±2φ, 0)
- (B, ±φ, ±2φ, 0)
- (−C, ±1, ±2φ, 0)
- (A, 0, ±1, ±√(3+4φ))
- (B, ±φ, ±1, ±√(3+4φ))
- (−C, ±φ, ±1, ±√(3+4φ))
- (A, 0, φ
^{2}, ±√(2+φ)) - (B, ±φ, φ
^{2}, ±√(2+φ)) - (−C, ±φ, φ
^{2}, ±√(2+φ)) - (0, 0, 0, ±2A)
- (0, 0, ±2φ, ±2B)
- (0, 0, ±2, ±2C)

where:

A | = | √((10+2√5)/5) |

B | = | √((5−√5)/10) |

C | = | √((5+2√5)/5) |

and φ=(1+√5)/2 is the Golden Ratio.