# Pentagonal magnabicupolic ring (EntityTopic, 17)

The pentagonal magnabicupolic ring is a CRF polychoron discovered by Quickfur. It is a member of the family of bicupolic rings, which contains eight other similar polychora. It is formed by attaching two pentagonal cupolae by their pentagonal faces, folding them into the fourth dimension with their decagonal ends connected by a decagonal prism, and then filling in the gaps with 5 triangular prisms and 5 square pyramids. For faces, it contains two decagons, one pentagon, 20 squares and 20 triangles.

It can also be constructed by the Stott expansion of the pentagonal prism pyramid.

## Dichoral angles

• Between pentagonal cupola and decagonal prism: 18° (exact)
• Between square pyramid and decagonal prism: atan(sqrt(5)-2) ≈ 13.28°
• Between triangular prism and decagonal prism: asin(sqrt((5-2*sqrt(5))/15)) ≈ 10.81°

## Cartesian coordinates

The coordinates of the pentagonal magnabicupolic ring are as follows:

```# Decagonal prism
<±2*phi, 0, ±1,  0>
<±1, ±√(3+4*phi), ±1,  0>
<±phi^2, ±√(2+phi), ±1,  0>

# Pentagon
<0, √((10+2*√5)/5),  0, √((5-2*√5)/5)>
<±phi, √((5-√5)/10), 0, √((5-2*√5)/5)>
<±1, -√((5+2*√5)/5), 0, √((5-2*√5)/5)>
```

## Equations

• Variables:
l ⇒ edge length
• The hypervolumes of a pentagonal magnabicupolic ring are given by:
total edge length = 45l
total surface area = Unknown
surcell volume = Unknown
bulk = Unknown
[!x,!y] ⇒ Unknown
[!z] ⇒ Unknown
[!w] ⇒ Unknown