|Bowers style acronym||Dip|
|Coxeter diagram||x x10o ()|
|Faces||10 squares, 2 decagons|
|Vertex figure||Isosceles triangle, edge lengths √2, √2, √(5+√5)/2|
|Measures (edge length 1)|
|Dihedral angles||4–4: 144°|
|Number of external pieces||12|
|Level of complexity||3|
|Abstract & topological properties|
|Symmetry||I2(10)×A1, order 40|
The decagonal prism, or dip, is a prismatic uniform polyhedron. It consists of 2 decagons and 10 squares. Each vertex joins one decagon and two squares. As the name suggests, it is a prism based on a decagon.
It is the highest convex polygonal prism to occur as cells in uniform polychora.
Vertex coordinates[edit | edit source]
A decagonal prism of edge length 1 has vertex coordinates given by:
Representations[edit | edit source]
A decagonal prism has the following Coxeter diagrams:
- x x10o (full symmetry)
- x x5x (as dipentagonal prism)
- s2s10x (as dipentagonal trapezoprism)
- xx10oo&#x (decagonal frustum)
- xx5xx&#x (dipentagonal frustum)
- xxxxx xFVFx&#xt (A1×A1 axial, square-first)
Semi-uniform variant[edit | edit source]
The decagonal prism has a semi-uniform variant of the form x y10o that maintains its full symmetry. This variant uses rectangles as its sides.
With base edges of length a and side edges of length b, its circumradius is given by and its volume is given by .
A decagonal prism with base edges of length a and side edges of length b can be alternated to form a pentagonal antiprism with base edges of length and side edges of lengths . In particular if the side edges are times the length of the base edges this gives a uniform pentagonal antiprism.
Variations[edit | edit source]
A decagonal prism has the following variations:
- Dipentagonal prism - prism with dipentagons as bases, and 2 types of rectangles
- Dipentagonal trapezoprism - isogonal with trapezoid sides
- Decagonal frustum
- Dipentagonal frustum
Related polyhedra[edit | edit source]
A number of Johnson solids can be formed by attaching various configurations of pentagonal cupolas and pentagonal rotundas to the bases of the decagonal prism:
- Elongated pentagonal cupola - Cupola attached to one base
- Elongated pentagonal rotunda - Rotunda attached to one base
- Elongated pentagonal orthobicupola - Cupolas in same orientation attached to both bases
- Elongated pentagonal gyrobicupola - Cupolas rotated by 36º attached to bases
- Elongated pentagonal orthocupolarotunda - Cupola attached to one base, rotunda with same pentagon orientation attached to other base
- Elongated pentagonal gyrocupolarotunda - Cupola attached to one base, rotunda with pentagon rotated by 36º attached to other base
- Elongated pentagonal orthobirotunda - Rotundas in same orientation attached to both bases
- Elongated pentagonal gyrobirotunda - Rotundas rotated by 36º attached to bases
The rhombisnub dodecahedron is a uniform polyhedron compound composed of 6 decagonal prisms.
External links[edit | edit source]
- Klitzing, Richard. "dip".
- Quickfur. "The Decagonal Prism".
- Wikipedia Contributors. "Decagonal prism".
- Hi.gher.Space Wiki Contributors. "Decagonal prism".