Decagonal prism

Decagonal prism
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Dip
Coxeter diagram	x x10o ()
Elements
Faces	10 squares, 2 decagons
Edges	10+20
Vertices	20
Vertex figure	Isosceles triangle, edge lengths √2, √2, √(5+√5)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{7+2\sqrt5}}{2} ≈ 1.69353$
Volume	$5\frac{\sqrt{5+2\sqrt5}}{2} ≈ 7.69421$
Dihedral angles	4–4: 144°
	4–10: 90°
Height	1
Central density	1
Number of external pieces	12
Level of complexity	3
Related polytopes
Army	Dip
Regiment	Dip
Dual	Decagonal tegum
Conjugate	Decagrammic prism
Abstract & topological properties
Flag count	120
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	I2(10)×A1, order 40
Convex	Yes
Nature	Tame

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

A decagonal prism of edge length 1 has vertex coordinates given by:

• $\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),$
• $\left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right).$

Representations

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

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 $\sqrt{a^2\frac{3+\sqrt5}{2}+\frac{b^2}{4}}$ and its volume is given by $5\frac{\sqrt{5+2\sqrt5}}{2}a^2b$.

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 $\sqrt{\frac{5+\sqrt5}{2}}a$ and side edges of lengths $\sqrt{a^2+b^2}$. In particular if the side edges are $\frac{1+\sqrt5}{2}$ times the length of the base edges this gives a uniform pentagonal antiprism.

Variations

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

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

• Klitzing, Richard. "dip".

• Quickfur. "The Decagonal Prism".

• Wikipedia Contributors. "Decagonal prism".
• Hi.gher.Space Wiki Contributors. "Decagonal prism".