Decagonal antiprism

{{Infobox polytope The decagonal antiprism, or dap, is a prismatic uniform polyhedron. It consists of 20 triangles and 2 decagons. Each vertex joins one decagon and three triangles. As the name suggests, it is an antiprism based on a decagon.
 * dim = 3
 * obsa = Dap
 * img=Decagonal antiprism.png
 * 3d=Decagonal antiprism.stl
 * off = Decagonal antiprism.off
 * type=Uniform
 * coxeter = s2s10s
 * symmetry = I2(20)×A1+, order 40
 * army=Dap
 * reg=Dap
 * faces = 20 triangles, 2 decagons
 * edges = 20+20
 * vertices = 20
 * circum = $\sqrt{(8+2√5+√50+22√5)/8}$ ≈ 1.67450
 * height = $\sqrt{(–4–2√5+{{rqadic|50+22√5})/2}} ≈ 0.86240
 * volume = 5{{radic|–2–2√5+2{{radic|650+290√5}}}}/6 ≈ 6.74929
 * verf = Isosceles trapezoid, edge lengths 1, 1, 1, √(5+√5)/2
 * dih= 3–3: acos((1–√10+2√5)/3) ≈ 159.18651º
 * dih2=10–3: acos(–{{radic|(11+4√5–2{{radic|(50+22√5}})/3}}) ≈ 95.24664º
 * dual = Decagonal trapezohedron
 * conjugate = Decagrammic antiprism
 * conv=Yes
 * orient=Yes
 * nat=Tame}$

Vertex coordinates
A decagonal antiprism of edge length 1 has vertex coordinates given by:
 * (±1/2, ±$\sqrt{(5+2√5)}$/2, H),
 * (±(3+$\sqrt{5}$)/4, ±$\sqrt{(5+√5)/8}$, H),
 * (±(1+$\sqrt{5}$)/2, 0, H),
 * (±$\sqrt{(5+2√5)}$/2, ±1/2, –H),
 * (±$\sqrt{(5+√5)/8}$, ±(3+$\sqrt{5}$)/4, –H),
 * (0, ±(1+$\sqrt{5}$)/2, –H),

where H = $\sqrt{(–4–2√5+√(50+22√5))/2}$/2 is the distance between the antiprism's center and the center of one of its bases.

Representations
A decagonal antiprism has the following Coxeter diagrams:


 * s2s10s
 * s2s20o (alternated icosagonal prism)
 * xo10ox&#x (bases considered separately)

Related polyhedra
A pentagonal cupola can be attached to a base of the decagonal antiprism to form the gyroelongated pentagonal cupola. If a pentagonal rotunda is attached instead the result is the gyroelongated pentagonal rotunda.

If two pentagonal cupolas are attached to the bases, the result is the gyroelongated pentagonal bicupola. If two rotundas are attached, the result is the gyroelongated pentagonal birotunda. If one cupola and one rotunda are attached, the result is the gyroelongated pentagonal cupolarotunda.