Talk:Antiprism

Symmetry groups

Surely these have "Rotation Symmetry Order n"? --Phil | Talk 13:01, Jun 14, 2004 (UTC)

There's 2n symmetric-rotation-things, 4n if mirroring is counted. Just wasn't sure what the weird name for that particular group was. (Just saw it on symmetry group.) Κσυπ Cyp   13:25, 14 Jun 2004 (UTC)

The uniform n-antiprism's symmetry group is D_nd. —Tamfang 03:42, 8 February 2006 (UTC)

More precise definition?

Do the bases of an antiprism have to be rotated so that the vertices of one are "above" the midpoints of the edges of the other, or can it be any rotation? In the first case, the triangles around the circumference of the bases will be isoceles, whereas they may be scalene under the second definition. Currently the definition in this article doesn't exclude, say, a cube-like thing where the top face is rotated 17° with respect to the bottom. —Bkell 19:57, 4 August 2005 (UTC)

Furthermore, would the definition of an antiprism include a theoretical configuration where the base faces are perfectly aligned, connected by pairs of right triangles rather than quadrilaterals? Erroramong (talk) 16:06, 4 May 2009 (UTC)

Cartesian coordinates

The coordinates given look like a prism, not an antiprism. I'll work out what they ought to be and come back. —Tamfang 03:39, 8 February 2006 (UTC)

There, I think that's right – someone please check me – and put it into pretty TeX format; I can't get the hang of the syntax yet. —Tamfang 07:42, 8 February 2006 (UTC)

They look OK to me now, points are OK but notsure on a. Write $S_{1}=\sin(\pi /n),S_{2}=\sin(2\pi /n),C_{1}=\cos(2\pi /n),C_{2}=\cos(2\pi /n)$ so first three points are

• k=0: $p_{0}=(0,1,a)$
• k=1: $p_{1}=(S_{1},C_{1},-a)$
• k=2: $p_{2}=(S_{2},C_{2},a)$

now distance between points is $l=|p_{1}-p_{0}|^{2}=|p_{2}-p_{0}|^{2}$

${\begin{matrix}l&=&(S_{1}-0)^{2}+(C_{1}-1)^{2}+4a^{2}\\&=&S_{1}^{2}+C_{1}^{2}-2C_{1}+1+4a^{2}\\&=&2-2C_{1}+4a^{2}\end{matrix}}$

${\begin{matrix}l&=&(S_{2}-0)^{2}+(C_{2}-1)^{2}\\&=&S_{2}^{2}+C_{2}^{2}-2C_{2}+1\\&=&2-2C_{2}\end{matrix}}$

Equating

$2-2C_{1}+4a^{2}=2-2C_{2}\;$

$2a^{2}=C_{1}-C_{2}\;$

expand $C_{2}=\cos(2\pi /n)=1-2\sin ^{2}(\pi /n)=1-2S_{1}^{2}$ gives

$2a^{2}=C_{1}+2S_{1}^{2}-1\;$

Hum seems to be a minus out. Actually I think

$2a^{2}=\cos(\pi /n)-\cos(2\pi /n)\;$

is a nicer way to express it. --Salix alba (talk) 11:36, 8 February 2006 (UTC)

Thus illustrating the proverb that the surest way to get a question answered on the Net is to post a wrong answer as fact. Good show! —Tamfang 20:12, 8 February 2006 (UTC)

Crossed antiprism

Crossed antiprism redirects here, yet the article says nothing about it. What is a crossed antiprism? I suspect it may be the case where the rotation of one face is 180° with respect to the other, causing the triangular faces to cross in the middle (based on a picture at the Prismatoid article, but I'm not sure. 128.232.228.174 (talk) 13:05, 22 May 2008 (UTC)

Hm, the article doesn't seem to cover stars at all. See Prismatic uniform polyhedron for a better treatment. The bases of a crossed antiprism must be stars (3/2 < p < 2) but they need not be out of phase. —Tamfang (talk) 05:13, 27 May 2008 (UTC)

Ah, thanks for the link to Prismatic uniform polyhedron. I couldn't remember where I'd seen written that part about rational numbers. —Tamfang (talk) 05:54, 27 September 2014 (UTC)

A crossed antiprism has retrograde bases instead of prograde ones. Double sharp (talk) 09:03, 25 April 2012 (UTC)

Tetrahedron?

Is this considered an antiprism? I don't see it. It is mentioned in the symmetry section. Baccyak4H (Yak!) 15:43, 7 August 2008 (UTC) Nevermind, I see it now (n=2). Baccyak4H (Yak!) 15:45, 7 August 2008 (UTC)

stars

There is a unique prograde star M/N-antiprism if 2×M < N AND √3 sin(θ) - cos(θ) > 1, where θ = π × M/N

The first condition looks wrong (I think it ought to be M > 2N) and I don't understand the second. —Tamfang (talk) 22:24, 28 September 2014 (UTC)

Possibly worth mentioning?

Is it worth mentioning in this article that a uniform pentagonal antiprism is an icosahedron with the top and bottom vertices sliced off? — 2A02:C7D:419:2500:C835:756:4ED7:8D71 (talk) 12:32, 22 May 2017 (UTC)

Mention that along with the gyroelongated Johnson solids … —Tamfang (talk) 20:45, 22 May 2023 (UTC)

I'm really confused about the uniform n-gonal antiprism surface area formula. For an edge length E, shouldn't it just be $2B+2n{\bigl (}{\tfrac {E^{2}{\sqrt {3}}}{4}}{\bigr )}$? Basically the area of the two bases plus 2n times the area of each equilateral triangle. I don't get where cotangent fits in. — Preceding unsigned comment added by 70.36.193.221 (talk) 13:05, 11 February 2024 (UTC) Also, what's the surface area formula for antiprisms with side length l and height h?

Short phrasing for captions & titles: « Example [???] »

@Steelpillow (talk · contribs): In the infobox, is the caption « Example uniform hexagonal antiprism » badly phrased, please? (Someone told me that this is bad English.) In advance, thank you very much for your answer! Cheers, --JavBol (talk) 16:36, 13 June 2022 (UTC)

Not so much bad English as inappropriate; too obvious to need mentioning. I have simplified it. — Cheers, Steelpillow (Talk) 19:38, 13 June 2022 (UTC)

existences

I remember writing somewhere that there is an uniform antiprism for every rational number >3/2, but the word rational appears nowhere in the present article; should it? —Tamfang (talk) 20:47, 22 May 2023 (UTC)

Snub antiprism

@David Eppstein. Sorry for pinging, but I see that you have recently edited this article.

Extended content

Similarly constructed, the ss{2,6} is a snub triangular antiprism (a lower symmetry octahedron), and result as a regular icosahedron. A snub pentagonal antiprism, ss{2,10}, or higher n-antiprisms can be similar constructed, but not as a convex polyhedron with equilateral triangles. The preceding Johnson solid, the snub disphenoid also fits constructionally as ss{2,4}, but one has to retain two degenerate digonal faces (drawn in red) in the digonal antiprism. Snub antiprisms Symmetry D2d, [2+,4], (2*2) D3d, [2+,6], (2*3) D4d, [2+,8], (2*4) D5d, [2+,10], (2*5) Antiprisms s{2,4} A2 (v:4; e:8; f:6) s{2,6} A3 (v:6; e:12; f:8) s{2,8} A4 (v:8; e:16; f:10) s{2,10} A5 (v:10; e:20; f:12) Truncated antiprisms ts{2,4} tA2 (v:16;e:24;f:10) ts{2,6} tA3 (v:24; e:36; f:14) ts{2,8} tA4 (v:32; e:48; f:18) ts{2,10} tA5 (v:40; e:60; f:22) Symmetry D2, [2,2]+, (222) D3, [3,2]+, (322) D4, [4,2]+, (422) D5, [5,2]+, (522) Snub antiprisms J84 Icosahedron J85 Concave sY3 = HtA3 sY4 = HtA4 sY5 = HtA5 ss{2,4} (v:8; e:20; f:14) ss{2,6} (v:12; e:30; f:20) ss{2,8} (v:16; e:40; f:26) ss{2,10} (v:20; e:50; f:32)

I'm recently improving the article Snub square antiprism, and I found the section with another type of antiprism, constructed by snubbing. I think I should leave it to you. Dedhert.Jr (talk) 17:09, 2 February 2024 (UTC)