Talk:Antiprism

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Mathematics rating: Start Class Low Priority  Field: Geometry

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 Dnd. —Tamfang 03:42, 8 February 2006 (UTC)

[edit] 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)

[edit] 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 S1 = sin(π / n),S2 = sin(2π / n),C1 = cos(2π / n),C2 = cos(2π / n) so first three points are

  • k=0: p0 = (0,1,a)
  • k=1: p1 = (S1,C1, − a)
  • k=2: p2 = (S2,C2,a)

now distance between points is l = | p1 − p0 | 2 = | p2 − p0 | 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)

[edit] 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)