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Dipyramid


Dipyramids

A dipyramid, also called a bipyramid, consists of two pyramids symmetrically placed base-to-base. The dipyramids are duals of the regular prisms. Their skeletons are the dipyramidal graphs.

The volume of a dipyramid of half-height h and base area A_b is equal to twice the volume of the corresponding pyramid, namely

 V=2/3A_bh.
(1)

The volume of a dipyramid whose base is a regular n-sided polygon with side a is therefore

 V_n=1/6ncot(pi/n)a^2h.
(2)
DipyramidsAndDuals

The canonical dipyramids are the duals of the right regular equilateral prisms and if the original prism has edge length a, the midradius of the dual dipyramid is

 rho_n=1/2acsc(pi/n).
(3)
DipyramidConvexHulls

An octagonal dipyramid is the convex hull of the second octahedron 2-compound.

CanonicalDipyramidFace

The angles of the isocelese triangular faces in a canonical n-dipyramid as illustrated above are given by

theta_1=cos^(-1)(sin^2(pi/n))
(4)
theta_2=cos^(-1)(cos((2pi)/n)+1/2sin^2((2pi)/n)).
(5)
CanonicalDipyramidNets

Nets for the canonical n-dipyramids are illustrated above for n=3 to 10.

Johnson solid J_(12) is a triangular dipyramid, the octahedron is a square dipyramid, and Johnson solid J_(13) is a pentagonal dipyramid.


See also

Deltahedron, Dipyramidal Graph, Elongated Pentagonal Dipyramid, Johnson Solid, Octahedron, Pentagonal Dipyramid, Prism, Pyramid, Square Dipyramid, Trapezohedron, Triangular Dipyramid

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References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 117, 1989.Pedagoguery Software. Poly. http://www.peda.com/poly/.Webb, R. "Prisms, Antiprisms, and their Duals." http://www.software3d.com/Prisms.html.

Referenced on Wolfram|Alpha

Dipyramid

Cite this as:

Weisstein, Eric W. "Dipyramid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dipyramid.html

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