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Pentagonal Dipyramid


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The pentagonal dipyramid is one of the convex deltahedra, and Johnson solid J_(13). It is also the dual polyhedron of the pentagonal prism U_(76) and is an isohedron.

It is implemented in the Wolfram Language as PolyhedronData[{"Dipyramid", 5{].

A pentagonal dipyramid appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).

For a pentagonal dipyramid having a base with unit edge lengths, the circumradius of the base pentagon is

 R=1/(10)sqrt(50+10sqrt(5)).
(1)

In order for the top and bottom edges to also be of unit length, the polyhedron must be of height

 h=sqrt(1-R^2)=1/(10)sqrt(50-10sqrt(5)).
(2)

The ratio of R/h is therefore given by

 R/h=phi,
(3)

where phi is the golden ratio.

The surface area and volume of a unit pentagonal dipyramid are

S=5/2sqrt(3)
(4)
V=1/(12)(5+sqrt(5)).
(5)

See also

Deltahedron, Dipyramid, Golden Ratio, Icosahedron, Isohedron, Johnson Solid, Rigidity Theorem, Triangular Dipyramid

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References

Escher, M. C. "Stars." Wood engraving. 1948. http://www.mcescher.com/Gallery/back-bmp/LW359.jpg.Forty, S. M.C. Escher. Cobham, England: TAJ Books, 2003.

Cite this as:

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

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