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The pentagonal dipyramid is one of the convex deltahedra, and Johnson solid 
. It is also the dual polyhedron
of the pentagonal prism 
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
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(1)
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In order for the top and bottom edges to also be of unit length, the polyhedron must be of height
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(2)
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The ratio of 
is therefore given by
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(3)
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where 
is the golden ratio.
The surface area and volume of a unit pentagonal dipyramid are
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(4)
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(5)
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