Elongated Dodecahedron


The elongated dodecahedron, also known as the extended rhombic dodecahedron and rhombo-hexagonal dodecahedron, is a space-filling polyhedron and primary parallelohedron bounded by eight rhombi of angle cos^(-1)(1/6) approx 80 degrees24^' (at the poles) and an equatorial zone of four equilateral hexagons having opposite angles of cos^(-1)(-2/3) approx 131 degrees49^', with the remaining four angles equal (Coxeter 1973, pp. 29-30 and 257).

The elongated dodecahedron can be constructed by stretching a rhombic dodecahedron until the middle ring of rhombi become regular hexagons. Note that in this case, the term "elongated" refers to stretching of some of the existing faces as opposed to insertion of additional faces, the latter meaning of which is used in the naming of Johnson solids.

It is implemented in the Wolfram Language as PolyhedronData["ElongatedDodecahedron"].

The elongated dodecahedron has surface area and volume given by


and moment of inertia tensor

 I=[(19)/(32)Ma^2 0 0; 0 (19)/(32)Ma^2 0; 0 0 7/(16)Ma^2]

for a uniform solid with mass M.

See also

Rhombic Dodecahedron

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Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.Fedorov, E. S. "Elemente der Gestaltenlehre." Mineralogicheskoe obshchestvo Leningrad (Verhandlungen der Russisch-Kaiserlichen Mineralogischen Gesellschaft zu St. Petersburg 21, 1-279, 1885.Tutton, A. E. H. Crystallography and Practical Crystal Measurement. London, pp. 567 (Fig. 448) and 723 (Fig. 585), 1922.

Cite this as:

Weisstein, Eric W. "Elongated Dodecahedron." From MathWorld--A Wolfram Web Resource.

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