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Rhombic Dodecahedron Stellations


RhombicDodecahedronStellations

There are four fully supported stellations of the rhombic dodecahedron including as usual the original solid in the count (Wells 1991; Webb). The three nontrivial ones (excluding the base solid) are illustrated above.

Applying Miller's rules gives one additional stellation, bringing the total to 5, all of which are reflexible (Webb).

RhombicDodecahedronStellationDiagram

The original rhombic dodecahedron, its facial planes, and the intersections of those planes with the facial plane of the "top" face are illustrated above.

RhombicDodecahedronStellation1

The first stellation is sometimes simply known as the stellated rhombic dodecahedron. It consists of 12 intersecting bowtie-shaped concave hexagons. Its outer boundary (concave hull) corresponds to Escher's solid and can be constructed by drawing diagonals across the square faces of a cuboctahedron and connecting centers of these diagonals with the vertices of neighboring squares.

The outer edges of the third stellation correspond with those of the truncated octahedron.

These stellations are implemented in the Wolfram Language as PolyhedronData["RhombicDodecahedronStellation", n] for n=1, 2, 3.


See also

Archimedean Dual Stellations, Cuboctahedron, Escher's Solid, Fully Supported Stellation, Miller's Rules, Rhombic Dodecahedron, Stellation, Truncated Octahedron

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References

Brill, D. "Double Star Flexicube." Brilliant Origami: A Collection of Original Designs. Tokyo: Japan Pub., pp. 98-103, 996.Cundy, H. and Rollett, A. "The Stellated Rhombic Dodecahedron" and "The Stellations of the Rhombic Dodecahedron." §3.9.5 and 3.13 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 127-128 and 149-151, 1989.Luke, D. "Stellations of the Rhombic Dodecahedron." Math. Gaz. 41, 189-194, 1957.Webb, R. "Enumeration of Stellations." http://www.software3d.com/Enumerate.php.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 215-216, 1991.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, p. 36, 1983.

Cite this as:

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

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