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faces of the rhombic dodecahedron, with each face containing the rhombus in the same plane, and that it has the same symmetries as the rhombic dodecahedron. It is the first stellation, meaning that no other self-intersecting polyhedron with the same face planes and the same symmetries has smaller
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For polyhedra formed only using faces in the same 12 planes and with the same symmetries, but with the faces allowed to become non-simple or with multiple faces in a single plane, additional possibilities arise. In particular, removing the inner rhombus from each hexagonal face of the stellation
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leaves four triangles, and the resulting system of 48 triangles forms a different non-convex polyhedron without self-intersections that forms the boundary of a solid shape, sometimes called Escher's solid. This shape appears in
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Escher's solid. This image does not depict the stellation, because different visible parts of a single hexagonal face of the stellation have different colors. However, the coloring is consistent with a depiction of the
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faces. Extending the faces outwards even farther in the same planes leads to two more stellations, if the faces are required to be
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of 20 − 36 + 12 = −4. Escher's solid instead has 48 triangular faces, 72 edges, and 26 vertices, yielding an
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The first stellation of the rhombic dodecahedron has 12 hexagonal faces, 36 edges, and 20 vertices, yielding an
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The first stellation of the rhombic dodecahedron has 12 faces, each of which is a non-convex hexagon. It is a
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515:"The compound of three octahedra and a remarkable compound of three square dipyramids, the Escher's solid"
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The 48 triangular faces of the solid are isosceles; if the longest edge of these triangles is length
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The vertices of the first stellation of the rhombic dodecahedron include the 12 vertices of the
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of the first stellation of the rhombic dodecahedron decomposed into 12 pyramids and 4 half-cubes
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between a cube and two copies of Escher's solid, is closely related to this tessellation.
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Geometry, games, graphs and education: the Joe
Malkevitch Festschrift
538:"Tessellations from group actions and the mystery of Escher's solid"
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Luke, Dorman (1957). "Stellations of the rhombic dodecahedron".
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with 12 faces, each of which is a non-convex hexagon. It is a
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Bridges: Mathematical
Connections in Art, Music, and Science
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369:. Six solids meet at each vertex. This honeycomb is
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58:Escher's solid is topologically equivalent to the
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79:first stellation of the rhombic dodecahedron
27:Self I intersecting polyhedron with 12 faces
426:. Comap, Inc., Bedford, MA. pp. 9–26.
567:"First stellation of rhombic dodecahedron"
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367:stellated rhombic dodecahedral honeycomb
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248:{\displaystyle {\tfrac {\sqrt {3}}{2}}s}
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18:First stellation of rhombic dodecahedron
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198:with 24 overlapping triangular faces.
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285:{\displaystyle 12{\sqrt {2}}s^{2}}
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292:and the volume of the solid is
120:Stellation, solid, and compound
496:"The Polyhedra of M.C. Escher"
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46:of three flattened octahedra.
345:of 26 − 72 + 48 = 2.
180:compound of three octahedra
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326:Vertices, edges, and faces
450:The Mathematical Gazette
607:Space-filling polyhedra
536:Mihăilă, Ioana (2005).
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564:Weisstein, Eric W.
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572:MathWorld
196:octahedra
184:Waterfall
165:Waterfall
162:'s works
151:STL model
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494:(1996).
415:(2008).
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