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234:, meaning that its faces are equilateral triangles and that it has a symmetry taking every face to every other face. There is one known infinite family of isohedral deltahedra, and 36 more that do not fall into this family; the compound of three octahedra is one of the 36 sporadic examples. However, its symmetry group does not take every vertex to every other vertex, so it is not itself a uniform polyhedron compound.
308:, della Francesca already includes a drawing of an octahedron circumscribed around a cube, with eight of the cube edges lying in the octahedron's eight faces. Three octahedra circumscribed in this way around a single cube would form the compound of three octahedra, but della Francesca does not depict the compound.
266:. A fourth isohedral deltahedron with the same face planes, also a stellation of the compound of three octahedra, has the same combinatorial structure as the tetrakis hexahedron but with the cube faces dented inwards into intersecting pyramids rather than attaching the pyramids to the exterior of the cube.
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of the tetrakis hexahedron. A different form of the tetrakis hexahedron, formed by using taller pyramids on each face of the cube, is non-convex but has equilateral triangle faces that again lie on the same planes as the faces of the three octahedra; it is another of the known isohedral deltahedra. A
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A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in this way from the three pairs of opposite cube squares form the compound of three octahedra. The eight
273:. Three of these reflection panes pass parallel to the sides of the cube, halfway between two opposite sides; the other six pass diagonally across the cube, through four of its vertices. These nine planes coincide with the nine equatorial planes of the three octahedra.
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has documented that Escher was familiar with Brückner's work and used it as the basis for many of the stellated polyhedra and polyhedral compounds that he drew. Earlier in 1948, Escher had made a preliminary woodcut with a similar theme,
336:, assuming that Escher rediscovered this shape independently, writes that "It is remarkable that Escher, without any knowledge of algebra or analytic geometry, was able to rediscover this highly symmetrical figure." However,
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cube vertices are the same as the eight points in the compound where three edges cross each other. Each of the octahedron edges that participates in these triple crossings is divided by the crossing point in the ratio 1:
353:, which can be seen as rhombic dodecahedron with shorter pyramids on the rhombic faces. The dual figure of the octahedral compound, the compound of three cubes, is also shown in a later Escher woodcut,
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349:(sometimes called Escher's solid), which can be formed as a compound of three flattened octahedra. This form as a polyhedron is topologically identical to the
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that pass through two opposite vertices of the starting octahedron. A third construction for the same compound of three octahedra is as the
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The compound of three octahedra can also be formed from three copies of a single octahedron by rotating each copy by an angle of
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symmetry, order 12. Each antiprism is rotated 40 degrees. The top and bottom planes can be seen to contain the compound
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The next appearance of the compound of three octahedra in the mathematical literature appears to be a 1900 work by
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The compound of three octahedra and a remarkable compound of three square dipyramids, the Escher's solid
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The compound of three octahedra re-entered the mathematical literature more properly with the work of
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The cube around which the three octahedra can be circumscribed has nine planes of
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The six vertices of one of the three octahedra may be given by the coordinates
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Bakos, T.; Johnson, Norman W. (1959), "Octahedra inscribed in a cube",
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to each face of the central cube. Thus, the compound can be seen as a
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third isohedral deltahedron sharing the same face planes, the
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Octahedron circumscribed around a cube, as drawn by
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526:(1968), "Some interesting octahedral compounds",
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301:De quinque corporibus regularibus
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241:with 14 vertices and 24 faces, a
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559:(1999), "Isohedral deltahedra",
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561:Periodica Mathematica Hungarica
298:In the 15th-century manuscript
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347:stellated rhombic dodecahedron
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22:Compound of three octahedra
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422:Compound of five octahedra
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364:Bakos & Johnson (1959)
256:compound of six tetrahedra
83:of order 48 (single color)
427:Compound of ten octahedra
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528:The Mathematical Gazette
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696:"Octahedron 3-Compound"
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438:References
330:chameleons
260:stellating
251:stellation
197:Symmetries
161:(0, 0, ±2)
701:MathWorld
396:enneagram
356:Waterfall
229:isohedral
216:Waterfall
713:Category
609:(1900),
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60:Vertices
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467:3608867
277:History
174:√
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148:of the
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536:JSTOR
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325:Stars
112:Stars
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