Related polyhedra Rhombic dodecahedron

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spherical rhombic dodecahedron



when projected onto sphere (see right), can seen edges make edges of 2 tetrahedra arranged in dual positions (the stella octangula). trend continues on deltoidal icositetrahedron , deltoidal hexecontahedron dual pairings of other regular polyhedra (alongside triangular bipyramid if improper tilings considered), giving shape alternative systematic name of deltoidal dodecahedron.


this polyhedron part of sequence of rhombic polyhedra , tilings [n,3] coxeter group symmetry. cube can seen rhombic hexahedron rhombi squares.




similarly relates infinite series of tilings face configurations v3.2n.3.2n, first in euclidean plane, , rest in hyperbolic plane.



stellations

like many convex polyhedra, rhombic dodecahedron can stellated extending faces or edges until meet form new polyhedron. several such stellations have been described dorman luke.



this animation shows construction of stellated rhombic dodecahedron inverting center-face pyramids of rhombic dodecahedron.


the first stellation, called stellated rhombic dodecahedron, known. can seen rhombic dodecahedron each face augmented attaching rhombic-based pyramid it, pyramid height such sides lie in face planes of neighbouring faces:





luke describes 4 more stellations: second , third stellations (expanding outwards), 1 formed removing second third, , adding original rhombic dodecahedron previous one.


great rhombic dodecahedron : 12 concave octagonal faces


great stellated rhombic dodecahedron : 12 concave dodecagonal faces








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