Related polytopes Rhombic dodecahedron

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in perfect vertex-first projection 2 of tesseract s vertices (marked in pale green) projected in center of rhombic dodecahedron


the rhombic dodecahedron forms hull of vertex-first projection of tesseract 3 dimensions. there 2 ways of decomposing rhombic dodecahedron 4 congruent rhombohedra, giving 8 possible rhombohedra projections of tesseracts 8 cubic cells. 1 set of projective vectors are: u=(1,1,-1,-1), v=(-1,1,-1,1), w=(1,-1,-1,1).


the rhombic dodecahedron forms maximal cross-section of 24-cell, , forms hull of vertex-first parallel projection 3 dimensions. rhombic dodecahedron can decomposed 6 congruent (but non-regular) square dipyramids meeting @ single vertex in center; these form images of 6 pairs of 24-cell s octahedral cells. remaining 12 octahedral cells project onto faces of rhombic dodecahedron. non-regularity of these images due projective distortion; facets of 24-cell regular octahedra in 4-space.


this decomposition gives interesting method constructing rhombic dodecahedron: cut cube 6 congruent square pyramids, , attach them faces of second cube. triangular faces of each pair of adjacent pyramids lie on same plane, , merge rhombuses. 24-cell may constructed in analogous way using 2 tesseracts.








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