Acronym  rad (old: rhode) 
Name 
rhombic dodecahedron, Voronoi cell of facecentered cubic (fcc) lattice, terminally chamfered cube, terminally chamfered octahedron, surtegmated cube, surtegmated octahedron 
© ©  
Inradius  sqrt(2/3) = 0.816497 
Vertex figure  [r^{4}], [R^{3}] 
Coordinates  
General of army  (is itself convex) 
Colonel of regiment  (is itself locally convex) 
Dual  co 
Dihedral angles
(at margins) 

Confer  chamfered cube chamfered oct 
External links 
The rhombs {(r,R)^{2}} have vertex angles r = arccos(1/3) = 70.528779° resp. R = arccos(1/3) = 109.471221°. Esp. rr : RR = sqrt(2). – Both, the actual shape of the rhombs and the dihedral angle given above, can easily be obtained from the Voronoi complex of the fcc honeycomb, radh, and its relation to the vertexinscribed primitive cubic honeycomb chon, cf. also the right first 2 pictures.
The last right picture displays that the vertexfirst projection of the tes happens to be nothing but rad. In fact, there are exactly two ways of dissecting rad into four congruent rhombohedra, giving a total of eight possible rhombohedra, each a projected cube. The remaining 2 vertices of tes, which are not visible at the surface of rad, thereby get both projected to its bodycenter. These two rhombohedral dissections then just represent nothing but the front and back cover of the projection preimage, i.e. of tes. – Thus, rerefering to the previous paragraph, radh happens to be nothing but a single sheet of cover from a likewise projection of test.
All a = rr and b = RR edges, as provided in the below description, only qualify as pseudo edges wrt. the full polyhedron. The true edge size used here is rR = x = 1.
Incidence matrix according to Dynkin symbol
o3m4o = ao3oo4ob&#zx → height = 0 a = rr = sqrt(8/3) = 1.632993 b = RR = 2/sqrt(3) = 1.154701 o.3o.4o.  6 *  4  4 [r^{4}] .o3.o4.o  * 8  3  3 [R^{3}] +++ oo3oo4oo&#x  1 1  24  2 +++ ao .. ob&#zx  2 2  4  12 {(r,R)^{2}}
m3o3m = aoo3oao3ooa&#zx → height = 0 a = rr = sqrt(8/3) = 1.632993 o..3o..3o..  4 * *  3 0  3 [R^{3}] .o.3.o.3.o.  * 6 *  2 2  4 [r^{4}] ..o3..o3..o  * * 4  0 3  3 [R^{3}] +++ oo.3oo.3oo.&#x  1 1 0  12 *  2 .oo3.oo3.oo&#x  0 1 1  * 12  2 +++ ... oao ...&#xt  1 2 1  2 2  12 {(r,R)^{2}}
© 20042020  top of page 