Acronym ...
Name octahedrally truncated rhombic dodecahedron,
chamfered cube
 
 ©    ©
Vertex figure [4,H2], [h3]
Coordinates
  1. (1/2 + 1/sqrt(3), 1/2 + 1/sqrt(3), 1/2 + 1/sqrt(3))       & all changes of sign
    (vertex inscribed cube)
  2. (1/2, 1/2, 1/2 + 2/sqrt(3))               & all permutations, all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Dihedral angles
(at margins)
  • between {4} and {(h,H,H)2}:   135°
  • between {(h,H,H)2} and {(h,H,H)2}:   120°
Confer rad   cube  
External
links
wikipedia   mcneill  

The non-regular hexahedra {(h,H,H)2} have vertex angles h = arccos(-1/3) = 109.471221° resp. H = arccos[-1/sqrt(3)] = 125.264390°.

Octahedral truncation applies to the octahedral vertices (vertex inscribed oct) only. Wrt. the rad this produces new square faces there. These are then face planes of a cutting cube. – The above transition shows a dynamical mutual scaling of cube and rad. The chamfered cube then is the instance, where all edges happen to have the same length.

Chamfering (or edge-only beveling – here being applied to the cube) flatens the former edges into new (non-regular hexagonal) faces.

There is a deeper, terminal chamfering of the cube too, which then reduces the original faces to nothing. Then the hexagons will become rhombs and the total figure becomes the rad. – When considering the below provided tegum sum Dynkin symbol, it becomes obvious that this figure also can be seen as a Stott expansion of the rad.


Incidence matrix according to Dynkin symbol

ax4oo3oc&#zx   → height = 0
                 a = (3+2 sqrt(3))/3 = 2.154701
                 c = sqrt(8/3) = 1.632993
(tegum sum of a-cube and (x,c)-sirco)

o.4o.3o.     | 8  * |  3  0 |  3 0  [h3]
.o4.o3.o     | * 24 |  1  2 |  2 1  [4,H2]
-------------+------+-------+-----
oo4oo3oo&#x  | 1  1 | 24  * |  2 0
.x .. ..     | 0  2 |  * 24 |  1 1
-------------+------+-------+-----
ax .. oc&#zx | 2  4 |  4  2 | 12 *  {(h,H,H)2}
.x4.o ..     | 0  4 |  0  4 |  * 6  {4}

© 2004-2019
top of page