Diamond cuboctahedron

3D view of a rhombic cuboctahedron ( animation )

Oldest printed representation of a diamond cuboctahedron from Leonardo da Vinci's Divina Proportione

Street lighting in front of the Mainz Cathedral

The (small) rhombic cuboctahedron is a polyhedron ( polyhedron ) that is one of the Archimedean solids . It is made up of 8 equilateral triangles and 18 squares . Three squares and one triangle each form a corner of the room .

Each 8 edges of the diamond cuboctahedron form the edges of a regular octagon . There are a total of six such independent, equilateral octagons in this polyhedron.

The name of the rhombicuboctahedron is based u. a. on the fact that 12 of the 18 squares are congruent with the 12 rhombuses of a circumscribed rhombic dodecahedron . The body that is dual to the rhombic cuboctahedron is the deltoidal icositetrahedron .

Formulas

Sizes of a rhombic cuboctahedron with edge length a
Volume ≈ 8.71 a ³	${\ displaystyle V = {\ frac {2} {3}} \, a ^ {3} \ left (6 + 5 {\ sqrt {2}} \ right)}$
Surface area ≈ 21.46 a ²	${\ displaystyle A_ {O} = 2 \, a ^ {2} \ left (9 + {\ sqrt {3}} \ right)}$
Umkugelradius ≈ 1.4 a	${\ displaystyle R = {\ frac {a} {2}} {\ sqrt {5 + 2 {\ sqrt {2}}}}}$
Edge ball radius ≈ 1.31 a	${\ displaystyle r = {\ frac {a} {2}} {\ sqrt {4 + 2 {\ sqrt {2}}}}}$
Diagonal angle ( square - square ) = 135 °	${\ displaystyle \ cos \, \ alpha _ {1} = - {\ frac {1} {2}} {\ sqrt {2}}}$
Diagonal angle (square - trine ) ≈ 144 ° 44 ′ 8 ″	${\ displaystyle \ cos \, \ alpha _ {2} = - {\ sqrt {\ frac {2} {3}}}}$
Corners solid angle ≈ 1.108 π	${\ displaystyle \ Omega = 2 \ pi - \ arccos \ left (- {\ frac {2} {3}} {\ sqrt {2}} \ right)}$