Beveled hexahedron

3D view of a beveled hexahedron ( animation )

The beveled hexahedron (also called cubus simus ) is a chiral polyhedron ( polyhedron ) that belongs to the Archimedean solids . It is composed of 38 surfaces, namely 6 squares and 32 equilateral triangles , and has 24 corners and 60 edges. Four triangles and one square each form a corner of the room .

The pairs of opposing squares are parallel and rotated by approx. 33 ° against each other (the axis of rotation runs through the center of the surface). The following pictures show two mirror images of beveled hexahedra.

Mirror variant 1
Mirror variant 2

The body that is dual to the beveled hexahedron is the pentagonikositetrahedron .

construction

"Leaning" decahedron with 8 pentagons and 2 squares as boundary surfaces

As the name suggests, this polyhedron is created by continuously chamfering a hexahedron , so that at the end six (smaller) squares remain that are coincident with the original boundary surfaces of the cube.

By placing small pyramids (with a pentagonal base and four equilateral triangles as well as a "half square" as a lateral surface ) on the eight pentagonal boundary surfaces of a special decahedron (see fig. Right) one also obtains a beveled hexahedron.

In the case of a rhombic cuboctahedron, if you twist those six squares that are coincident with the boundary surfaces of a circumscribed cube (and face each other in pairs) by the angle ω (see formula below) and each insert a diagonal into the remaining, now distorted squares, there is also a beveled hexahedron.

Formulas

Section of a room corner of the beveled hexahedron

In the following, the term t denotes the cosine of the smaller central angle ζ in the chordal pentagon (the white lines in the graphic on the right) with the side lengths a and d (square diagonal).

{\ displaystyle t = \ cos (\ zeta) = {\ frac {1} {6}} \ left ({\ sqrt [{3}] {19 + 3 {\ sqrt {33}}}} + {\ sqrt [{3}] {19-3 {\ sqrt {33}}}} - 2 \ right)}

If the number 1 is added to twice the value of t, the Tribonacci constant is obtained, which represents the limit of the ratio (≈ 1.84) of two consecutive numbers in this sequence.

Sizes of a beveled hexahedron with edge length a
volume	${\ displaystyle V \, = {\ frac {a ^ {3}} {3 {\ sqrt {1-2t}}}} \ left (3 {\ sqrt {2t}} + 4 {\ sqrt {2 + 2t }} \ right)}$
Surface area	${\ displaystyle A_ {O} = 2a ^ {2} \ left (3 + 4 {\ sqrt {3}} \ right)}$
Umkugelradius	${\ displaystyle R = {\ frac {a} {2}} {\ sqrt {\ frac {2-2t} {1-2t}}}}$
Edge ball radius	${\ displaystyle r = {\ frac {a} {2 {\ sqrt {1-2t}}}}}$
1. Diagonal angle ( trine - trigon) ≈ 153 ° 14 ′ 5 ″	${\ displaystyle \ cos \ alpha _ {1} = - {\ frac {1} {3}} \ left (1 + 4t \ right)}$
2nd dihedral angle (square trigon) ≈ 142 ° 59 ′	${\ displaystyle \ cos \ alpha _ {2} = {\ frac {1-2 {\ sqrt {1 + t}}} {\ sqrt {3}}}}$
Surface-edge-angle (square-trine) ≈ 126 ° 24 ′ 12 ″	${\ displaystyle \ cos \ beta = {\ frac {1-2 {\ sqrt {t}}} {{\ sqrt {2}} \ left (1 - {\ sqrt {t}} \ right)}}}$
3D edge angle (trine – trine) ≈ 114 ° 48 ′ 43 ″	${\ displaystyle \ cos \ gamma = \, - t}$
Square rotation angle ≈ 32 ° 56 ′ 6 ″	${\ displaystyle \ sin \ omega = {\ frac {2t} {(1-2t) (1 + 2t) ^ {2}}} - 1}$
Corners solid angle ≈ 1.1426 π	${\ displaystyle \ Omega = \, 3 \ alpha _ {1} +2 \ alpha _ {2} -3 \ pi}$

Web links

Commons : Beveled hexahedron - collection of images, videos, and audio files

Wiktionary: beveled hexahedron - explanations of meanings, word origins, synonyms, translations

Eric W. Weisstein : Beveled hexahedron . In: MathWorld (English).