Increased Truncated Cube

Increased Truncated Cube
Increased Truncated Cube
	; ( 3D model )
Type of	Johnson's polyhedron
The properties	convex
Combinatorics
22 facets; 48 ribs; 28 peaks	Χ = 2
Facets	12 triangles; 5 squares; 5 octagons
Vertex configuration	2x4 + 8 (3.8 2 ); 4 (3.4 3 ); 8 (3.4.3.8)
	Scan
Classification
Designations	J 66 , M 11 + M 5
Symmetry group	C 4v

The expanded truncated cube ^[1] is one of Johnson's polyhedra ( J ₆₆ , according to Zalgaller - M ₁₁ + M ₅ ).

Composed of 22 faces: 12 regular triangles , 5 squares and 5 regular octagons . Among the octagonal faces, 1 is surrounded by four octagonal and four triangular, the remaining 4 - by three octagonal and five triangular; among the square faces 1 is surrounded by four square ones, the remaining 4 - square and three triangular; among triangular 4 faces are surrounded by three octagonal, 4 faces - two octagonal and square, the remaining 4 - octagonal and two square.

It has 48 edges of the same length. 8 edges are located between two octagonal faces, 24 edges - between the octagonal and triangular, 4 edges - between two square, the remaining 12 - between square and triangular.

The extended truncated cube has 28 vertices. At 16 vertices, two octagonal faces and one triangular converge; 8 vertices meet octagonal, square and two triangular faces; at 4 vertices, three square and triangular faces converge.

An extended truncated cube can be obtained from two polyhedra - a truncated cube and a four-sloping dome ( J ₄ ) - by attaching them to each other with octagonal faces.

Content

1 Metric
2 In coordinates
3 notes
4 References

Metric

If an extended truncated cube has an edge of length $a$ ${\ displaystyle a}$ $a$ , its surface area and volume are expressed as

S=\left(15+10{\sqrt {2}}+3{\sqrt {3}}\right)a^{2}\approx 34{,}3382880a^{2},

{\ displaystyle S = \ left (15 + 10 {\ sqrt {2}} + 3 {\ sqrt {3}} \ right) a ^ {2} \ approx 34 {,} 3382880a ^ {2},}

S=\left(15+10{\sqrt {2}}+3{\sqrt {3}}\right)a^{2}\approx 34{,}3382880a^{2},

V=\left(8+{\frac {16{\sqrt {2}}}{3}}\right)a^{3}\approx 15{,}5424723a^{3}.

{\ displaystyle V = \ left (8 + {\ frac {16 {\ sqrt {2}}} {3}} \ right) a ^ {3} \ approx 15 {,} 5424723a ^ {3}.}

In coordinates

The expanded truncated cube can be positioned in a Cartesian coordinate system so that its vertices have coordinates

$(\pm ({\sqrt {2}}-1);\;\pm 1;\;\pm 1),$ ${\ displaystyle (\ pm ({\ sqrt {2}} - 1); \; \ pm 1; \; \ pm 1),}$
$(\pm 1;\;\pm ({\sqrt {2}}-1);\;\pm 1),$ ${\ displaystyle (\ pm 1; \; \ pm ({\ sqrt {2}} - 1); \; \ pm 1),}$
$(\pm 1;\;\pm 1;\;\pm ({\sqrt {2}}-1)),$ ${\ displaystyle (\ pm 1; \; \ pm 1; \; \ pm ({\ sqrt {2}} - 1)),}$
$(\pm ({\sqrt {2}}-1);\;\pm ({\sqrt {2}}-1);\;3-{\sqrt {2}}).$ ${\ displaystyle (\ pm ({\ sqrt {2}} - 1); \; \ pm ({\ sqrt {2}} - 1); \; 3 - {\ sqrt {2}}).}$

In this case, the axis of symmetry of the polyhedron will coincide with the axis Oz, and two of the four planes of symmetry - with the planes xOz and yOz.

Notes

↑ Zalgaller V. A. Convex polyhedra with regular edges / Zap. scientific sem. LOMI, 1967. - V. 2. - Page. 23.

Links

Weisstein, Eric W. The expanded truncated cube on the Wolfram MathWorld website.

[1] Zalgaller V. A. Convex polyhedra with regular edges / Zap. scientific sem. LOMI, 1967. - V. 2. - Page. 23.