Thrice-extended hexagonal prism

Thrice-extended hexagonal prism
Thrice-extended hexagonal prism
	; ( 3D model )
Type of	Johnson's polyhedron
The properties	convex
Combinatorics
17 faces; 30 ribs; 15 peaks	Χ = 2
Facets	12 triangles; 3 square; 2 hexagons
Vertex configuration	3 (3 4 ); 12 (3 2 .4.6)
	Scan
Classification
Designations	J 57 , P 6 + 3M 2
Symmetry group	D 3h

The triple-enlarged hexagonal prism ^[1] is one of Johnson's polyhedra ( J ₅₇ , according to Zalgaller - P ₆ + 3M ₂ ).

Composed of 17 faces: 12 regular triangles , 3 squares and 2 regular hexagons . Each hexagonal face is surrounded by three square and three triangular; each square face is surrounded by two hexagonal and two triangular; among the triangular faces, 6 are surrounded by a hexagonal and two triangular, the other 6 - square and two triangular.

It has 30 edges of the same length. 6 edges are located between the hexagonal and square faces, 6 edges - between the hexagonal and triangular, 6 edges - between the square and triangular, the remaining 12 - between two triangular.

A triple-stacked hexagonal prism has 15 peaks. At 12 vertices, the hexagonal, square and two triangular faces converge; at 3 peaks - four triangular.

A triple-extended hexagonal prism can be obtained from four polyhedra - three square pyramids ( J ₁ ) and a regular hexagonal prism , all edges of which are the same length - by attaching the base of the pyramids to three pairwise non-adjacent square faces of the prism.

Content

1 Metric
2 In coordinates
3 notes
4 References

Metric

If a triple-stacked hexagonal prism has an edge of length $a$ ${\ displaystyle a}$ , its surface area and volume are expressed as

S=\left(3+6{\sqrt {3}}\right)a^{2}\approx 13{,}3923048a^{2},

{\ displaystyle S = \ left (3 + 6 {\ sqrt {3}} \ right) a ^ {2} \ approx 13 {,} 3923048a ^ {2},}

V={\frac {1}{2}}\left({\sqrt {2}}+3{\sqrt {3}}\right)a^{3}\approx 3{,}3051830a^{3}.

{\ displaystyle V = {\ frac {1} {2}} \ left ({\ sqrt {2}} + 3 {\ sqrt {3}} \ right) a ^ {3} \ approx 3 {,} 3051830a ^ {3}.}

In coordinates

Tri-extended hexagonal prism with rib length $2$ ${\ displaystyle 2}$ can be placed in a Cartesian coordinate system so that its vertices have coordinates

$\left(\pm 1;\;\pm 1;\;\pm {\sqrt {3}}\right),$ ${\ displaystyle \ left (\ pm 1; \; \ pm 1; \; \ pm {\ sqrt {3}} \ right),}$
$\left(\pm 2;\;\pm 1;\;0\right),$ ${\ displaystyle \ left (\ pm 2; \; \ pm 1; \; 0 \ right),}$
$\left(0;\;0;\;{\sqrt {2}}+{\sqrt {3}}\right),$ ${\ displaystyle \ left (0; \; 0; \; {\ sqrt {2}} + {\ sqrt {3}} \ right),}$
$\left(\pm {\frac {3+{\sqrt {6}}}{2}};\;0;\;-{\frac {{\sqrt {2}}+{\sqrt {3}}}{2}}\right).$ ${\ displaystyle \ left (\ pm {\ frac {3 + {\ sqrt {6}}} {2}}; \; 0; \; - {\ frac {{\ sqrt {2}} + {\ sqrt { 3}}} {2}} \ right).}$

In this case, one of the four symmetry axes of the polyhedron will coincide with the Oy axis, and two of the four planes of symmetry will coincide with the xOz and yOz planes.

Notes

↑ Zalgaller V. A. Convex polyhedra with regular edges / Zap. scientific sem. LOMI, 1967. - V. 2. - Page. 22.

Links

Weisstein, Eric W. Three-fold Hexagonal Prism at Wolfram MathWorld .

[1] Zalgaller V. A. Convex polyhedra with regular edges / Zap. scientific sem. LOMI, 1967. - V. 2. - Page. 22.