Twisted elongated quadrangular bipyramid

Twisted elongated quadrangular bipyramid
Twisted elongated quadrangular bipyramid
	; ( 3D model )
Type of	Johnson's polyhedron
Properties	convex deltahedron
Combinatorics
16 facets; 24 ribs; 10 peaks	Χ = 2
Verge of	triangles
Vertex configuration	2 (3 4 ); 8 (3 5 )
Dual Polyhedron
	Scan
Classification
Designations	J 17 , M 2 + A 4 + M 2
Symmetry group	D 4d

The twisted elongated tetragonal bipyramid ^[1] is one of the Johnson polyhedra ( J ₁₇ ; according to Zalgaller it is M ₂ + A ₄ + M ₂ ), the deltahedron .

Composed of 16 regular triangles ; has 24 edges of the same length and 10 vertices. In two vertices, four faces converge, in the remaining 8 (located as vertices of a regular quadrangular anti-prism ), five faces each.

A twisted elongated quadrangular bipyramid can be obtained from two square pyramids ( J ₁ ) and a regular quadrangular antiprism, all edges of which are of the same length, by attaching the bases of the pyramids to the bases of the antiprism.

Content

Metric characteristics

If a twisted elongated quadrangular bipyramid has an edge length $a$ ${\ displaystyle a}$ its surface area and volume are expressed as

S=4{\sqrt {3}}\;a^{2}\approx 6{,}9282032a^{2},

{\ displaystyle S = 4 {\ sqrt {3}} \; a ^ {2} \ approx 6 {,} 9282032a ^ {2},}

V={\frac {1}{3}}\left({\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\approx 1{,}4284045a^{3}.

{\ displaystyle V = {\ frac {1} {3}} \ left ({\ sqrt {2}} + {\ sqrt {4 + 3 {\ sqrt {2}}}} \ right) a ^ {3} \ approx 1 {,} 4284045a ^ {3}.}

Coordinates

Twisted elongated quadrangular bipyramid with rib length $2$ ${\ displaystyle 2}$ can be placed in the Cartesian coordinate system so that its vertices have coordinates

$\left(0;\;0;\;\pm \left({\sqrt {2}}+{\frac {1}{\sqrt[{4}]{2}}}\right)\right),$ ${\ displaystyle \ left (0; \; 0; \; \ pm \ left ({\ sqrt {2}} + {\ frac {1} {\ sqrt [{4}] {2}}} \ right) \ right),}$
$\left(\pm 1;\;\pm 1;\;{\frac {1}{\sqrt[{4}]{2}}}\right),$ ${\ displaystyle \ left (\ pm 1; \; \ pm 1; \; {\ frac {1} {\ sqrt [{4}] {2}} \ right),}$
$\left(\pm {\sqrt {2}};\;0;\;-{\frac {1}{\sqrt[{4}]{2}}}\right),$ ${\ displaystyle \ left (\ pm {\ sqrt {2}}; \; 0; \; - {\ frac {1} {\ sqrt [{4}] {2}}} \ right),}$
$\left(0;\;\pm {\sqrt {2}};\;-{\frac {1}{\sqrt[{4}]{2}}}\right).$ ${\ displaystyle \ left (0; \; \ pm {\ sqrt {2}}; \; - {\ frac {1} {\ sqrt [{4}] {2}}} \ right).}$

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

Notes

↑ Zalgaller V. A. Convex polyhedra with regular faces / Zap. scientific this LOMI, 1967. - Vol. 2. - Pp. 20.

Links

Weisstein, Eric W. Twisted elongated quadrangular bipyramid (Eng.) On Wolfram MathWorld .

[1] Zalgaller V. A. Convex polyhedra with regular faces / Zap. scientific this LOMI, 1967. - Vol. 2. - Pp. 20.