Elongated triangular pyramid

Elongated triangular pyramid
Elongated triangular pyramid
	; ( 3D model )
Type of	Johnson's polyhedron
The properties	convex
Combinatorics
7 faces; 12 ribs; 7 peaks	Χ = 2
Facets	4 triangles; 3 square
Vertex configuration	1 (3 3 ); 3 (3.4 2 ); 3 (3 2 .4 2 )
Dual polyhedron
	Scan
Classification
Designations	J 7 , M 1 + P 3
Symmetry group	C 3v

The elongated triangular pyramid ^[1] is one of Johnson's polyhedra ( J ₇ , according to Zalgaller - M ₁ + P ₃ ).

Composed of 7 faces: 4 regular triangles and 3 squares . Each square face is surrounded by two square and two triangular; Among the triangular faces, 1 is surrounded by three square, the remaining 3 - square and two triangular.

It has 12 edges of the same length. 3 edges are located between two square faces, 6 edges - between square and triangular, the remaining 3 - between two triangular.

The elongated triangular pyramid has 7 peaks. At 3 vertices, two square faces and one triangular converge; at 3 vertices, two square and two triangular faces converge; three triangular faces converge at 1 vertex.

An elongated triangular pyramid can be obtained from two polyhedra - a regular tetrahedron and a regular triangular prism , all the edges of which are the same length - by attaching them to each other with triangular faces.

Content

Metric

If an elongated triangular pyramid has an edge of length $a$ ${\ displaystyle a}$ , its surface area and volume are expressed as

S=\left(3+{\sqrt {3}}\right)a^{2}\approx 4{,}7320508a^{2},

{\ displaystyle S = \ left (3 + {\ sqrt {3}} \ right) a ^ {2} \ approx 4 {,} 7320508a ^ {2},}

V=\left({\frac {\sqrt {2}}{12}}+{\frac {\sqrt {3}}{4}}\right)a^{3}\approx 0{,}5508638a^{3}.

{\ displaystyle V = \ left ({\ frac {\ sqrt {2}} {12}} + {\ frac {\ sqrt {3}} {4}} \ right) a ^ {3} \ approx 0 {, } 5508638a ^ {3}.}

In coordinates

Elongated triangular pyramid with rib length $2$ ${\ displaystyle 2}$ can be placed in a Cartesian coordinate system so that its vertices have coordinates

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

In this case, the symmetry axis of the polyhedron will coincide with the Oz axis, and one of the three planes of symmetry will coincide with the yOz plane.

Space Fill

With the help of elongated triangular pyramids, square pyramids ( J ₁ ) and / or octahedrons, it is possible to bridge a three-dimensional space without gaps and overlays ( see illustration ).

Notes

↑ Zalgaller V. A. Convex polyhedra with regular edges / Zap. scientific sem. LOMI, 1967. - V. 2. - Page. 20.

Links

Weisstein, Eric W. An elongated triangular pyramid on the Wolfram MathWorld website.

[1] Zalgaller V. A. Convex polyhedra with regular edges / Zap. scientific sem. LOMI, 1967. - V. 2. - Page. 20.