Right multidimensional polyhedra

A regular n- dimensional polytope is a polyhedron of an n- dimensional Euclidean space that is most symmetric in a sense. Regular three-dimensional polyhedra are also called platonic solids .

Content

Definition

The n- dimensional polyhedron flag $P$ ${\ displaystyle P}$ called a set of its faces $F=(F_{0},F_{1},\dots ,F_{n-1})$ ${\ displaystyle F = (F_ {0}, F_ {1}, \ dots, F_ {n-1})}$ where $F_{i}$ ${\ displaystyle F_ {i}}$ there is $i$ ${\ displaystyle i}$ -dimensional facet of the polyhedron P, and $F_{i}\subseteq F_{n-1}$ ${\ displaystyle F_ {i} \ subseteq F_ {n-1}}$ for $i=1,2,\dots ,n-1$ ${\ displaystyle i = 1,2, \ dots, n-1}$ .

A regular n- dimensional polytope is a convex n- dimensional polyhedron $P$ ${\ displaystyle P}$ which for any two of its flags $F$ ${\ displaystyle F}$ and $F^{'}$ ${\ displaystyle F '}$ there is movement $P$ ${\ displaystyle P}$ translating $F$ ${\ displaystyle F}$ at $F^{'}$ ${\ displaystyle F '}$ .

Classification

Dimension 4

There are 6 regular 4-dimensional polyhedra (polycells):

Title	Symbol Schläfli	Cell	Number cells	Number facets	Number ribs	Number tops
Pyatichnik	{3.3,3}	right tetrahedron	five	ten	ten	five
Tesseract	{4.3,3}	cube	eight	24	32	sixteen
16th month	{3.3,4}	right tetrahedron	sixteen	32	24	eight
Twenty Quarter	{3,4,3}	octahedron	24	96	96	24
One hundred and fifty	{5.3,3}	dodecahedron	120	720	1200	600
Six Cell	{3,3,5}	right tetrahedron	600	1200	720	120

Dimensions 5 and above

In each of the higher dimensions, there are 3 regular polyhedra ( polytope ):

Title	Schläfli Symbol
n- dimensional correct simplex	{3; 3; ...; 3; 3}
n- dimensional hypercube	{4; 3; ...; 3; 3}
n- dimensional hyperoctahedron	{3; 3; ...; 3; 4}

Geometric Properties

Angles

The dihedral angle between (n-1) -dimensional adjacent faces of a regular n-dimensional polyhedron defined by its Schläfli symbol $\{p_{1},p_{2},p_{3},\dots ,p_{N-3},p_{N-2},p_{N-1}\}$ ${\ displaystyle \ {p_ {1}, p_ {2}, p_ {3}, \ dots, p_ {N-3}, p_ {N-2}, p_ {N-1} \}}$ is determined by the formula ^[1] ^[2] ^[3] :

\sin ^{2}\beta ={\frac {\cos ^{2}{\frac {\pi }{p_{n-1}}}}{1-{\frac {\cos ^{2}{\frac {\pi }{p_{n-2}}}}{1-{\frac {\cos ^{2}{\frac {\pi }{p_{n-3}}}}{\frac {\ddots }{1-{\frac {\cos ^{2}{\frac {\pi }{p_{3}}}}{1-{\frac {\cos ^{2}{\frac {\pi }{p_{2}}}}{1-\cos ^{2}{\frac {\pi }{p_{1}}}}}}}}}}}}}}

{\ displaystyle \ sin ^ {2} \ beta = {\ frac {\ cos ^ {2} {\ frac {\ pi} {p_ {n-1}}}} {1 - {\ frac {\ cos ^ { 2} {\ frac {\ pi} {p_ {n-2}}} {1 - {\ frac {\ cos ^ {2} {\ frac {\ pi} {p_ {n-3}}}} { \ frac {\ ddots} {1 - {\ frac {\ cos ^ {2} {\ frac {\ pi} {p_ {3}}} {1 - {\ frac {\ cos ^ {2} {\ frac {\ pi} {p_ {2}}} {1- \ cos ^ {2} {\ frac {\ pi} {p_ {1}}}}}}}}}}}}

Where $\beta$ ${\ displaystyle \ beta}$ - half the angle between (n-1) -dimensional adjacent faces of a regular n-dimensional polyhedron

Radii, Volumes

The radius of the inscribed N-dimensional sphere:

r_{N}=r_{N-1}\operatorname {tg} {\beta },

{\ displaystyle r_ {N} = r_ {N-1} \ operatorname {tg} {\ beta},}

Where $r_{N-1}$ ${\ displaystyle r_ {N-1}}$ Is the radius of the inscribed (N-1) -dimensional sphere of the face.

The volume of an N-dimensional polyhedron:

V_{N}={\frac {1}{N}}V_{N-1}A_{N-1}r_{N},

{\ displaystyle V_ {N} = {\ frac {1} {N}} V_ {N-1} A_ {N-1} r_ {N},}

Where $V_{N-1}$ ${\ displaystyle V_ {N-1}}$ - the volume of (N-1) -dimensional face, $A_{N-1}$ ${\ displaystyle A_ {N-1}}$ - the number of (N-1) -dimensional faces.

Tiles

In dimension n = 4

Tesseract Honeycomb
Hex Cell Cellular
Twenty Four Cell Cells

In dimension n ≥ 5

Hypercubic Honeycombs

Notes

Om Sommerville DMY Dimensions . - London, 1929. - p. 189. - 196 p.
↑ Coxeter HSM Regular Polytoopes . - London, 1948. - p. 134. - 321 p.
↑ B. Rosenfeld Multidimensional spaces . - Science, 1966. - p. 193.

Links

Visual example on YouTube

Regular Polytopes (Platonic solids) in 4D (Neopr.) (2003). The date of circulation is January 30, 2011. Archived May 4, 2012.

E. Yu. Smirnov. Reflection groups and regular polyhedra . - M .: MTSNMO, 2009. - 48 p. - ISBN 978-5-94057-525-2 .

E. B. Winberg, O. V. Schwartzman. Discrete groups of motions of spaces of constant curvature // Itogi Nauki i Tekhn. Ser. Let's do it. problems mat. Funda. directions. - 1988. - T. 29 . - p . 147–259 .

[1] Om Sommerville DMY Dimensions . - London, 1929. - p. 189. - 196 p.

[2] Coxeter HSM Regular Polytoopes . - London, 1948. - p. 134. - 321 p.

[3] B. Rosenfeld Multidimensional spaces . - Science, 1966. - p. 193.