16-cell
(4-orthoplex)
Schlegel diagram
(vertices and edges)
TypeConvex regular 4-polytope
4-orthoplex
4-demicube
Schläfli symbol{3,3,4}
Coxeter diagram
Cells16 {3,3}
Faces32 {3}
Edges24
Vertices8
Vertex figure

Octahedron
Petrie polygonoctagon
Coxeter groupB4, [3,3,4], order 384
D4, order 192
DualTesseract
Propertiesconvex, isogonal, isotoxal, isohedral, regular
Uniform index12

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.[1] It is also called C16, hexadecachoron,[2] or hexdecahedroid [sic?] .[3]

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes, and is analogous to the octahedron in three dimensions. It is Coxeter's ${\displaystyle \beta _{4))$ polytope.[4] Conway's name for a cross-polytope is orthoplex, for orthant complex. The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure. The 16-cell has 16 cells as the tesseract has 16 vertices.

## Geometry

The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[a]

Each of its 4 successor convex regular 4-polytopes can be constructed as the convex hull of a polytope compound of multiple 16-cells: the 16-vertex tesseract as a compound of two 16-cells, the 24-vertex 24-cell as a compound of three 16-cells, the 120-vertex 600-cell as a compound of fifteen 16-cells, and the 600-vertex 120-cell as a compound of seventy-five 16-cells.

Regular convex 4-polytopes
Symmetry group A4 B4 F4 H4
Name 5-cell

Hyper-tetrahedron
5-point

16-cell

Hyper-octahedron
8-point

8-cell

Hyper-cube
16-point

24-cell

24-point

600-cell

Hyper-icosahedron
120-point

120-cell

Hyper-dodecahedron
600-point

Schläfli symbol {3, 3, 3} {3, 3, 4} {4, 3, 3} {3, 4, 3} {3, 3, 5} {5, 3, 3}
Coxeter mirrors
Mirror dihedrals 𝝅/2 𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2 𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2 𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2
Graph
Vertices 5 8 16 24 120 600
Edges 10 24 32 96 720 1200
Faces 10 triangles 32 triangles 24 squares 96 triangles 1200 triangles 720 pentagons
Cells 5 tetrahedra 16 tetrahedra 8 cubes 24 octahedra 600 tetrahedra 120 dodecahedra
Tori 1 5-tetrahedron 2 8-tetrahedron 2 4-cube 4 6-octahedron 20 30-tetrahedron 12 10-dodecahedron
Inscribed 120 in 120-cell 675 in 120-cell 2 16-cells 3 8-cells 25 24-cells 10 600-cells
Great polygons 2 𝝅/2 squares x 3 4 𝝅/2 rectangles x 3 4 𝝅/3 hexagons x 4 12 𝝅/5 decagons x 6 50 𝝅/15 dodecagons x 4
Petrie polygons 1 pentagon 1 octagon 2 octagons 2 dodecagons 4 30-gons 20 30-gons
Isocline polygons 1 {8/2}=2{4} x {8/2}=2{4} 2 {8/2}=2{4} x {8/2}=2{4} 2 {12/2}=2{6} x {12/6}=6{2} 4 {30/2}=2{15} x 30{0} 20 {30/2}=2{15} x 30{0}
Long radius ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
Edge length ${\displaystyle {\sqrt {\tfrac {5}{2))}\approx 1.581}$ ${\displaystyle {\sqrt {2))\approx 1.414}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle {\tfrac {1}{\phi ))\approx 0.618}$ ${\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2))))\approx 0.270}$
Short radius ${\displaystyle {\tfrac {1}{4))}$ ${\displaystyle {\tfrac {1}{2))}$ ${\displaystyle {\tfrac {1}{2))}$ ${\displaystyle {\sqrt {\tfrac {1}{2))}\approx 0.707}$ ${\displaystyle {\sqrt {\tfrac {\phi ^{4)){8))}\approx 0.926}$ ${\displaystyle {\sqrt {\tfrac {\phi ^{4)){8))}\approx 0.926}$
Area ${\displaystyle 10\left({\sqrt {\tfrac {8}{9))}\right)\approx 9.428}$ ${\displaystyle 32\left({\sqrt {\tfrac {3}{16))}\right)\approx 13.856}$ ${\displaystyle 24}$ ${\displaystyle 96\left({\sqrt {\tfrac {3}{16))}\right)\approx 41.569}$ ${\displaystyle 1200\left({\tfrac {\sqrt {3)){8\phi ^{2))}\right)\approx 99.238}$ ${\displaystyle 720\left({\tfrac {25+10{\sqrt {5))}{8\phi ^{4))}\right)\approx 621.9}$
Volume ${\displaystyle 5\left({\tfrac {5{\sqrt {5))}{24))\right)\approx 2.329}$ ${\displaystyle 16\left({\tfrac {1}{3))\right)\approx 5.333}$ ${\displaystyle 8}$ ${\displaystyle 24\left({\sqrt {\tfrac {2}{9))}\right)\approx 11.314}$ ${\displaystyle 600\left({\tfrac {1}{3\phi ^{3}{\sqrt {8))))\right)\approx 16.693}$ ${\displaystyle 120\left({\tfrac {2+\phi }{2\phi ^{3}{\sqrt {8))))\right)\approx 18.118}$
4-Content ${\displaystyle {\tfrac {\sqrt {5)){24))\left({\tfrac {\sqrt {5)){2))\right)^{4}\approx 0.146}$ ${\displaystyle {\tfrac {2}{3))\approx 0.667}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle {\tfrac ((\text{Short))\times {\text{Vol))}{4))\approx 3.863}$ ${\displaystyle {\tfrac ((\text{Short))\times {\text{Vol))}{4))\approx 4.193}$

### Coordinates

Disjoint squares
xy plane
( 0, 1, 0, 0) ( 0, 0,-1, 0)
( 0, 0, 1, 0) ( 0,-1, 0, 0)
wz plane
( 1, 0, 0, 0) ( 0, 0, 0,-1)
( 0, 0, 0, 1) (-1, 0, 0, 0)

The 16-cell is the 4-dimensional cross polytope, which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system.

The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is 2.

The vertex coordinates form 6 orthogonal central squares lying in the 6 coordinate planes. Squares in opposite planes that do not share an axis (e.g. in the xy and wz planes) are completely disjoint (they do not intersect at any vertices).[b]

The 16-cell constitutes an orthonormal basis for the choice of a 4-dimensional reference frame, because its vertices exactly define the four orthogonal axes.

### Structure

The Schläfli symbol of the 16-cell is {3,3,4}, indicating that its cells are regular tetrahedra {3,3} and its vertex figure is a regular octahedron {3,4}. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. Its edge figure is a square. There are 4 tetrahedra and 4 triangles meeting at every edge.

The 16-cell is bounded by 16 cells, all of which are regular tetrahedra.[c] It has 32 triangular faces, 24 edges, and 8 vertices. The 24 edges bound 6 orthogonal central squares lying on great circles in the 6 coordinate planes (3 pairs of completely orthogonal[d] great squares). At each vertex, 3 great squares cross perpendicularly. The 6 edges meet at the vertex the way 6 edges meet at the apex of a canonical octahedral pyramid.[e]

### Rotations

 A 3D projection of a 16-cell performing a simple rotation A 3D projection of a 16-cell performing a double rotation

Rotations in 4-dimensional Euclidean space can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.[6] The 16-cell is a simple frame in which to observe 4-dimensional rotations, because each of the 16-cell's 6 great squares has another completely orthogonal great square (there are 3 pairs of completely orthogonal squares).[b] Many rotations of the 16-cell can be characterized by the angle of rotation in one of its great square planes (e.g. the xy plane) and another angle of rotation in the completely orthogonal great square plane (the wz plane).[f] Completely orthogonal great squares have disjoint vertices: 4 of the 16-cell's 8 vertices rotate in one plane, and the other 4 rotate independently in the completely orthogonal plane.[h]

In 2 or 3 dimensions a rotation is characterized by a single plane of rotation; this kind of rotation taking place in 4-space is called a simple rotation, in which only one of the two completely orthogonal planes rotates (the angle of rotation in the other plane is 0). In the 16-cell, a simple rotation in one of the 6 orthogonal planes moves only 4 of the 8 vertices; the other 4 remain fixed. (In the simple rotation animation above, all 8 vertices move because the plane of rotation is not one of the 6 orthogonal basis planes.)

In a double rotation both sets of 4 vertices move, but independently: the angles of rotation may be different in the 2 completely orthogonal planes. If the two angles happen to be the same, a maximally symmetric isoclinic rotation takes place.[i] In the 16-cell an isoclinic rotation by 90 degrees of any pair of completely orthogonal square planes takes every square plane to its completely orthogonal square plane.[j]

### Constructions

#### Octahedral dipyramid

Octahedron ${\displaystyle \beta _{3))$ 16-cell ${\displaystyle \beta _{4))$
Orthogonal projections to skew hexagon hyperplane

The simplest construction of the 16-cell is on the 3-dimensional cross polytope, the octahedron. The octahedron has 3 perpendicular axes and 6 vertices in 3 opposite pairs (its Petrie polygon is the hexagon). Add another pair of vertices, on a fourth axis perpendicular to all 3 of the other axes. Connect each new vertex to all 6 of the original vertices, adding 12 new edges. This raises two octahedral pyramids on a shared octahedron base that lies in the 16-cell's central hyperplane.[10]

Stereographic projection of the 16-cell's 6 orthogonal central squares onto their great circles. Each circle is divided into 4 arc-edges at the intersections where 3 circles cross perpendicularly. Notice that each circle has one Clifford parallel circle that it does not intersect. Those two circles pass through each other like adjacent links in a chain.

The octahedron that the construction starts with has three perpendicular intersecting squares (which appear as rectangles in the hexagonal projections). Each square intersects with each of the other squares at two opposite vertices, with two of the squares crossing at each vertex. Then two more points are added in the fourth dimension (above and below the 3-dimensional hyperplane). These new vertices are connected to all the octahedron's vertices, creating 12 new edges and three more squares (which appear edge-on as the 3 diameters of the hexagon in the projection).

Something unprecedented has also been created. Notice that each square no longer intersects with all of the other squares: it does intersect with four of them (with three of the squares crossing at each vertex now), but each square has one other square with which it shares no vertices: it is not directly connected to that square at all. These two separate perpendicular squares (there are three pairs of them) are like the opposite edges of a tetrahedron: perpendicular, but non-intersecting. They lie opposite each other (parallel in some sense), and they don't touch, but they also pass through each other like two perpendicular links in a chain (but unlike links in a chain they have a common center). They are an example of Clifford parallel polygons, and the 16-cell is the simplest regular polytope in which they occur.[g] Clifford parallelism emerges here and occurs in all the subsequent 4-dimensional convex regular polytopes, where it can be seen as the defining relationship among disjoint regular 4-polytopes and their co-centric parts. It can occur between congruent (similar) polytopes of 2 or more dimensions.[11] For example, as noted above all the subsequent convex regular 4-polytopes are compounds of multiple 16-cells; those 16-cells are Clifford parallel polytopes.

#### Tetrahedral constructions

The 16-cell has two Wythoff constructions from regular tetrahedra, a regular form and alternated form, shown here as nets, the second represented by tetrahedral cells of two alternating colors. The alternated form is a lower symmetry construction of the 16-cell called the demitesseract.

Wythoff's construction replicates the 16-cell's characteristic 5-cell in a kaleidoscope of mirrors. Every regular 4-polytope has its characteristic 4-orthoscheme, an irregular 5-cell.[k] There are three regular 4-polytopes with tetrahedral cells: the 5-cell, the 16-cell, and the 600-cell. Although all are bounded by regular tetrahedron cells, their characteristic 5-cells (4-orthoschemes) are different tetrahedral pyramids, all based on the same characteristic irregular tetrahedron. They share the same characteristic tetrahedron (3-orthoscheme) and characteristic right triangle (2-orthoscheme) because they have the same kind of cell.[l]

The characteristic 5-cell of the regular 16-cell is represented by the Coxeter-Dynkin diagram , which can be read as a list of the dihedral angles between its mirror facets. It is an irregular tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. The regular 16-cell is subdivided by its symmetry hyperplanes into 384 instances of its characteristic 5-cell that all meet at its center.

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 16-cell).[m] If the regular 16-cell has unit radius edge and edge length ${\displaystyle {\sqrt {2))}$, its characteristic 5-cell's ten edges have lengths ${\displaystyle {\sqrt {\tfrac {1}{6))))$, ${\displaystyle {\sqrt {\tfrac {1}{2))))$, ${\displaystyle {\sqrt {\tfrac {2}{3))))$ (the exterior right triangle face, the characteristic triangle), plus ${\displaystyle {\sqrt {\tfrac {3}{4))))$, ${\displaystyle {\sqrt {\tfrac {1}{4))))$, ${\displaystyle {\sqrt {\tfrac {1}{12))))$ (the other three edges of the exterior 3-orthoscheme facet, the characteristic tetrahedron), plus ${\displaystyle 1}$, ${\displaystyle {\sqrt {\tfrac {1}{2))))$, ${\displaystyle {\sqrt {\tfrac {1}{3))))$, ${\displaystyle {\sqrt {\tfrac {1}{4))))$ (edges that are the characteristic radii of the regular 16-cell).[13] The 4-edge path along orthogonal edges of the orthoscheme is ${\displaystyle {\sqrt {\tfrac {1}{2))))$, ${\displaystyle {\sqrt {\tfrac {1}{6))))$, ${\displaystyle {\sqrt {\tfrac {1}{4))))$, ${\displaystyle {\sqrt {\tfrac {1}{4))))$, first from a 16-cell vertex to a 16-cell edge center, then turning 90° to a 16-cell face center, then turning 90° to a 16-cell tetrahedral cell center, then turning 90° to the 16-cell center.

#### Helical construction

Net and orthogonal projection

A 16-cell can be constructed from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each bent in the fourth dimension into a ring. The two circular helixes spiral around each other, nest into each other and pass through each other forming a Hopf link. The 16 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex. The purple edges represent the Petrie polygon of the 16-cell.

Thus the 16-cell can be decomposed into two similar cell-disjoint circular chains of eight tetrahedrons each, four edges long. This decomposition can be seen in a 4-4 duoantiprism construction of the 16-cell: or , Schläfli symbol {2}⨂{2} or s{2}s{2}, symmetry 4,2+,4, order 64.

### As a configuration

This configuration matrix represents the 16-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 16-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

${\displaystyle {\begin{bmatrix}{\begin{matrix}8&6&12&8\\2&24&4&4\\3&3&32&2\\4&6&4&16\end{matrix))\end{bmatrix))}$

## Tessellations

One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol {3,3,4,3}. Hence, the 16-cell has a dihedral angle of 120°.[14] Each 16-cell has 16 neighbors with which it shares a tetrahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point. Twenty-four 16-cells meet at any given vertex in this tessellation.

The dual tessellation, the 24-cell honeycomb, {3,4,3,3}, is made of regular 24-cells. Together with the tesseractic honeycomb {4,3,3,4} these are the only three regular tessellations of R4.

## Projections

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph
Dihedral symmetry [12/3] [4]
Projection envelopes of the 16-cell. (Each cell is drawn with different color faces, inverted cells are undrawn)

The cell-first parallel projection of the 16-cell into 3-space has a cubical envelope. The closest and farthest cells are projected to inscribed tetrahedra within the cube, corresponding with the two possible ways to inscribe a regular tetrahedron in a cube. Surrounding each of these tetrahedra are 4 other (non-regular) tetrahedral volumes that are the images of the 4 surrounding tetrahedral cells, filling up the space between the inscribed tetrahedron and the cube. The remaining 6 cells are projected onto the square faces of the cube. In this projection of the 16-cell, all its edges lie on the faces of the cubical envelope.

The cell-first perspective projection of the 16-cell into 3-space has a triakis tetrahedral envelope. The layout of the cells within this envelope are analogous to that of the cell-first parallel projection.

The vertex-first parallel projection of the 16-cell into 3-space has an octahedral envelope. This octahedron can be divided into 8 tetrahedral volumes, by cutting along the coordinate planes. Each of these volumes is the image of a pair of cells in the 16-cell. The closest vertex of the 16-cell to the viewer projects onto the center of the octahedron.

Finally the edge-first parallel projection has a shortened octahedral envelope, and the face-first parallel projection has a hexagonal bipyramidal envelope.

## 4 sphere Venn diagram

A 3-dimensional projection of the 16-cell and 4 intersecting spheres (a Venn diagram of 4 sets) are topologically equivalent.

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## Symmetry constructions

The 16-cell's symmetry group is denoted B4.

There is a lower symmetry form of the 16-cell, called a demitesseract or 4-demicube, a member of the demihypercube family, and represented by h{4,3,3}, and Coxeter diagrams or . It can be drawn bicolored with alternating tetrahedral cells.

It can also be seen in lower symmetry form as a tetrahedral antiprism, constructed by 2 parallel tetrahedra in dual configurations, connected by 8 (possibly elongated) tetrahedra. It is represented by s{2,4,3}, and Coxeter diagram: .

It can also be seen as a snub 4-orthotope, represented by s{21,1,1}, and Coxeter diagram: or .

With the tesseract constructed as a 4-4 duoprism, the 16-cell can be seen as its dual, a 4-4 duopyramid.

Name Coxeter diagram Schläfli symbol Coxeter notation Order Vertex figure
Regular 16-cell {3,3,4} [3,3,4] 384
Demitesseract
Quasiregular 16-cell
=
=
h{4,3,3}
{3,31,1}
[31,1,1] = [1+,4,3,3] 192
Alternated 4-4 duoprism 2s{4,2,4} [[4,2+,4]] 64
Tetrahedral antiprism s{2,4,3} [2+,4,3] 48
Alternated square prism prism sr{2,2,4} [(2,2)+,4] 16
Snub 4-orthotope = s{21,1,1} [2,2,2]+ = [21,1,1]+ 8
4-fusil
{3,3,4} [3,3,4] 384
{4}+{4} or 2{4} [[4,2,4]] = [8,2+,8] 128
{3,4}+{ } [4,3,2] 96
{4}+2{ } [4,2,2] 32
{ }+{ }+{ }+{ } or 4{ } [2,2,2] 16

## Related complex polygons

The Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in ${\displaystyle \mathbb {C} ^{2))$ shares the same vertices as the 16-cell. It has 8 vertices, and 8 3-edges.[15][16]

The regular complex polygon, 2{4}4, , in ${\displaystyle \mathbb {C} ^{2))$ has a real representation as a 16-cell in 4-dimensional space with 8 vertices, 16 2-edges, only half of the edges of the 16-cell. Its symmetry is 4[4]2, order 32.[17]

 In B4 Coxeter plane, 2{4}4 has 8 vertices and 16 2-edges, shown here with 4 sets of colors. The 8 vertices are grouped in 2 sets (shown red and blue), each only connected with edges to vertices in the other set, making this polygon a complete bipartite graph, K4,4.[18]

## Related uniform polytopes and honeycombs

The regular 16-cell and tesseract are the regular members of a set of 15 uniform 4-polytopes with the same B4 symmetry. The 16-cell is also one of the uniform polytopes of D4 symmetry.

The 16-cell is also related to the cubic honeycomb, order-4 dodecahedral honeycomb, and order-4 hexagonal tiling honeycomb which all have octahedral vertex figures.

It belongs to the sequence of {3,3,p} 4-polytopes which have tetrahedral cells. The sequence includes three regular 4-polytopes of Euclidean 4-space, the 5-cell {3,3,3}, 16-cell {3,3,4}, and 600-cell {3,3,5}), and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space.

It is first in a sequence of quasiregular polytopes and honeycombs h{4,p,q}, and a half symmetry sequence, for regular forms {p,3,4}.

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds

## Notes

1. ^ The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content[5] within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 16-cell is the 8-point 4-polytope: second in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope.
2. ^ a b c d In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or completely orthogonal to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.
3. ^ The boundary surface of a 16-cell is a finite 3-dimensional space consisting of 16 tetrahedra arranged face-to-face (four around one). It is a closed, tightly curved (non-Euclidean) 3-space, within which we can move straight through 4 tetrahedra in any direction and arrive back in the tetrahedron where we started. We can visualize moving around inside this tetrahedral jungle gym, climbing from one tetrahedron into another on its 24 struts (its edges), and never being able to get out (or see out) of the 16 tetrahedra no matter what direction we go (or look). We are always on (or in) the surface of the 16-cell, never inside the 16-cell itself (nor outside it). We can see that the 6 edges around each vertex radiate symmetrically in 3 dimensions and form an orthogonal 3-axis cross, just as the radii of an octahedron do (so we say the vertex figure of the 16-cell is the octahedron).
4. ^ Two flat planes A and B of a Euclidean space of four dimensions are called completely orthogonal if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.[b]
5. ^ Each vertex in the 16-cell is the apex of an octahedral pyramid, the base of which is the octahedron formed by the 6 other vertices to which the apex is connected by edges. The 16-cell can be deconstructed (four different ways) into two octahedral pyramids by cutting it in half through one of its four octahedral central hyperplanes. Looked at from inside the curved 3 dimensional volume of its boundary surface of 16 face-bonded tetrahedra, the 16-cell's vertex figure is an octahedron. In 4 dimensions, the vertex octahedron is actually an octahedral pyramid. The apex of the octahedral pyramid (the vertex where the 6 edges meet) is not actually at the center of the octahedron: it is displaced radially outwards in the fourth dimension, out of the hyperplane defined by the octahedron's 6 vertices. The 6 edges around the vertex make an orthogonal 3-axis cross in 3 dimensions (and in the 3-dimensional projection of the 4-pyramid), but the 3 lines are actually bent 90 degrees in the fourth dimension where they meet in an apex.
6. ^ a b Each great square vertex is 2 distant from two of the square's other vertices, and 4 distant from its opposite vertex. The other four vertices of the 16-cell (also 2 distant) are the vertices of the square's completely orthogonal square.
7. ^ a b c Clifford parallels are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.[7] A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the 3-sphere.[8] In the 16-cell the corresponding vertices of completely orthogonal great circle squares are all 2 apart, so these squares are Clifford parallel polygons.[f] Note that only the vertices of the great squares (the points on the great circle) are 2 apart; points on the edges of the squares (on chords of the circle) are closer together.
8. ^ Completely orthogonal great squares are non-intersecting and rotate independently because the great circles on which their vertices lie are Clifford parallel.[g] The two squares cannot intersect at all because they lie in planes which intersect at only one point: the center of the 16-cell.[b] Because they are perpendicular and share a common center, the two squares are obviously not parallel and separate in the usual way of parallel squares in 3 dimensions; rather they are connected like adjacent square links in a chain, each passing through the other without intersecting at any points, forming a Hopf link.
9. ^ In an isoclinic rotation, all 6 orthogonal planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted sideways by that same angle. An isoclinic displacement (also known as a Clifford displacement) is 4-dimensionally diagonal. Points are displaced an equal distance in two orthogonal directions at once, and displaced a total Pythagorean distance equal to the square root of twice that distance. For example, when the unit-radius 16-cell rotates isoclinically 90° in a great square invariant plane and 90° in its completely orthogonal great square invariant plane, each vertex is displaced to another vertex one 2 edge distant, moving 1 unit-radius in two orthogonal coordinate directions.
10. ^ The 90 degree isoclinic rotations of two completely orthogonal planes take them to each other. In the 16-cell all 6 orthogonal planes rotate by 90 degrees, and also tilt sideways by 90 degrees to their Clifford parallel[g] plane.[9]
11. ^ An orthoscheme is a chiral irregular simplex with right triangle faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own facets (its mirror walls). Every regular polytope can be dissected radially into instances of its characteristic orthoscheme surrounding its center. The characteristic orthoscheme has the shape described by the same Coxeter-Dynkin diagram as the regular polytope without the generating point ring.
12. ^ A regular polytope of dimension k has a characteristic k-orthoscheme, and also a characteristic (k-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.[12] The interior tetrahedra and triangles thus formed will also be orthoschemes.
13. ^ The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.

## Citations

1. ^ Coxeter 1973, p. 141, §7-x. Historical remarks.
2. ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
3. ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
4. ^ Coxeter 1973, pp. 120=121, §7.2. See illustration Fig 7.2B.
5. ^ Coxeter 1973, pp. 292–293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.
6. ^ Kim & Rote 2016, p. 6, §5. Four-Dimensional Rotations.
7. ^ Tyrrell & Semple 1971, pp. 5–6, §3. Clifford's original definition of parallelism.
8. ^ Kim & Rote 2016, pp. 7–10, §6. Angles between two Planes in 4-Space.
9. ^ Kim & Rote 2016, pp. 8–10, Relations to Clifford Parallelism.
10. ^ Coxeter 1973, p. 121, §7.21. See illustration Fig 7.2B: "${\displaystyle \beta _{4))$ is a four-dimensional dipyramid based on ${\displaystyle \beta _{3))$ (with its two apices in opposite directions along the fourth dimension)."
11. ^
12. ^ Coxeter 1973, p. 130, §7.6; "simplicial subdivision".
13. ^ Coxeter 1973, pp. 292–293, Table I(ii); "16-cell".
14. ^ Coxeter 1973, p. 293.
15. ^ Coxeter 1991, pp. 30, 47.
16. ^
17. ^ Coxeter 1991, p. 108.
18. ^ Coxeter 1991, p. 114.

## References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H.S.M. Coxeter:
• Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
• Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge University Press.
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Coxeter, H.S.M.; Shephard, G.C. (1992). "Portraits of a family of complex polytopes". Leonardo. 25 (3/4): 239–244. doi:10.2307/1575843. JSTOR 1575843. S2CID 124245340.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Kim, Heuna; Rote, Günter (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
• Tyrrell, J. A.; Semple, J.G. (1971). Generalized Clifford parallelism. Cambridge University Press. ISBN 0-521-08042-8.