16-cell

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.[3]

Regular hexadecachoron
(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

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 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).[lower-alpha 1]

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
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 1 16-cell 2 16-cells 3 8-cells 5 24-cells x 5 5 600-cells x 2
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 1 1 1 1 1 1
Edge length √5/√2 ≈ 1.581 √2 ≈ 1.414 1 1 1/ϕ ≈ 0.618 1/√2ϕ2 ≈ 0.270
Short radius 1/4 1/2 1/2 √2/2 ≈ 0.707 1 - (√2/2√3φ)2 ≈ 0.936 1 - (1/2√3φ)2 ≈ 0.968
Area 10•√8/3 ≈ 9.428 32•√3/4 ≈ 13.856 24 96•√3/4 ≈ 41.569 1200•√3/8φ2 ≈ 99.238 720•25+10√5/8φ4 ≈ 621.9
Volume 5•5√5/24 ≈ 2.329 16•1/3 ≈ 5.333 8 24•√2/3 ≈ 11.314 600•1/3√8φ3 ≈ 16.693 120•2 + φ/2√8φ3 ≈ 18.118
4-Content √5/24•(√5/2)4 ≈ 0.146 2/3 ≈ 0.667 1 2 Short∙Vol/4 ≈ 3.907 Short∙Vol/4 ≈ 4.385

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).[lower-alpha 2]

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.[lower-alpha 3] 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[lower-alpha 4] 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.[lower-alpha 5]

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).[lower-alpha 2] 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).[lower-alpha 6] 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.[lower-alpha 8] The two squares cannot intersect at all because they lie in planes which intersect at only one point: the center of the 16-cell.[lower-alpha 2] 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.</ref>

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.[lower-alpha 9] 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.[lower-alpha 10]

Octahedral dipyramid
Octahedron 16-cell
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.[8]

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.[lower-alpha 7] 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.[9] 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 4-orthoscheme in a kaleidoscope of mirrors. An orthoscheme is a chiral irregular simplex 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 into instances of its characteristic orthoscheme.

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 4-orthoschemes are all different tetrahedral pyramids, each based on a different characteristic irregular tetrahedron. For example, since the 5-cell is dimensionally analogous to the tetrahedron, the characteristic simplex of the 5-cell is a 4-pyramid based on the characteristic tetrahedron of the regular tetrahedron. Since the 16-cell is dimensionally analogous to the octahedron, the characteristic simplex of the 16-cell is a 4-pyramid based on the characteristic tetrahedron of the regular octahedron.

To construct the 16-cell from its dimensionally analogous polyhedron the octahedron, we raised two 4-pyramids on the octahedron: an octahedral dipyramid. To construct the characteristic 4-orthoscheme of the 16-cell (an irregular 5-cell) on its 3-orthoscheme base (an irregular tetrahedron), we raise a 4-pyramid on the characteristic tetrahedron of the regular octahedron. As with any tetrahedral pyramid, this entails adding four new edges from the base tetrahedron's vertices to a fifth vertex, the apex of the 4-pyramid. Since the base is irregular (a 3-orthoscheme with edges of 4 different lengths), and the four new cells that will be created as "sides" of the 4-pyramid must be congruent 3-orthoschemes (because this 16-cell is a regular 4-polytope), the 4 new edges must be of those same 4 different lengths. Provided they are, the resulting 4-pyramid will be the characteristic 4-orthoscheme that we can expect to fit and fill each of the two 4-pyramids of the regular 16-cell.

The characteristic orthoscheme of the unit-radius octahedron has edges of lengths √2, √2/2, √3/2 (the exterior right triangle face), plus 1, 1, √2/2 (edges that are radii of the octahedron). The path along 3 orthogonal edges is √2/2, √2/2, 1 (first along an octahedron edge to its midpoint, then turning 90° to the octahedron center, then turning 90° again to another octahedron vertex).

To construct the 4-orthoscheme add four new edges, one of each length, and one new vertex (an apex at which all the new edges meet). Attach the new (16-cell) edge of length √2 to the first vertex in the path. Attach the new edge of length √3/2 to the second vertex in the path (the vertex that is an octahedron mid-edge). Attach the new edge of length 1 (a 16-cell long radius) to the third vertex in the path (the vertex at the center of the octahedron, which will also be the center of the 16-cell). Attach the (shortest) new edge of length √2/2 to the fourth vertex in the path (the only vertex that does not already have such an edge attached). The path has been extended from 3 orthogonal edges to 4 and is now √2/2, √2/2, 1, √2/2 with the new apex vertex as the fifth vertex in the path. The fifth path vertex, like the second path vertex, is a mid-edge of the 16-cell rather than a 16-cell vertex.

32 of these 4-orthoschemes will fit in the 16-cell , 16 above the hyperplane of the octahedral base and 16 below it.[lower-alpha 11] All 32 surround one axis of the 16-cell and meet at its center. Because the 16-cell has four orthogonal axes, there are four ways to fill it with 4-orthoschemes.

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.

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°.[10] 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.

The 16 cells ordered by number of intersecting spheres (from 0 to 4)     (see all cells and k-faces)
4 sphere Venn diagram and 16-cell projection in the same orientation

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

The Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in shares the same vertices as the 16-cell. It has 8 vertices, and 8 3-edges.[11][12]

The regular complex polygon, 2{4}4, , in 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.[13]

Orthographic projections of 2{4}4 polygon

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.[14]

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.

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

It is in a sequence to three regular 4-polytopes: the 5-cell {3,3,3}, 600-cell {3,3,5} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have tetrahedral cells.

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}.

See also
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List 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. 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.[lower-alpha 2]
  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 vertex figure is an octahedron. In 4 dimensions, the 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. 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. 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. 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. In the 16-cell the corresponding vertices of completely orthogonal great circle squares are all √2 apart, so these squares are Clifford parallel polygons.<ref name='completely orthogonal squares' group='lower-alpha'>
  8. Completely orthogonal great squares are non-intersecting and rotate independently because the great circles on which their vertices lie are Clifford parallel.[lower-alpha 7] 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.
  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 (as in the unit-radius 16-cell the edge length is √2).
  10. The 90 degree rotations of two completely orthogonal planes take them to each other. All 6 orthogonal planes rotate by 90 degrees, and also tilt sideways by 90 degrees to their Clifford parallel[lower-alpha 7] plane.[7]
  11. 8 of the 16 4-orthoschemes in each octahedral pyramid will need to be left-handed orthoschemes, and the other 8 right-handed, so each 4-orthoscheme can be cell-bonded to five surrounding 4-orthoschemes of the opposite chirality (an alternating pattern). But in practice it does not matter whether you construct left-handed or right-handed 4-orthoschemes, because in 4 dimensional space you can turn each one inside out after construction, if necessary to give it the chirality needed, simply by rotating it, without folding, stretching, or deforming it in any way! Möbius was apparently the first to realize the possibility of a double rotation in four dimensional space, which turns a rotated rigid object inside out as well as "around".

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. Kim & Rote 2016, pp. 8–10, Relations to Clifford Parallelism.
  8. Coxeter 1973, p. 121, §7.21. See illustration Fig 7.2B: " is a four-dimensional dipyramid based on (with its two apices in opposite directions along the fourth dimension)."
  9. Tyrrell & Semple 1971.
  10. Coxeter 1973, p. 293.
  11. Coxeter 1991, pp. 30, 47.
  12. Coxeter & Shephard 1992.
  13. Coxeter 1991, p. 108.
  14. 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
      • (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.
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