16cell (4orthoplex)  

Type  Convex regular 4polytope 4orthoplex 4demicube 
Schläfli symbol  {3,3,4} 
Coxeter diagram  
Cells  16 {3,3} 
Faces  32 {3} 
Edges  24 
Vertices  8 
Vertex figure  Octahedron 
Petrie polygon  octagon 
Coxeter group  B_{4}, [3,3,4], order 384 D_{4}, order 192 
Dual  Tesseract 
Properties  convex, isogonal, isotoxal, isohedral, regular, Hanner polytope 
Uniform index  12 
In geometry, the 16cell is the regular convex 4polytope (fourdimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid19th century.^{[1]} It is also called C_{16}, hexadecachoron,^{[2]} or hexdecahedroid [sic?] .^{[3]}
It is a part of an infinite family of polytopes, called crosspolytopes or orthoplexes, and is analogous to the octahedron in three dimensions. It is Coxeter's polytope.^{[4]} Conway's name for a crosspolytope is orthoplex, for orthant complex. The dual polytope is the tesseract (4cube), which it can be combined with to form a compound figure. The 16cell has 16 cells as the tesseract has 16 vertices.
The 16cell is the second in the sequence of 6 convex regular 4polytopes (in order of size and complexity).^{[a]}
Each of its 4 successor convex regular 4polytopes can be constructed as the convex hull of a polytope compound of multiple 16cells: the 16vertex tesseract as a compound of two 16cells, the 24vertex 24cell as a compound of three 16cells, the 120vertex 600cell as a compound of fifteen 16cells, and the 600vertex 120cell as a compound of seventyfive 16cells.^{[b]}
Regular convex 4polytopes  

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
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/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 𝝅/2  
Graph  
Vertices  5 tetrahedral  8 octahedral  16 tetrahedral  24 cubical  120 icosahedral  600 tetrahedral  
Edges  10 triangular  24 square  32 triangular  96 triangular  720 pentagonal  1200 triangular  
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 5tetrahedron  2 8tetrahedron  2 4cube  4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  675 in 120cell  2 16cells  3 8cells  25 24cells  10 600cells  
Great polygons  2 squares x 3  4 rectangles x 4  4 hexagons x 4  12 decagons x 6  100 irregular hexagons x 4  
Petrie polygons  1 pentagon  1 octagon  2 octagons  2 dodecagons  4 30gons  20 30gons  
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
Disjoint squares  

 

The 16cell is the 4dimensional 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).^{[c]}
The 16cell constitutes an orthonormal basis for the choice of a 4dimensional reference frame, because its vertices exactly define the four orthogonal axes.
The Schläfli symbol of the 16cell 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 16cell is bounded by 16 cells, all of which are regular tetrahedra.^{[e]} 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^{[f]} 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.^{[d]}
A 3D projection of a 16cell performing a simple rotation 
A 3D projection of a 16cell performing a double rotation 
Rotations in 4dimensional Euclidean space can be seen as the composition of two 2dimensional rotations in completely orthogonal planes.^{[6]} The 16cell is a simple frame in which to observe 4dimensional rotations, because each of the 16cell's 6 great squares has another completely orthogonal great square (there are 3 pairs of completely orthogonal squares).^{[c]} Many rotations of the 16cell 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).^{[j]} Completely orthogonal great squares have disjoint vertices: 4 of the 16cell's 8 vertices rotate in one plane, and the other 4 rotate independently in the completely orthogonal plane.^{[g]}
In 2 or 3 dimensions a rotation is characterized by a single plane of rotation; this kind of rotation taking place in 4space 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 16cell, 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.^{[q]} In the 16cell an isoclinic rotation by 90 degrees of any pair of completely orthogonal square planes takes every square plane to its completely orthogonal square plane.^{[r]}
Octahedron  16cell 

Orthogonal projections to skew hexagon hyperplane 
The simplest construction of the 16cell is on the 3dimensional 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 16cell's central hyperplane.^{[10]}
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 3dimensional hyperplane). These new vertices are connected to all the octahedron's vertices, creating 12 new edges and three more squares (which appear edgeon as the 3 diameters of the hexagon in the projection), and three more octahedra.^{[h]}
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 nonintersecting. 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 planes, and the 16cell is the simplest regular polytope in which they occur. Clifford parallelism^{[l]} of objects of more than one dimension (more than just curved lines) emerges here and occurs in all the subsequent 4dimensional regular polytopes, where it can be seen as the defining relationship among disjoint regular 4polytopes and their concentric parts. It can occur between congruent (similar) polytopes of 2 or more dimensions.^{[11]} For example, as noted above all the subsequent convex regular 4polytopes are compounds of multiple 16cells; those 16cells are Clifford parallel polytopes.
The 16cell 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 16cell called the demitesseract.
Wythoff's construction replicates the 16cell's characteristic 5cell in a kaleidoscope of mirrors. Every regular 4polytope has its characteristic 4orthoscheme, an irregular 5cell.^{[s]} There are three regular 4polytopes with tetrahedral cells: the 5cell, the 16cell, and the 600cell. Although all are bounded by regular tetrahedron cells, their characteristic 5cells (4orthoschemes) are different tetrahedral pyramids, all based on the same characteristic irregular tetrahedron. They share the same characteristic tetrahedron (3orthoscheme) and characteristic right triangle (2orthoscheme) because they have the same kind of cell.^{[t]}
Characteristics of the 16cell^{[13]}  

edge^{[14]}  arc  dihedral^{[15]}  
𝒍  90°  120°  
𝟀  60″  60°  
𝝓  45″  45°  
𝟁  30″  60°  
60°  90°  
45°  90°  
30°  90°  
The characteristic 5cell of the regular 16cell is represented by the CoxeterDynkin 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 16cell is subdivided by its symmetry hyperplanes into 384 instances of its characteristic 5cell that all meet at its center.
The characteristic 5cell (4orthoscheme) has four more edges than its base characteristic tetrahedron (3orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4orthoscheme, at the center of the regular 16cell).^{[u]} If the regular 16cell has unit radius edge and edge length 𝒍 = , its characteristic 5cell's ten edges have lengths , , (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁), plus , , (the other three edges of the exterior 3orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus , , , (edges which are the characteristic radii of the regular 16cell). The 4edge path along orthogonal edges of the orthoscheme is , , , , first from a 16cell vertex to a 16cell edge center, then turning 90° to a 16cell face center, then turning 90° to a 16cell tetrahedral cell center, then turning 90° to the 16cell center.
A 16cell can be constructed (three different ways) from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each bent in the fourth dimension into a ring.^{[16]}^{[17]} 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 16cell. The eightcell ring of tetrahedra contains three octagrams of different colors, eightedge circular paths that wind twice around the 16cell on every third vertex of the octagram. The orange and yellow edges are two fouredge halves of one octagram, which join their ends to form a Möbius strip.
Thus the 16cell can be decomposed into two celldisjoint circular chains of eight tetrahedrons each, four edges long, one spiraling to the right (clockwise) and the other spiraling to the left (counterclockwise). The lefthanded and righthanded cell rings fit together, nesting into each other and entirely filling the 16cell, even though they are of opposite chirality. This decomposition can be seen in a 44 duoantiprism construction of the 16cell: or , Schläfli symbol {2}⨂{2} or s{2}s{2}, symmetry [4,2^{+},4], order 64.
Three eightedge paths (of different colors) spiral along each eightcell ring, making 90° angles at each vertex. (In the Boerdijk–Coxeter helix before it is bent into a ring, the angles in different paths vary, but are not 90°.) Three paths (with three different colors and apparent angles) pass through each vertex. When the helix is bent into a ring, the segments of each eightedge path (of various lengths) join their ends, forming a Möbius strip eight edges long along its singlesided circumference of 8𝝅, and one edge wide.^{[p]} The six fouredge halves of the three eightedge paths each make four 90° angles, but they are not the six orthogonal great squares: they are openended squares, fouredge 360° helices whose open ends are antipodal vertices. The four edges come from four different great squares, and are mutually orthogonal. Combined endtoend in pairs of the same chirality, the six fouredge paths make three eightedge Möbius loops, helical octagrams. Each octagram is both a Petrie polygon of the 16cell, and the helical track along which all eight vertices rotate together, in one of the 16cell's distinct isoclinic rotations.^{[v]}
Five ways of looking at the same skew octagram^{[w]}  

Edge path  Petrie polygon^{[18]}  16cell  Discrete fibration  Isocline chords 
Octagram_{{8/3}}^{[19]}  Octagram_{{8/1}}  Coxeter plane B_{4}  Octagram_{{8/2}=2{4}}  Octagram_{{8/4}=4{2}} 
The eight √2 edges of the edgepath of an isocline.^{[x]}  Skew octagon of eight √2 edges. The 16cell has 3 of these 8vertex circuits.  All 24 √2 edges and the four √4 orthogonal axes.^{[y]}  Two completely orthogonal (disjoint) great squares of √2 edges.^{[g]}  Eight √4 chords of an isocline (doubled).^{[z]} 
Each eightedge helix is a skew octagram_{{8/3}} that winds twice around the 16cell and visits every vertex before closing into a loop. Its eight edges are the circular pathnearedges of an isocline, a geodesic arc on which vertices move during an isoclinic rotation.^{[q]} The isoclines connect opposite vertices of facebonded tetrahedral cells,^{[o]} which are also opposite vertices (antipodal vertices) of the 16cell, so the isoclines have √4 chords.^{[z]} The isocline winds around the 16cell twice (720°) the way the edges of the octagram_{{8/3}} wind around twice, passing alongside each of the √2 edges once,^{[x]} and alongside each of the √4 orthogonal axes of the 16cell twice.^{[y]} The isocline makes a circle of circumference 8𝝅.^{[p]}
The eightcell ring is chiral: there is a righthanded form which spirals clockwise, and a lefthanded form which spirals counterclockwise. The 16cell contains one of each, so it also contains a left and a right isocline; the isocline is the circular axis around which the eightcell ring twists. Each isocline visits all eight vertices of the 16cell.^{[ac]} Each eightcell ring contains half of the 16 cells, but all 8 vertices; the two rings share the vertices, as they nest into each other and fit together. They also share the 24 edges, though left and right octagram helices are different eightedge paths.^{[ad]}
Because there are three pairs of completely orthogonal great squares,^{[c]} there are three congruent ways to compose a 16cell from two eightcell rings. The 16cell contains three leftright pairs of eightcell rings in different orientations, with each cell ring containing its axial isocline.^{[v]} Each leftright pair of isoclines is the track of a leftright pair of distinct isoclinic rotations: the rotations in one pair of completely orthogonal invariant planes of rotation.^{[g]} At each vertex, there are three great squares and six octagram isoclines that cross at the vertex and share a 16cell axis chord.^{[ae]}
This configuration matrix represents the 16cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 16cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
One can tessellate 4dimensional Euclidean space by regular 16cells. This is called the 16cell honeycomb and has Schläfli symbol {3,3,4,3}. Hence, the 16cell has a dihedral angle of 120°.^{[21]} Each 16cell 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. Twentyfour 16cells meet at any given vertex in this tessellation.
The dual tessellation, the 24cell honeycomb, {3,4,3,3}, is made of regular 24cells. Together with the tesseractic honeycomb {4,3,3,4} these are the only three regular tessellations of R^{4}.
Coxeter plane  B_{4}  B_{3} / D_{4} / A_{2}  B_{2} / D_{3} 

Graph  
Dihedral symmetry  [8]  [6]  [4] 
Coxeter plane  F_{4}  A_{3}  
Graph  
Dihedral symmetry  [12/3]  [4] 
The cellfirst parallel projection of the 16cell into 3space 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 (nonregular) 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 16cell, all its edges lie on the faces of the cubical envelope.
The cellfirst perspective projection of the 16cell into 3space has a triakis tetrahedral envelope. The layout of the cells within this envelope are analogous to that of the cellfirst parallel projection.
The vertexfirst parallel projection of the 16cell into 3space 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 16cell. The closest vertex of the 16cell to the viewer projects onto the center of the octahedron.
Finally the edgefirst parallel projection has a shortened octahedral envelope, and the facefirst parallel projection has a hexagonal bipyramidal envelope.
A 3dimensional projection of the 16cell and 4 intersecting spheres (a Venn diagram of 4 sets) are topologically equivalent.


The 16cell's symmetry group is denoted B_{4}.
There is a lower symmetry form of the 16cell, called a demitesseract or 4demicube, 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 4orthotope, represented by s{2^{1,1,1}}, and Coxeter diagram: or .
With the tesseract constructed as a 44 duoprism, the 16cell can be seen as its dual, a 44 duopyramid.
Name  Coxeter diagram  Schläfli symbol  Coxeter notation  Order  Vertex figure 

Regular 16cell  {3,3,4}  [3,3,4]  384  
Demitesseract Quasiregular 16cell 
= = 
h{4,3,3} {3,3^{1,1}} 
[3^{1,1,1}] = [1^{+},4,3,3]  192  
Alternated 44 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 4orthotope  =  s{2^{1,1,1}}  [2,2,2]^{+} = [2^{1,1,1}]^{+}  8  
4fusil  
{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 16cell. It has 8 vertices, and 8 3edges.^{[22]}^{[23]}
The regular complex polygon, _{2}{4}_{4}, , in has a real representation as a 16cell in 4dimensional space with 8 vertices, 16 2edges, only half of the edges of the 16cell. Its symmetry is _{4}[4]_{2}, order 32.^{[24]}
In B_{4} Coxeter plane, _{2}{4}_{4} has 8 vertices and 16 2edges, 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, K_{4,4}.^{[25]} 
The regular 16cell and tesseract are the regular members of a set of 15 uniform 4polytopes with the same B_{4} symmetry. The 16cell is also one of the uniform polytopes of D_{4} symmetry.
The 16cell is also related to the cubic honeycomb, order4 dodecahedral honeycomb, and order4 hexagonal tiling honeycomb which all have octahedral vertex figures.
It belongs to the sequence of {3,3,p} 4polytopes which have tetrahedral cells. The sequence includes three regular 4polytopes of Euclidean 4space, the 5cell {3,3,3}, 16cell {3,3,4}, and 600cell {3,3,5}), and the order6 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}.