The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|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{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {
p,q,r} in four dimensions|ps=; 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.}} within the same radius. The 4-simplex (5-cell) is the 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.|name=polytopes ordered by size and complexity|group=}} 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.
Coordinates 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 . 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). These planes are
completely orthogonal. 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. 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 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. The 6 orthogonal central planes of the 16-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an
octahedron with 3 orthogonal great squares.
Rotations Rotations in 4-dimensional Euclidean space can be seen as the composition of two 2-dimensional rotations in
completely orthogonal planes. 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). 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). 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. 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. 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.
Constructions Octahedral dipyramid 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. 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), and three more octahedra. 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 parallels, and the 16-cell is the simplest regular polytope in which they occur.
Clifford parallelism of objects of more than one dimension (more than just curved
lines) emerges here and occurs in all the subsequent 4-dimensional regular polytopes, where it can be seen as the defining relationship
among disjoint concentric regular 4-polytopes and their corresponding parts. It can occur between congruent (similar) polytopes of 2 or more dimensions. 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. 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. 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). If the regular 16-cell has unit radius edge and edge length 𝒍 = \sqrt{2}, its characteristic 5-cell's ten edges have lengths \sqrt{\tfrac{2}{3}}, \sqrt{\tfrac{1}{2}}, \sqrt{\tfrac{1}{6}} around its exterior right-triangle face (the edges opposite the
characteristic angles 𝟀, 𝝉, 𝟁), plus \sqrt{\tfrac{3}{4}}, \sqrt{\tfrac{1}{4}}, \sqrt{\tfrac{1}{12}} (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the
characteristic radii of the regular tetrahedron), plus 1, \sqrt{\tfrac{1}{2}}, \sqrt{\tfrac{1}{3}}, \sqrt{\tfrac{1}{4}} (edges which are the characteristic radii of the regular 16-cell). The 4-edge path along orthogonal edges of the orthoscheme is \sqrt{\tfrac{1}{2}}, \sqrt{\tfrac{1}{6}}, \sqrt{\tfrac{1}{4}}, \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 , bounded by three eight-edge circular paths of different colors, cut and laid out flat in 3-dimensional space. It contains an
isocline axis (not shown), a helical circle of circumference 4𝝅 that twists through all four dimensions and visits all 8 vertices. The two blue-blue-yellow triangles at either end of the cut ring are the same object. A 16-cell can be constructed (three different ways) from two
Boerdijk–Coxeter helixes of eight chained tetrahedra, each bent in the fourth dimension into a ring.{{Failed verification|date=April 2025|reason=As an initial matter, Coxeter 1970 never uses the symbol [3,3,4]_4. The following discussion assumes that {3,3,4}_4 was meant. Section 11 of Coxeter 1970 describes that {3,3,4}_4 is a honeycomb that is generated from the 16-cell {3,3,4} by identifying antipodal points on the 3-sphere, i.e., it is a honeycomb in elliptic 3-space that is topologically different from a 16-cell. It has half the cells (8), faces (16), edges (12), and vertices (4) as a 16-cell (see Table 2 in Coxeter 1970). Coxeter 1970 does not state anywhere that the 16-cell is composed of two rings of eight tetrahedral cells and it does not follow from the comment in the source immediately before this note.}} 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. The eight-cell ring of tetrahedra contains three
octagrams of different colors, eight-edge circular paths that wind twice around the 16-cell on every third vertex of the octagram. The orange and yellow edges are two four-edge halves of one octagram, which join their ends to form a
Möbius strip. Thus the 16-cell can be decomposed into two cell-disjoint circular chains of eight tetrahedrons each, four edges long, one spiraling to the right (clockwise) and the other spiraling to the left (counterclockwise). The left-handed and right-handed cell rings fit together, nesting into each other and entirely filling the 16-cell, even though they are of opposite chirality. 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. Three eight-edge paths (of different colors) spiral along each eight-cell 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 eight-edge path (of various lengths) join their ends, forming a Möbius strip eight edges long along its single-sided circumference of 4𝝅, and one edge wide. The six four-edge halves of the three eight-edge paths each make four 90° angles, but they are
not the six orthogonal great squares: they are open-ended squares, four-edge 360° helices whose open ends are
antipodal vertices. The four edges come from four different great squares, and are mutually orthogonal. Combined end-to-end in pairs of the same
chirality, the six four-edge paths make three eight-edge Möbius loops,
helical octagrams. Each octagram is both a
Petrie polygon of the 16-cell, and the helical track along which all eight vertices rotate together, in one of the 16-cell's distinct isoclinic
rotations. Each eight-edge helix is a
skew octagram {8/3} that
winds three times around the 16-cell and visits every vertex before closing into a loop. Its eight edges are chords of an
isocline, a helical arc on which the 8 vertices circle during an isoclinic rotation. All eight 16-cell vertices are apart except for opposite (antipodal) vertices, which are apart. A vertex moving on the isocline visits three other vertices that are apart before reaching the fourth vertex that is away. The eight-cell ring is
chiral: there is a right-handed form which spirals clockwise, and a left-handed form which spirals counterclockwise. The 16-cell contains one of each, so it also contains a left and a right isocline; the isocline is the circular axis around which the eight-cell ring twists. Each isocline visits all eight vertices of the 16-cell. Each eight-cell 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 eight-edge paths. Because there are three pairs of completely orthogonal great squares, there are three congruent ways to compose a 16-cell from two eight-cell rings. The 16-cell contains three left-right pairs of eight-cell rings in different orientations, with each cell ring containing its axial isocline. Each left-right pair of isoclines is the track of a left-right pair of distinct isoclinic rotations: the rotations in one pair of completely orthogonal invariant planes of rotation. At each vertex, there are three great squares and six octagram isoclines that cross at the vertex and share a 16-cell axis chord.
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. \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 ==