16-cell

In geometry, the 16-cell is the regular convex 4-polytope with Schläfli symbol. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C₁₆, hexadecachoron, or hexdecahedroid.
It is the 4-dimensional member of an infinite family of polytopes called cross-polytopes, orthoplexes, or hyperoctahedrons which are analogous to the octahedron in three dimensions. It is Coxeter's polytope. The dual polytope is the tesseract, which it can be combined with to form a compound figure. The cells of the 16-cell are dual to the 16 vertices of the tesseract.

Geometry

The 16-cell is the second in the sequence of 6 convex regular 4-polytopes.
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 16-cell is the 4-dimensional cross polytope, which means its vertices lie in opposite pairs on the 4 axes of a Cartesian coordinate system.
The eight vertices are,,,. 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 are completely disjoint. 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, indicating that its cells are regular tetrahedra and its vertex figure is a regular octahedron. 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. 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 each forming an octahedron with 3 orthogonal great squares.

Rotations

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. Many rotations of the 16-cell can be characterized by the angle of rotation in one of its great square planes and another angle of rotation in the completely orthogonal great square 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. 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 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. 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.
The octahedron that the construction starts with has three perpendicular intersecting squares. 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. These new vertices are connected to all the octahedron's vertices, creating 12 new edges and three more squares, 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, 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 are like the opposite edges of a tetrahedron: perpendicular, but non-intersecting. They lie opposite each other, and they don't touch, but they also pass through each other like two perpendicular links in a chain. They are an example of Clifford parallel planes, and the 16-cell is the simplest regular polytope in which they occur. Clifford parallelism of objects of more than one dimension 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 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 are different tetrahedral pyramids, all based on the same characteristic irregular tetrahedron. They share the same characteristic tetrahedron and characteristic right triangle 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 has four more edges than its base characteristic tetrahedron, joining the four vertices of the base to its apex. If the regular 16-cell has unit radius edge and edge length ? =, its characteristic 5-cell's ten edges have lengths,, around its exterior right-triangle face, plus,, , plus,,, . The 4-edge path along orthogonal edges of the orthoscheme is,,,, 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

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. 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 and the other spiraling to the left. 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 ⨂ or ss, symmetry , order 64.
Three eight-edge paths spiral along each eight-cell ring, making 90° angles at each vertex. Three paths pass through each vertex. When the helix is bent into a ring, the segments of each eight-edge path join their ends, forming a Möbius strip eight edges long along its single-sided circumference of 4?, 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 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 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.