In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}.

[3] It is the 4-dimesional member of an infinite family of polytopes called cross-polytopes, orthoplexes, or hyperoctahedrons which are analogous to the octahedron in three dimensions.

[4] The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure.

[b] Hyper-tetrahedron 5-point Hyper-octahedron 8-point Hyper-cube 16-point 24-point Hyper-icosahedron 120-point Hyper-dodecahedron 600-point The 16-cell is the 4-dimensional cross polytope (4-orthoplex), which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system.

[c] 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.

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.

If the two angles happen to be the same, a maximally symmetric isoclinic rotation takes place.

The octahedron has 3 perpendicular axes and 6 vertices in 3 opposite pairs (its Petrie polygon is the hexagon).

This raises two octahedral pyramids on a shared octahedron base that lies in the 16-cell's central hyperplane.

[10] The octahedron that the construction starts with has three perpendicular intersecting squares (which appear as rectangles in the hexagonal projections).

They are an example of Clifford parallel planes, and the 16-cell is the simplest regular polytope in which they occur.

Clifford parallelism[k] 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.

[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.

[s] 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.

The regular 16-cell is subdivided by its symmetry hyperplanes into 384 instances of its characteristic 5-cell that all meet at its center.

around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),[t] plus

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.

[16][17] The two circular helixes spiral around each other, nest into each other and pass through each other forming a Hopf link.

The orange and yellow edges are two four-edge halves of one octagram, which join their ends to form a Möbius strip.

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.

Three eight-edge paths (of different colors) spiral along each eight-cell ring, making 90° angles at 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.

[o] 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.

Combined end-to-end in pairs of the same chirality, the six four-edge paths make three eight-edge Möbius loops, helical octagrams.

[v] 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.

[ab] Because there are three pairs of completely orthogonal great squares,[c] there are three congruent ways to compose a 16-cell from two eight-cell rings.

This octahedron can be divided into 8 tetrahedral volumes, by cutting along the coordinate planes.

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

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 .

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.