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.

[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.

[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).

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.

They also share the 24 edges, though left and right octagram helices are different eight-edge paths.

[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.