The Hexadecachoron: The Sixteen-Cell

The 4D cross-polytope and dual of the tesseract—16 tetrahedral cells forming the spikiest regular polychoron.

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Schläfli Symbol
{3,3,4}
Cells
16 tetrahedra
Faces
32 triangles
Vertices
8

Understanding the 16-Cell

The hexadecachoron, commonly called the 16-cell, is the four-dimensional cross-polytope—the dual of the tesseract. Where the tesseract extends cubic symmetry to 4D, the 16-cell extends octahedral symmetry. With 16 tetrahedral cells meeting in sharp points, it's the "spikiest" of the regular polychora.

The Cross-Polytope Pattern

The 16-cell follows a pattern that works in any dimension:

  • 1D: 2 points at (±1) — a line segment
  • 2D: 4 points at (±1, 0) and (0, ±1) — a square
  • 3D: 6 points at (±1, 0, 0), (0, ±1, 0), (0, 0, ±1) — an octahedron
  • 4D: 8 points at (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1) — a 16-cell

Each vertex lies on a coordinate axis, at distance 1 from the origin.

The 16 Tetrahedral Cells

The 16-cell is bounded by 16 tetrahedral cells. Each tetrahedron uses 4 vertices that include:

  • 2 opposite pairs (one pair from each of two coordinate axes)
  • No two vertices from the same axis

Duality with the Tesseract

The 16-cell and tesseract are duals:

  • The 16-cell has 8 vertices; the tesseract has 8 cells
  • The 16-cell has 16 cells; the tesseract has 16 vertices
  • Edges (24 each) and faces (32 vs 24) swap roles

If you place a point at the center of each tetrahedral cell of a 16-cell and connect adjacent ones, you get a tesseract. And vice versa.

Mathematical Structure

The Schläfli Symbol {3,3,4}

The symbol {3,3,4} encodes the 16-cell:

  1. {3} = Faces are triangles
  2. {3,3} = Cells are tetrahedra (3 triangles at each vertex)
  3. {3,3,4} = Four tetrahedra meet around each edge

Compare to the tesseract {4,3,3}: the 16-cell's symbol is the reverse, reflecting their duality.

Symmetry: The B₄ Group

The 16-cell shares the tesseract's B₄ symmetry group (384 elements). Both are part of the hypercube/orthoplex family, and their symmetries are interchangeable.

Coordinates

The 8 vertices of a 16-cell with circumradius 1:

(±1, 0, 0, 0)
(0, ±1, 0, 0)
(0, 0, ±1, 0)
(0, 0, 0, ±1)

These are the simplest possible vertex coordinates for a regular polychoron.

Measurements

For a 16-cell with edge length 1:

Measure Value
Circumradius1/√2 ≈ 0.707
Inradius1/4
Hypervolume1/6
Surface volume8/3 × √2 ≈ 3.771
Dihedral anglearccos(1/3) ≈ 70.53°

The 16-cell has a smaller circumradius than the tesseract, making it more "compact."

Unique Properties

Tiling 4D Space

Like the tesseract, the 16-cell can tile 4D Euclidean space. The 16-cell honeycomb is a regular tiling where 24 cells meet at each vertex and 8 cells surround each edge. This is one of only three regular 4D honeycombs.

The Demihypercube Connection

The 16-cell is also known as the demitesseract—it can be obtained by taking alternating vertices of a tesseract. If you select half the tesseract's 16 vertices (those with an even number of minus signs in their coordinates), you get the 8 vertices of a 16-cell.

Sharp and Spiky

The 16-cell is the "sharpest" regular polychoron:

  • Its cells are tetrahedra, the simplest and most pointed cells
  • Four cells meet at each edge (the maximum for any regular polychoron)
  • Its dihedral angle (70.53°) is the smallest among the six regulars

The Rectified 16-Cell

When you rectify a 16-cell (truncate it until edge midpoints become vertices), you get the 24-cell! This remarkable fact connects three of the six regular polychora.

The Quaternion Connection

The 8 vertices of a 16-cell map beautifully to the 8 unit quaternions:

  • (±1, 0, 0, 0) ↔ ±1
  • (0, ±1, 0, 0) ↔ ±i
  • (0, 0, ±1, 0) ↔ ±j
  • (0, 0, 0, ±1) ↔ ±k

This makes the 16-cell fundamental to quaternion geometry.

Visualizing the 16-Cell

Stereographic Projection

In stereographic projection, the 16 tetrahedral cells appear as distorted tetrahedra. 4 cells cluster at each of the 8 vertex positions. The overall shape has an octahedral flavor, reflecting its cross-polytope nature.

4D Rotation Effects

During 4D rotation (especially in planes like XW or YW):

  • The "spikes" of the 16-cell seem to poke in and out
  • Tetrahedral cells flip between inner and outer positions
  • The projection oscillates between different octahedral views

Citation: The hexadecachoron (16-cell), denoted {3,3,4}, is the four-dimensional cross-polytope and the dual of the tesseract. Discovered by Ludwig Schläfli in 1852, it consists of 16 tetrahedral cells, 32 triangular faces, 24 edges, and 8 vertices located on the coordinate axes. The 16-cell can tile 4D Euclidean space and is equivalent to the demitesseract. Its 8 vertices correspond to the 8 unit quaternions.

Related Polychora