The Hexagonal Cell: Symmetry Beyond the Frame

Three regular hexagons meet at a point: $3\times120°=360°$. Their prisms also meet four at a time along the horizontal edges, where the dihedral angle is $90°$. The geometry closes in both directions. This gives the Infinity Cell a new exact Euclidean home.

There is another structure inside the frame. Join the hexagon’s center to its six vertices and it becomes six equilateral triangles. Each triangle has $60°$ corners, and reflections in its sides generate the triangular tiling. These are the fundamental chambers of the affine reflection group $\widetilde{A}_2$.

The hexagonal room contains six chambers. Several reflection sequences can reach the same geometric room with different internal orientations. The wireframe still agrees; an asymmetric object or colored pattern can appear in several orientations. Exact room geometry does not require every colored reflection to look identical.

The triangular prism in Mirrors has the proportions of one of these six chambers. Each cell is framed independently by the camera. Unfold shows their actual relationship, then extends the rooms through the plane and into three layers of space.

The hexagonal mirror-room construction appears in Miles Reid and Balázs Szendrői’s Geometry and Topology, §6.6.

The Lift view follows an exact inclusion of three root systems:

$A_2 \subset D_3 \subset D_4.$

The six roots of $A_2$ form a regular hexagon. Six more vertices complete the cuboctahedron, the twelve-neighbor arrangement in a face-centered cubic lattice. Twelve more complete the regular four-dimensional 24-cell. The same original hexagon belongs to every stage.

One coordinate rule produces the whole structure. In four dimensions take every vector with two nonzero coordinates, each equal to $+1$ or $-1$. These are the twenty-four roots of $D_4$. Keep only those with fourth coordinate zero to obtain $D_3$. Also require the first three coordinates to sum to zero to obtain $A_2$. Joining pairs at distance $\sqrt{2}$ gives, respectively, 96, 24, and 6 edges.

The 4D rotation turns the two directions perpendicular to the original hexagon. Its six vertices therefore stay fixed while the surrounding structure moves. The drawing is a projection: apparent edge lengths vary, but the lengths in the original geometry remain equal. The transition reveals additional vertices at their exact positions.

The 24-cell is a regular four-dimensional polytope made of twenty-four octahedral cells. This dimensional extension is a separate construction from the Euclidean mirror room. Both begin with an exact hexagon and expose a larger order in which it participates.

John Baez’s Lie Theory Through Examples develops the connections between these root systems, lattices, and regular polytopes.

Ordinary diamond has cubic crystal symmetry and a tetrahedral bonding network. Its structure includes puckered six-membered rings. Cubic and hexagonal diamond retain tetrahedral coordination while arranging related layers differently.

A regular hexagon can be folded into a chair-shaped ring while keeping every edge the same length. At a particular fold, its bond angles become the tetrahedral angle $\arccos(-1/3)\approx109.47°$. The top view remains hexagonal while a tilted view reveals the alternating heights. This is a physical branch of the same geometric question: how can one exact arrangement belong to a larger structure?

Diamond’s hardness reflects its bonding framework; it does not rank the crystal’s symmetry above other shapes. The crystal structure is also distinct from the close-packed lattice associated with the cuboctahedron in Lift.

See Murri and colleagues, Quantifying hexagonal stacking in diamond.