geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...

defines architectonic and catoptric tessellations as the uniform tessellations (or

honeycombs A honeycomb is a mass of hexagonal prismatic cells built from beeswax by honey bees in their nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. H ...

) of Euclidean 3-space with prime space groups and their

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, ...

, as three-dimensional analogue of the Platonic, Archimedean, and Catalan tiling of the plane. The singular

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

of an architectonic tessellation is the dual of the

cell Cell most often refers to: * Cell (biology), the functional basic unit of life * Cellphone, a phone connected to a cellular network * Clandestine cell, a penetration-resistant form of a secret or outlawed organization * Electrochemical cell, a de ...

of the corresponding

catoptric Catoptrics (from ''katoptrikós'', "specular", from ''katoptron'' "mirror") deals with the phenomena of reflected light and image-forming optical systems using mirrors. A catoptric system is also called a ''catopter'' (''catoptre''). Histor ...

tessellation, and vice versa. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual. There are other uniform honeycombs constructed as gyrations or prismatic stacks (and their duals) which are excluded from these categories.

Enumeration

The pairs of architectonic and catoptric tessellations are listed below with their

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

. These tessellations only represent four symmetry

space group In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that ...

s, and also all within the

cubic crystal system In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties o ...

. Many of these tessellations can be defined in multiple symmetry groups, so in each case the highest symmetry is expressed.

Vertex Figures

The vertex figures of all architectonic honeycombs, and the dual cells of all catoptric honeycombs are shown below, at the same scale and the same orientation: Architectonic Vertex Figures and Dual Cells

Symmetry

These four symmetry groups are labeled as:

Enumeration

Vertex Figures

Symmetry

References

Further reading