This project displays stereographically projected Bravais lattice. 

The Bravais lattices are infinite arrays of points generated by translations. Two Bravais lattices are considered identical if their symmetry groups coincide, so they classify the symmetry groups of translational symmetries. There are 14 of them in three dimensions. For more intuition about what these lattice look like, check this table on Wikipedia.

The stereographic projection is a way of projecting three-dimensional unit vectors (forming a sphere) to the plane. A point p on the sphere is mapped to the intersection of the line through p and the north pole with a plane cutting the sphere through the equator. The points in the southern hemisphere are mapped inside the equator, while the points in the northern hemisphere are mapped outside.

To draw the stereographic projection of a lattice, keep only the half of the lattice on the side of the plane containing the south pole. Project the lattice points radially on the southern hemisphere of the unit sphere, and map them to a disk in the plane through the stereographic projection. Each vector is then represented by a dot in the plane by a point whose size is inversely proportional to the distance of the corresponding lattice point to the origin.

In the pictures above, the hexagonal base of the fundamental cell lie in the stereographic plane. If instead we arrange for the stereographic plane to be perpendicular to the base of the fundamental cell, the Wulff diagrams of the same lattices look as follows.

The cubic Bravais lattice has no parameter, but comes in three discrete centering types. The type P (primitive) lattice is the standard cubic lattice. The type I lattice (body-centered) has in addition a point at the center of each cube, and the type F lattice (face-centered) has a point at the center of each face. Here are the Wulff nets of the P, I and F cubic lattices.

The rhombohedral lattice have a fundamental cell whose edges have all the same lengths, and each pair of edges intersect with the same angle alpha. Alpha parameterize the family. Here are rhombohedral lattices for alpha = 75°, 60° and 45°.

Tetragonal

The tetragonal lattice is a one-parameter family, coming both in primitive (P) and body-centered (I) types. The fundamental cell is a right prism with a square base, and the parameter is the height of the prism c relative to the side length of the base a. When they coincide, we get cubic lattices. The two series of three pictures below show lattices with c > a, c = a (cubic case) and c < a, first for the primitive type and then for the body-centered type.

The fundamental cell of orthorhombic lattices have orthogonal edges, without restrictions on their relative lengths. They come in the P (primitive), S (base-centered), I (body-centered) and F (face centered) centering types. The rows below respectively show lattices of type P, S, I and F. For each type, we show a row of three lattices whose width a, height b and depth c satisfy respectively a < b < c, c < a < b and b < c < a.

The monoclinic Bravais lattices form a family parameterized by the edge lengths a, b, and c of the fundamental cell, as well as one angle gamma. They comes in two centering types, the primitive one (P) and the base-centered (S). In the following three images, we see P-type monoclinic Bravais lattices with gamma taking the values 45°, 80° and 110°.

P-type monoclinic Bravais lattices with gamma = 65° and the edge lengths satisfying respectively a < c < b, b < a < c, c < b < a.

Same two series of three images as above, but with S-type monoclinic Bravais lattices.

The triclinic Bravais lattices are parameterized by three angles and two fundamental cell edge length ratios. They have no symmetry beyond the translations along the edges of the fundamental cell. Here are nine triclinic lattice with randomly generated parameters.