The images below picture root systems of Lie algebras. Lie groups are the mathematical objects formalising the physical concept of a continuous symmetry. The Lie group transformations "infinitely close to the identiy" form an algebra, the Lie algebra of the Lie group. In turn a (sufficiently nice) Lie algebra can be fully characterised by a more primitive structure, its root system.

Root systems are configurations of vectors satisfying very constraining geometrical properties, which can be found on the Wikipedia page linked above. It turns out that root systems come into four infinite families labeled by integers n: An, Bn, Cn and Dn, as well as five exceptional root systems, named E6, E7, E8, F4 and G2. The integer labels the dimension of the vector space in which the vectors forming the roots system live.

It should be emphasized that there are many other mathematical objects classified by root systems, see for instance Dynkin diagram or ADE classification for more information.

One can get pictures of root systems by projecting the vectors of a given root system on a 2-dimensional plane, and by linking the tips of the vectors by edges. Usually, this is done choosing the 2-dimensional plane to be in the most symmetric configuration, which results in the rather boring pictures you can see on the Wikipedia pages linked above. But it turns out that one can get a rich variety of interesting images by simply projecting along a random 2-dimensional plane. Each projection offers a glimpse into the intricate geometry of the higher dimensional root systems.

In the pictures below, the tip of each vector are linked by an edge whose transparency is proportional to their distance (with no edge between vectors whose distance exceed a threshold). This provide a visual hierarchy into picture that would otherwise be too complex.

Root systems are among the most regular and constrained objects in mathematics. The growing chaos when projecting higher and higher dimensional root system two dimensions is quite fascinating.

The A series is associated to the special unitary group SU(n+1) and its quotients. A2 is 2-dimensional, so no projection is necessary to picture the root system as it is. The origin is at the center of the picture, and the corners of the hexagon are the six vectors in the A2 root system.

The vectors of the A3 root system form a cuboctahedron.

A4. A4 is associated to the Lie group SU(5). SU(5) is the smallest simple group containing the standard model symmetry group SU(3) x SU(2) x U(1), and gives rise to the simplest Grand Unified Theories, the Georgi-Glashow model.

A5

A6

A7

A8

A16

The B series is associated to the odd-dimensional orthogonal groups SO(2n+1). Again B2 is 2-dimensional so no need of a projection to picture it.

B3 is a cuboctahedron with vertices at the center of its square faces, which form an inscribed octahedron.

B4

B6

B16

The C series is associated to the symplectic groups Sp(n). C2 is identical to B2. 

C3 is an octahedron with vertices in the middle of each of its edges, forming an inscribed cuboctahedron (notice the duality with B3 above).

C4

C6

C7

C8

The D series is associated to the even-dimensional orthogonal groups SO(2n). D2 and D3 are isomorphic to A2 and A3 respectively. Here is D4.

D5, associated to SO(10) is the basis of another Grand Unified Theory, see the note on A4 above.

D8

D16 is associated to SO(32) famously known to be one of the possible gauge group of 10-dimensional Type I supergravity/superstring. The root lattice generated by the D16 root system is one of the two even self-dual lattices in 16-dimension, which is the reason for its appearance in one of the two flavours of the heterotic string (the other one is E8 x E8).

G2 is two dimensional, so again, no need for a projection.

F4

E6

E7

E8, the largest exceptional root system appears in many places in mathematics. The lattice it generates is the lowest-dimensional even unimodular lattice (all the scalar products between its basis vectors are even, yet the volume of its fundamental cell is 1). It is a solution to the sphere packing problem in dimension 8. The product of two E8 lattices yield an even unimodular lattice in dimension 16, providing the second flavour of the heterotic string (see the discussion about D16 above).

Look at the left picture with your right eye and vice versa to see projections of root systems to three-dimensions.

B8

C10

D16

E6

E7

E8

Dall-e's interpretation of root systems and Lie algebras