The planned layout for the Bott clock, which I never finished sketching. The fundamental groups of the infinite orthogonal group are periodic with period 8, a fact known as Bott periodicty. This is enshrined as the large clock in the center of the image, showing the 8 different groups: $\mathbb{Z}_2, \mathbb{Z}_2, 0, \mathbb{Z}, 0, 0, 0, \mathbb{Z}$. There are many other objects in math with a similar 8-fold classification. This drawing shows how these objects are made from one another, in a Seussian factory. Some of the mathematical gadgets include:

• The Cartan classification of symmetric spaces (top right)
• The K-theory groups measuing vector bundles on spheres (bottom left)
• Clifford algebras and clifford modules (center)
• Classifications of topological quantum materials (bottom right)

A zoom in on the central core of the Bott clock, the creation of clifford algebras. We start with a vat of square roots of negative one, ajoined one by one to a vector space to form the cliffod algerbas. From the clifford algebras, we carve out the spin groups (that’s the buzz saw!). The clifford algebras pass through the representation screen and act on $\mathbb{R}^n$ by reflection. The spin group inside acts as a double cover of the orthogonal group (the two sheets form the double cover, and the orthogonal group is represented by the gimble around the central vector space). Some of the Spin groups are siphoned off, to act as spinors.

The Bott clock also connects to geometry. The spin groups are put atop manifolds as spinors, representations of the local rotation of a manifold. Using these spinors, we form differential operators called dirac operators. These take in a vector bundle with a spin structure. The indexifier has hands made of these dirac operators. It grabs a space, squeezes it out, and extracts the kernel and cokernel of the dirac operator. We give it a space, it drips out a vector bundle. Whenever the dirac operators come in familiees, we get a single vector space for each point in the family, building a vector bundle. This is the fundamental cycle, building vector bundles from other vector bundles. This machine underlies Atiyah’s K-theoretic proof of Bott periodicity.

The Wheel of dirac operators signifies the space of fredholm operators. THese are disconnected, and their connected components are indexed by the index of the Fredholm operator (the kernel minus the cokernel). The first fundamental group of each of these spaces gives the corresponding group on the Bott clock. This is discussed in the appendix of Atiyah’s book on K-theory