The Standard Model and Polytope Shadows

In order to classify the hundreds of - then thought to be elementary - hadrons discovered in collider experiments during the ‘particle-zoo’ era in the 1960s, Gell-Mann recognized that all the known hadrons can be organized in highly geometrical patterns, like triangles and hexagons. Most crucially, he realized that all those shapes could be mathematically understood if the hadrons were not elementary, but rather composed of ‘quarks’! Gell-Mann’s work led to the current formulation of the Standard Model of Particle Physics, which, at its most fundamental level, is nothing but a statement about symmetries, and those can be represented as geometrical shapes. This talk is all about the story of our ongoing attempts to apply Gell-Mann's spirit to today’s Standard Model. After revising his original work and the connection between symmetries and geometrical shapes, we will first focus on the electroweak sector. Then, we will discuss how we can go beyond that by including colour in this context, and - hopefully - the local Lorentz symmetry associated with gravity.