# The Penrose dodecahedron and the Witting polytope are identical in CP(3)

## TL;DR

This paper shows that the Penrose dodecahedron and the Witting polytope are essentially the same in CP(3), linking different geometric structures used in quantum contextuality proofs and highlighting their diverse proof types.

## Contribution

It establishes the unitary equivalence of the Penrose dodecahedron and Witting polytope in CP(3), connecting complex and real proofs of the Kochen-Specker theorem.

## Key findings

- Penrose dodecahedron and Witting polytope are unitarily equivalent in CP(3)
- Witting polytope appears in different projective spaces with distinct proof types
- Provides complex and real proofs of the Kochen-Specker theorem using these structures

## Abstract

It is demonstrated that the set of 40 states of a spin-3/2 particle used by Zimba and Penrose to give proofs of the Kochen-Specker and Bell theorems is identical (i.e., unitarily equivalent) in CP(3) to the set of 40 rays derived from the vertices of the Witting polytope, which is a regular complex polytope in C(4). The Witting polytope actually has two different apparitions in projective spaces of different dimensions: it appears in CP(3) as the Penrose dodecahedron and in RP(7) (after an initial inflation into R(8)) as a set of rays associated with the root vectors of the Lie algebra E8. The interest of these apparitions is that they provide proofs of the Kochen-Specker theorem, but of very different types: while the proofs provided by the Penrose dodecahedron are complex (in both senses of the word), those provided by the E8 system are real and easy to grasp (being parity proofs that take no more than simple counting to verify). The different proofs it provides in different settings would seem to justify calling the Witting polytope a "quantum chameleon", and we raise (but leave unanswered) the question of whether it is the only object of this type.