The Platonic solids is the name traditionally given to the five regular convex polyhedra, namely the tetrahedron, the octahedron, the cube, the icosahedron and the dodecahedron. Perhaps strongly boosted by the towering historical influence of their namesake, these beautiful solids have, in well over two millennia, transcended traditional boundaries and entered the stage in a range of disciplines. Examples include natural philosophy and mathematics from classical antiquity, scientific modeling during the days of the European scientific revolution and visual arts ranging from the renaissance to modernity. Motivated by mathematical beauty and a rich history, we consider the Platonic solids in the context of modern quantum mechanics. Specifically, we construct Bell inequalities whose maximal violations are achieved with measurements pointing to the vertices of the Platonic solids. These Platonic Bell inequalities are constructed only by inspecting the visible symmetries of the Platonic solids. We also construct Bell inequalities for more general polyhedra and find a Bell inequality that is more robust to noise than the celebrated Clauser-Horne-Shimony-Holt Bell inequality. Finally, we elaborate on the tension between mathematical beauty, which was our initial motivation, and experimental friendliness, which is necessary in all empirical sciences.
 Sabine Hossenfelder, Lost in Math: How Beauty Leads Physics Astray, Basic Book 2018.
 Encyclopedia of Ancient Greece, N. W. Wilson. Taylor & Francis 2010.
 A History of Mathematics, U. C. Merzbach and C. B. Boyer. John Wiley & Sons, Third edition 2011.
 A Commentary on the First Book of Euclid's Elements, Proklos Diadochos. Princeton University Press, Reprint edition 1992.
 A History of Mechanical Inventions, A. P. Usher. Harvard University Press, Revised Edition 2011.
 Measuring Heaven: Pythagoras and His Influence on Thought and Art in Antiquity and the Middle Ages, C. L. Joost-Gaugier. Cornell University Press 2007.
 The Republic, VII, Plato.
 Timaeus, Plato. Hackett Publishing Company, Second edition 2000.
 The Golden Ratio: The Story of Phi, the World's Most Astonishing Number, M. Livio. Broadway Books; Reprint edition 2003.
 The Magic Mirror of M. C. Escher, B. Ernst. Ballantine Books, 1976.
 E. Aiton, Johannes Kepler and the 'Mysterium Cosmographicum', Sudhoffs Archiv, Bd. 61, H. 2 (1977 2. QUARTAL), pp. 173-194.
 D. Monroe, Focus: Nobel Prize-Discovery of Quasicrystals, Phys. Rev. Focus 28, 14 (2011).
 History of the Parallel Postulate, F. P. Lewis, The American Mathematical Monthly, Vol. 27, No. 1. (Jan., 1920), pp. 16-23.
 N. Gisin, Quantum Chance, Springer 2014.
 J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics Vol 1, 3 pp.195-200 (1964).
 B. Hensen et. al., Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres, Nature 526, 682 (2015); L. K. Shalm et. al., Strong Loophole-Free Test of Local Realism, Phys. Rev. Lett. 115, 250402 (2015); M. Giustina et. al., Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons, Phys. Rev. Lett. 115, 250401 (2015).
 N. Gisin, Bell inequalities: many questions, a few answers, The Western Ontario Series in Philosophy of Science, pp 125-140, Springer 2009.
 Armin Tavakoli, Ingemar Bengtsson, Nicolas Gisin, and Joseph M. Renes, "Compounds of symmetric informationally complete measurements and their application in quantum key distribution", Physical Review Research 2 4, 043122 (2020).
 Armin Tavakoli, "Semi-Device-Independent Certification of Independent Quantum State and Measurement Devices", Physical Review Letters 125 15, 150503 (2020).
 H. Chau Nguyen, Sébastien Designolle, Mohamed Barakat, and Otfried Gühne, "Symmetries between measurements in quantum mechanics", arXiv:2003.12553.
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