The Platonic solids and fundamental tests of quantum mechanics Armin Tavakoli and Nicolas Gisin Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland The Platonic solids is the name tra- history of these solids and then use them to de- ditionally given to the five regular con- rive simple Bell inequalities tailored to be max- vex polyhedra, namely the tetrahedron, imally violated for measurement settings point- the octahedron, the cube, the icosahedron ing towards the vertices of the Platonic solids. and the dodecahedron. Perhaps strongly In this way, we connect beautiful mathematics boosted by the towering historical influ- with foundational quantum physics. However, ence of their namesake, these beautiful these Platonic Bell inequalities do not distinguish solids have, in well over two millennia, themselves with regard to experimental friendli- transcended traditional boundaries and ness: quantum theory predicts that their viola- entered the stage in a range of disciplines. tions are less robust to noise than the much sim- Examples include natural philosophy and pler Clauser-Horne-Shimony-Holt (CHSH) Bell mathematics from classical antiquity, sci- inequality [1]. In fact, Platonic Bell inequali- entific modeling during the days of the Eu- ties require more measurement settings - as many ropean scientific revolution and visual arts as the number of vertices of the platonic solid - ranging from the renaissance to moder- than the CHSH Bell inequality, which requires nity. Motivated by mathematical beauty only the absolute minimum of two settings per and a rich history, we consider the Platonic side. We also construct Bell inequalities tailored solids in the context of modern quantum to another class of elegant polyhedra, namely the mechanics. Specifically, we construct Bell Archimedean solids, i.e. the semi-regular polyhe- inequalities whose maximal violations are dra. In particular we consider the famous Buck- achieved with measurements pointing to yball, a polyhedron which corresponds to the the vertices of the Platonic solids. These carbon-60 molecule used in the first molecular in- Platonic Bell inequalities are constructed terferometer [2] , which requires even more mea- only by inspecting the visible symmetries surement settings. However, we find that these of the Platonic solids. We also construct Bell inequalities also do not offer notable experi- Bell inequalities for more general poly- mental advantages. Finally, we depart from Bell hedra and find a Bell inequality that is inequalities motivated by mathematical beauty more robust to noise than the celebrated and instead focus our research on finding experi- Clauser-Horne-Shimony-Holt Bell inequal- mentally friendly Bell inequalities: starting from ity. Finally, we elaborate on the tension the Buckyball we iteratively search for noise ro- between mathematical beauty, which was bust Bell inequalities. This leads us to a Bell our initial motivation, and experimental inequality that is somewhat more noise tolerant arXiv:2001.00188v3 [quant-ph] 4 Jun 2020 friendliness, which is necessary in all em- than the CHSH Bell inequality. However, it is pirical sciences. remarkably inelegant. We conclude with a dis- cussion of the danger for theoretical physics to become - and remain - too focused on mathe- 1 Introduction matical beauty [3] at the expense of developing connections with experiments. Which physicist has never been attracted by mathematical beauty? And what is more beau- tiful than the Platonic solids; the five regular polyhedra in our three-dimensional space (see Fig.1)? Here, we first present the fascinating Accepted in Quantum 2020-06-04, click title to verify. Published under CC-BY 4.0. 1 tries to a tangibly sacred status. The perhaps most famous example of such metaphysical be- liefs is due to the Pythagoreans2 [8]. Their ideas of sacred geometries were influential, notably also on key figures such as Plato in the fifth century BC. In The Republic, Plato writes that "geome- try will draw the soul towards truth, and create the spirit of philosophy" [9]. In Timaeus, Plato makes concrete the link between geometry and natural philosophy; he discusses the five regu- lar polyhedra, i.e. the polyhedra whose vertices Figure 1: The five Platonic solids inscribed in spheres. are identical and whose faces are identical regu- From left to right: the tetrahedron, the octahedron, the lar polygons, namely the tetrahedron, the octahe- cube, the icosahedron and the dodecahedron. dron, the cube, the icosahedron and the dodec- ahedron. Today, these five solids are known as 2 A brief history of the Platonic solids the Platonic solids (see Fig.1). Plato assigned in arts, philosophy and science four of the solids to the four classical elements thought to be the fundamental form of all mat- This section provides a broader context for the ter; the tetrahedron to fire, the octahedron to air, Platonic solids. Readers interested exclusively in the cube to earth and the icosahedron to water. Bell inequalities may jump to the next section. To the remaining fifth solid, Plato left the follow- The ancient Greek civilisations laid the founda- ing mysterious comment [10] "A fifth regular solid tions of western natural philosophy. The develop- still exists, namely the dodecahedron, which does ment of the latter is permeated by a fascination not form the element of any substance; but God for geometry. The magnum opus of Greek ge- used it as a pattern for dividing the zodiac into its ometry, Euclid’s Elements, remained a standard twelve signs." Later, his pupil Aristotle added a textbook until the 20th century [4]. First printed fifth element to the original four elements, namely in Venice in 1482 as one of the earliest mathemat- the aether3. It historically became associated to ics books set in type, it has since been re-printed the dodecahedron, perhaps due to its relevance in at least a thousand editions1 and is certainly for the golden ratio. From a purely mathematical the most influential mathematical work in history standpoint, the Platonic solids were the focus of [5]. Geometry allowed the early natural philoso- the 13’th book of Euclid’s Elements which stud- phers to describe, understand and make predic- ies their construction and their proportions when tions about, the physical world. In the sixth inscribed in a sphere. century BC, Thales of Miletus, often hailed as The Platonic solids can be appreciated by mod- the first scientific philosopher in western civili- ern mathematicians for their appealing geometric sation, likely used his knowledge of geometry to properties, by modern natural scientists for their measure the height of the pyramids of Egypt [6]. occurrence in nature, historical scientific models Centuries later, in the Hellenistic period, Eratos- and metaphysical ideas, and by a broader modern thenes accurately calculated the circumference of audience for their historical appearance in west- the Earth and Hipparcus discovered the preces- ern visual arts and natural philosophy, as well sion of the equator. Archimedes’ geometry led as their sheer beauty. It appears reasonable to him to the Law of the Lever [7], still taught to every pupil in physics class. 2For instance, the number three was an ideal number Geometry was often ascribed a deeper mean- as it was the number of vertices in a triangle, which was a symbol of Apollo. The number ten was termed a perfect ing, beyond pure mathematics and its applica- number due to the number of vertices in a geometry called tions. This entails attributing spiritual, religious a tetractys. The number was therefore honoured by the or philosophical meaning to certain proportions, Pythagoreans not gathering in groups of more than ten planar shapes and solids, elevating the geome- people. 3Aether theories persisted in science until the strong 1Ref. [5], authored in 1968, suggests that the Elements negative evidence put forward by the Michelson-Morley is only outdone in number of editions by the Bible. experiment, performed in 1887. Accepted in Quantum 2020-06-04, click title to verify. Published under CC-BY 4.0. 2 say that the historical interest in the Platonic are often the dominating mineral in various forms solids was substantially aided by the fact they of soil. Perhaps, had Plato ascribed the tetrahe- were so strongly endorsed by a character as ti- dron rather than the cube as the manifestation tanic as Plato. of earth, his metaphysical ideas might have bet- ter withstood the test of time. Almost two millennia after Plato, the main- tained appreciation for the Platonic solids could for instance be seen in Luca Pacioli’s mathemat- 3 A brief history of Bell inequalities ics book De Divina Proportione. Published in 1509, it spends its first section motivating the di- This section provides a non-technical introduc- vinity of the golden ratio; in particular by em- tion to Bell inequalities. Readers interested phasising that the golden ratio appears in the mainly in the technical considerations may pro- dodecahedron, which is a representation of the ceed immediately to the next section. aether [11]. The book’s lasting success even out- Modern science, with its emphasis on empiri- side mathematics circles may in part be due to cism, has for long left behind ideas of Euclidean its masterful illustrations of the Platonic solids geometry being fundamental to describing na- and various other geometries, in drawings signed ture. The 19th century saw the development of Leonardo da Vinci. In fact, the works of many curved (non-euclidean) geometry4 which in the artists feature the Platonic solids; ranging from early 20th century found a fundamental role in the renaissance mosaics in the cathedral of San Einstein’s theory of gravity. The 20th century Marco in Venice to the 20th century works of also brought with it the perhaps most radical Maurits Escher, who incidentally also kept a cov- change of scientific paradigm since the days of eted model of the nested Platonic solids in his Newton, namely the theory of quantum mechan- office [12].