Platonic Solids Eloise Lardet The University of Edinburgh What are Platonic solids? History Applications You might have heard of “regular polygons”. Examples include The name “Platonic" comes from the Greek mathematician Although they may not be the elements of the Universe, as the equilateral triangle, square, regular pentagon. They have , who lived around 400 BC. He may not have been the Plato believed, the Platonic solids do appear all over nature, the same number of edges as vertices, and all angles are the first to discover all five solids, but he had a theory about the and have many uses. same. Universe based on these shapes. Many viruses have the shape of an , such as HIV. In 2D there are an infinite number of regular polygons. • There are tiny single-celled organisms called Radiolaria which But what about regular shapes in 3D? • live in the ocean. All five platonic solids can be seen in They must satisfy these conditions: Radiolaria skeletons. All the faces must be the same • The same number of faces must meet at each vertex (corner) Figure: Plato’s Elements [2] • How many polyhedra satisfy these conditions? Only 5! He believed that the Platonic solids represented the four basic These are called Platonic solids. Let’s meet them... elements (fire, , air, and water), with the last one representing the whole Universe. Figure: Icosahedral HIV virus [4] Figure: Icosahedral Radiolarian [5] Euclid was another Greek mathematician who researched Regular polyhedra are common among molecules due to Platonic solids. The last book of Euclid’s Elements is all about • their strong structures and symmetries. them. He is thought to be the first person to prove that only , tetrahedra, and octahedra can all be seen in naturally five Platonic solids exist. • occurring crystal structures.

Tetrahedron In the 16th century, Johannes 4 triangles 6 squares 6 squares had a new theory • • • 4 vertices 8 vertices 8 vertices about the Universe and the • • • 6 edges 12 edges 12 edges Platonic solids. Figure: Fool’s Gold (Pyrite) Cube [6] Figure: Fluorite Octahedron [7] • • • He imagined placing a sphere Most dice are platonic solids, due to the fairness of each face around each of the Platonic • being rolled. solids. The ratios between the sizes of these spheres were [1] Platonic Solids: , Cube, [4] HIV virus attacking cell. 3D render. Octahedron, Icosahedron, . User martynowi.cz, Shutterstock remarkably similar to the The original uploader was Cyp at English [5] Circogonia Icosahedra. Figure: Kepler’s Mysterium distances between the orbits Wikipedia. Ernst Haeckel, 1904. Kunstformen der Cosmographicum [3] Natur. Dodecahedron Icosahedron round the of the six https://commons.wikimedia.org/ wiki/Platonic_solid. https://upload.wikimedia.org/ 12 pentagons 20 triangles known planets at the time. [2] Platonic solids as elements. wikipedia/commons/0/02/ • 20 vertices • 12 vertices Originally by , Circogoniaicosahedra_ekw.jpg He outlined this theory in his book “Mysterium [6] Isolated Pyrite cubic crystal. • 30 edges • 30 edges , 1619. Cosmographicum". This theory was proven to be wrong, but it https://mathigon.org/course/ User Myriam B, Shutterstock • • . Figures: Platonic Solids [1] shows how Platonic solids have had strong influences in polyhedra/platonic [7] Piece of blue Fluorite mineral from Seilles, [3] Kepler’s Mysterium Cosmographicum. Belgique. science. Johannes Kepler, 1597. Mysterium User Myriam B, Shutterstock Cosmographicum. 1/1