#Solid shapes vertices series#

Two common arguments below demonstrate no more than five Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question-one that requires an explicit construction. The classical result is that only five convex regular polyhedra exist. These coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as

The coordinates for the tetrahedron, dodecahedron, and icosahedron are given in two positions such that each can be deduced from the other: in the case of the tetrahedron, by changing all coordinates of sign ( central symmetry), or, in the other cases, by exchanging two coordinates ( reflection with respect to any of the three diagonal planes). The Greek letter φ is used to represent the golden ratio 1 + √ 5 / 2 ≈ 1.6180. He also discovered the Kepler solids, which are two nonconvex regular polyhedra.įor Platonic solids centered at the origin, simple Cartesian coordinates of the vertices are given below. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of which was that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy. The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. The six spheres each corresponded to one of the planets ( Mercury, Venus, Earth, Mars, Jupiter, and Saturn). Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of Saturn.

#Solid shapes vertices series#

In Mysterium Cosmographicum, published in 1596, Kepler proposed a model of the Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. In the 16th century, the German astronomer Johannes Kepler attempted to relate the five extraterrestrial planets known at that time to the five Platonic solids. Much of the information in Book XIII is probably derived from the work of Theaetetus. Andreas Speiser has advocated the view that the construction of the five regular solids is the chief goal of the deductive system canonized in the Elements. In Proposition 18 he argues that there are no further convex regular polyhedra. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. Įuclid completely mathematically described the Platonic solids in the Elements, the last book (Book XIII) of which is devoted to their properties.

Aristotle added a fifth element, aither (aether in Latin, 'ether' in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, '.the god used for arranging the constellations on the whole heaven'. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. in which he associated each of the four classical elements ( earth, air, water, and fire) with a regular solid. Plato wrote about them in the dialogue Timaeus c. The Platonic solids are prominent in the philosophy of Plato, their namesake.

In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.

Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. Some sources (such as Proclus) credit Pythagoras with their discovery. The ancient Greeks studied the Platonic solids extensively. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertices of the dodecahedron, and the arrangement of the knobs was not always symmetric. The Platonic solids have been known since antiquity. Assignment to the elements in Kepler's Mysterium Cosmographicum