The Platonic Solids

The Five Platonic Solids

The transparent figure that Fra Luca Paccioli,in the HISTORY page, is studying is a composite of the platonic solids. The 5 Platonic solids are illustrated above with some of their properties. Euclid, 300 BCE and the Ancient Greeks, in their love for geometry, called these five solids, the atoms of the Universe. In the same way that we today believe that all matter, is made up of combinations of atoms so the Ancient Greeks (also) believed that all physical matter is made up of the atoms of the Platonic Solids and that all matter also has a mystical side represented by their connection with earth, air, fire, water and aether. Similar to our modern atom which shows a nucleus surrounded by electrons in orbits creating spheres of energy, the Greeks felt that these Platonic solids also have a spherical property, where one Platonic Solid fits in a sphere, which alternately fits inside another Platonic Solid, again fitting in another sphere. Paccioli is studying in a transparency the way in which one solid fits inside another. The concept of one sphere fitting inside another sphere is surprisingly frequently seen in different cultures but manifesting in uniquely different ways, as will be accurately described in the author's forth coming book.
The Pythagoreans knew that there were only five regular convex solids, the tetrahedon, cube, octahedron, icosohedron and dodecahedron and each one could be accurately circumscribed by a sphere. The dodecahedron had twelve regular faces, which corresponded to the twelve signs of the Zodiac; it was therefore a symbol of the universe for the Pythagoreans. Moreover, each one of these faces is a pentagon. Euclid described these five regular solids in Book Thirteen of the Elements. They are associated with the name of Plato because of his efforts to relate them to the important entities of which he supposed the world to be made. Plato discusses them in his various dialogues.
The corners of the octohedron fit in the centre of thr cube faces. The icosohedron can be inscribed in an octohedron, so that each vertex of the former divides an edge of the latter into the Golden Proportion The icosohedron and the dodecohedron are also uniquely connected with the Golden Proportion by virtue of three intersecting golden rectangles which fit into both. In modern times it has been discovered that the shape of many of the viruses is either icosohedron or cube.

The pentagon (the five-sided figure) is closely related to the icosohedron and its complement the dodecohedron. The diagram shows clearly how the dodecahedron is made up of twelve pentagonal faces. The Icosohedron is made up of triangular faces, but also grouped in 5 as seen at the vertex where 5 triangular faces come together.
Whereas these later 2 mentioned platonic solids are seen in the 5 pointed star the remaining three platonic solids can be found in the 6 pointed star.