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. |