I thought I would start adding content and discussions about Scientific, Spiritual, and Esoteric subject matter pertaining to my art work, One of the things I am going to be addressing, is sacred Geometry and how it appears within Art, Music, Movement. This first post is just to get some information down about Platonic Solids as a foundation and building blocks of Geometry.
Assignment to the elements in Kepler’s Mysterium Cosmographicum
Platonic solid
From Wikipedia, the free encyclopedia
JIn three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent (identical in shape and size) regular (all angles equal and all sides equal) polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria:
| Tetrahedron | Cube | Octahedron | Dodecahedron | Icosahedron |
| Four faces | Six faces | Eight faces | Twelve faces | Twenty faces |
(Animation) (3D model) | (Animation) (3D model) | (Animation) (3D model) | (Animation) (3D model) | (Animation) (3D model) |
Classification
The classical result is that only five convex regular polyhedra exist. 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.
Geometric proof
{3,3} Defect 180° | {3,4} Defect 120° | {3,5} Defect 60° | {3,6} Defect 0° |
{4,3} Defect 90° | {4,4} Defect 0° | {5,3} Defect 36° | {6,3} Defect 0° |
| A vertex needs at least 3 faces, and an angle defect. A 0° angle defect will fill the Euclidean plane with a regular tiling. By Descartes’ theorem, the number of vertices is 720°/defect. |
The following geometric argument is very similar to the one given by Euclid in the Elements:
- Each vertex of the solid must be a vertex for at least three faces.
- At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be less than 360°. The amount less than 360° is called an angle defect.
- The angles at all vertices of all faces of a Platonic solid are identical: each vertex of each face must contribute less than 360°/3 = 120°.
- Regular polygons of six or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. For these different shapes of faces the following holds:
- Triangular faces: Each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
- Square faces: Each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.
- Pentagonal faces: Each vertex is 108°; again, only one arrangement of three faces at a vertex is possible, the dodecahedron.
Altogether this makes 5 possible Platonic solids.
I just Took this little excerpt from Wikipedia, follow thier links to look a little more deeply, was this helpful? would you like to see More like this? I’ll be getting a discussion page going as well, as we start to delve deeper, and look at the relationships between geometry and the Ethos
