Platonic Solid Limited Number Of Platonic Polyhedra

That there are only five such three-dimensional solids is easily demonstrated. To create a vertex, at least three faces must meet at a point and the total of their angles must be less than 360°, i.e the corners of the face must be less than 360°/3=120°. The only polygons meeting these requirements are the triangle, square, and pentagon.

2 Dual polyhedra

Name and picture	Face polygon	Faces	Edges	Vertices	Faces meeting at each vertex	Symmetry group
tetrahedron ()	triangle	4	6	4	3	T_d
cube (hexahedron) ()	square	6	12	8	3	O_h
octahedron ()	triangle	8	12	6	4	O_h
dodecahedron ()	pentagonThis is an article about the geometrical shape. See The Pentagon for an article about the building near Washington, DC. See also: Pentagon (disambiguation). In geometry, a pentagon is any five-sided polygon. However, the term is commonly used to mean a re	12	30	20	3	I_h
icosahedronAn icosahedron [aiks'hidrn] noun (plural: -drons, -dra [-dr]) is a polyhedron having 20 faces. The faces of a regular icosahedron are equilateral triangles. Etymology 16th Century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], "icosa'hedra ()	triangle	20	30	12	5	I_h

Note that if you connect the centers of the faces of a tetrahedron, you get another tetrahedron. If you connect the centers of the faces of an octahedron, you get a cube, and vice versa. If you connect the centers of the faces of a dodecahedron, you get an icosahedron, and vice versa. These pairs are said to be dual polyhedraIn geometry, polyhedra are associated into pairs called duals where the vertices of one correspond to the faces of the others. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces,.

3 Origins of name

The Platonic solids are named after PlatoFor the computing technology, see PLATO System. Plato ( Greek: Platon (c. 427 BC c. 347 BC) was an immensely influential classical Greek philosopher, student of Socrates, teacher of Aristotle, writer, and founder of the Academy in Athens. Plato, who is be, who wrote about them in TimaeusTimaeus is a theoretical treatise of Plato, written circa 360 B. which conjectures on the composition of the four elements which the ancient Greeks thought made up the universe: earth, water, air, and fire. Plato conjectured each of these elements to be m. Plato learned about these solids from his friend Theaetetus. The constructions of the solids are included in Book XIII of EuclidEuclid of Alexandria ( Greek: Eukleides (circa 365 275 BC) was a Greek mathematician, now known as "the father of geometry". His most famous work is the Elements widely considered to be history's most successful textbook. Within it, the properties of geom's Elements. Proposition 13 describes the construction of the tetrahedron, proposition 14 of the octahedron, proposition 15 of the cube, proposition 16 of the icosahedron, and proposition 17 of the dodecahedron.

4 Ancient symbolism

Plato conceived the four classical elementSeveral ancient classical element ideas exist. The Greek version of these ideas persisted throughout the Middle Ages and into the Renaissance, deeply influencing European thought and culture. Classical elements in Greece The Greek classical elements are fs as atoms with the geometrical shapes of four of the five platonic solids that had been discovered by the Pythagoreans (in the TimaeusTimaeus is a theoretical treatise of Plato, written circa 360 B. which conjectures on the composition of the four elements which the ancient Greeks thought made up the universe: earth, water, air, and fire. Plato conjectured each of these elements to be m). These are, of course, not the true shapes of atoms; but it turns out that they are some of the true shapes of packed atoms and molecules, namely crystals: The mineral salt sodium chlorideProperties General Name Sodium chloride Chemical formula Na Cl Appearance White or clear solid CAS-number 7647-14-5 Physical Formula weight 58. 4 amu Melting point 1074 K (801 °C) Boiling point 1738 K (1465 °C) Density 2. 2 ×103 kg/ m3 Crystal structure f occurs in cubic crystals, fluorite ( calcium floride ) in octahedra, and pyrite in dodecahedra (see uses below).

This concept linked fire with the tetrahedron, earth with the cube, air with the octahedron and water with the icosahedron. There was logical reasoning behind these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and breaks when picked up, in stark difference to the smooth flow of water.

The fifth Platonic Solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven" (Timaeus 55). He didn't really know what else to do with it. Aristotle added a fifth element, aithêr (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.

5 Other symbolism

Historically, Johannes Kepler followed the custom of the Renaissance in making mathematical correspondences, and identified the five platonic solids with the five planets – Mercury, Venus, Mars, Jupiter, Saturn which themselves represented the five classical elements.

6 Inscribed Platonic polyhedra

When the Platonic polyhedra are inscribed in a sphere, they occupy the following percentages of that sphere's volume:

Tetrahedron: 12.2518%
Cube: 36.7553%
Octahedron: 31.8310%
Dodecahedron: 66.4909%
Icosahedron: 60.5461%

Note that, despite the common assumption, the dodecahedron occupies significantly more of the sphere's volume than the apparently more spherical icosahedron. The corners of the dodecahedron are less sharp than the corners of the icosahedron, and therefore fit closer to the circumscribing sphere.

7 Uses

The shapes are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as D followed by the number of faces (d8, d20 etc.).

The tetrahedron, cube, and octahedron, are found naturally in crystal structures. The dodecahedron is combinatorially identical to the pyritohedron (in that both have twelve pentagonal faces), which is one of the possible crystal structures of pyrite. However, the pyritohedron is not a regular dodecahedron, but rather has the same symmetry as the cube.

8 External links

Stella: Polyhedron Navigator Tool for exploring polyhedra
Paper Models of Polyhedra Many links
The Uniform Polyhedra
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
London South Bank University Water structure and behavior
Book XIII of Euclid's Elements.

Platonic solids

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1 Limited number of Platonic polyhedra