8/16/2015 Polygons Polygons are simple closed plane figures made with three or more line segments. Polygons cannot be made with any curves. Polygons are named according to their number of line segments, or sides.

A regular polygon is a polygon with all sides congruent and all angles congruent such as equilateral triangle, square, regular pentagon, regular hexagon, ………..

8/16/2015 Regular Polygons Namenumber of sidesAngle at vertex triangle360 quadrilateral490 pentagon5108 hexagon6120 heptagon7~128.6 octagon8135 nonagon9140 decagon gon11~147.3 dodecagon12150 n-gonn{(n-2)180}/n

A polyhedron is called a regular polyhedron or Platonic Solids if the faces of the polyhedron are congruent regular polygonal regions, and if each vertex is the intersection of the same number of edges. Polyhedra is the plural for polyhedron.

Relationship Between Polygons and Polyhedrons A polyhedron and polygon share some of the same qualities. A regular polyhedron’s face is the shape of a regular polygon. For example: A tetrahedron has a face that is an equilateral triangle. This means that every face that makes the tetrahedron is an equilateral triangle. Around all the vertices and every edge is the same equilateral triangle.

Relationship Between Polygons and Polyhedrons A polyhedron is made of a net which is basically like a layout plan. It is flat and made of all the faces that you will see on the polyhedron. For example: A cube has six faces all of them are squares. When you open the cube up and lay it out flat you see all of the six squares that make up the cube.

Platonic Solids Tetrahedron

Platonic Solids Octahedron Tetrahedron

Platonic Solids Octahedron Tetrahedron Icosahedron

Platonic Solids Cube Octahedron Tetrahedron Icosahedron

Platonic Solids Hexahedron Octahedron Dodecahedron Tetrahedron Icosahedron ~ There are only five platonic solids ~

Five “Regular” Polyhedra

Platonic solids were known to humans much earlier than the time of Plato. There are carved stones (dated approximately 2000 BC) that have been discovered in Scotland. Some of them are carved with lines corresponding to the edges of regular polyhedra.

Icosahedral dice were used by the ancient Egyptians.

Ancient Roman Dice ivory stone

Evidence shows that Pythagoreans knew about the regular solids of cube, tetrahedron, and dodecahedron. A later Greek mathematician, Theatetus ( BC) has been credited for developing a general theory of regular polyhedra and adding the octahedron and icosahedron to solids that were known earlier.

Nets of Platonic Solids: Nets of Platonic Solids:

The name “Platonic solids” for regular polyhedra comes from the Greek philosopher Plato ( BC) who associated them with the “elements” and the cosmos in his book Timaeus. “Elements,” in ancient beliefs, were the four objects that constructed the physical world; these elements are fire, air, earth, and water. Plato suggested that the geometric forms of the smallest particles of these elements are regular polyhedra. Fire is represented by the tetrahedron, air the octahedron, water the icosahedron, cube the earth, and the almost-spherical dodecahedron the universe.

Harmonices Mundi Johannes Kepler Symbolism from Plato: Octahedron = air Tetrahedron = fire Cube = earth Icosahedron = water Dodecahedron = the universe

We define the dual of a regular polyhedron to be another regular polyhedron, which is formed by connecting the centers of the faces of the original polyhedron

Among the most important of M.C. Escher's works from a mathematical point of view are those dealing with the nature of space itself. His woodcut Three Intersecting Planes is wonderful example of his work with space because it exemplifies the artist’s concern with the dimensionality of space, and with the mind's ability to discern three-dimensionality in a two-dimensional representation. Escher’s Work with Polygons and Polyhedrons

M.C. Escher ( ) Stars, 1948

M.C. Escher Double Planetoid, 1949

M.C. Escher Waterfall, 1961

M.C. Escher Reptiles, 1943

Cube with Ribbons (lithograph, 1957)

Euler’s Formula and Platonic Solids FacesVerticesEdges Tetrahedron4 triangular Hexahedron (Cube)6 square Octahedron8 triangular Dodecahedron12 pentagonal Icosahedron20 triangular,

Leonhard Euler Euler = “Oiler”

See the fabulous TOOL for investigating Platonic Solids Platonic Solids Rock Video Platonic Solids on Wikipedia Solids on Wikipedia Platonic Solids - Wolfram Math World Platonic Solids - Wolfram Math World Math is Fun Platonic Solids Math is Fun Platonic Solids Interactive Models of Platonic and Archimedean Solids Interactive Models of Platonic and Archimedean Solids

Mathematics Enclyclopedia Platonic Solids Platonic Solids The Mathematical Art of M.C. Escher Polyhedra Platonic Solids and Plato's Theory of Everything The Geometry Junkyard