Jess Armstrong Erica Freehoff Two dimensional polygons Regular - PowerPoint PPT Presentation

Jess Armstrong Erica Freehoff

 Two dimensional polygons ◦ Regular polygon- all sides and all angles congruent ◦ Infinitely many can be constructed  Three dimensional polyhedron ◦ Regular polyhedron- all faces are congruent regular polygons and all its vertices are similar ◦ Only 5 exist

 Shapes and symmetry important to Pythagoreans  Plato’s Timaeus represented 5 elements of physical world ◦ Fire - tetrahedron ◦ Earth - hexahedron ◦ Air - octahedron ◦ Water – icosahedron ◦ Universe- dodecahedron  Proved by Euclid in Elements

 At least 3 polygonal faces must meet to form a vertex  The situation at each vertex is the same  Sum of face angles at each vertex must be <360°  Angle sum at each vertex must divide evenly into the number of faces meeting at it

 3 triangles = 180°  4 triangles = 240°  5 triangles = 300°  6 triangles = 360°(not possible, flat surface)

 3 squares = 270°  3 pentagons = 324°  4 squares = 360° (not  4 pentagons = way possible, flat surface) too much!

 Regular hexagon angles measures 120°, 3 would be 360°  too much!  Other regular polygons would have angles measuring over 120°  too much!

 Polyhedrons constructed of regular polygons but not necessarily all the same kind  Johannes Kepler proves that there are only 13

 Crystalline structures of chemical compounds ◦ Tetrahedral- silicates ◦ Hexahedral- lead ore and rock salt ◦ Octahedral – fluorite ◦ Dodecahedral- garnet ◦ Icosahedral (truncated) – “ buckyball ”

 500-400 BC Pythagoreans  350 BC Plato, Timaeus  250 BC Archimedes  1600 AD Johannes Kepler Berlinghoff, William and Fernando Gouvea . “In Perfect Shape: The Platonic Solids.” Math through the Ages. Farmington: Oxton House, 2004. 163-168. Dunham, William. “Euclid and the Infinitude of Primes.” Journey Through Genius. New York: Penguin Books, 1990. 78-80.