SLIDE 1
Jess Armstrong Erica Freehoff
SLIDE 2 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
SLIDE 3
SLIDE 4 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
SLIDE 5
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
SLIDE 6
3 triangles = 180° 4 triangles = 240° 5 triangles = 300° 6 triangles = 360°(not possible, flat surface)
SLIDE 7
3 squares = 270° 4 squares = 360° (not
possible, flat surface)
3 pentagons = 324° 4 pentagons = way
too much!
SLIDE 8
Regular hexagon angles measures 120°,
3 would be 360° too much!
Other regular polygons would have
angles measuring over 120° too much!
SLIDE 9
Polyhedrons constructed of regular polygons
but not necessarily all the same kind
Johannes Kepler proves that there are only 13
SLIDE 10 Crystalline structures of chemical compounds
- Tetrahedral- silicates
- Hexahedral- lead ore and rock salt
- Octahedral – fluorite
- Dodecahedral- garnet
- Icosahedral (truncated) – “buckyball”
SLIDE 11 500-400 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.
