Transcript regular
Platonic Solids Greek concept of Symmetry Seen in their art, architecture and mathematics Greek Geometry The most symmetric polygons are regular Regular polygons have all sides and angles congruent Ex. Regular triangle is an equilateral triangle Ex. Regular quadrilateral is a square Regular polygons exist for any number of sides Ex. Pentagon, Hexagon, Heptagon, etc. Whoa its in 3-D!! A polyhedron is regular if Its faces are congruent regular polygons Its vertices are similar A cube is a regular polyhedron Euclid says… Only a few regular polyhedra exist In contrast there are infinite regular polygons Actually there are exactly five types of regular polyhedra And they are… Tetrahedron 4 Faces Each face is a triangle Associated by Pythagoreans with the element fire Octahedron 8 faces Each face is a triangle Associated with air Icosahedron 20 faces – associated with water Each face is Drumroll please…………….. A TRIANGLE!! Hexahedron 6 faces- associated with earth The sides are squares, this is a cube Associated with water Dodecahedron 12 faces Each face is a pentagon To the Pythagoreans, it represents the universe Vertices At least three polygonal faces must meet at any vertex, forming a peak. The sum of these angles is < 360 degrees Otherwise the faces overlap or form a flat surface In a regular polyhedra the situation at one vertex is the same as at any other 6 triangles, 3 hexagons = 360 degrees Archimedean Solids Not so Regular Allow different types of regular solids to be used to create a 3-D object Archimedes discovered 13 such solids Best known example The truncated icosahedron In Europe they use it to play football In America it is called a soccer ball Misnomers Plato used this knowledge to found an elaborate theory where all things are composed of right triangles Renaissance mathematicians learn the regular solids from Plato, but must relearn Archimedean solids. Kepler Johannes Kepler rediscovered all Archimedean solids Originally believed planetary orbits were in spheres Later realized planetary obits were elliptical “If one inscribed a cube into the sphere of Saturn the faces of the cube would be tangent to the sphere of Jupiter.” Examples in the Elements Hexahedron: shape of the crystalline structures of lead ore and rock salt Octahedron: shape of crystals formed by fluorite Dodecahedron: shape of crystals formed by garnet All three of above: shape of crystals formed by iron pyrite Tetrahedron: shape of basic crystalline form of the silicates Truncated Icosahedron: outlines the vertices where sixty carbon atoms form the molecule known as “buckyball”