Platonic Solids

Greek concept of Symmetry Seen in their art, architecture and mathematics Seen in their art, architecture and mathematics Greek Geometry Greek Geometry The most symmetric polygons are regular The most symmetric polygons are regular Regular polygons have all sides and angles congruent Regular polygons have all sides and angles congruent Ex. Regular triangle is an equilateral triangle Ex. Regular triangle is an equilateral triangle Ex. Regular quadrilateral is a square Ex. Regular quadrilateral is a square Regular polygons exist for any number of sides Regular polygons exist for any number of sides Ex. Pentagon, Hexagon, Heptagon, etc. Ex. Pentagon, Hexagon, Heptagon, etc.

Whoa its in 3-D!! A polyhedron is regular if A polyhedron is regular if Its faces are congruent regular polygons Its faces are congruent regular polygons Its vertices are similar Its vertices are similar A cube is a regular polyhedron A cube is a regular polyhedron

Euclid says… Only a few regular polyhedra exist Only a few regular polyhedra exist In contrast there are infinite regular polygons In contrast there are infinite regular polygons Actually there are exactly five types of regular polyhedra Actually there are exactly five types of regular polyhedra And they are… And they are…

Tetrahedron 4 Faces 4 Faces Each face is a triangle Each face is a triangle Associated by Pythagoreans with the element fire Associated by Pythagoreans with the element fire

Octahedron 8 faces 8 faces Each face is a triangle Each face is a triangle Associated with air Associated with air

Icosahedron 20 faces – associated with water Each face is Drumroll please…………….. A TRIANGLE!!

Hexahedron 6 faces- associated with earth 6 faces- associated with earth The sides are squares, this is a cube The sides are squares, this is a cube Associated with water Associated with water

Dodecahedron 12 faces 12 faces Each face is a pentagon Each face is a pentagon To the Pythagoreans, it represents the universe To the Pythagoreans, it represents the universe

Vertices At least three polygonal faces must meet at any vertex, forming a peak. At least three polygonal faces must meet at any vertex, forming a peak. The sum of these angles is < 360 degrees The sum of these angles is < 360 degrees Otherwise the faces overlap or form a flat surface 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 In a regular polyhedra the situation at one vertex is the same as at any other 6 triangles, 3 hexagons = 360 degrees 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 Allow different types of regular solids to be used to create a 3-D object Archimedes discovered 13 such solids Archimedes discovered 13 such solids Best known example Best known example The truncated icosahedron The truncated icosahedron In Europe they use it to play football In Europe they use it to play football In America it is called a soccer ball 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 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. Renaissance mathematicians learn the regular solids from Plato, but must relearn Archimedean solids.

Kepler Johannes Kepler rediscovered all Archimedean solids Johannes Kepler rediscovered all Archimedean solids Originally believed planetary orbits were in spheres Originally believed planetary orbits were in spheres Later realized planetary obits were elliptical 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.” “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 Hexahedron: shape of the crystalline structures of lead ore and rock salt Octahedron: shape of crystals formed by fluorite Octahedron: shape of crystals formed by fluorite Dodecahedron: shape of crystals formed by garnet Dodecahedron: shape of crystals formed by garnet All three of above: shape of crystals formed by iron pyrite All three of above: shape of crystals formed by iron pyrite Tetrahedron: shape of basic crystalline form of the silicates Tetrahedron: shape of basic crystalline form of the silicates Truncated Icosahedron: outlines the vertices where sixty carbon atoms form the molecule known as “buckyball” Truncated Icosahedron: outlines the vertices where sixty carbon atoms form the molecule known as “buckyball”