Ch 9 - Polyhedra

polyhedron def

A closed figure in 3D space whose boundary consists of a finite number of non-parallel polygonal faces. The intersection of two faces must either be an edge, a vertex, or empty (so they don't intersect)

Regular polygon def

A polygon is regular if all sides are same length and all angles are the same

Convex polyhedron def

Convex if the line joining any two distinct points on the boundary is contained in or on the boundary of the shape

Polygon def and our restriction on parallel lines

Cyclic path in the plane consisting of line segments where no two successive lines are parallel

Euler's formula about polyhedra

Let P be a polyhedron that can be deformed into a sphere (rather than say a doughnut shape), with v vertices, e edges, and f faces. Then v-e+f=2

How do we prove there are only the 5 platonic solids

Suppose we have a regular convex polyhedron with p-gon faces where q meet at each vertex. It is convex so the angles at a vertex sum to less than 2π, then q(p-2)π/p <2π double implies q(p-2)<2p as we know p>0 for it to be the number of sides of a polygon. Then we rearrange until we get to (p-2)(q-2)<4. Since p and q are integers greater than zero, and to have a polygon at all each must be ≥3, we know (p-2) and (q-2) must be integers ≥1, and multiplying them together to still be less than 4 means they must be 1, 2, or 3. If 1 then p-2=q-2=1, then p and q are 3, tetrahedron. If 2 then one of p-2, q-2 is 2 and other is 1, giving p=4, q=3 (cube) or p=3, q=4 (octahedron). If 3 then we have p=5, q=3 (dodecahedron) or p=3, q=5 (icosahedron)

Platonic polyhedron Def

a polyhedron is platonic if it is convex and regular

Regular Polyhedron Def

a polyhedron is regular if app faces are congruent regular polygons and the number of faces meeting at a vertex is the same for any vertex

Names of all platonic solids, and what it's made up of [5]

tetrahedron, 4 triangles meeting at 4 vertices with 3 faces at each vertex. Cube. Octahedron, 8 triangular faces with 4 triangles at each vertex, 6 vertices. Dodecahedron, 12 pentagonal faces, 3 at each of the 20 vertices, icosahedron 20 triangular faces with 5 faces at each of 12 vertices.

A polygon is convex if...

the line joining any two points on the path lies inside the polygon or on edge