Polyhedra: Plato, Archimedes, Euler

35 Pages·2014·2.49 MB·English

Polyhedra: Plato, Archimedes, Euler
Robert L. Benedetto
Amherst College
MathPathatMountHolyokeCollege
Tuesday, July 15, 2014
Regular Polygons
Definition
A polygon is a planar region R bounded by a finite number of
straight line segments that together form a loop.
If all the line segments are congruent, and all the angles between
the line segments are also congruent, we say R is a regular
polygon.
Polyhedra
Definition
A polyhedron is a spatial region S bounded by a finite number of
polygons meeting along their edges, and so that the interior is a
single connected piece.
So these are not polyhedra:
But these are:
Regular Polyhedra
Definition
Let S be a polyhedron. If
(cid:73) all the faces of S are regular polygons,
(cid:73) all the faces of S are congruent to each other, and
(cid:73) all the vertices of S are congruent to each other,
we say S is a regular polyhedron.
Cube (Regular Hexahedron) Regular Octahedron
Another Regular Polyhedron
Regular Tetrahedron
Two More Regular Polyhedra
Regular Dodecahedron Regular Icosahedron
And They’re Dual:
Some Ancient History (of Regular Polyhedra)
(cid:73) Pythagoras of Samos, c.570–c.495 BCE: Knew about at
least three, and possibly all five, of these regular polyhedra.
(cid:73) Theaetetus (Athens), 417–369 BCE: Proved that there are
exactly five regular polyhedra.
(cid:73) Plato (Athens), c.426–c.347 BCE: Theorizes four of the solids
correspond to the four elements, and the fifth (dodecahedron)
to the universe/ether.
(cid:73) Euclid (Alexandria), 3xx–2xx BCE: Book XIII of The Elements
discusses the five regular polyhedra, and gives a proof
(presumably from Theaetetus) that they are the only five.
For some reason, the five regular polyhedra are often called the
Platonic solids.
There Can Be Only Five: Setup
Given a regular polyhedron S whose faces are regular n-gons, and
with k polygons meeting at each vertex:
1. k ≥ 3 and n ≥ 3.
2. S is completely determined by the numbers k and n.
3. S must be convex: For any two points in S, the whole line
segment between the two points is contained in S.
4. Since S is convex, the total of angles meeting at a vertex of S
is less than 360 degrees.
(cid:16)180(n−2)(cid:17)◦
5. Each angle of a regular n-gon has measure .
n
[Triangle: 60◦, Square: 90◦, Pentagon: 108◦, Hexagon: 120◦.]
There Can Be Only Five: Payoff
From Previous Slide: The regular polyhedron S is determined by
n ≥ 3 (faces are regular n-gons) and k ≥ 3 (number at each
vertex). But total angles at each vertex must be less than 360◦.
(cid:73) Triangles (n = 3):
(cid:73) k =3 triangles at each vertex: Tetrahedron
(cid:73) k =4 triangles at each vertex: Octahedron
(cid:73) k =5 triangles at each vertex: Icosahedron
(cid:73) k ≥6 triangles at each vertex: Angles total ≥360◦. NO!
(cid:73) Squares (n = 4):
(cid:73) k =3 squares at each vertex: Cube
(cid:73) k ≥4 squares at each vertex: Angles total ≥360◦. NO!
(cid:73) Pentagons (n = 5):
(cid:73) k =3 pentagons at each vertex: Dodecahedron
(cid:73) k ≥4 pentagons at each vertex: Angles total >360◦. NO!
(cid:73) n-gons (n ≥ 6):
(cid:73) k ≥3 n-gons at each vertex: Angles total ≥360◦. NO!