Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Fall 2005 MATH025.1001
This web page refers to selected material from the book.
First, Section 4.5 of the book, on the Platonic solids.
Second, Section 4.1 on the Pythagorean Theorem.
Third, Section 4.6 on the shortest distances on a sphere.
Platonic Solids
Section 4.5 of the book discusses the platonic solids.
- There are just five of them, the tetrahedron, the cube, the octahedron,
the dodecahedron, and the icosahedron.
- To precisely define a platonic solid, it is not sufficient to say that
all its surfaces must be identical regular polygons (can you give an
interesting example of a solid that satisfies this weaker definition, but that
is not a platonic solid?).
- Besides all surfaces being identical regular polygons, one may add that
the vertices must lie on a sphere.
For a silly reason this still does not suffice.
We must also add that the volume is positive, that is, a platonic solid is not
flat.
- Naturally one must verify that the five objects mentioned above
really satisfy the required properties.
But how do we show that there are no others?
One of the tools which helps is the Euler invariant:
If a solid (without holes) has V many vertices, E many edges, and S many
surfaces, then 2 + E = V + S
- When we take the midpoints of all the surfaces of a platonic solid, and
then connect these midpoints with their nearest neighbor midpoints, and fill in
the surfaces, we get a new platonic solid, called the dual of the
original one.
As the book shows, the tetrahedron is its own dual.
The cube and the octahedron are one another's duals.
The dodecahedron and the icosahedron are one another's duals.
- The five platonic solids were considered so special, that for a long
time researchers, often influenced by Pythagorean philosophy, wondered whether
there were connections between them and some deeper properties of the universe,
and beyond.
Originally Kepler (some time after 1600) was among them.
Experimental observations forced him to abolish his `platonic' model of the
solar system, to be replaced by a model following the so-called Kepler laws.
After about 2 millennia, the Pythagorean philosophy lost its support in the
intellectual community.
The Pythagorean Theorem
Section 4.1 of the book discusses the Pythagorean Theorem.
- If the hypotenuse of a right triangle has length c, and the lengths of
the other sides are a and b, then a^2 + b^2 = c^2.
- In class we add the classical `behold' proof to the one from the book.
This proof was considered so clear and easy, that by looking at the picture, any
competent mathematician can see the proof: Behold the picture.
Rather than starting with a square with sides of length c, one should work with
a square with sides of length a+b.
Then take 4 times the right triangle from a corner, as we did in class.
Shortest Distance on a Sphere
Section 4.6 of the book includes a discussion on shortest travel distances on
a sphere.
- The interactive earth sphere illustration on the accompanying CD disk is
quite informative.
Use it to check out some shortest routes between distant cities on the globe.
- The shortest distances are exactly parts of the so-called great circles,
the `weird' lines, of the non-Euclidean geometry without parallel postulate.
Last updated: November 2005
Comments & suggestions:
wimr@mscs.mu.edu