The Elements of Euclid

Euclid was a student of Plato's students at the Academy prior to 300 BC. He taught and founded a school at Alexandria, Egypt in the time Ptolemy I, who reigned from 306 to 287 BC. Archimedes and Apollonias both may have studied at Euclid's school at some time early in their careers. His principal contribution to mathematics was as a teacher and textbook writer. His Elements consist of 13 books (or chapters), which organized most of the known mathematics at the time into a deductive theory, survived intact and has been used as a text and research reference since that time. Young Abraham Lincoln studied out of the first book by hearthlight. The Elements is included in the Great Books collection of the Encyclopedia Brittanica, where it runs to 400 closely written pages. According to Boyer, Euclid's successors referred to him as The Elementator!

Brief description of the Elements

Books I and II deal mostly with the works of the Pythagoreans on plane geometry. The next to last proposition (prop 47) of Book I is Pythagoras's theorem. The last propostion of Book II, prop 14, shows how to square an arbitrary polygonal figure.

Books III and IV with the work of Hipprocrates of Chios on circles. Book III ends with this proposition:

Proposition 37

If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut the circle, and the other fall on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the straight line which falls on circle, the straight line which falls on it will touch the circle .

Modern translation : In the diagram below, the square of the length of the blue segment is equal to the product of the length of the magenta segment with the length of the segment which is magenta and orange, ie

A brief discussion of the Platonic solids

A regular polyhedron of type (n,m) (aka platonic solid) is a solid figure having the property that each face is a regular n-gon and each vertex is common to m faces. The last argument in Euclid's Element is a plausibility argument that there are only five regular polyhedra, namely those of type (3,3) tetrahedron, (3,4) octohedron, (3,5) icosahedron, (4,3) cube, and (5,3) dodecahedron. Basically, Euclid 'notes' that it takes at least three faces to determine a vertex. Then he states that more than 5 equilateral triangles, more than 3 squares, or more than 3 regular pentagons cannot joined edge to edge around a vertex to form a solid angle. Then he states that even 3 regular n-gons (n > 5) cannot be joined edge to edge around a vertex to form a solid angle. Although what he says is perfectly plausible and if accepted proves the claim, he does not elevate it to 'Proposition' status. It was not until the 18th century that Euler gave a rigorous proof of a much more general statement about maps on a sphere.

Definition . A connected network or map drawn on the surface of a sphere divides the surface into countries . If two countries share a boundary, we assume they are joined along an arc (called an edge ). The endpoints of the edges are the points that are on the border of more than 2 countries and are called vertices .

Lemma. In any map on a sphere, the number of countries plus the number of vertices exceeds the number edges by 2 .

Outline of Proof . Start with the simplest nontrivial case: three countries. In this case, there are two vertices and three edges, and Euler's relation holds. Now suppose the relation is true for all maps with N countries. Take a map with N + 1 countries and remove an edge. The reduced map has 1 fewer countries, so in the reduced map

N + vertices = 2 + edges.

Now there are 3 cases:

First case: By removing the edge, no vertex is removed. Then the reduced map has 1 fewer edges and the same number of vertices, and so Euler's relation hold when the edge is reinserted.

(N+1) + vertices = 2 + (edges +1)

Exercise: finish this proof.

Definition . A map drawn on the surface of a sphere is of regular type (n,m) if each country borders n other countries and each point which is on the border of more than two countries is on the border of m countries.

Theorem (Euler) The regular types of maps are precisely (3,3), (3,4), (3,5), (4,3), and (5,3).

Outline of Proof. In a map of type (n,m), by counting we see that the number of countries is , and also . Now use this and Eulers relation to write the number of countries in terms of n and m. Then show that this function achieves integer values only at the 5 pairs listed in the theorem.

These polyhedra form the source of many interesting problems. They can be drawn quickly using the plottools package in Maple.

Problems with regular polyhedra.

Problem . Draw a cube in Maple using only plots[polygonplot3d] and plots[display]. Repeat for the remaining 4.

Problem. Each regular polyhedron has associated with it a dihedral angle, the angle at which each pair of adjacent faces meet. It is easy to see that the dihedral angle of a cube is 90 degrees. Compute the angle for the regular tetrahedron. Repeat for each of the remaining 3.

Problem . Suppose that a regular tetrahedron is inscribed in a unit sphere. Find the length of each edge, and the surface are and volume of the tetrahedron. Repeat for each of the remaining 4.

Problem. Is there an expanding sequence of the five regular polyhedra each inscribed in the next? (I think this is the problem that occupied a good deal of Kepler's mathematical life in the 1600s.)