German Democratic Republic
This country, once known as East Germany, is now re-united with West Germany. The stamp has a picture of the regular solid called the icosahedron, together with a portrait of Leonhard Euler and the famous formula named after him:
e-k+f = 2, where e is the number of vertices, k the number of edges and f the number of faces. For the icosahedron, e = 12, k = 30, and f = 20, giving 12-30+20 = 2 as the formula insists. Try it for the cube, tetrahedron, octahedron and dodecahedron.
In fact, Euler's Formula holds for ALL convex polyhedra, or for
ANY map of
vertices and joining edges drawn on a sphere. Indeed, it holds for any
map drawn on the plane provided you count the outside as a face.
(See the article with a proof of this in Zimaths 2.1, page 12, but
beware! - the letters used there represent the English words: V-E+F = 2.)
The result is often described as saying that Euler's number, for any surface topologically equivalent to a sphere, is TWO. For surfaces of different `genus', like the doughnut ( º the teacup), or the teapot, Euler's number is different. Can you work out what it is in those cases?
This stamp has a diagram of a logarithmic spiral inside a golden rectangle. This was the favourite curve of Jacob Bernoulli (1654-1705), who came of a family which produced many mathematicians; they lived in Basle, Switzerland, to where they had fled in 1583 from persecution of Protestants in Antwerp.
The spiral as depicted on the stamp flows beautifully through the corners of a sequence of nested golden rectangles, and Jacob asked that such a diagram be engraved on his tombstone, together with the words ``Eadem mutato resurgo'', meaning: ``Though changed I shall arise the same''. He was enraptured with the mathematical properties of the `Spira Mirabilis'- or miraculous spiral (as he called it), many of which he discovered himself. For example, it has the equi-angular property: every line drawn from the inner `pole' meets the curve at the same angle; and it is self-similar: each remainder-curve as you travel inwards is similar to the whole. These properties are clearly displayed in the structure of snail shells and spiral horns, whose growth and form incarnates the mathematics of golden rectangles and logarithmic spirals.
I think Jacob Bernoulli must also have seen the curve as a symbol of the RESURRECTION. To see why, draw the curve on a piece of cardboard, with the pole at the centre, push a pin through the pole and spin the card on the pin.
TO CONSTRUCT THE SPIRAL: First, construct a golden rectangle (as described in Zimaths 3.1, page 29). Start with the base-line AA¢, then construct the square ABEF. Bisect AF at G, mark point D on AA¢ such that GD = GE, and then complete the golden rectangle ABCD.
Next, take the smaller golden rectangle ECDF and chop off a square ECHI from it, to get a third golden rectangle HDFI. Chop off a square from this to get a fourth golden rectangle KFIJ, and continue this process for ever.... You will be spiralling in toward the pole, which is the point of intersection of the diagonals BD and CF. All of this can be seen on the stamp, but upside down. Now draw the spiral through points A,E,H,K, ¼
Notice that those diagonals BD and CF cut the spiral at four points between A and I (and then infinitely more points), and each time at the same angle; so will any other line through the pole. Another amazing property of the spiral is shown on the stamp: a line from A to E will be reflected off the curve through a right-angle bend along EH, then again along HK, and so forever, towards the pole. Imagine what you'd see if you walked down a corridor made of a mirror surface curved like the spiral! And next time you find a large (empty!) snail shell of this form, put it to your ear and listen to the roaring sound of an infinity of echoes ¼.
Why is this curve called the logarithmic spiral? Because its constantly
changing scale reflects that of the linear logarithmic scale - and for a
very good reason: the equation of the curve in polar coordinates, with the
origin at the pole, is
|r = k1ek2q, or lnr = k2q+k3.|
Here the constants ki may be chosen arbitrarily. You can use this
equation (with scientific calculator or natural log
tables) to sketch a spiral, and check the equiangular property.
Here is a proof of the property: Let P be the point (r,q) on the spiral, and let a be the angle made with the curve, by the radial line OP. Looking at the elementary triangle formed by taking the point P¢(r+dr, q+dq), we have:
The third stamp shows Pythagoras' theorem, or rather, its converse: if the sides a, b, c of a triangle ABC satisfy the condition that a2+b2 = c2 (as indeed 32+42 = 52) then the angle C will be a right angle. A deductive proof of Pythagoras'theorem can be found in Zimaths 1.4, page 24, and a proof of the converse on page 37. A dissection proof-at-a-glance of Pythagoras can be found in Zimaths 1.3, page 15.