Finding even perfect numbers is comparatively easy. There is a formula for them, which essentially appears in Euclid’s Elements. The formula is 2n–1(2n – 1), provided that the term inside the brackets is a prime number.
All the perfect numbers that have so far been discovered are even; an odd perfect number, if it exists, remains to be found. This is the oldest unsolved problem in mathematics.
Certainly there are no odd perfect numbers up to 10300 (1 followed by 300 zeros). They may not exist, but if one is ever found, mathematicians will already know a lot about it: that it has at least nine prime factors, for example.
6th century BC Greece
Regular Polygons
A regular polygon has equal angles and equal sides.
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Examples of regular polygons are the equilateral triangle (all sides equal, all angles equal to 60°) and the square (all sides equal, all angles equal to 90°). Then comes a pentagon, then a hexagon, and so on.
How do you draw these shapes? Greek mathematicians were very particular about exactness in geometry, and required exact constructions. They would not allow a protractor to measure and draw angles, because one cannot do so exactly. They did not allow a ruler to measure and set out lengths, as one cannot be sure one has the exact length. These constructions had to be made with two instruments only – a straight edge and compasses.
Constructions of an equilateral triangle and a square are part of school mathematics. The construction of a triangle is shown.
The line AB is drawn, then arcs of the same length as AB are drawn to intersect at C. Notice that a straight edge has been used to draw the lines, and compasses to draw the arcs. We do not need to use a protractor to measure an angle of 60°.
With a lot more work, it is possible to construct a regular pentagon. Hexagons and octagons are straightforward. Heptagons (7 sides) and nonagons (9 sides) had to wait!
The diagram shows the construction of an equilateral triangle.
6th century BC Greece
Platonic Solids
There are precisely five Platonic solids.
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For a regular or Platonic solid, all the faces are equal regular polygons.
A regular polygon, such as a square or an equilateral triangle, has equal angles and equal sides. The best-known Platonic solid is the cube, whose six faces are equal squares.
The proof that there are no more than five such solids appears as the very last proposition in Euclid’s Elements.
They are called Platonic solids from their appearance in Plato’s Timaeus (dated about 350 BC). This is an obscure and ambiguous book, however, with many possible interpretations. It contains what could be described as an atomic theory in which the four elements of matter – fire, air, water, and earth – consist of these solids. They look as follows:
| fire | tetrahedron |
| air | octahedron |
| water | icosahedron |
| earth | cube |
Fire, for example, consists of countless atoms, each of which is a tiny tetrahedron. The sharp points of this solid explain why fire is painful.
Earth (or solid matter in general) consists of atoms, each of which is a tiny cube. The fact that cubes can be densely stacked together explains why earth is heavy.
The dodecahedron represents star and planet matter, which was believed to be different from matter on the Earth.
The five solids were known before Plato. They are attributed to the Pythagoreans, who reportedly sacrificed one hundred oxen to celebrate the discovery of the dodecahedron.
See: The Pythagoreans, page 17; Regular Polygons, page 21; Euclid’s Elements, page 35.
Tetrahedron: four triangular faces.
Octahedron: eight triangular faces.
Cube: six square faces.
Dodecahedron: 12 pentagonal faces.
Icosahedron: 20 triangular faces.
6th century BC Global
The Golden Ratio
A ratio of lengths that occurs in mathematics, nature, and art.
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The sides of a rectangle are in the golden ratio if, when you remove a square, the new rectangle is similar to the original one. The golden ratio is a number that is defined geometrically but which occurs in many other contexts.
The diagram shows the situation. The sides are r and 1. If we remove a square of side 1 by cutting along the dotted line, we now have a rectangle which is 1 by r – 1.
This is similar to the original rectangle. Hence the ratios r/1 and
Putting
r2 – r – 1 = 0.
The positive solution of this quadratic equation is the value of r.
This ratio is called the golden ratio, or golden section, and it is written as φ (pronounced phi). Like so many other things, its discovery is credited to the Pythagoreans.
This illustrates how the golden rule is defined.
The exact value of φ is
Here are some of the occurrences of the ratio:
• In mathematics, it occurs in the pentagon and the pentagram (five-pointed star) and the Penrose tiling. The ratio of successive terms of the Fibonacci sequence tends to φ
• In nature, the shell of the Nautilus snail and the pattern of sunflower petals are said to exhibit the ratio.
• In art, the façade of the Parthenon in