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 Athens is reputed to be a rectangle in the golden ratio though this is controversial. Renaissance artists were very interested in the ratio and it appears in many paintings. Luca Pacioli, the inventor of double-entry bookkeeping, wrote a book on the ratio called De divina proportione, illustrated by Leonardo da Vinci. Composers such as Béla Bartok and Claude Debussy deliberately used the ratio in their music.
The ratio of 1 mile to 1 kilometer is 1.609, very close to 1.618, but that is probably just a coincidence.
See: Fibonacci Numbers, pages 54–55; Tessellations, pages 68–69.
5th century BC Greece
Trisecting the Angle
The problem of dividing an angle into three equal parts.
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We can bisect an angle, but can we trisect it?
Greek mathematicians set many problems. This and the next two topics contain the three most important problems, having had a great influence on the progress of Greek mathematics and, indeed, on all mathematics.
All three problems are geometrical. They involve performing a geometrical construction. For Greek mathematicians, constructions had to be exact, using straight edge and compasses only. You were not allowed to use a ruler to measure distances nor a protractor to measure angles.
Suppose we are given an angle. The problem is to trisect it or, in other
