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 words, to cut it into three equal angles. Can this be done with straight edge and compasses only? You are not meant to measure the angle with a protractor and then divide by three.
To bisect it (or to cut it into two equal parts) is straightforward. To bisect ∠ABC:
Put the point of the compasses on B. Draw an arc cutting the lines at D and E. Put the point of the compasses at D, draw an arc. Put the point of the compasses at E, draw an arc. These arcs meet at F. FB bisects the angle.
Is there a similar construction to trisect the angle? The Greeks found many clever methods but they all required more instruments than the basic straight edge and compasses.
This matter has been a very popular problem among amateur mathematicians and is possibly one for which the greatest number of false proofs have been proposed.
5th century BC Greece
Doubling the Cube
Given a cube, construct a cube with exactly twice its volume.
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The problem of doubling the cube is equivalent to constructing a line with a certain length, using straight edge and compasses only.
In ancient times, the island of Delos in the Aegean Sea was troubled by a plague. When the people consulted an oracle, it told them that the fault lay in the altar to Apollo which was shaped like a cube. The god was offended because it was too small and it should be made twice as large.
The Delians reconstructed the altar, doubling its height, width, and depth, but the plague continued unabated. A new appeal to the oracle revealed the reason for this: the altar was now too large.
Suppose the side of the original cube was 1 unit. As all three dimensions had been doubled, they were each 2 units. The volume of the altar was now 2 x 2 x 2 = 8 times the original volume. The god Apollo wanted the volume of the altar, not its sides, to be twice as large.
Suppose each side becomes k units. Then the volume of the altar is k x k x k, which is k3. If the volume is now doubled, then k3 = 2. Therefore, k itself must be 3√2, the cube root of 2.
The problem now becomes the following: given a length of one unit, construct a length of 3√2 units. Greek mathematicians (and Greek gods, presumably) were very particular about exactness. The Greeks tried all kinds of ingenious methods to construct this length, but all of the methods used something other than straight edge and compasses. It would be many years before the answer to this conundrum was finally found.
See: Doubling the Cube and Trisecting the Angle Revisited, page 118
5th century BC Greece
Squaring the circle
This is another of the three Greek problems (the others being Trisecting the Angle and Doubling the Cube): Given a circle, construct a square of equal area.
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The problem of squaring the circle reduces to the following: given a line of length 1, construct a line of length π. As always in Greek geometry, the only instruments you are allowed are a straight edge and compasses.
The phrase “squaring the circle” has entered ordinary language to mean a task that is inherently impossible. The original meaning was subtler and less clear cut, however. It means to find a method of constructing a square exactly equal in area to a given circle. It is far from obvious that this is impossible.
Suppose, for simplicity, that the circle has a radius of 1 unit. Then its area is π x 12, which is π. If the equivalent square has side x units, then its area is x2.
The problem now becomes to find a length x such that x2 = π. The side x of the square must be equal to √π, the square root of π.
The square root is not a problem. If you can construct a length k, then you can construct a length √k. The problem is π. Ingenious methods were invented to draw a line of length π, but they all used moving parts, or required curves that could not be drawn exactly, such as spirals.
The three Greek problems have all been resolved, but only over 2,000 years since they were originally posed.
Area π
Area x2
The above illustration shows a circle and square of equal area.
See: Squaring the Circle Revisited, page 133
5th century BC Greece
Zeno’s Paradoxes
Zeno (c. 490–c. 430 BC)
These paradoxes concern the infinite divisibility of space and time, and suggest that motion is impossible.
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Matter cannot be divided indefinitely. There are particles that cannot be cut up any further. Is the same true of space and time, or can they be divided indefinitely? Zeno’s paradoxes concern this question.
Zeno’s paradoxes include the following two puzzles:
1. For the Dichotomy, suppose that you want to cross a field. Before you reach the other side you must get half way across. Before you reach the halfway point you must get a quarter of the way across and so on. To cross the field you must travel an infinite number of smaller distances, and so it is impossible to get across at all.
2. Achilles and the