I
n the land of the pharaohs in ancient Egypt, the builders of the majestic pyramids and temples that adorn the surroundings of the Nile river made use of an amazing low- tech, biodegradable, zero-radiation, inexpensive, and low-maintenance computer to find solutions to a variety of geometric problems that they encountered in their daily lives: theknotted rope. As the name suggests, this computer consists of a rope of suitable length tied
together at the ends, and interspersed with a preselected fixed number of equally spaced knots. There is evidence that one popular model of this computer comprised 12 knots as shown in Figure 25.1, in two configurations: loose (left) and taut (center). The users of this computer knew from experience in the field that if three people holding the rope at the knots numbered one, five, and eight walked away from each other until all three strands of rope between them were taut, the final shape of the rope would be a triangle.* Furthermore,
and this is the crucial point, they were confident that the triangle would be a right-triangle: the angle made between the two short sides of lengths three and four is 90° angle. One of the most useful applications of this computer was, therefore, the construction of 90° angles in architecture. To illustrate this application, assume, for example, that the workers had to build a new wall that made a 90° angle with another wall already built, and refer to
* Ross, C. (2007), p. 157. For an alternate view, see Rowe, D. E. (2006), p. 52, who argues that the use of the knotted rope
for laying out right angles in ancient Egypt is speculation, and that there is little reliable evidence that the builders were aware of the Pythagorean theorem. Imhausen, A. (2006) discusses additional ancient Egyptian mathematics.
9 8 7 12 11 10 9 8 7 6 5 4 3 2 1 C A B 6 5 4 3 2 1 12 11 10
176 ◾ The Geometry of Musical Rhythm
Figure 25.1 (right), where the old wall is shown in the horizontal position. Assume further that this new wall is required to meet the old wall at the point marked A. First, one worker holds the rope at the fifth knot and stands at position A. The second worker then takes knot number one, and walks away from the first worker along the old wall until the rope between them is taut, ending at position B. Finally, the third worker takes knot number eight, and walks away from the other two workers adjusting his or her position until both strands are taut. The engineers were convinced that if the new wall was built in a straight line from A to C, the angle BAC would be a 90° angle.
You may ask yourself how the workers came to be so assured that in a triangle with side lengths consisting of 3, 4, and 5 units (called a 3-4-5 triangle), the angle between the two short sides was a right angle. In those days, the workers followed algorithms outlined in manuals issued by the pharaohs. Very likely, in one such manual, the procedure outlined above for constructing walls at 90° angles with the knotted rope was followed by an official statement of the form: If this algorithm is followed, then the correct solution is guaranteed
to be found. Next to this statement would usually be stamped the seal of approval of the
pharaoh. This process might be called a proof by governmental decree. It would be sev- eral thousand years later before the correctness of this algorithm was formally established using a more democratic method: a mathematical proof.
In the sixth century BC, the great mathematician Pythagoras of Samos in ancient Greece proved a wonderful theorem about right-angled triangles, of which the 3-4-5 triangle used in the knotted rope computer is a special case.* This theorem, now called Pythagoras’ theo-
rem, is taught to every child in school. The theorem states that in a right-angled triangle, the area of the square with side equal to the longest side of the triangle (called the hypot- enuse) is equal to the sum of the areas of the squares with sides equal to the other two sides of the triangle. The proof of Pythagoras’ theorem that appears in Euclid’s Elements is rather lengthy and detailed. Since then, hundreds of different proofs have been discovered, some of them quite elegant, short, and simple. One example that falls in the latter category is illustrated in Figure 25.2. The right-angled triangle in question is the dark gray triangle with sides a, b, and c, which occurs four times in the left diagram. It is required to be proved that the area of the square with side a plus the area of the square with side b (both drawn in light gray on the right diagram) equals the area of the large white square with side
c (on the left diagram). The two complete squares, on the left and right diagrams, both have
sides of length a + b, and are therefore equal. Also, each complete square contains four copies of the dark gray right-angled triangle, arranged in different ways. Therefore, remov- ing these four triangles from both complete squares leaves equal areas. The remaining area on the left is the white square of side c, and the remaining area on the right consists of the two light-gray squares of sides a and b, thus proving the theorem.
This proof of Pythagoras’ theorem holds for any positive values of a, b, and c, as long as the given triangle has a right angle. For the special case a = 3, b = 4, and c = 5, a simple and
* An ancient Babylonian clay tablet, dating back to 1800 BC, known as Plimpton 322 at Columbia University, contains
other such Pythagorean triples of numbers, leading some authors to suggest that the Pythagorean theorem may have been known well before Pythagoras. See Robson, E. (2001, 2002) and Polster, B. (2004), p. 4. However, knowledge of examples that satisfy a theorem does not by itself imply knowledge of the theorem; for that, a proof is required.
Deep Rhythms ◾ 177
original proof illustrated in Figure 25.3 may be found in the ancient Chinese classic book
Chou Pei Suan Ching, probably written during the Han dynasty period dated between 500
BC and AD 200. Given a triangle with a 90° angle flanked by sides of lengths three and four, it must be proved that the area of the square with side length eh is 32 + 42 = 9 + 16 = 25.
First, four copies of the triangle (shaded in light gray) are embedded in a 7 × 7 square array consisting of 49 unit squares, as shown in the figure. This leaves out four identical triangles at the corners of the array, as well as one small dark gray square in the center. The area of the entire array is 49. Therefore, the area of the gray and white triangles is 48. But the area of the white triangles equals the area of the gray triangles. Therefore, the area of the gray triangles is 24. The area of the square with side eh is equal to the area of the four gray triangles plus the small unit square in the middle, and is thus 25, as was required to prove.
When students are asked how they would prove that the knotted-rope algorithm used by the ancient Egyptians to construct a right angle is correct, some are quick to answer: apply
the theorem of Pythagoras. They are then surprised to discover that Pythagoras’ theorem
does not do the job. The theorem of Pythagoras proves that if we are given a right-angled triangle, then the square of the hypotenuse equals the sum of the squares of the other two sides. The theorem required to prove the correctness of the knotted-rope algorithm would have to state that if we are given a triangle in which the square of the hypotenuse equals the sum of the squares of the other two sides, then the angle opposite the hypotenuse must be a right angle. In other words, we require the converse of Pythagoras’ theorem. It is the
a c c b a a c b b
FIGURE 25.2 An elegant, simple, and short proof of Pythagoras’ theorem.
A e B
h
D g C
f
178 ◾ The Geometry of Musical Rhythm
difference between the logical statements if and only if. The good news is that a proof of this converse theorem also appears in the Elements of Euclid. In fact, it appears just after the proof of Pythagoras’ theorem.
Pythagoras was also a great music theorist, and laid some of the foundations of scales, chords, and tuning in music. The quotation There is geometry in the humming of strings is attributed to him. Pythagoras was referring to the pitch aspects of music in relation to the plucking of strings of different lengths. However, he could just as well have added that there is geometry in the drumming of drums. Pythagoras experimented with strings of different lengths that had the same tension and discovered that they sounded well together when the ratios of their lengths were related by small integers such as 1:2 and 2:3. In the ubiquitous diatonic scale used today, the major and minor three-note chords (triads) are considered to be the most stable and resonant because they are made up of the three most stable and resonant intervals: three, four, and five, the same integers that make up the smallest integers that satisfy the Pythagorean theorem. The C major and minor chords are shown as triangles in Figure 25.4. Interestingly, even though the major and minor chords have exactly the same intervals, the fact that the order of the intervals is reversed in going from one chord to the other makes the chords sound quite different from each other.
There are many ways in which 12 may be divided into three intervals, and the partition used by the major and minor chords is very special. The most even division would yield a triangle with three edges of the same length of four. Then, there are many partitions that would use two intervals of the same length, such as [3-3-6]. In addition, there are partitions that have very small and very large intervals such as [1-2-9]. The major chord on the other hand is the partition closest to [4-4-4] that has three distinct distances.
More than 2500 years after Pythagoras, in the twentieth century, another great and pro- lific mathematician, Paul Erdős, asked a very simple geometry question that also concerns distinct distances, musical rhythms, and scales, and that still has mathematicians baffled. Erdős asked if one can arrange n points in the plane with the restrictions that no three of the points should lie on a line, and no four of them may lie on a circle, so that for every value of i = 1, 2, . . ., n − 1, there is a distance determined by pairs of these points that occurs
C C# D D# A# B E F F#
C major chord C minor chord G G# 4 3 5 4 3 5 A C C# D D# A# B E F F# G G# A
Deep Rhythms ◾ 179
exactly i times. In other words, each distance realized by pairs of points should occur a
unique (distinct) number of times. We call such a set a deep set. The corner points of a rect-
angle of width a and height b does not yield unique distances because all three distances, a,
b, and c, where c is the diagonal of the rectangle, occur the same number of times, namely
twice. Furthermore, all four points lie on a circle, and thus violate one of the constraints. Therefore, the corners of a rectangle do not constitute a deep set.
You may ask yourself why Erdős disallowed the points to lie on a line or a circle. The answer is that the problem is too easy and uninteresting when these allowances are made. Consider, for example, the seven points on a line separated by unit distances, pictured in Figure 25.5. As the picture makes clear, distance six occurs once, distance five occurs twice, distance four three times, distance three four times, distance two five times, and distance one six times. In other words, no distance occurs the same number of times as any other distance. In other words, the multiplicity of each distance is unique.
A similar situation arises with points evenly spaced out on a circle. Consider the circle with 12 pulses and seven onsets at pulses zero through six, and refer to Figure 25.6. Again, each circular arc-length (geodesic distance) occurs a unique number of times. This remains true for any number of pulses and onsets as long as the onsets span at most one semicircle.
However, let us return for a moment to the original problem that Erdős posed for the two-dimensional plane. An example of a valid solution for n = 4 is illustrated in Figure 25.7. Two of the four points are located on the horizontal axis at coordinates 1 and –1, and the other two points are located on the vertical axis at coordinates −1 and 3. The histo- gram of the three distinct distances that occur between the points is shown on the right. The smallest distance d1 = 2 occurs twice, the next largest distance d2 = 2 occurs three
times forming an equilateral triangle, and the largest distance d3 = 1 + 3 occurs once
between the two points on the vertical axis.
So far, the only solutions that have been found for this problem are for n = 2, 3, 4, 5, 6, 7, and 8. The solution for n = 8 illustrated in Figure 25.8 is quite elegant. The first diagram shows the arrangement of the eight points, along with all their pairwise distances. Since the eight points are located at the vertices of a triangular lattice, it helps to see the distances
0 1 2 3 4 5 6 6 × 1 5 × 2 4 × 3 3 × 4 2 × 5 1 × 6
FIGURE 25.5 For evenly spaced out points on a line a unit distance apart, every distance occurs a
180 ◾ The Geometry of Musical Rhythm
embedded in the triangulation. The remaining seven diagrams show each of the seven distinct distances by themselves in order of increasing multiplicity. The distance equal to one side of a triangle occurs once. The distance equal to two sides of a triangle occurs five times. The distance equal to two heights of a triangle occurs four times. For the remaining distances, it helps to view every pair of adjacent triangles as a lozenge, and combine adja- cent lozenges into parallelograms. Then, the distance corresponding to the long diagonal
4 × 3 1 2 3 4 5 6 0 7 8 9 6 × 1 5 × 2 10 11 1 5 4 3 2 6 0 7 8 9 10 11 1 2 3 4 5 6 0 7 8 9 10 11 1 2 3 4 5 6 0 7 8 9 10 11 1 5 4 3 2 6 0 7 8 9 10 11 1 2 3 4 5 6 0 7 8 9 10 11 1 × 6 3 × 4 2 × 5
FIGURE 25.6 For evenly spaced out points in a semicircle, every distance occurs a unique number of times. 2 1 –1 d1 d2 d3 –2 –1 1 2
FIGURE 25.7 A set of four points with distinct distance multiplicities (left) and their histogram