7. A MEETING OF SQUARES

Pythagorean trees, Pythagoras theorem, Pythogorean triples
                    




  1. Pythagoras’ theorem
The Pythagorean tree fractal is based on Pythagoras’ theorem. However, the fronds of the illustrated fern tree can be modelled very realistically using fractals.

























Pythagoras’ theorem

Probably the most famous of all the theorems left to us by the ancient Greeks is the theorem of Pythagoras. It concerns the areas of squares constructed on the three sides of a right-angled triangle.

If ABC is right-angled at C, and squares are constructed on the three sides of the triangle, then the sum of the areas of the two smaller squares equals the area of the largest square. In algebraic terms we have

AB2 = BC2 + CA2.

We do not know how Pythagoras proved his result, but it is likely that he used a dissection method like that illustrated below. We let a, b, c denote the side-lengths of sides BC, CA, AB of &Mac198;ABC above.
1. Noting that in the figure below the eight triangles are congruent (have the same size and shape), can you deduce Pythagoras’ theorem?

Incidentally we observe that for a right-angled triangle with sides of length a, b and hypotenuse (longest side) of length c, Pythagoras’ theorem can be simply expressed as

a2 + b2 = c2.

We shall come back to this expression a little later.


























The Pythagorean tree

The Pythagorean tree is a fascinating fractal based on the Pythagoras construction. We give here the version where the right-angled triangle is isosceles. This gives rise to a symmetric tree. The Logo program corresponds to the illustrated drawing sequence. The roof of this ‘house’ is right-angled, and the tree construction is reproduced outwards from the sloping ‘roof’ arrows.

to PythagTree :size :level
if :level = 0 [fd :size rt 90 stop]
repeat 4 [fd :size lt 90]
lt 90
fd :size
rt 45
PythagTree :size * 0.707 :level - 1
PythagTree :size * 0.707 :level - 1
lt 45
fd :size
end
pu setx -25 sety -85 seth 90 pd
PythagTree 50 6
The resulting fractal is pictured below.


























Real or rational?

The real numbers are the labels we give to points on the real number line. Examples are –2, –2/3, 0, 1, 2. Rational numbers (or fractions) are numbers which can be expressed as the ratio of two integers.

2. Let the large square in the Pythagorean tree have side-length 1. What is the size of the first few smaller squares, in decreasing order? Can you see a pattern? You should obtain a sequence which starts

1, 1/2, 1/2, ... .

The discovery of 2 (in a right-angled triangle) quickly led to problems for the early Greeks. For they believed that all real numbers are rational – expressible as fractions.
3. (a) Suppose 2 is a fraction: i.e. let 2 = a/b. Dividing through, we may assume that a and b have no common factors (other than 1). Can you see that a2 = 2b2?

(b) Show that a must be even. (What would happen if a were odd?) If a is even, we can write a = 2d.

(c) Show that the equation a2 = 2b2 now becomes 4d2 = 2b2, i.e., 2d2 = b2.

(d) Show that now b must be even.

The Pythagoreans saw that if a and b are both even, then the common factor of 2 can be cancelled out. This gives a contradiction, as we have assumed that a and b have no such common factor. These days we argue that our original assumption must have been wrong, i.e., it is not true that 2 is a fraction. But the early Greeks were reluctant to believe this, as their whole theory of numbers was based on the belief that all numbers on the number line were expressible as fractions.






















Pythagorean triples

As we have seen, Pythagoras’ result can be written as

a2 + b2 = c2.

Of special interest is the case where a, b, and c are all integers. A simple example is

32 + 42 = 52.

We say that (3, 4, 5) is a Pythagorean triple. Are there any others? In fact a general formula for Pythagorean triples has been known since the time of Diophantus and the early Greeks.
4. (a) Let m, n be any positive integers. We set a = m2 – n2, b = 2mn, c = m2 + n2. Complete the table below.
m n a = m2- n2 b = 2mn c = m2 + n2
2
3
4
4
5
5
6
 1
2
1
3
2
3
1
 3
5		 4 5

(b) Can you show that if (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc), for any positive integer k?

A little calculation gives:

(m2 – n2)2 + (2mn)2 = (m2 + n2)2.

It is known that the formulae in Question 4 give rise to all Pythagorean triples.

























Further investigations

5. Use your library to discover more about Pythagoras. There are many interesting stories about him and his discoveries.

6. Pythagoras’ theorem for an isosceles right-angled triangle is often seen illustrated in floor mosaics. See if you can find examples of this.

7. When the Egyptians built the pyramids, they may have used a loop of rope on which there were 12 equally spaced knots to ensure that the blocks were placed ‘on the square’. Explain how they would have used this rope.
8. It is possible to use the method of Q 3 to show that 3 is not a fraction. Try to adapt the proof given there. [Hint: in the above proof, integers are even because they are divisible by 2. Here you will need to consider divisibility by 3.]

9. Look back at your table of Pythagorean triples. Notice that for (3, 4, 5) and (5, 12, 13) the last two integers are just 1 apart. Notice how m and n are related here. Can you find other Pythagorean triples in which the last two integers differ by 1?

10. We constructed the Pythagorean tree based on an isosceles right-angled triangle. If we choose a non-isosceles triangle, we obtain a lop-sided Pythagorean tree. See if you can modify the program to generate such a tree.