Pythagoras and The Sunlit Uplands

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Pythagoras and The Sunlit UplandsWilliam A. EwbankTaylor UniversityUpland, Indiana 46989

^. . . Pythagoras, using square and triangular slatesand rocks . . . might have made his well known dis-covery in about 540 BC. . ."

In teaching your students mathematics, do you "lead them down the gardenpath," or do they feel they are in a long tunnel, with no daylight to be seenat the far end? Perhaps we all sometimes feel we are in a long tunnel. It isworth taking the time, once in a while, to climb to the sunlit uplands�on aclear day (the kind that would appeal to Doris Glass)�and lift up your eyesto see where we have come from, and where we are going.A long look backwards, not to the dawn of mathematics, but to the early

chapters in the history of mathematics, will show us one of the giants in ourdiscipline, and one of the greatest landmarks�Pythagoras and his famoustheorem. When I was a schoolboy, I heard of a scheme to communicate withintelligent beings in other planets by laying out in the Sahara Desert a hugeEuclidean diagram of the theorem, in the hopes that the "beings" with theirtelescopes would see that rational living persons inhabited this globe. I doubtif the scheme was ever carried out, and we now have much moresophisticated means of communication and travel.

Dr. J. Bronowski, in his famous film series made for the BBC (Bronowski,1975), demonstrated vividly how Pythagoras, using square and triangularslates or rocks at Crotona, in southern Italy, might have made hiswell-known discovery in about 540 BC (according to Hirschy, there is "nodoubt that this result was known prior to the time of Pythagoras" (Hirschy,1969)).Now, looking around, let us see how other ideas developed from the

theorem. Let us indulge in a little "what if-ing."First, what if the triangle is not a right triangle? The adjustment for this

situation is learned by all students of trigonometry as the "law of cosines";

The final term of this formula represents an addition or subtraction to the(b2 + c2) part according as to whether angle A is greater than or less than90° respectively. When a is 90°, cosA is zero, leaving the familiar

a2 == b2 + c2

This adaptation, in equivalent form, appears in Euclid’s Elements, Book II,

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repositions 12 and 13 (Hayes, 1969), written some 200 years after Pythagoras.Next, what if the shapes on the arms of the right triangle are not squares? It can belown (Ewbank, 1973) that the theorem of Pythagoras applies to any similar shapes,hich are similarly placed, and this can be demonstrated using regular and isometriceoboards (Figure 1). We can say, for example:

the area of the semi-circle on the hypotenuse of a right triangle is equal to thesum of the areas of the semi-circles on the other two sides, and

the area of the regular hexagon on the hypotenuse of a right triangle is equal tothe sum of the areas of the regular hexagons on the other two sides.

FIGURE 1: Regular and isometric geoboards illustrate how the Pythagoreantheorem can be adapted to non-square shapes


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Still in a geometric mode, we can ask what if the basic shape is not atriangle. Is it possible, for instance, for the area of the square on one side ofa quadrilateral to be equal to the sum of the area on the other three sides? oris it possible for the aggregate of the areas of squares on two of the sides toequal the aggregate of the areas on the other two sides? These questions canbe answered in the affirmative, and can best be illustrated by numericalexamples (we restrict ourselves to whole numbers). Two well-knownPythagorean triples are:

32 +42 = 52and 52 + 122 = 132,

which implies, by substitution, that

S2^- 42+ 122 = 132 (Figure 2)

Thus a quadrilateral of sides 3, 4, 12, and 13 fulfills these conditions. Aninfinite number of quadrilaterals meet the specifications of this extension ofPythagoras.

FIGURE 2: The Pythagorean theorem adapted to a quadrilateral of sides 3,4, 12, 13

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A series of numerical equalities, based on the simplest of Pythagoreantriples, goes like this:

32 + 42 = 52 (triangle)102 + II2 + 122 = 132 + 142 (pentagon)212 + 222 + 232 + 242 = 252 + 262 + 272 (heptagon)362 + 372 + 382 + 392 + 402 = 412 + 422 + 432 + 442 (enneagon)

and so on.

The first number is given by (2n + l)n, where n is the number of terms onthe right-hand side, a beautiful extension of the theorem of Pythagoras toother basic shapes.The famous theorem is an excellent example of the unity of arithmetic,

algebra and geometry, a unity unfortunately fragmented in the typical highschool curricula. Continuing our gaze into the realm of numbers, we mightask what if we examine the cubes of numbers instead of their squares. Arethere any solutions to

a3 + b3 = c3

where a, b, and c are integers?or in general to

a" + b" = c" ?

This question brings us to another episode in the history of mathematics, theso-called ’Termat’s last theorem." Pierre de Fermat (1601-1665) was aFrench amateur mathematician who worked with Pascal on the calculus ofprobabilities. It was his custom to write notes in the margins of his books,and a note, one of the last he wrote, in the margin of his copy ofDiophantus* Arithmetica, reads as follows:

To divide a cube into two cubes, a fourth power, or in general any powerwhatever into two powers of the same denomination above the second isimpossible, and I have assuredly found an admirable proof of this, but the marginis too narrow to contain it.

Unfortunately Fermat died before he could publish his proof, and theconjecture is still unproven. According to Fey, it is now known that theconjecture is true for all n < 4003, and for many special values of n (Fey,1969; Christopher, 1983).

In practical terms, if we take a series of cubes of integral edge lengths,such as the 1 cm, 2 cm, 3 cm, etc, cubes in the set of Cuisenaire MetricBlocks, and imagine an infinite series of cubes of increasing size, one couldnever find any two blocks whose combined volume matched that of a singleblock. However, it is possible to find two cubes whose combined volumematches that of two other different cubes. The following equations comprisesome examples:


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I3 + 123 = 93 + 10323 + 163 = 93 + 15323 + 343 = 153 + 333

In 1756-7 the noted Swiss mathematician Leonhard Euler (1707-1783) showedthat the general solution of

is given byx = r [(a2 + 3b2)2 - (a - 3b)]y = r [-(a2 + 3b2)2 + (a + 3b)]u == r [(a2 + 3b2) (a + 3b) - 1]v = r [-(a2 + 3b2) (a - 3b) + 1]

where a, b, r ^ 0 are rational numbers.Other combinations of cubes are possible, such as:

53 + 53 + 103 = I3 + 23 + 83 + 9313 + 63 4- 73 4- 173 + 183 + 233 = 23 + 33+ II3 + 133 + 213 + 223

and in higher powers:

14 + 64 + 74 + 174 + 184 + 234 = 24 + 34 + II4 + 134 + 214 + 22415 + 65 + 75 + 175 + 185 + 235 = 25 + 35 + II5 + 135 + 215 + 225

Our gaze has taken in fields of geometry, algebra, trigonometry, andarithmetic. It is a change, and instructive, to emerge from what often seem tous teachers as narrow tunnels of endeavor to look around at the endeavors ofothers. How wonderful it is when we ourselves, our students, do a littleexploring in the sunlit uplands!

References

Bronowski, Jacob. Music of the Spheres, in film series Ascent of Man. B.B.C.London: 1975

Christopher, William. Pierre de Fermat and Gerd Fallings. California Mathematics. 8.2 (November 1983)

Ewbank, William A. If Pythagoras Had a Geoboard. Mathematics Teacher. 66. 3.(March 1973)

Fey, James. Format’s Last Theorem, in Historical Topics for the MathematicsClassroom. Thirty-first Yearbook, National Council of Teachers of Mathematics,N.C.T.M. Washington, D.C.: 1969

Hayes, Eleanor. Trigonometric Identities, in Historical Topics for the MathematicsClassroom.

Hirschy, Harriet D. The Pythagorean Theorem, in Historical Topics for theMathematics Classroom.

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Pythagoras and The Sunlit Uplands