An Amusing Emergence of the Pythagorean Theorem from Circular Planetary Motions
Amitava Biswas1, Abhishek Bisaria2
2Spectral D &T. Bridgewater, New Jersey, USA.
We present here an unusual example where a simple geometric theorem emerges from a complex mechanism. Even if the reader is familiar with the Pythagorean theorem [1-3] and how to calculate the area of a circle, this example may provide a little surprise. We have utilized a basic rule from mechanics, that two points on any arbitrarily moving rigid body always remain at a fixed distance apart, and their relative motion is always tangential and never radial . We have also utilized a basic planetary phenomenon that Earth must complete 366 rotations about its axis when circling around the Sun in 365 days . That is because, one additional rotation about its axis must be completed for going around the Sun, with reference to a fixed reference such as a distant star. Likewise, with reference to a fixed reference, the Moon must complete one rotation about its axis when going around Earth, although the near side of the Moon remains towards Earth during the motion.
In the left half of Figure 1, the line BC is tangential to the inner circle of the gray ring, and ABC is a right triangle. Whereas in the right half of Figure 1, the line BC is NOT tangential to the inner circle of the gray ring, and ABC is an obtuse angle. For both arrangements, the gray ring is swept by the revolving line BC, like a satellite with its near side always remaining inwards. In this context, motion of the line BC is determined by motions of the two points B and C, keeping a fixed distance BC between them.
Consider the right half of Figure 1. One complete rotation of the line BC around the point A is the combination of one complete rotation of the point B around the point A and one complete rotation of the point C around the point B. The direction of the instantaneous motion of the point B is tangential to the inner circle and does NOT coincide with the line BC. With this instantaneous motion of the point B, the line BC tends to sweep along a strip of finite non-zero width at any particular instant, as indicated by two parallel arrows in the right half of Figure 1.
Now consider the left half of Figure 1. Similar to the above situation, one complete rotation of the line BC around the point A is the combination of one complete rotation of the point B around the point A and one complete rotation of the point C around the point B. However, in this case, the direction of the instantaneous motion of the point B always coincides with the line BC itself. With this instantaneous motion of the point B, the line BC does not sweep any area, just like a knife can not spread butter when sliding along its edge. Therefore, the gray ring is swept entirely by one rotation of the point C around the point B. This implies that the area of the gray ring is equal to the area of a circle of radius BC. The gray ring is also the difference between the circle of radius AC and the circle of radius AB, leading to the Pythagorean theorem for the right triangle ABC.
This article derived the Pythagorean theorem from circular planetary motions, attesting to the inseparable bondage between mechanics and geometry. We hope this presentation will be a little refreshing for students, teachers, and others, using the Pythagorean theorem for rectangular objects.
This work has benefited from faculty start-up funds granted by the University of Southern Mississippi to Dr. Amitava Biswas.
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2. Yanney BF and Calderhead JA (1897). New and Old Proofs of the Pythagorean Theorem. The American Mathematical Monthly, 4 (3) pp. 79-81.
3. Peterson BE (2009). Teaching the Pythagorean Theorem for Understanding. The Mathematics Teacher, 103, (2) pp. 160-161.
4. DenHartog JP (1961). Mechanics. Dover, Mineola, New York. ISBN-10: 9780486607542.
5. Chaisson E and McMillan S (2017). Astronomy Today (9th Edition). Pearson, New York. ISBN-10: 0134580559.
Received on 18.12.2019 Accepted on 26.12.2019
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Int. J. Tech. 2019; 9(2):43-44.