Pythagoras Sensei

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Posted by admin | Posted in Bujinkan Budo Taijutsu, Mental, Physical | Posted on 28-09-2012

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Too often have I encountered a Bujinkan class where gata are summarily dismissed, and “henka” are “studied” instead.  (I have actually heard “teachers” refer to gata derisively, actively advocating against their study.)

Although I believe I understand the source of such an opinion, I find myself vehemently disagreeing, and look to Pythagoras Sensei to support my differing opinion.

Pythagoras Sensei?

Pythagoras’ most memorable contribution to mathematics is the Pythagorean Theorem, a2 + b2 = c2, which we use to determine the length of the hypotenuse of a right triangle.

If ever asked to use the Pythagorean Theorem, a2 + b2 = c2is blurted out almost immediately.  However, if asked to prove the Pythagorean Theorem, a2 + b2 = c2is also blurted out almost immediately, which is incorrect.

I will readily admit that up until I was exposed to the proof of the Pythagorean Theorem—which took me aback with its simplicity—I also would have blurted a2 + b2 = c2.

For most of us, just knowing the Pythagorean Theorem has limited value, as determining the length of the side of a right triangle may not be an active part of our lives.

However, taking the time to prove and understand the Pythagorean Theorem instead of just using its conclusion can potentially yield many insights beyond the basic properties of right triangles.

We all typically associate the Pythagorean Theorem with triangles, but its proof requires two unequal squares, or “equilateral rectangles:”

We know from fundamental geometry that the area of a square is the length of a side multiplied by itself, or squared; the area square with a lateral of length x is x2, thus the area of the smaller square is c2, and the area of the larger square is d2.

If we take the smaller square and place it within the larger square in such a manner that each small square vertex is also a point on each large square lateral, we have:

From this perspective, it should be obvious that the area of the larger square is equal to the area of the smaller square plus the areas of the four triangles.  However, at this point, knowing only the length of the hypotenuse of any triangle, we are unable to define the triangles in any meaningful way.

If we use such vertices on the laterals to define the points in which each lateral is segmented into two, we can state that d = a + b, and once again use fundamental geometrical principles, we can assert:

Fundamental geometry also gives us the area of a right triangle by multiplying both laterals on each side of the right angle, and dividing by two, thus the area of each triangle depicted is ab/2.  Using geometry, we have been able to fully define each square and triangle, and can assert relationships between them.  Knowing that the area of the large square is equal to the area of the small square plus the area of the four triangles, we can craft the algebraic equality:

d2 = c2 + ab/2 + ab/2 + ab/2 + ab/2 or d2 = c2 + 4(ab/2)

Since we have already established that d2 = a2 + b2, we can rewrite the equation:

(a + b)2 = c2 + 4(ab/2)

Which expands into:

a2 + ab + ab + b2 = c2 + 2ab

Algebraically reducing:

a2 + 2ab + b2 = c2 + 2ab

a2 + b2 = c2

For the above proof of the Pythagorean Theorem, we needed both geometry and algebra, two distinct mathematical disciplines.  In order to use such disciplines, an understanding of arithmetic is also required.  To use the symbols a, b, c and d to identify the unknown lengths of the laterals requires the use of abstraction, which ironically, does not necessarily require the understanding of abstraction…but now you know of its existence.

The difference between understanding the proof of the Pythagorean Theorem versus using the Pythagorean Theorem is that simply using it requires no understanding, but is also severely limited in its use, while understanding it exposes us to the fundamental “disciplines” that allow it to be true, which also significantly increase its potential value in determining more than just the length of the hypotenuse of a right triangle.

The Egyptians effectively reversed the Pythagorean Theorem to ensure the corners of their structures were indeed right angles, effectively demonstrating the foundational basis of the Pythagorean Theorem.

Certainly, there must be other proofs of the Pythagorean Theorem that require understanding of other mathematical disciplines, such as trigonometry, et al.  It is conceivable that by using the Pythagorean Theorem as merely a seed, understanding or at least exposure of advanced mathematical disciplines is possible.

As presented to us, the Pythagorean Theorem is merely a concise and simple codified proscription of sequences that must be done in order to obtain a single answer.

Any insights that proving the Pythagorean Theorem yields will be individual and likely differ as such; each of us will understand what we are ready to understand.  Returning to study the proof periodically will also likely yield additional insights.

Just like a kata.

What’s also interesting is the Pythagoras Sensei did not even know he was a teacher; he was merely studying.  But that’s another topic…