Pythagoras Sensei


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…


Raw notes from flying to Japan…


Posted by admin | Posted in Bujinkan Budo Taijutsu | Posted on 25-09-2012

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I wrote these notes on the plane during my last trip to Japan, with the intention of sometime going over them and restructuring them into a semi-coherent post.  I leave in 8 hours for my next trip to Japan, and find myself having not done anything with them yet.  I will likely be inspired to write more notes down during my new journey, but I am compelled to share my thoughts from the last one, although they will be in “raw” form, with basically little editorializing.  Please be aware that they’re not all in their “nicest” form.  Also realize that these were spontaneous thoughts at the time, and reflect my opinions; feel free to challenge them, but I will likely not entertain derision, since I’ll think you’re an asshat and thus not worth entertaining.

  • The role of uke is not about winning; it is about helping tori.
  • Kata are not fighting techniques; they are teaching techniques developed from battlefield insights.
  • Although tori is scripted to win, tori should mindfully enter kata with the purpose of studying the lessons inherent within the kata.
  • Although it may be conceivable to view kata as a viable fighting technique, it is important to realize that kata are essentially codified patters of movement, and thus present a predictable sequence; predictability is easily defeated.
  • If kata are can be easily defeated, then why study kata?
  • When done purposefully and mindfully—taking the time for proper inspection—it is not difficult to not only identify fundamental techniques, but even the fundamentals within those fundamentals, the “meta-fundamentals” or kiso.
  • The proper study of kata inherently implies a deep understanding of fundamentals.
  • A potential analogy for a kata is to equate it to a cargo vessel.  Cargo vessels contain crates, and each crate it contains in turn contains pallets, which in turn contain a collection of smaller items, ad nauseum.  If the kata is the cargo vessel, the crates are the fundamentals—kihon—within the kata, and the containers within the crates are the fundamentals of the fundamentals.
  • Properly studying kata should make each and every one of us question our basics, and such questioning should in turn make us study said basics.
  • Big things are made up of little things.  I often hear how as individuals we wish to improve the “whole,” whether that “whole” is an organization we may belong to, family, humanity, or even universe, depending on our current level of existentialism.  Due to ego, most of us don’t realize that as “knots on the net,” we as individuals are part of the “whole,” and thus by simply improving ourselves, we can in turn improve the “whole.”
  • When we study kata, and get to a point within the kata we don’t “get” or understand, instead of succumbing to the ego-driven response to move through the sequence faster and stronger, in an attempt to “ensure” our “victory,” we should instead realize that the failure of the kata is likely due to a failure in our basics, and that by improving our basics, we will improve the kata, and in the process perhaps get a glimpse of what secrets and treasures the kata contains.
  • Too often I encounter a “class” where we will spend 5-10 minutes on a kata before moving into “henka.”  (Sometimes, a “class” will just “do henka,” but rarely is it explained what kata it is a “henka” of.  I’m starting to realize that to many “teachers,” “henka” just really means “I’m pulling this right out of my ass.”  Learn to discern this and avoid such “teachers,” as your time would be better spent drilling basics than trying to understand the horseshit they’re peddling.
  • How worthless are kata?  About as worthless as you want to make them.
  • Beyond being a vessel containing static fundamentals, kata also provide not only contexts to such fundamentals, but transitions between them.  So in addition to providing us with the opportunity to self-inspect the fundamentals within our fundamentals, kata also give us the opportunity to self-inspect the fundamentals within our transitions.  (Are we in kamae between our kamae?  Are we breaking namba?  Etcetera…)
  • Perhaps the inyo of kata is that by providing a repetitive and codified pattern of movements—predictable and defeatable—kata also provide us the tools and incentives to better understand our art and even ourselves.  Perhaps it is only through kata that we can become free of kata.
  • Over the centuries, the only constant to our art is the shoden within the densho, since kuden and taiden are and will always be open to individual perspectives, interpretations, and “filters,” and shinden is personal.  It is my personal belief that our ultimate goal is to discover the “treasures” of the densho, and we do this by “learning” from those that have been “studying” longer than us.  At the risk of sounding silly, from the kuden and taiden from our “teachers,” we strive of the shinden of the shoden.