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What is the deep idea in the Pythagorean theorem?

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  • Andrius Kulikauskas
    Hi Pamela, Kiyavilo and all! I m getting great response to a letter that I wrote to the Math Future group. I share my second letter. Perhaps we have
    Message 1 of 1 , Feb 11, 2011
      Hi Pamela, Kiyavilo and all! I'm getting great response to a letter
      that I wrote to the Math Future group. I share my second letter.
      Perhaps we have thoughts? Andrius

      I'm replying to the thread "Doubling the Area and Discovery" at
      http://groups.google.com/group/mathfuture/, and yet starting a new
      thread, which might be interesting to authors Eli Maor and Mark Levi,
      and so I CC them.

      Linda, going back to your original question about "fourths" of a
      rectangle. I found it interesting, as you said, that if you cut up a
      rectangle by its diagonals, then all of the fourths have the same area.

      I think a straightforward proof is to cut up the rectangle into
      "eighths". For example, you can cut up the rectangles into four smaller
      rectangles, then slash those in half along the diagonals. Then you can
      see that there are eight right triangles, all of the same size. They
      make up the fourths.

      Alexander, what a fantastic site you have! I appreciate your page of
      proofs of the Pythagorean theorem:
      I just finished looking through "The Pythagorean theorem: a 4,000-year
      history" by Eli Maor, which I found quite inspiring.

      I'll take the opportunity to explain the kind of book I'd like to write
      in the Public Domain for adult self-learners of basic math. I'm trying
      to explain teach math in terms of the deepest principles, and thus the
      fewest principles, but also classic problems that illustrate them.

      Here's a classic problem: Your usual sandwich at your Usual Cafe costs
      $9. Across the street, at the Unusual Cafe, the same sandwich costs
      one-third more. But today you have a coupon good at the Unusual Cafe
      with which everything is one-third off. Where is it cheaper to buy the

      I've had stock traders and professors tell me "the price is the same".
      But actually, one-third more than $9 is $12, and one-third off of $12 is
      $8. So today it's cheaper across the street. The lesson is a deep
      idea: "Algebra is thinking step-by-step". You can't solve this problem
      by doing it in one step. That's because the meaning of "one-third"
      changes. This may be the simplest problem to illustrate this point.
      And it's a real life problem that builds math intuition, not a contrived
      problem that destroys it. I think that several dozen such "classic
      problems" could illustrate all of the deep ideas in basic math.

      Some more deep ideas: "Every answer is an amount and a unit".
      (Otherwise, we don't know how to interpret it, we don't have the
      context.) "If the units are the same, then addition means combine."
      "If the units are different, then addition means list." In other words,
      2 feet + 3 feet is 5 feet, but 2 feet + 3 inches isn't 5 anything. And
      similarly, we can add 2 / 7 + 3 /7, or 2 % + 3 %, or 2 X + 3 X, or 2
      million + 3 million (noting that a number can serve as a unit). We use
      a single unit to calculate, but we use multiple units to make final
      answers understandable, as in running a marathon in 2 hours + 9 minutes
      + 36 seconds. This is the spirit that I want to teach in. I wrote a
      set of notes for precalculus where I was able to cover all of it in 10
      weeks at a rapid clip by just focusing on the deep ideas:
      I want to write a short, punchy paperback (something like Darrell Huff's
      "How to Lie With Statistics" which is about 150 pages, or George Polya's
      "How to Solve It") that covers counting, arithmetic, algebra, geometry,
      precalculus. Here are my notes:

      For example, I teach trigonometry from the deep idea "A Right Triangle
      is Half of a Rectangle". I think that's much more easy and meaningful
      to memorize than, say, the formula for the area of a triangle, which
      follows directly from this deep idea, yet means nothing of itself. Now,
      this deep idea lets us think of a shape in two ways (it's a map, a
      morphism) so that (in the world of right triangles) the shape is given
      by one angle, whereas (in the world of rectangles) the shape is given by
      the ratio. So immediately we see that there is a one-to-one
      correspondence between an angle and a ratio. And we see that no formula
      is involved! And that we can calculate, that is, approximate the angle
      and the ratio by constructions. So much learned, so quickly, with
      nothing to memorize. (Except that "A Right Triangle is Half of a
      Rectangle" is a very deep idea! which we're never taught.)

      So I ask myself, what's the main point of the Pythagorean theorem a**2 +
      b**2 = c**2. I have a Ph.D. in math, but I didn't learn how to prove
      the Pythagorean theorem until a couple of years ago. At that time I saw
      the picture of the Chinese proof (Alexander's proof #4) and I thought it
      through as a principle "Four times a right triangle is the difference of
      two squares", which, if you see the picture, makes sense to a small
      child, and gives a straightforward algebraic derivation: 4 x 1/2 ab =
      (a + b)**2 - c**2. The diagram also suggests the relationship between
      two different coordinate systems.
      Yet it didn't seem central enough of an idea.

      So I found Eli Maor's book from the Chicago Public Library. It has a
      wonderful variety of proofs and insights. What struck me? And what
      strikes us?

      The deepest idea that I found was "A right triangle is two copies of
      itself". Actually, that's a consequence of another deep idea, which is
      that "A triangle is two right triangles" which we see by drawing the
      altitude, and which yields the formula for the area of a triangle.

      So if your initial triangle happens to be a right triangle, then you
      draw the altitude and see that it's two right triangles. Then note that
      they all have a right angle. And each piece shares an acute angle with
      the whole. But then, for each piece, the third angle must be the same
      as the third angle of the whole. So the pieces and the whole must have
      the same shape.

      Then another deep idea is "Any area with a dimension of length X is
      proportional to a square of area X**2".

      Then we see, as in Eli Maor's own noteworthy "instantaneous" proof of
      the Pythagorean theorem, that the whole right triangle's area equals the
      two pieces' area, and so their areas A + B = C relate also as k * a**2 +
      k * b**2 = k * c**2 for some proportion k. That means that a**2 + b**2
      = c**2, but much more importantly, that k * a**2 + k * b**2 = k * c**2
      and A + B = C for ANY three similar shapes with some dimension a, b, c!

      At this point, I start to see, Why are we learning a**2 + b**2 = c**2 ?
      Why are we fixating on that? It's just as meaningless as a*b/2.

      Whereas the idea "A right triangle is two copies of itself" is a
      beautiful, fractal, self similar idea with great implications. (And I
      never knew this idea!) It's saying that a right triangle works as a
      mechanism that can split anything into two copies of itself and yet
      preserve the total area. That's the point of Euclid's classic theorem
      (Alexander's proof #1) except that Euclid, following the Greek
      aesthetic, wants to construct it for a square, and so the square gets
      split along the altitude into two pieces, two rectangles, then moving
      along (as halves, which is to say, two right triangles), preserving
      their areas, because their widths don't change, until they are shown to
      be halves of the desired squares.

      "A right triangle is two copies of itself" is, I suppose, at the heart
      of "Four times a right triangle is the difference of two squares"
      because if you rotate a right triangle, then after four rotations you
      get back to where you started.

      But how to truly think about this? There's something going on here by
      which, thanks to the self-similarity of the right triangle, two
      dimensions become independent, and yet, I suppose, comparable. What is
      it all about!

      Here's another proof that's so simple:
      which says, look at a tesellation, an infinite tiling of squares of
      lengths a and b, and see on the diagonal the squares of length c they
      form, and note that there is a one-to-one correspondence between the
      squares of lengths a, b and c, and you can mark out a region as large as
      you like, and it will be more and more clear that the a**2 and the b**2
      must end up covering the same ground as the c**2.

      Mark Levi has several proofs of the Pythagorean theorem in his book "The
      Mathematical Mechanic" which I have yet to fathom. They are very simple
      but require some physics insights. He's playing slick with physical
      units, setting them equal to 1 as needed, as physicists do. And then
      he's able to show, for example, that the Pythagorean theorem follows
      immediately from the fact that a water-filled right triangular fish
      tank, free to rotate, won't. Or that it follows from the areas swept
      out by a, b, c if they are rotated around a circle. Or he proves it
      with springs balancing forces.

      I think you see how I'm thinking, how I'm trying to find and share the
      gist, which I was never taught. We're given piles of textbooks with
      thousands of pages when perhaps fifty problems could communicate all of
      basic math.

      I hope to write such a book. I think that's one way I should try to
      make a living. I think it would be honest work. I'm impressed that Tom
      Henderson was so successful raising money at http://www.kickstarter.com
      https://www.kickstarter.com/projects/1541803748/punk-mathematics to
      write a book "Punk Math". He certainly presents his math viscerally.
      So I thought that "Gospel Math" would represent my aesthetic.

      I appreciate thoughts, how do we like such an approach? and what do we
      think is at the heart of the Pythagorean theorem? And also, how might I
      try to make a living from creating such books or learning materials in
      the Public Domain? There seems to be a real need and there must be a
      way to earn at least a few months pay.

      Wow! I really appreciate the great responses to my post from yesterday.
      I hope to reply tomorrow.

      Thank you,


      Andrius Kulikauskas
      +1 (773) 306-3807
      Twitter: @selflearners
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