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The Ontario Science Centre once had a "proof" that was made using wooden strips that were about an inch wide and perhaps a quarter of an inch deep. There was a

The Ontario Science Centre once had a "proof" that was made using wooden strips that were about an inch wide and perhaps a quarter of an inch deep. There was a right angled triangle with squares on each side. It looked just like typical diagrams except that (a) it was mounted on the wall, (b) the triangle and the squares were covered with glass, (c) there was exactly enough coloured liquid inside to fill the square on the hypotenuse. Every minute or so, the triangle rotated in such a way as to allow all the water from the square on the hypotenuse to flow into the squares on the other two sides. Of course, the fit was exact. I remember watching the exhibit, thinking about it, and suddenly feeling a greater depth of understanding, not of geometry, but of trigonometry! Of course, not a proof, but a brilliant exhibit. (Here is a similar demo on youtube: https://www.youtube.com/watch?v=CAkMUdeB06o https://www.youtube.com/watch?v=CAkMUdeB06o)

abtinf 9y ago
That is a great demo. Why isn't it a proof? In a sense, isn't a concrete fact of reality better than an abstract proof?
- nerdponx 9y ago
  It doesn't show that it's true in all cases, just that one.
  - muro 9y ago
    It could leave the hypotenuse intact and let you change the size of other sides (while enforcing the right angle).
    - dagw 9y ago
      Also, imagine if a^2+b^2 was actually equal to c^1.9998724... Your physical model wouldn't be accurate enough to detect that discrepancy.
- immad 9y ago
  You could use it as a starting point to a proof by induction maybe, but it currently only proves one case of the theorem.
  - tuukkah 9y ago
    Further, it doesn't prove exact mathematical equality. Depending on the scale of the demonstration and the accuracy of the observations, it demonstrates approximate equality up to a certain precision. (You could also demonstrate π=3.14.)
- cyphar 9y ago
  It only shows that the Pythagorean theorem holds for that one particular set of side lengths. A mathematical proof requires that you show it is true for all possible side lengths. An analogy would be that I take two equal-length sticks and say "given any two sticks they will be the same length". I have an example (the two sticks I'm holding) but this does not amount to a proof of my statement (and the statement is obviously incorrect).
  - jacobolus 9y ago
    It’s interesting to wonder if a contraption might be built whereby some slider adjusts the sizes of the two smaller squares, while maintaining the right angle (i.e. keeps the corner on a circle with the hypotenuse as diameter). It would be much more mechanically complicated and harder to build, but pretty awesome.
    - srtjstjsj 9y ago
      That's easy enough, but it changes the total area, which means some kind of drainage is required, which complicates the demonstration of equal-area squares.
      - jacobolus 9y ago
        As long as you don’t change the size of the large square, the total area of the two smaller squares must be the same. That’s just a restatement of the Pythagorean Theorem.
- soVeryTired 9y ago
  Demonstrations based on physical objects aren't mathematically precise. Who's to say that the wood doesn't flex, or that the fluid doesn't compress? You can see that the areas are equal to within measurement tolerance, but not that they're exactly equal. Here's a classic 'physical' pictorial proof that 31.5 = 32.5: https://jeremykun.files.wordpress.com/2011/07/31equals32.png https://jeremykun.files.wordpress.com/2011/07/31equals32.png
  - deleted 9y ago
    [deleted]
- adrianN 9y ago
  Besides the issue that a physical demonstration only shows that the theorem is true for one particular triangle, reality is sometimes more tricky than one thinks. See for example this https://en.wikipedia.org/wiki/Missing_square_puzzle https://en.wikipedia.org/wiki/Missing_square_puzzle
kybernetikos 9y ago
the thing that bugs me about that is that its using volume to illustrate a fact about area. Now of course it's legitimate assuming all the rectangular prisms have the same depth but it still feels a bit like some sleight of hand is taking place.
- srtjstjsj 9y ago
  Constant depth volume is mathematically identical to area.
  - kybernetikos 9y ago
    That's why I said > Now of course it's legitimate assuming all the rectangular prisms have the same depth But I still consider it to add an extra element to the demonstration that may potentially confuse people who look at it (or have them wondering 'are they all the same depth or not?'), which is something you don't want in an educational demo.

abtinf 9y ago

That is a great demo. Why isn't it a proof? In a sense, isn't a concrete fact of reality better than an abstract proof?

nerdponx 9y ago
It doesn't show that it's true in all cases, just that one.
- muro 9y ago
  It could leave the hypotenuse intact and let you change the size of other sides (while enforcing the right angle).
  - dagw 9y ago
    Also, imagine if a^2+b^2 was actually equal to c^1.9998724... Your physical model wouldn't be accurate enough to detect that discrepancy.
immad 9y ago
You could use it as a starting point to a proof by induction maybe, but it currently only proves one case of the theorem.
- tuukkah 9y ago
  Further, it doesn't prove exact mathematical equality. Depending on the scale of the demonstration and the accuracy of the observations, it demonstrates approximate equality up to a certain precision. (You could also demonstrate π=3.14.)
cyphar 9y ago
It only shows that the Pythagorean theorem holds for that one particular set of side lengths. A mathematical proof requires that you show it is true for all possible side lengths. An analogy would be that I take two equal-length sticks and say "given any two sticks they will be the same length". I have an example (the two sticks I'm holding) but this does not amount to a proof of my statement (and the statement is obviously incorrect).
- jacobolus 9y ago
  It’s interesting to wonder if a contraption might be built whereby some slider adjusts the sizes of the two smaller squares, while maintaining the right angle (i.e. keeps the corner on a circle with the hypotenuse as diameter). It would be much more mechanically complicated and harder to build, but pretty awesome.
  - srtjstjsj 9y ago
    That's easy enough, but it changes the total area, which means some kind of drainage is required, which complicates the demonstration of equal-area squares.
    - jacobolus 9y ago
      As long as you don’t change the size of the large square, the total area of the two smaller squares must be the same. That’s just a restatement of the Pythagorean Theorem.
soVeryTired 9y ago
Demonstrations based on physical objects aren't mathematically precise. Who's to say that the wood doesn't flex, or that the fluid doesn't compress? You can see that the areas are equal to within measurement tolerance, but not that they're exactly equal. Here's a classic 'physical' pictorial proof that 31.5 = 32.5: https://jeremykun.files.wordpress.com/2011/07/31equals32.png https://jeremykun.files.wordpress.com/2011/07/31equals32.png
- deleted 9y ago
  [deleted]
adrianN 9y ago
Besides the issue that a physical demonstration only shows that the theorem is true for one particular triangle, reality is sometimes more tricky than one thinks. See for example this https://en.wikipedia.org/wiki/Missing_square_puzzle https://en.wikipedia.org/wiki/Missing_square_puzzle

kybernetikos 9y ago

the thing that bugs me about that is that its using volume to illustrate a fact about area. Now of course it's legitimate assuming all the rectangular prisms have the same depth but it still feels a bit like some sleight of hand is taking place.

srtjstjsj 9y ago
Constant depth volume is mathematically identical to area.
- kybernetikos 9y ago
  That's why I said > Now of course it's legitimate assuming all the rectangular prisms have the same depth But I still consider it to add an extra element to the demonstration that may potentially confuse people who look at it (or have them wondering 'are they all the same depth or not?'), which is something you don't want in an educational demo.