Hacker News Reader

Comment by herodotus | Hacker News Reader

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)

View on HN

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?

It doesn't show that it's true in all cases, just that one.

It could leave the hypotenuse intact and let you change the size of other sides (while enforcing the right angle).

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.

You could use it as a starting point to a proof by induction maybe, but it currently only proves one case of the theorem.

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.)

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).

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.

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.

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.

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

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

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.

Constant depth volume is mathematically identical to area.

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.

Reply on news.ycombinator.com