
Proof Without Words: Gregory’s Theorem - JohnHammersley
https://divisbyzero.com/2018/09/28/proof-without-word-gregorys-theorem/
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herodotus
My favourite proof without words is the proof that the sum of the interior
angles of any triangle is 180 degrees. Proof: draw an arbitrary triangle in
chalk on the ground. Stand on one of the lines at one of the corners and note
the direction you are facing. Now back up until you are at the corner of the
triangle behind you. Rotate your self by shuffling inside the angle to the
next line. Walk to the next corner, repeat. Walk (backwards this time) to the
last corner, repeat. Now shuffle back to your starting position. You are
facing exactly 180 degrees in the other direction!

Of course it is way better to just do this than write it down! But obviously I
can't do it on HN.

~~~
whatshisface
Does that prove that every triangle has angles adding up to 180, or just that
one?

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Corvus
“draw an arbitrary triangle” means this works with any triangle.

~~~
whatshisface
But you'd have to try an infinite number of triangles before you were
convinced that it worked for all of them.

~~~
justinpombrio
This is an interesting confusion.

OP said "draw an arbitrary triangle in chalk on the ground", and from there
gave a set of instructions that showed you how to check that the interior
angles of _that_ triangle added up to 180 degrees. You're right that that
concrete set of instructions only proves it for that one triangle.

But that's not the whole proof. If I followed these instructions for a right
triangle with angles 45,45,90, I'd only have proved that _that_ triangle's
angles added up to 180 degrees.

The full proof is that if you _imagine_ doing this, for _any_ triangle, it's
clear that it will always work. When I picture this in my head, it's clear
that I'm going to end up having turned 180 degrees, regardless of the
measurements of the triangle.

This leap: starting with a set of concrete instructions that you can do on a
particular object, but then verifying that they would work no matter what
object you started with, is common in proofs.

EDIT: Nevermind, totally unrelated.

~~~
whatshisface
Why is it clear that it will always work? It would be a proof if that part was
written down. Since we all already know that the angles add up to 180, it
might just be the intuition taught to us by another proof of the statement
leaking backwards into the implicit step in this attempt at proving the
statement. The part that's left off (the proof that it always works) is
actually the bigger part of the contraption.

~~~
justinpombrio
Because you end up back on the line you started at, facing in the opposite
direction. That's a 180 degree turn.

~~~
whatshisface
Suppose I found a triangle whose internal angles added up to 190 degrees. If I
did the experiment on it, I would end up 10 degrees away from where I was
predicted to be. How can this scenario be ruled out?

~~~
justinpombrio
Both of these things are true:

(i) Like you said, if you found a triangle whose internal angles added up to
190 degrees, and you followed the procedure, you would have turned 190
degrees, rather than 180 degrees. This is true because during the procedure,
you turned three times, each by one of the angles of the triangle, so the
total amount you turned was the sum of the angles.

(ii) You would end up back at the line you started with, facing in exactly the
opposite direction. This is true because the last step of the instructions is
to turn until you're facing in the direction of this line. Thus you have
turned 180 degrees.

Now of course this is nonsense: you can't have turned 180 degrees, but also
turned 190 degrees. How did we arrive at nonsense like this? The logic is
sound, so it must have been one of the assumptions. Which assumption is
questionable? Oh, right, the triangle whose angles added up to 190 degrees.

This is a proof by contradiction, that shows that a triangle whose angles add
up to 190 degrees cannot exist.

~~~
whatshisface
Inside of that is the assumption that the sum of the inner angles that I
rotate by as I walk around a closed loop is equal to the angle between my
initial and final directions at the starting point, so that having turned 180
means that my feet have shuffled by a total of 180. That isn't true outside of
Euclidean geometry, which indicates that its proof might not be as trivial as
it seems.

~~~
justinpombrio
Bah, I _almost_ talked about what would happen if you did this on a sphere.
Yes, there is an assumption that rotations and translations are commutative
and associative. We're so used to this that our intuitions sensibly hide it.

The fact that there are hidden assumptions doesn't invalidate the proof,
though. There are _always_ hidden assumptions. Even if I give you formal
axioms to reason with, you need a system in which to interpret those axioms.

~~~
whatshisface
Although I see the reasoning, I'm still not comfortable with the proof. It
sounds to me like "plug something in to the Zeta function, observe that it
isn't a nontrivial zero, conclude that there are no nontrivial zeros." Even if
it seems completely intuitve to me I still wouldn't consider it proven.

~~~
justinpombrio
Interesting. For me, there is no clear gap between intuitive proofs and formal
proofs. Sure, intuition can lead you astray, but as you do mathematics, you
develop your intuition so that you stop being intuitively certain of false
things. Contrariwise, formal proofs are more likely to contain dumb algebraic
errors, but as you do mathematics you learn to be exceedingly careful in your
calculations.

But the wider point I want to make is that there's no gap between the two.
_Real proofs aren't fully expanded._ If you've worked with a theorem prover
like CoQ, it becomes painfully clear how many steps even the simplest proof
skips. For example, the proof that the sqrt of 2 is irrational is really easy:

[https://www.math.utah.edu/~pa/math/q1.html](https://www.math.utah.edu/~pa/math/q1.html)

But look at how many steps this skips, if you want to get close to actual
axioms and definitions:

\- You squared both sides of an equation. In this case, that's fine because
you're only doing forward reasoning, but if you wanted to reason backwards
you'd also have to check that both sides had the same sign to start.

\- You multiplied by q^2. That's only valid if q^2 is nonzero. Now intuitively
we know that q^2 is nonzero, since q is nonzero. But it needs a proof.

\- You deduced that p was even from the fact that p^2 was even. How do you
prove that? My first thought is to use the fundamental theorem of arithmetic.
I don't know about you, but my intuition completely glossed over the fact that
the proof that sqrt(2) is irrational made use of the fundamental theorem of
arithmetic when I first read it. Either that, or there's another way to prove
this; what is it?

Now, you could expand this proof to include all the steps. But we don't
bother, because it's painstaking and not actually that likely to catch flaws
in the proof, because we intuitively know these things. Likewise, I feel like
the triangle proof is the same way: it skips over some things, but we
intuitively know it's okay and you could expand it (to talk about
commutativity of translation and rotation) but there's usually not much reason
to bother.

Although it's a lot more obvious how to go about expanding the proof that
sqrt(2) is irrational, I'll give you that.

~~~
whatshisface
There is an implicit, widely-held sense that the detail in proofs should scale
with the expected training of the people who will be reading them. If your
intended audience has internalized their field so well that the expansion and
checking happens completely automatically and subconsciously, more power to
them - but on the other end of the scale you have the proof that the square
root of two is irrational, or this one about interior angles. Other than
machine-aided proofs the most thoroughly expanded you ever see anything is in
highschool geometry!

So, what justifies this scheme? I would say that once you have seen a
technique used in full detail, you don't need to see the detail elsewhere
because there's a meta-theorem in your head that applies to every case where
things line up in a pattern where the technique works. Slowly this replaces
your natural intuition.

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zazen
I haven't previously come across the idea of trying to make wordless proofs. I
guess this is the mathematical equivalent of code golf: a clever game for
insiders to play, but basically the opposite of good practice from a
readability perspective.

~~~
laurentl
I didn’t get this feeling from the proof in the article. I thought there was a
kind of poetry to it. The diagram and colors make it reasonably easy to
understand and they emphasize the geometrical nature of the proof, which uses
only elementary math: the formula for the area of a triangle and Thales’
theorem (If my memory is correct, I haven’t used that name since high
school!).

On the other hand, a “proof with words” would likely have derived the explicit
formulas for I_n and C_n in order to prove the relationship and would have
been much more clanky.

If I had to make a comparison, this felt like reading good functional code.
Concise, a bit mysterious at first, but once you understand what it does you
feel enlightened. (Vs eg. wading through for loops and index arithmetic, which
also does the job but is more tedious to read)

~~~
whatshisface
I think it wasn't so much a proof as it was an image that made you realize the
proof with some thought. Still, that's about what all published proofs are in
the end, isn't it?

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heinrichhartman
Here is another proof without words for the Pythagorean Theorem:
[https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/Fil...](https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/File:Pythagoras-
proof-anim.svg)

~~~
JBiserkov
Nice. I also like the proof which uses the only the first image a calculates
the area of the square twice:

    
    
        c^2 + 4*a*b/2 = c^2 + 2*a*b
    
        (a+b)^2 = a^2 + b^2 + 2*a*b

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espeed
Evidently Oliver Selfridge [1] said a puzzle based on deriving the radius like
this once tripped Feynman up...

"Puzzles from last week"
[http://web.media.mit.edu/~walter/MAS-A12/week11.html](http://web.media.mit.edu/~walter/MAS-A12/week11.html)

[1]
[https://en.wikipedia.org/wiki/Oliver_Selfridge](https://en.wikipedia.org/wiki/Oliver_Selfridge)

~~~
abecedarius
What's the expected mistaken answer?

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louis_
Hi there. Just a precision: the fact that the sum of the interior angles of a
triangle is 180 degrees is not really provable, it is a postulate of the
Euclidean geometry. It is an intuitive and accepted property, it reflects the
fact that the geometric space usually considered is flat. Proving this
statement would be possible provided that you change basic postulates. At that
moment, it would become a proposition of these new mathematics and would
certainly be provable starting from the new postulates. Changing these
postulates is possible, that's how we got the parabolic and hyperbolic
geometries, see wiki!

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gcb0
great. now draw a few hundred more n. and don't forget the min-max values.

