
Einstein's boyhood proof of the Pythagorean theorem (2015) - S4M
https://www.newyorker.com/tech/elements/einsteins-first-proof-pythagorean-theorem
======
herodotus
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
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?

~~~
cyphar
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
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
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
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.

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chx
Since we are on HN and talking of the Pythagorean theorem, I must mention E.
W. Dijkstra's proof.
[https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/E...](https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD975.html)
he proves the generalization sgn(α + β - γ) = sgn(a² + b² - c²) also by using
similar triangles. Incredibly neat.

~~~
elzr
Whoa, that IS neat. Thanks for sharing! A key part of his reformulation is π =
α + β + γ (the sum of the internal angles of a triangle equal 2 right angles).
That statement is, funny enough, equivalent to the parallel postulate.

Going on a tangent now: lately I've been thinking that the "dead horse" of the
Pythagorean theorem is actually trying to tell us that flat (as opposed to
fractal!) dimensions come from composable self-similarity (squares).

~~~
chx
Yeah if the parallel postulate is out then this obviously doesn't stand.
Starting from the North Pole, walking down the prime meridian to the equator
then turning right and walking along the equator finally turning right to the
north again at the right point so you walk through New Orleans creates a
triangle on a sphere with three right angles...

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logicallee
This is the most succcinct proof, it's an ancient Chinese visual proof:

[https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras...](https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras/pythagoras/pythagoras7.gif)

It's not labelled this way but notice that the BIG squares are identical,
having sides of a+b. Only the four triangles are rearranged. Just subtract the
four identical triangles.

In the second arrangement, a c^2 area is present in addition to the same four
triangles. In the first arrangement this had been rearranged to an a^2 and a
b^2 area.

It's important to assure yourself the triangles are the same and the big
square is the same - there is no tiny hidden sliver or something, and right
angles are preserved, there are no shenanigans.

second explanation of same:

[http://www.math.toronto.edu/colliand/notes/pythagoras.html](http://www.math.toronto.edu/colliand/notes/pythagoras.html)

~~~
ndr
This is beautiful.

To be honest the second image with the tilted c² is enough. From that picture
alone you can figure the outer square has the same area as the inner square +
4 times the triangle area:

(a+b)² = c² + 4(ab/2)

a² + b² + 2ab = c² + 2ab

a² + b² = c²

~~~
logicallee
Thanks. Your proof works, I could follow it.

Yours does use some elementary algebra[1], which wasn't used in the Chinese
geometric proof. I wonder if ancient Chinese mathematicians could simplify
(a+b)^2 symbolically, or even did symbolic algebra this way?

(When I referred to "ancient Chinese visual proof" I wasn't bullshitting, but
I didn't find a source with the exact 2-part picture, though this makes same
claim using same pictures:
[http://www.researchhistory.org/2012/10/24/earliest-
evidence-...](http://www.researchhistory.org/2012/10/24/earliest-evidence-of-
pythagoras-theorem/) )

I'm sure they knew that the area of triangles, which you also use, is (ab/2) -
but the purely visual proof needs nothing more than the knowledge that the
area of a square is the square of the length of its sides. (And I guess some
obvious facts like that no matter how you divide an area the sum of the areas
of its parts will be same - the reason I mention tiny slivers is in some fake
geometric proofs this intuitive knowledge is abused.

For example, see this excellent description:

[https://en.m.wikipedia.org/wiki/Missing_square_puzzle](https://en.m.wikipedia.org/wiki/Missing_square_puzzle)

Before you open the solution, you can look at the trick for as long as you
want, you won't figure it out.)

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

------
pash
Einstein’s 1949 article in the Saturday Review is “Notes for an Autobiography”
[0]. It’s worth reading.

0\.
[https://archive.org/details/EinsteinAutobiography](https://archive.org/details/EinsteinAutobiography)

------
signa11
> Einstein’s predictions, during a solar eclipse in 1919—was asked if it was
> really true that only three people in the world understood the theory, he
> said nothing. “Don’t be so modest, Eddington!” his questioner said. “On the
> contrary,” Eddington replied. “I’m just wondering who the third might be.”

for another side of eddington, see 'empire of the stars'
[[https://www.amazon.com/Empire-Stars-Obsession-Friendship-
Bet...](https://www.amazon.com/Empire-Stars-Obsession-Friendship-
Betrayal/dp/061834151X)] quite a fun read, i picked it up on a whim, and could
not just put it down over the course of a 8hr train ride :)

~~~
mjburgess
You may wish to include the entire sentence:

> When Arthur Eddington—the British astrophysicist who led the team that
> confirmed Einstein’s predictions, during a solar eclipse in 1919—was asked
> if it was really true that only three people in the world understood the
> theory, he said nothing. “Don’t be so modest, Eddington!” his questioner
> said. “On the contrary,” Eddington replied. “I’m just wondering who the
> third might be.”

It's hard to understand without it.

~~~
mayankkaizen
If you are alluding to some nuance, I probably missed it.

~~~
skj
In the first quote it is implied that Einstein is being asked the question.
Then it isn't clear who Eddington was. The questioner, I thought at first.
Then no, a phone typo of Einstein? I found it confusing and the larger quote
cleared it up.

------
Hasz
Every time Pythagoras is brought up, I can't help but think of the cult of
Pythagoras as well.

One of the tenants is the refutation of beans. For what reason, I have no
idea. I've always found it strange that a mind so capable was also equally
capable of such folly.

------
SebNag_
Trying to figure out the proof of a² + b² = c² by myself, without looking up
the solution, was somehow exiting. Being exposed to a riddle and trying to
find the solution is kinda cool.

However, not solving it after 10 minutes left me feeling a bit dumb... :)

~~~
jacobolus
This is a general problem with mathematics education. See
[http://toomandre.com/travel/sweden05/WP-SWEDEN-
NEW.pdf](http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf)

Real (non-trivial, non-obvious) problems that someone hasn’t seen before can
take hours, days, weeks, years, whole careers, or sometimes centuries to
solve. Some of them later turn out to be impossible (and for many we still
just don’t know).

 _Real_ math education would have students grappling with relatively open-
ended problems that take significant amounts of rumination and some cleverness
to solve. It would explicitly encourage/reward close critical reading,
creative brainstorming, planning, strategic thinking, generalization and
specialization, executive control (e.g. time management), error checking, and
clarity of exposition (including when asking for help after being stuck).
There would be no shame in throwing out incorrect hypotheses, asking for
clarification, getting stuck on a problem, making subtle mistakes which could
serve as good examples for future improvement, etc. But skill and stamina at
such work must be trained slowly, starting from an early age.

The problem is that current (US) math education instead pre-chews everything,
assigns students lists of exercises almost identical to what they saw someone
solve before, and mostly tests memorization/recall and willingness to do the
same trivial task over and over for hours despite being terribly bored, under
purely extrinsic motivation.

For people used to such math homework, the standard response to a single
problem which takes >5 minutes to work through is to give up.

~~~
dagw
_The problem is that current (US) math education..._

Worth noting that the US tried the kind of math education you are suggesting
(called the New Math initiative) and it failed miserably. The math education
we are seeing today is largely born out of a counter reaction to that failure.

~~~
jacobolus
The New Math was something fairly different than what I am suggesting. It was
an attempt at an alternate curriculum for primary/secondary school based on
higher-level / more abstract mathematical topics, partially displacing study
of arithmetic.

The New Math curriculum per se wasn’t so terrible, though it certainly had
flaws (like anything invented from scratch out of context and not slowly
developed and tweaked over time in response to feedback in a real-world
setting). The bigger problem was that the proponents of the New Math didn’t
have much buy-in from students, parents, teachers, school administrators, or
the broader society, didn’t really do any outreach or teacher training, didn’t
really produce enough supporting materials, and just dumped the curriculum on
schools without support.

Parents and teachers didn’t know what to make of the curriculum (were
unqualified to teach with or assess it), and didn’t feel involved in the
process, and as a result there was a lot of opposition.

But what I’m talking about is not teaching different subjects per se, but
teaching whatever subject in a different way, focused more on solving problems
and thinking than on precisely mimicking teacher’s demonstrations or
memorizing formulas. The current typical math pedagogy is patronizing,
emphasizes memorization/recall and very careful attention to details
(sometimes irrelevant details about formatting), teaches students that they
shouldn’t try to think for themselves and teaches them to conflate getting the
right answer with being “smart” or “good at math” and that anyone who makes a
mistake or doesn’t know how to get the answer is “stupid” or inherently
incapable.

------
moomin
A friend of mine used to describe this as a “one-line proof”. Never heard it
ascribed to Einstein before, though.

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tempestn
Had to stop reading long enough to see if I could prove it myself using
similar triangles. This is definitely cleaner than what I came up with though,
which involved creating a similar triangle with side b1 set equal to side c of
the original.

------
garyrob
Can anyone point to a proof that for similar non-isosceles right triangles,
the area is proportional to the square of a side? Can that be generalized to
other shapes?

~~~
jonsen
When you scale a figure, say by doubling all distances, the area quadruples.
Scale by factor k and the area goes up by a factor k^2. So if the area
proportion between the triangle and the square is T:S, and we scale both
figures by the same factor k, we get areas k^2* T and k^2* S respectively.

But k^2* T:k^2* S = T:S.

------
jungletime
I remember watching a documentary once, and it kind of made the case that
Einstein wasn't an isolated genius, but more like Mark Zuckerberg. Basically
organizing and putting together other scientists work,then unfairly getting
all the credit. Anyone seen this documentary before? Its been years since I
watched it, so memory is vague.

~~~
malkarouri
For what it's worth, organizing other scientists work from a new perspective
and having a unified framework is the highest form of genius a scientist can
achieve. That is why for example Einstein is credited with relativity rather
than Poincare.

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jordigh
I find this style a bit frustrating to read. I just want to know what his
proof is, but there's so much fluff in the article that the actual meat is
kind of hidden. So anyway, "his" proof is to drop a height to the hypotenuse
and use similar triangles. Ah, right, I know this proof and I remember it as
the canonical proof my grade 9 geometry textbook presented.

~~~
mlevental
this is the New Yorker not an maa article. it's supposed to be entertaining
for a typical New Yorker reader. I swear man some people are so tone deaf it's
frustrating: the whole world doesn't cater to people like us. at least try to
make at attempt to empathize.

edit:

besides it's literally labeled steps 1-5 (i had no trouble skipping the
"fluff" and finding the proof).

~~~
jordigh
I'm sure other people like this style of article. I'm just telling you I
don't.

Even the steps 1-5 are a bit fluffy for me. If you had just told me "drop a
height to the the hypotenuse and use similar triangles" I would have known
what was meant, more succinctly.

Do you have a suggestion on where to get unfluffed math news? I sometimes look
at the AMS notices or Terry Tao's blog, but they don't cover quite the same
breadth as things like Quanta magazine. I don't know where to get the big news
for major events from an unfluffed source. Usually by the time the news has
hit Quanta magazine or the NYT, it has already been circulating somewhere else
long enough to be old news. I just don't know where else to look.

~~~
mlevental
[http://www.scimagojr.com/journalrank.php?category=2603](http://www.scimagojr.com/journalrank.php?category=2603)

~~~
jordigh
I guess, but that's a bit of a firehose. Is there really nothing intermediate
between the NYT and just scouring all the journals? Something like the AMS
notices or Terry Tao's blog, but slightly more frequently updated?

~~~
nileshtrivedi
Check out Quanta magazine

------
jwilk
(2015)

~~~
S4M
Thank you. I edited the title.

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thethirdone
If you use noscript, the images of the proof will not appear making it nearly
impossible to follow.

~~~
tyingq
Sympathetic, but noscript compatibility has been out the window for a decade
at least. You can't reliably book travel, interact socially, read email, buy
basic retail items, etc, without being extremely selective about providers.
I'm with you philosophically, but that ship sailed long ago.

~~~
S4M
The thing is that there is no particular reason for non animated images to
require javascript.

~~~
philtar
Lazy loading

~~~
deathanatos
They're <10KiB a piece, most of them.

If the New Yorker would use the proper format of SVG here (or even PNG — the
diagrams are all JPEGs), they would likely compress even better; most of them
also include a ton of whitespace either side of the actual image. Testing one
of them, the size falls by ~28% if you correct for all this.

On mobile, where my connection is less reliable, I also have a strong distaste
for lazy loading: it squanders all the time available from when I started
reading until now. Then, when the image is finally in view, then we start the
long haul of fetching it, and risk that I've again lost good connectivity.

~~~
NamTaf
Conversely, I like that it doesn't waste my data if I only get 1/2 way through
an article and have to close it because I'm out and about and the world around
me may interrupt me reading something on my phone.

