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Visual walk through of Euclid's proof of the Pythagorean theorem

97 points by vicapow 13y ago

symmetricsaurus 13y ago
Very nice visualization. Reading these old proofs is quite tedious but having this visualization makes it much easier to follow. Still, trying to understand Euclid makes you thankful for the more than 2000 years of advancements in mathematical notation and theory.
- Jtsummers 13y ago
  > Still, trying to understand Euclid makes you thankful for the more than 2000 years of advancements in mathematical notation and theory. The importance of notation is greatly underestimated, IMO. Even when programming, an arguably mathematical activity, many people still resist improvements to notation that allow for clearer expressions. The contortions some of my colleagues have gone through in Fortran or C++ (because they refuse to use libraries that would clean up their code significantly, they roll their own versions) to express their ideas.
justinpombrio 13y ago
Or, if you want a proof with fewer words, http://isomorphismes.tumblr.com/image/790452593 http://isomorphismes.tumblr.com/image/790452593
- fryguy 13y ago
  This is showing a^2 + b^2 + 2ab (left image) = c^2 + 2ab (right image), since they both have area (a + b)^2. This implies that a^2 + b^2 = c^2, since 2ab is constant across sides.
- csense 13y ago
  This is probably the easiest, most intuitive proof of the Pythagorean Theorem. Euclid's proof, by contrast, is a bit of a slog.
  - psykotic 13y ago
    Euclid has two proofs. The first is the one submitted here. The second one uses the complicated theory of proportion from Book V. However, it's very simple and intuitive when recast in modern terms. Take a right triangle ABC, drop an altitude onto the hypotenuse, note the two smaller right triangles are similar to ABC, calculate the area of ABC in two different ways using this decomposition, and compare the results.
- ryannevius 13y ago
  I always liked this one... http://25.media.tumblr.com/a7ac256b1db7bbb775f559bfa736b962/tumblr_mrsl3goCez1s6r1vho1_400.gif http://25.media.tumblr.com/a7ac256b1db7bbb775f559bfa736b962/...
  - zodiac 13y ago
    That's not a mathematical proof, for example it doesn't show why the theorem holds for all a, b and c, and not just the values used in the demonstration.
    - floatrock 13y ago
      But it gets the target audience to perk up and be interested for more.
commlink 13y ago
Well done, but it would have been better if the graphics depicted a right scalene triangle -- the more general case -- rather than a right isosceles triangle.
- SamReidHughes 13y ago
  You can drag the vertices.
- mannykannot 13y ago
  My thoughts exactly - then I found you can drag point C left and right. Undocumented software!
netrus 13y ago
If you enjoy visualizations of Euclidean geometry, I highly recommend the work of Byrne: http://www.math.ubc.ca/~cass/Euclid/byrne.html http://www.math.ubc.ca/~cass/Euclid/byrne.html Example: http://www.math.ubc.ca/~cass/Euclid/book1/images/bookI-prop1.html http://www.math.ubc.ca/~cass/Euclid/book1/images/bookI-prop1... (I found this to be one of the most impressive examples used by Edward Tufte in his books).
- fuzionmonkey 13y ago
  Byrne's Elements is pretty remarkable. It's an interesting juxtaposition of old 19th century typesetting (complete with initials) and comparatively "modern" graphic design. Some pages look like they could be right out of De Stijl. Archive.org hosts a nice PDF copy, which I find more convenient than the jpegs: https://archive.org/details/firstsixbooksofe00byrn https://archive.org/details/firstsixbooksofe00byrn
agumonkey 13y ago
Funny, few days ago I wanted to write it from scratch starting through the geometric 'interpretation', just to see if I remembered. I had to use `square of sum` identity though. http://imgur.com/umPQpeu http://imgur.com/umPQpeu ps: vector editor is not my website.
onedognight 13y ago
In case it's not obvious, you can drag the points in the diagram to change the size of triangles and the associated squares.
- sesqu 13y ago
  Excellent! I did want different-sized squares.
- clintonc 13y ago
  Also, if you hover over the geometric objects in the proof (like an AB or ABC), the corresponding object is highlighted in the figure.
philtodd 13y ago
I tried several times to explain the essence of Euclid's proof to a non-mathematician (my wife). Here is the explanation which clicked. http://euclidsmuse.com/app?id=344 http://euclidsmuse.com/app?id=344
avmich 13y ago
Not very clean. It's not immediately obvious how some statements are derived from initial axioms.
gravity13 13y ago
Interesting. What is Setosa?
kljohnson 13y ago
Here's another visual proof: http://mobius.maplesoft.com/maplenet/mobius/application.jsp?appId=9468543 http://mobius.maplesoft.com/maplenet/mobius/application.jsp?...
Houshalter 13y ago
I feel stupid. This is very difficult to follow.
mishmax 13y ago
Cool. What made you choose Angular over D3.js for this btw?
- vicapow 13y ago
  It's using Angular with D3.
graeme 13y ago
A bit OT, but I'm interesting in going through Euclid by drawing it with a compass and ruler. Is there any good guide to doing this? I have Byrne's copy of Euclid. But I found there were multiple points I got stuck when trying to draw it myself.
MarkTee 13y ago
Here's a nice (physical) visualization that uses water: http://imgur.com/gallery/1ZGJGD6 http://imgur.com/gallery/1ZGJGD6

97 points by vicapow 13y ago

Very nice visualization. Reading these old proofs is quite tedious but having this visualization makes it much easier to follow. Still, trying to understand Euclid makes you thankful for the more than 2000 years of advancements in mathematical notation and theory.

Jtsummers 13y ago
> Still, trying to understand Euclid makes you thankful for the more than 2000 years of advancements in mathematical notation and theory. The importance of notation is greatly underestimated, IMO. Even when programming, an arguably mathematical activity, many people still resist improvements to notation that allow for clearer expressions. The contortions some of my colleagues have gone through in Fortran or C++ (because they refuse to use libraries that would clean up their code significantly, they roll their own versions) to express their ideas.

justinpombrio 13y ago

Or, if you want a proof with fewer words, http://isomorphismes.tumblr.com/image/790452593 http://isomorphismes.tumblr.com/image/790452593

fryguy 13y ago
This is showing a^2 + b^2 + 2ab (left image) = c^2 + 2ab (right image), since they both have area (a + b)^2. This implies that a^2 + b^2 = c^2, since 2ab is constant across sides.
csense 13y ago
This is probably the easiest, most intuitive proof of the Pythagorean Theorem. Euclid's proof, by contrast, is a bit of a slog.
- psykotic 13y ago
  Euclid has two proofs. The first is the one submitted here. The second one uses the complicated theory of proportion from Book V. However, it's very simple and intuitive when recast in modern terms. Take a right triangle ABC, drop an altitude onto the hypotenuse, note the two smaller right triangles are similar to ABC, calculate the area of ABC in two different ways using this decomposition, and compare the results.
ryannevius 13y ago
I always liked this one... http://25.media.tumblr.com/a7ac256b1db7bbb775f559bfa736b962/tumblr_mrsl3goCez1s6r1vho1_400.gif http://25.media.tumblr.com/a7ac256b1db7bbb775f559bfa736b962/...
- zodiac 13y ago
  That's not a mathematical proof, for example it doesn't show why the theorem holds for all a, b and c, and not just the values used in the demonstration.
  - floatrock 13y ago
    But it gets the target audience to perk up and be interested for more.

commlink 13y ago

Well done, but it would have been better if the graphics depicted a right scalene triangle -- the more general case -- rather than a right isosceles triangle.

SamReidHughes 13y ago
You can drag the vertices.
mannykannot 13y ago
My thoughts exactly - then I found you can drag point C left and right. Undocumented software!

netrus 13y ago

If you enjoy visualizations of Euclidean geometry, I highly recommend the work of Byrne: http://www.math.ubc.ca/~cass/Euclid/byrne.html http://www.math.ubc.ca/~cass/Euclid/byrne.html Example: http://www.math.ubc.ca/~cass/Euclid/book1/images/bookI-prop1.html http://www.math.ubc.ca/~cass/Euclid/book1/images/bookI-prop1... (I found this to be one of the most impressive examples used by Edward Tufte in his books).

fuzionmonkey 13y ago
Byrne's Elements is pretty remarkable. It's an interesting juxtaposition of old 19th century typesetting (complete with initials) and comparatively "modern" graphic design. Some pages look like they could be right out of De Stijl. Archive.org hosts a nice PDF copy, which I find more convenient than the jpegs: https://archive.org/details/firstsixbooksofe00byrn https://archive.org/details/firstsixbooksofe00byrn

agumonkey 13y ago

Funny, few days ago I wanted to write it from scratch starting through the geometric 'interpretation', just to see if I remembered. I had to use `square of sum` identity though. http://imgur.com/umPQpeu http://imgur.com/umPQpeu ps: vector editor is not my website.

onedognight 13y ago

In case it's not obvious, you can drag the points in the diagram to change the size of triangles and the associated squares.

sesqu 13y ago
Excellent! I did want different-sized squares.
clintonc 13y ago
Also, if you hover over the geometric objects in the proof (like an AB or ABC), the corresponding object is highlighted in the figure.

philtodd 13y ago

I tried several times to explain the essence of Euclid's proof to a non-mathematician (my wife). Here is the explanation which clicked. http://euclidsmuse.com/app?id=344 http://euclidsmuse.com/app?id=344

avmich 13y ago

Not very clean. It's not immediately obvious how some statements are derived from initial axioms.

gravity13 13y ago

Interesting. What is Setosa?

kljohnson 13y ago

Here's another visual proof: http://mobius.maplesoft.com/maplenet/mobius/application.jsp?appId=9468543 http://mobius.maplesoft.com/maplenet/mobius/application.jsp?...

Houshalter 13y ago

I feel stupid. This is very difficult to follow.

mishmax 13y ago

Cool. What made you choose Angular over D3.js for this btw?

vicapow 13y ago
It's using Angular with D3.

graeme 13y ago

A bit OT, but I'm interesting in going through Euclid by drawing it with a compass and ruler. Is there any good guide to doing this? I have Byrne's copy of Euclid. But I found there were multiple points I got stuck when trying to draw it myself.

MarkTee 13y ago

Here's a nice (physical) visualization that uses water: http://imgur.com/gallery/1ZGJGD6 http://imgur.com/gallery/1ZGJGD6