Visual walk through of Euclid's proof of the Pythagorean theorem | Hacker News Reader

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

setosa.io

97 pointsvicapow27 comments

If you enjoy visualizations of Euclidean geometry, I highly recommend the work of Byrne:

http://www.math.ubc.ca/~cass/Euclid/byrne.html

Example: 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).

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

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

Excellent! I did want different-sized squares.

Also, if you hover over the geometric objects in the proof (like an AB or ABC), the corresponding object is highlighted in the figure.

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

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.

This is probably the easiest, most intuitive proof of the Pythagorean Theorem. Euclid's proof, by contrast, is a bit of a slog.

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.

I always liked this one...

http://25.media.tumblr.com/a7ac256b1db7bbb775f559bfa736b962/...

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.

But it gets the target audience to perk up and be interested for more.

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.

> 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.

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

ps: vector editor is not my website.

I feel stupid. This is very difficult to follow.

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

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.

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

Here's another visual proof:

http://mobius.maplesoft.com/maplenet/mobius/application.jsp?...

Here's a nice (physical) visualization that uses water:

http://imgur.com/gallery/1ZGJGD6

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.

You can drag the vertices.

My thoughts exactly - then I found you can drag point C left and right.

Undocumented software!

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

It's using Angular with D3.

Interesting. What is Setosa?

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