July 18, 2011 9:59 AM Subscribe

A thread full of proofs without words at MathOverflow and quite a lot more of them courtesy of Google Books.

posted by Wolfdog (22 comments total) 43 users marked this as a favorite

posted by Wolfdog (22 comments total) 43 users marked this as a favorite

Proofs without words always have a sense of parity, symmetry and minimalism to me; remind me most strongly of the way I was shown the Pythagorean triples as a child.

posted by malusmoriendumest at 10:30 AM on July 18, 2011

posted by malusmoriendumest at 10:30 AM on July 18, 2011

Since I've looked unsuccessfully for one, and we're on topic, does anyone know of a picture proof for this? If you don't have one, have you ever seen one?

posted by klausman at 11:00 AM on July 18, 2011

posted by klausman at 11:00 AM on July 18, 2011

This is an absurdly clever one on the volume of the frustum of a pyramid, from the second link.

posted by Wolfdog at 11:25 AM on July 18, 2011 [1 favorite]

posted by Wolfdog at 11:25 AM on July 18, 2011 [1 favorite]

For those with a visual bent, I cannot recommend Visual Complex Analysis highly enough. Really a revelation if you've only studied complex analysis using traditional textbooks.

posted by jcruelty at 11:29 AM on July 18, 2011 [4 favorites]

posted by jcruelty at 11:29 AM on July 18, 2011 [4 favorites]

I studied a lot of math and am scratching my head at what a few of these are trying to prove (or how they proved it)

And the comments have no explanation. ARRGH!

posted by schmod at 11:32 AM on July 18, 2011

And the comments have no explanation. ARRGH!

posted by schmod at 11:32 AM on July 18, 2011

Not quite wordless, but lovely (and beloved of Tufte): Byrne's 1847 edition of Euclid's Elements substitutes the geometrical objects themselves for their names.

posted by RogerB at 11:53 AM on July 18, 2011 [1 favorite]

posted by RogerB at 11:53 AM on July 18, 2011 [1 favorite]

How do you read the text form of the equations? E.g., $1^2 + 2^2 + \dots + n^2 = \frac13n(n+1)(n+\frac12)$

posted by Joe in Australia at 12:40 PM on July 18, 2011

posted by Joe in Australia at 12:40 PM on July 18, 2011

A lot of these seem less like proofs and more like examples? For instance, the graphic showing that the cardinality of the real numbers is the same as the cardinality of a finite open real interval? Still, very fun.

Oh my lord Visual Complex Analysis looks awesome. I've often wondered why all my other math texts weren't like that.

posted by invitapriore at 12:54 PM on July 18, 2011

Oh my lord Visual Complex Analysis looks awesome. I've often wondered why all my other math texts weren't like that.

posted by invitapriore at 12:54 PM on July 18, 2011

If you're not familiar with reading TeX, you can paste a string like that into TeXify and it will render it for you as intended. I would include a link to a rendered version of that very formula, but MetaFilter helpfully changes the backslashes in the URL to forward slashes.

posted by Wolfdog at 1:12 PM on July 18, 2011

It's not just an example, although the proof is open to quibbling:

The vertical lines connect every point on the interval to a point on an arc of a circle which is the same width as the interval. The angled lines are the radii of the lower half of that circle and they are extended to their intersection with line tangent to the circle.

For every line drawn vertically from the upper interval there is precisely one intersection with the lower half of the circle.

For every point on the lower arc of the circle, there is precisely one radiated line that can be drawn from the centre of the circle and through that point.

Each radiated line intersects with the lower line at precisely one point, which means that every point on the upper interval matches one radiated line which matches one point on the lower parallel line.

As you approach the ends of the upper interval the radiated lines associated with each point get more and more parallel with the lower line. At the end of the interval the radiated lines are actually parallel.

Parallel lines meet only at infinity.

This means that every point on the upper interval matches precisely one point on an infinite line.

Therefore, an interval has as many points as an infinite line.

posted by Joe in Australia at 1:18 PM on July 18, 2011

The vertical lines connect every point on the interval to a point on an arc of a circle which is the same width as the interval. The angled lines are the radii of the lower half of that circle and they are extended to their intersection with line tangent to the circle.

For every line drawn vertically from the upper interval there is precisely one intersection with the lower half of the circle.

For every point on the lower arc of the circle, there is precisely one radiated line that can be drawn from the centre of the circle and through that point.

Each radiated line intersects with the lower line at precisely one point, which means that every point on the upper interval matches one radiated line which matches one point on the lower parallel line.

As you approach the ends of the upper interval the radiated lines associated with each point get more and more parallel with the lower line. At the end of the interval the radiated lines are actually parallel.

Parallel lines meet only at infinity.

This means that every point on the upper interval matches precisely one point on an infinite line.

Therefore, an interval has as many points as an infinite line.

posted by Joe in Australia at 1:18 PM on July 18, 2011

Right, but your proof has been extrapolated from what the image implies, rather than having been directly transcribed, agreed? Maybe I'm just being media-ist but I don't get the feeling of the conjecture having been proved by the image the same way I get that feeling when I read your proof.

posted by invitapriore at 1:25 PM on July 18, 2011

posted by invitapriore at 1:25 PM on July 18, 2011

Can someome explain to me how the '32.5 = 31.5' one works?

(or does 32.5 actually equal 31.5?)

posted by memebake at 1:43 PM on July 18, 2011

(or does 32.5 actually equal 31.5?)

posted by memebake at 1:43 PM on July 18, 2011

It's very subtle, but neither of those figures are actually triangles.

posted by kmz at 1:52 PM on July 18, 2011 [1 favorite]

Thanks kmz, conincidentally, googling the problem gives this page at math.stackexchange.com, a sortof sister site to mathoverflow.

posted by memebake at 1:55 PM on July 18, 2011

posted by memebake at 1:55 PM on July 18, 2011

Joe in Australia: formulas *should* automatically render, if you turn javascript on.

posted by madcaptenor at 2:13 PM on July 18, 2011

posted by madcaptenor at 2:13 PM on July 18, 2011

Joe In Australia your proof is off a bit. In Euclidean geometry(standard high school geometry and what we are dealing with here) parallel lines do not meet at infinity*. That only occurs in non-Euclidean geometry. Since this is an open interval it will never generate a parallel line. Assuming that the middle of the line is x the interval is (x-r,x+r) it will never have the value x-r or x+r because that is not in the set. If this was a closed interval [x-r,x+r] that would mess up the entire proof since it would generate a parallel line that would never intersect.

Anyways I agree that the visual proof is somewhat lacking because it hard to understand that real numbers are dense(mathematical definition) visually.

*Explanation of lines at infinity

posted by roguewraith at 4:15 PM on July 18, 2011

Anyways I agree that the visual proof is somewhat lacking because it hard to understand that real numbers are dense(mathematical definition) visually.

*Explanation of lines at infinity

posted by roguewraith at 4:15 PM on July 18, 2011

klausman, perhaps this [pdf] will please you to some extent. It's not without words, but it is with pictures.

posted by stebulus at 8:42 PM on July 19, 2011

posted by stebulus at 8:42 PM on July 19, 2011

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posted by Obscure Reference at 10:15 AM on July 18, 2011