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Proofs without Words and Beyond (maa.org)
34 points by adgasf 6 months ago | hide | past | web | favorite | 12 comments



If anyone is interested in proofs without words, my undergrad advisor has published a bunch of them

https://community.plu.edu/~edgartj/#pwws


What exactly is the first diagram supposed to be proving about angle trisection?

If it's trying to show a way to solve the problem, it's kind of missing the point of doing it with just a compass and straightedge. There are much simpler tools you can add that let you trisect an angle. And the device depicted doesn't actually work, because when you try to align the ends the middle points will poke through the paper.

If it's trying to argue that you need more tools than compass and straightedge it's just nonsense because you could make an analogous picture with a bisector yet angle bisection is trivial.


All “proofs without words” require some imagination (and they can be extra frustrating if one doesn't see the point quickly). In that sense, each one is like a puzzle. I quite enjoyed this one.

All it's trying to prove is that the shown device works, and correctly trisects any given angle. (The proof, if I had to put it into words, is something like: let the outer bars be A and B, and let the middle bars be X and Y. Because of the hinges, X is exactly halfway between A and Y, and Y is exactly halfway between X and B, so X and Y trisect the angle.)

Here is a 33-second video: https://www.youtube.com/watch?v=sxwMGcshJI8 — though it's an animation and the device is different, you can see there won't be any problem with poking through the paper or having to bend it out of shape. I find the depicted device elegantly simple and find it hard to imagine something significantly simpler; what are some of the much simpler tools you mention?


The device in your video is designed differently and would work without any fudging. It's key that the writing point is at a 90 degree angle to the device spines, and each writing point has a different length.

> I find the depicted device elegantly simple and find it hard to imagine something significantly simpler; what are some of the much simpler tools you mention?

If you're starting off from the basis of "compass and straightedge" (and pen), then all you need is to upgrade your straightedge with a fixed pair of marks. If you're starting from nothing, all you need is a square of paper; origami can trisect an angle.


> It's key that the writing point is at a 90 degree angle to the device spines

But the device in the picture is exactly like that one! The three main spines will always occupy exactly the same positions in either the device in the picture or in the device in the video; only the interior hinges used to enforce it are different. The device lies flat on the paper in both cases, and the writing point (your pencil or whatever you stick in the holes) will of course be at 90° angle to the device spines.

Actually a better search found names for these devices: see terms like “Sylvester's Link Fan” and Laisant's Link #2 here: http://www.takayaiwamoto.com/Greek_Math/Trisect/Linkage/Link... — and a much better video showing almost exactly the device in the picture: https://www.youtube.com/watch?v=uTQVi7zskGU (except for quadrisection instead of trisection). (And the article does mention the Laisant reference; hadn't noticed it earlier: see p.38 here: https://files.eric.ed.gov/fulltext/ED058058.pdf#page=49 )

> [...] straightedge with a fixed pair of marks [...] origami

Fair enough; these devices are indeed simpler to construct, but the actual process of trisection of a given angle is not as simple as “plonk the device on the paper, open it up to the angle, and there, it's already trisected”. Nor is the proof that you've successfully trisected as straightforward.


I don't know what it's trying to prove about angle trisection, but the device depicted isn't intended to be stood upright. That wouldn't work at all. It's intended to be laid down with the main hinge at the intersection of the lines.

So the device does work, but it's certainly not doing compass and straight-edge anything.


As designed, even tilted you need to bend it out of shape to make a mark. And if you're bending it out of shape you're not guaranteed an accurate result, so why even use it? You can approximate a trisection without it.


I think you're not understanding how to use this. I've played with one, and it works just fine, subject to the theoretical aspects, which also exist with compass and straight-edge constructions.

As designed you lay it flat on the paper with the hinge at the intersection. You open out the arms so the points on the outer arms lie on the lines. You mark where the two inner points lie, and there you are, angle trisected.

But all that seems fairly obvious to me, and I thought it would be fairly obvious to you. So if that's not what you're thinking then I don't know where our thinking diverges.

Not that it matters - it's a device to trisect and angle. It works when used properly, but it's of no real mathematical interest.


The points will dangle over the paper. This keeps you from making a mathematically precise point. It works to approximate a trisection in practice, but in the realm of pure geometry it can't give you an exact result.

You can make a precisely curved version to overcome this, but that loses a lot of the elegance. Especially when all you need to trisect an angle is a straightedge marked to measure a single fixed distance.


In the case of the actual device, one of which I've held in my hand and actually used, no, the points don't dangle over the paper. The points descend to make contact with the paper. Perhaps that's what you mean by:

> You can make a precisely curved version to overcome this, but that loses a lot of the elegance.

For me, it doesn't lose elegance, but that's a matter of taste.

> Especially when all you need to trisect an angle is a straightedge marked to measure a single fixed distance.

Although that, in turn, requires sliding the straight-edge along and jiggling it about to match a distance to the one marked on the edge. But we're arguing about nothing. The device exists, works, is precise, in my opinion is elegant.

Feel free to disagree.


I don't know what you mean by "middle points" but I think you are too quick to dismiss this as unbuildable. To my eye, it's buildable if you don't consider every joint as a pin that digs into the paper.

It's presumably intended as a reminder that there are mechanisms that can trisect an angle, some of them very beautiful and entirely symmetrical. Dare I say it, the message is that math is pluralistic–so don't go too hard on those trisectors and visual thinkers who are working with different assumptions from the symbol-shufflers embedded in mainstream academic mathematical culture.

http://mathworld.wolfram.com/NeusisConstruction.html etc.


Agreed. It seems an odd inclusion for a visual proof. And afaik there's no simple visual way to demonstrate the impossibility of angle trisection. The proofs use the complex machinery of abstract algebra.




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