4 ms·

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

by svat 8y ago

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 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?

Dylan16807 8y ago
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.
- svat 8y ago
  > 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_Laisant.html 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 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 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.

by svat 8y ago

Dylan16807 8y ago

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.

svat 8y ago
> 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_Laisant.html 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 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 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.