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What other mathematical construction tools or methods do exist apart from compass and straightedge?

I know of folding and neusis (incl. tomahawk), also perhaps calipers (dividers) for plastic ratio.

One question mentioned "3D ruler": Are there numbers that we can't get with a usual compass and ruler, but can get with 3D compass and ruler?

Is there any other? Perhaps some classification of them?

I'm looking for simple methods, nothing complicated.

curious_metazoan
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  • Notice that the classical tool is not a ruler but a straightedge. I’ve been told that with a straightedge marked with two distinguished points, you can, for instance, trisect an angle. – Lubin Jul 24 '19 at 22:24
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    @Lubin, see Arthur Baragar, Constructions using a compass and twice-notched straightedge, http://baragar.faculty.unlv.edu/papers/TwiceNotch.pdf – Gerry Myerson Jul 25 '19 at 00:03

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One such tool is the ellipsograph (trammel of Archimedes). It's related to the Tusi couple. Here is a demonstration: https://www.youtube.com/watch?v=7Fn-26Jmi5E

A related toy is the spirograph.

Another tool that uses mechanical links is the pantograph. It's used for copying and scaling.

The harmonograph employs pendulums to draw curves.

curious_metazoan
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I very much enjoyed David Richeson's Tales of Impossibility (MAA review link), which is a history of the famous geometrical problems of antiquity including angle trisection and circle squaring.

In interstitials between chapters on the book Richeson reviews other tools to be added to the geometer's toolkit in addition to the neusis, including:

Richeson also discusses other interesting construction techniques in addition to origami, such as with toothpick geometry. I learned of an interesting angle trisection with toothpicks, discovered by the author of the children's book Harold and the Purple Crayon.

Mark S
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