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Thread: Tomahawk angle trisector - photo

1. The Following 3 Users Say Thank You to Altair For This Useful Post:

mwmkravchenko (Jan 16, 2021), Philip Davies (Jan 28, 2021), Toolmaker51 (Jan 19, 2022)

2. Unclear where these are popular; we don't all have 3D printers yet, but not difficult to fabricate.

3. It's ingenious as hell but I have to admit that I've never had occasion to need to trisect an angle.

4. It's a puzzle to me, how the 'rigid' contours operate to TRISECT (what a typo!) angles whether obtuse or acute. I don't see it in drawing near as useful, where all kinds of geometry can do this. That said, in duplication, it would be on material (sheet metal, wood), or a completed object, that presents unwieldy size

5. Originally Posted by Toolmaker51
It's a puzzle to me, how the 'rigid' contours operate to bisect angles whether obtuse or acute. I don't see it in drawing near as useful, where all kinds of geometry can do this. That said, in duplication, it would be on material (sheet metal, wood), or a completed object, that presents unwieldy size
Go to the Wikipedia article on the tomahawk...

https://en.wikipedia.org/wiki/Tomahawk_(geometry)

and page down to the "Trisection" portion. That will show you how the tool geometry is exploited to trisect angles.

In Euclidian geometry there was a strong concentration on using only a compass and (uncalibrated) straightedge to solve various geometric problems. Since bisection was easy with these tools, it was expected that trisection should be possible. Only later was it proved that trisection with those tools was impossible.

All these attempts led to an interest in trisection and numerous searches for simplistic ways to do it. One result of this concentration of effort was the tomahawk. (Mathematicians, it seems, often beat to death problems that have only limited practical application.)

All this noodling went on in the days before our high accuracy goniometers, digital computers, and other precision approaches to solving the problem. Still the simplicity of the tomahawk is a tribute to the ingenuity of the human mind.

6. The Following User Says Thank You to mklotz For This Useful Post:

Toolmaker51 (Jan 21, 2022)

7. I see two significant results in MKlotz's offering.....unsatisfied by google (and certainty of imperfect search term), I failed to dive far enough to see Wikipedia pop up.

Originally Posted by mklotz
Go to the Wikipedia article on the tomahawk...

https://en.wikipedia.org/wiki/Tomahawk_(geometry) <<< SNIPPED>>> show you how the tool geometry is exploited to trisect angles.

1. (Mathematicians, it seems, often beat to death problems that have only limited practical application.)
2. Still the simplicity of the tomahawk is a tribute to the ingenuity of the human mind.
I'd say "beat to death" is unfair. Mathematician I am not; but we all appreciate those who can see what is not entirely tangible, what they'd like to accomplish, and pursue that until a solution OR proved unsolvable.

Better still, new minds with different perspectives and means at hand, remedy what had flummoxed the predecessors.

Speaking of remedies, who would rather trust a physician than a mathematician?

holey-moley get a load of this little monster!
https://en.m.wikipedia.org/wiki/Arbelos

8. I've had numerous left-side-of-the-mean types complain about the compass and straightedge restriction. (Why not just measure the angle with a protractor and divide by three?)

What they fail to realize that this restriction is the mathematics version of having a metalworker trainee make perfect parts using only a square and file. You learn to focus on the features of the part and not on the available tools.

Not many math tools existed in the time of Euclid (~300 BC) but that's not the reason for the restriction. It's more related to his idea of deriving mathematical relationships from the very simplest of assumptions.

More here...

https://en.wikipedia.org/wiki/Straig...s_construction

9. The Following User Says Thank You to mklotz For This Useful Post:

Toolmaker51 (Jan 21, 2022)

10. Originally Posted by mklotz
<<<<snipped>>>>

Again 1] You learn to focus on the features of the part and not on the available tools.

And 2] Not many math tools existed in the time of Euclid (~300 BC) but that's not the reason for the restriction. It's more related to his idea of deriving mathematical relationships from the very simplest of assumptions.

More here...

https://en.wikipedia.org/wiki/Straig...s_construction
Had that very discussion yesterday; a friend on verge going overboard tooling up a humble garage shop, eyes glazed over on tool catalog. Hyperventilating (not really just illustrating) over digital gimcracks and those red anodized 'tools' I despise. Losing battle until the history card I threw down, even with the unrelated processes.
I have two (three) overwhelming favorite salvos; *1 Eratosthenes of Cyrene, determined size of Earth, with surprisingly good measurement using a simple scheme that combined geometrical calculations with physical observations. *2 is Harrison and his succession of prizewinning chronometers; decades in the making, and removed one entry that excelled with one even better. *3 is Charles Proteus Steinmetz, who solved untold number of calculations, but pinnacle is sorting out Henry Ford's ailing generator, apparently with no measuring instruments at all!

1] https://en.wikipedia.org/wiki/Eratosthenes
2] https://en.wikipedia.org/wiki/Chronometer_watch
3] https://en.wikipedia.org/wiki/Charles_Proteus_Steinmetz

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