Trisect an Angle with Archimedean Spiral (visual proof)

This is a short, animated visual proof demonstrating how to trisect any angle using the Archimedean spiral. #manim #math​​ #mathshorts​ #mathvideo​ #trisect #trisectangle #impossible #geometry #chords #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #inequality
This animation is based on a visual proof from the wonderful book Charming Proofs by Claudi Alsina and Roger Nelsen: (Affiliate link; I may get small commissions) amzn.to/3LbcpgG
To learn more about animating with manim, check out:
manim.community

[@romilrh]

This is perfect for when I want to trisect an angle but all I have with me is my trusty Archimedian spiral, thank you

[@MathVisualProofs]

Hah! You can probably make one out of string and a nail or tack :)

[@PhilBoswell]

Nicely done. It would be even nicer if you gave some kind of hint as to how to draw the Archimedean Spiral itself 🤔

[@jakobr_]

You can’t draw the archimedean spiral with just a straight edge and compass, that’s the whole point of including it as a “new tool”

[@MathVisualProofs]

You can draw one by tying a pencil to a string and winding the string around a nail or tack :)

[@mihailmilev9909]

@@MathVisualProofs omg

[@mihailmilev9909]

@@MathVisualProofs and why can't this or a mechanism mimicing it be incorporated into modern compasses?

[@ultracreador]

Sería más práctico utilizar una regla Tomahawk que utilizar la espiral. Sería aún más práctico utilizar una regla recta con medidas

[@wendolinmendoza517]

There is a very simple and beautiful method for trisecting any angle, based on origami. That means anybody can try it out with a piece of sheet. In fact the method (and the theorem that proves it works) is from a Japanese and is pretty recent, I think about 1980. You can find a very good exposition in the book "Project Origami" by Thomas Hull

[@MathVisualProofs]

Yes I know about the Beloch fold - very cool way to do it too :) thanks!

[@FromTheMountain]

Nice one! It would be cool if you could give sone exposition on the proof of why trisecting an angle is not possible in general, but I guess this might be somewhat outside the scope of the channel.

[@MathVisualProofs]

Wouldn’t be visual probably. Impossibility proofs are generally difficult to do visually. I’ll see what I can come up with :)

[@wendolinmendoza517]

@@MathVisualProofs Visual Galois Theory, that's definitely a challenge

[@MathVisualProofs]

@@wendolinmendoza517 for sure. There are some nice explanations out there that get at it. But it’s beyond my animating skill level yet :)

[@ksbalaji1287]

Just beautiful! Thanks.

[@MathVisualProofs]

Thank you too!

[@AlbertTheGamer-gk7sn]

Using the spiral, you can construct a regular heptagon (and nonagon) as well.

[@temirkhansmagulov1188]

Mind-blowing! Cool. Never thought about that

[@MathVisualProofs]

Glad you liked it!

[@temirkhansmagulov1188]

@@MathVisualProofs i remember once i heard in high school that you cannot trisect an angle with a straight line and compass. Since then this is the second method i learned how to trisect an angle but with different methods (the first one was with origami)

[@MathVisualProofs]

@@temirkhansmagulov1188 Yes! Not possible with just classical tools, but it is possible with others. As you mention, Origami works. So does neusis construction. You might like this one too: kzbin.info/www/bejne/ZnWWfJx5otqgb5I

[@UnbreakablePickaxe]

mark two equidistant points on both arms of the angle, join them with a line, trisect the line, join the points of trisection of line with the vertex of angle, and angle is trisected.

[@MathVisualProofs]

Are you sure those angles are equal?

[@UnbreakablePickaxe]

@@MathVisualProofs hmm it appears I was wrong, I was just guessing...

[@RobSteel117]

Questions: 1) What if the angle vertex is not at the origin? 2) What if the angle points off into another quadrant, do we just draw the appropriate Archimedes spiral to spin off into that quadrant? 3) Could we use this technique to split an angle into any number of sections? I use Geogebra for my study. Geogebra won't do polar, as far as I know, so I create the spiral in Geogebra with this a=Curve(t cos(t),t sin(t),t,0,2 π) and I modify the signs of the first two args to get the spiral in the quadrant where I need it. 4) any ideas about how to get the spiral to use some other point as it's origin?

[@SuperYoonHo]

Thank you!

[@MathVisualProofs]

Thanks for watching :)

[@SuperYoonHo]

@@MathVisualProofs No problem:)

[@SherlockSage]

Loved this video! :D

[@MathVisualProofs]

Glad to hear it! Thanks :)

[@rotemperi-glass4825]

beautiful

[@MathVisualProofs]

Thank you! 😊

[@ojas3464]

👍Courant in his book What is Mathematics gives another trisection tool. Hope you find interest and time to make a Visual of it☺

[@MathVisualProofs]

Yes! There are other ways for sure - I think Courant uses a Neusis construction (with marked ruler). Richeson's book Tales of Impossibility has some other ideas too :)

[@SherlockSage]

@@MathVisualProofs Another trisection tool, surprisingly enough, is origami. Not sure how to represent that in Manim though 🤔

[@MathVisualProofs]

@@SherlockSage I have thought about using the beloch fold to do it. I’ll see if I can.

[@muhammadaryasaputra6754]

If we divide a line from origin to P into n equal parts, we can divide any angle into n equal parts, right?

[@MathVisualProofs]

Yes! Very cool ;)

[@tylerduncan5908]

Euclid could have gotten so much more done if all he had was a string and a post to wrap it around. I wonder what impossible constructions with straightedge and compass become possible by adding string. I know that bisecting an angle is equivalent to finding a cube root, maybe you could do a video on that?

[@MathVisualProofs]

I’ll see if I can figure something out. Thanks!

[@paradiselost9946]

ellipses spring to mind...

[@_yukulele]

Great, now we just need to know how to create an Archimedean spiral with a graduated ruler and a compass.

[@MathVisualProofs]

String and a nail?

[@user-kn6sw2jl2p]

WoW

[@matejcataric2259]

nice

[@MathVisualProofs]

Thanks!

[@kevinbihari]

You can just use a lattice right? Just straight lines. Pick a gridpoint matching 1/3rd the slope.

[@ayushsharma8804]

That would only give you (tan (theta))/3

[@rgfs71]

You skipped over how to trisect r. Would this not be as complicated as just splitting the arc?

[@MathVisualProofs]

Here is dividing a line into 7 equal length parts: kzbin.info/www/bejne/l5e9mnR9lrB4j5Y ; same technique works for cutting segment into n parts for any n.

[@KaliFissure]

how did you cut that section into thirds? what tool?

[@MathVisualProofs]

This video kzbin.info/www/bejne/l5e9mnR9lrB4j5Y shows how to divide a line segment into 7 pieces. Same technique works for 3.

[@KaliFissure]

@@MathVisualProofs thanks

[@yuriakahumanity]

How does one draw an Archimedian spiral?

[@MathVisualProofs]

Could wrap a string around a nail or a tack?

[@phiarchitect]

Can you construct an Archimedean spiral with a straight-edge and compass?

[@MathVisualProofs]

Nope. If you could then you could trisect the angle with straightedge and compass :)

[@phiarchitect]

@@MathVisualProofs That is your slight of hand in this beautiful video :)

[@phiarchitect]

@@MathVisualProofs Maybe a better question is - can the intersection points be expressed algebraically?

[@MathVisualProofs]

@@phiarchitect depends on the starting angle I would guess. But straight edge and compass can't even get you cube roots.

[@phiarchitect]

@@MathVisualProofs true - let's square the circle instead.

[@PASHKULI]

How is that when P has θ = α (angle), so r = α (length)?

[@MathVisualProofs]

The definition of the spiral is that angle and distance from origin are equal.

[@PASHKULI]

@@MathVisualProofs I do not get that. An angle = to distance?!

[@PASHKULI]

@@MathVisualProofs A spiral is a function for radius changing with the change of the angle + a constant (offset): R = C + aθ. The greater the angle, the further the radius reaches. When the coefficient 'a' of that change of angle is 1 (the constant C is 0), we get the "circular" spiral (Archimedean spiral).

[@MathVisualProofs]

@@PASHKULI that’s the definition of the spiral. In polar coordinates we have r=theta. So as theta changes so does the distance from the origin. When theta is 0 so is r. When theta is pi/2, pointing straight up y-axis, then the distance from origin is pi/2. Etc.

[@PASHKULI]

@@MathVisualProofs If the angle θ (we need to trisect) from centre of circle with R = 1, cuts an arc length from the same circle = A (not α), how is that arc with length A equal to the radius r of the spiral when it reached the same angle θ? So, if the spiral r = C + aθ (C = 0, a = 1), then r = θ. A length can not be in degrees. So it must be approximated by radians (π component in it). The problem is, there is no tool to draw spirals. We can only approximate it with a circle type arcs with a compass. I am trying to understand the method to draw a spiral, hence its r = θ (in rad). How?

[@Itoyokofan]

Even a kid can trisect an angle. Just use a ruler, a compass and a sharpie, just use a neusis technique. It's not that hard, really. Just start teaching that in schools already.

[@MathVisualProofs]

Yes a ruler will do it. But there is a lot of history and deep mathematics that comes from this problem as well.

[@xicad1533]

it's easier to use origami

[@agrajyadav2951]

Holy shit

[@MathVisualProofs]

😁