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June 23rd, 2014, 06:47 AM   #1
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Angle Trisection by Geometry, $100 USD Prize

Hello everyone,

Someone please help me understand angle trisection by using geometry, NOT algebra. If I understand it by geometry, I will remember it forever.

Help me understand this. This is the trisection of 60 degrees. It works for any degrees.



Please explain it by using geometry in this style.

First, Proof of the Pythagorean Theorem



Cubing by geometry.



Cube root by geometry.



That is my invention by the way, reversing the direction of Lill's method.

Neusis cube root by Beloch fold (an improvement of reversing Lill's method), and Euclidean (in 3D space) cube root by parabola.

(a parabola is Euclidean in 3D space). Here is Y = X².



Neusis and Euclidean cube roots of Phi = ³√ [ (√5 -1) ÷ 2] ≈ 0.85179964207

Geometry is precise and exact, while Mechanics is not
La Géométrie (1637) - René Descartes



Angle trisection by neusis.



Explanation:

https://www.math.lsu.edu/~verrill/origami/trisect/

Thank you.

My name is Long Quach. I am an amateur origami artist. This is my Flickr page.

https://www.flickr.com/photos/85937466@N02/

Last edited by skipjack; February 13th, 2017 at 07:00 AM.
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June 23rd, 2014, 07:13 AM   #2
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Before I attempt to work through the geometry let me make this note.

An angle of 20 degrees = $\pi/9$ cannot be constructed with ruler and straightedge since the minimal polynomial of $\sin(\pi/9)$ is $64x^6-96x^4+36x^2-3$ with degree 6 which is not a power of 2. (That is: no proper sextic number is constructible.) Arbitrary angles can be trisected with origami. So if the technique uses origami/neusis or any equivalent technique then it could trisect not just this but any angle; on the other hand if it avoids these techniques it can't trisect even this (60-degree) angle.

Your use of a parabola makes it seem that you are using neusis. It has a long history -- angles have been trisected by neusis for over two thousand years.
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June 23rd, 2014, 07:28 AM   #3
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A parabola can be constructed by Euclidean methods in 3D space. I did it in GeoGebra.

Origami/neusis can only trisect to the magnification of what your eye can see.

I know neusis has been around.





This can be proven, in the style that reversing the direction of Lill's method can be proven, geometrically AND algebraically.
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June 23rd, 2014, 07:50 AM   #4
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I'm only a high school graduate. Keep that in mind in your explanation.
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June 23rd, 2014, 08:32 AM   #5
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My thoughts on 3D geometry

I think 3D geometry is Euclidean like 2D geometry.

A sphere is Euclidean in 3D like a circle is in 2D.

If we spin a circle we get a sphere.



Euclidean sphere maker:



From the sphere, we can construct perpendicular planes, like circles can construct perpendicular lines.



A compass makes a circle in 2D space. A compass makes a cone in 3D space.
If we spin a triangle we have a cone.



3 points determine a plane.


The intersection of a plane and cone makes a parabola, 3D Euclidean.




Last edited by long_quach; June 23rd, 2014 at 08:42 AM.
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June 23rd, 2014, 08:43 AM   #6
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Believe it or not, I learned all geometry from Donald Duck in Mathemagic Land.
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June 23rd, 2014, 08:58 AM   #7
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Post questions one by one, try to use minimum number of images and make it simple.....
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June 23rd, 2014, 09:26 AM   #8
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There's only one question.

Explain this construction logically, so I can understand it, recreate it from memory from understanding it, so I will remember it forever.

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June 23rd, 2014, 09:55 AM   #9
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Quote:
Originally Posted by long_quach View Post
A parabola can be constructed by Euclidean methods in 3D space. I did it in GeoGebra.
Obviously (?) a parabola cannot be constructed with compass and straightedge without the use of neusis.

Quote:
Originally Posted by long_quach View Post
Origami/neusis can only trisect to the magnification of what your eye can see.
Actually they can give exact trisections -- though not exact quinquesections.
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June 23rd, 2014, 09:59 AM   #10
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Quote:
Originally Posted by long_quach View Post
I'm only a high school graduate. Keep that in mind in your explanation.
You don't need to know Galois theory -- just know that there is some branch of math (not usually taught until after you graduate from college) which explains rather precisely what can be constructed by various techniques.

In fact to understand these results (but not their proofs) high-school algebra and geometry should suffice. To construct an angle $\theta$ it must be the case that you can write $\sin(\theta)$ as an expression involving only integers using addition, subtraction, multiplication, division, and square roots (no other operations).

Last edited by CRGreathouse; June 23rd, 2014 at 10:02 AM.
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