Challenge: splitting an angle into three equal parts

In summary, the conversation discusses the possibility of solving geometric problems using origami instead of traditional ruler and compass methods. It is mentioned that origami allows for the doubling of the cube, trisection of angles, and solving polynomial equations. The potential impact of Japanese scholars on geometry and the concept of squaring the circle is also briefly touched upon. The conversation concludes with a mention of Kawasaki's theorem, which states that a folding with specific angles can be flattened if and only if those angles add up to pi.
  • #1
MartinV
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I recently decided to take a whack at this problem. Came up with an interesting approach, thought it would make a good conversation topic.

Anyone else tried to do this? What were your results?
 
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  • #2
What do you use for splitting? Computers?
With compass and ruler it's impossible (proven) unless you have some certain angles for which it can be done.
You might always get some close approximations, but that's what you get with a computer, too.
 
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  • #3
Curiously, it can be done with origami. It's a very neat question which problems can be solved with origami as opposed to simple ruler and compass.
 
  • #4
micromass said:
Curiously, it can be done with origami. It's a very neat question which problems can be solved with origami as opposed to simple ruler and compass.
Wow! Never heard about it. The classical three induced a lot of mathematics. Do you know whether anyone has explored origami methods in greater detail, will say which objects allowed transformations lead to?
 
  • #5
fresh_42 said:
Wow! Never heard about it. The classical three induced a lot of mathematics. Do you know whether anyone has explored origami methods in greater detail, will say which objects allowed transformations lead to?

Yes, it has been explored in a lot of details. Origami allows the doubling of the cube, the trisection of an angle and solving cubic and quartic polynomial equations. http://www.cs.mcgill.ca/~jking/papers/origami.pdf
 
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  • #6
I'm going to love ##\mathbb{O}##. What would have happened to geometry etc., if Euclid and Archimedes had been Japanese scholars? This would be an interesting topic for a science fiction novel.
 
  • #7
Squaring the circle would still be impossible though
 
  • #8
micromass said:
Squaring the circle would still be impossible though
Yes, but this is cool: A folding with a center where all folds meet by angles ##α_1, \dots , α_{2n}## can be flattened if and only if
$${\displaystyle \alpha _{1}+\alpha _{3}+\cdots +\alpha _{2n-1}=\pi }$$
(Kawasaki's theorem - a version of)
 

1. How can I split an angle into three equal parts?

There are several methods to split an angle into three equal parts. One common method is to use a compass to draw arcs from each side of the angle, intersecting at the vertex. Then, draw a line connecting the two points of intersection. This line will divide the angle into three equal parts.

2. Can I use a protractor to split an angle into three equal parts?

Yes, you can use a protractor to measure and divide an angle into three equal parts. Place the protractor on the angle with the center at the vertex. Mark the angles at 1/3 and 2/3 of the original angle measure. Draw lines from the vertex to each mark to divide the angle into three equal parts.

3. What is the mathematical formula for splitting an angle into three equal parts?

The mathematical formula for splitting an angle into three equal parts is (1/3)*angle measure. For example, if the angle measure is 90 degrees, each angle part would be (1/3)*90 = 30 degrees.

4. Can I use a ruler to split an angle into three equal parts?

No, a ruler alone cannot divide an angle into three equal parts. A ruler can only measure length, not angles. You will need to use a compass or protractor to accurately divide the angle into three equal parts.

5. Are there any online tools or calculators to help me split an angle into three equal parts?

Yes, there are many online tools and calculators available to help you split an angle into three equal parts. Some may require you to enter the angle measure, while others may offer step-by-step instructions. It is important to double-check the accuracy of the tool or calculator before using the results in any calculations.

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