Cube root of a line

Question

Well this may be simple but I am not getting it.

Give a line segment (of length $l$)(and a segment of unit length if you require) how to construct a line of length $l^{1/3}$ with only a straight edge and compass?

I know how to draw a line with length $\sqrt l$ (through similarity process) but am at a loss at the cube root one. Can someone help?

in general, cubic root is not constructible by compass and straightedge alone. That's how people finally figure out classical geometry problem like doubling the cube or trisecting an angle is unsolvable. — achille hui, Jun 13 '16 at 07:28
@achillehui Is there any proof or so? Just asking.. (But cubing a line is possible I know so I thought...) — Qwerty, Jun 13 '16 at 07:31
Look at wiki page of constructible number and refs there as a start. — achille hui, Jun 13 '16 at 07:33
Or the the part showing $\sqrt[3]{2}$ is not constructible in doubling the cube — Henry, Jun 13 '16 at 07:36
@Qwerty : however you can construct $2^{1/3}$ by a de neusis construction, using a marked ruler. — Watson, Jun 13 '16 at 10:00
@Henry and Watson : So I see that a given line segment of unit length is not the same as a marked ruler. Have I got this correct? — Qwerty, Jun 13 '16 at 12:41
Given a marked ruler, you can freely move the ruler against your graph until a certain condition is met. In this process, you solved a cubic equation. However, you can't achieve "continuous motion" with a line segment and unmarked ruler. — Trebor, Jul 09 '21 at 06:44

score 1 · Accepted Answer · answered May 31 '21 at 17:59

In terms of rational root theorem, it's not possible to draw cubed root of 2. However, I have found folks who figured things out using Euclidean geometry.

This one is the simplest: https://demonstrations.wolfram.com/ConstructingTheCubeRootOfTwo/#more

Also, I came up with a hack myself. My approach is clunky and not precise past the thousandths decimal. I made this hack based upon some simple facts that

cubed root of 2 is 1.25992105...
sin(39.045) = 0.629930566
2x(sin(39.045)) = 1.25986113

So if you had an isosceles triangle with the apex 78.09 degrees and the two sides are length 1, then the base should be 1.259.

Step 1: Create an angle that is .46875 degrees Construct a right angle. Trisect the right angle. Take one of the 30 degree angles and bisect it 6 times. You will get an angle that is .46875 degrees.

Step 2: Create an angle that is 5.625 degrees. Construct a right angle. Bisect the right angle 4 times. You will get an angle that is 5.625 degrees

Step 3: Construct an angle that is 72 degrees. Create a Golden Triangle (two sides equal to phi and the base is 1). One of the base angles will be 72 degrees.

Step 4: Join angles from Step 1 - 3 together (Euclid Book 1 proposition 23). You will get an angle that is 78.09 degrees. Make a an isosceles triangle with sides = 1. The base will be very close to the cubed root of 2.

Cube root of a line

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