Undergrad Doubling a Cube: Can 3D Geometry Help?

[Thecla]

In plane geometry it is impossible to construct a line equal to the (cube root of 2) times the length of
a side of a cube, making it impossible to double a cube with a compass and straight edge. Maybe plane geometry needs one more dimension.
What happens if we extend the geometry to 3D(solid geometry). Is it possible to double a cube in solid geometry using the basic construction tools of solid geometry(whatever is equivalent to a compass, straight-edge)?

 

[fresh_42]

In plane geometry it is impossible to construct a line equal to the (cube root of 2) times the length of
a side of a cube, making it impossible to double a cube with a compass and straight edge. Maybe plane geometry needs one more dimension.
What happens if we extend the geometry to 3D(solid geometry). Is it possible to double a cube in solid geometry using the basic construction tools of solid geometry(whatever is equivalent to a compass, straight-edge)?

The proof, that it is impossible to double the cube doesn't use dimensions. The restrictions are alone due to the allowed means. Therefore the answer is No.

 

[dagmar]

Is it possible to double a cube in solid geometry using the basic construction tools of solid geometry(whatever is equivalent to a compass, straight-edge)?

Be careful how you phrase your question. The only allowed means are compass and ruler like @fresh_42 said.

Using a marked ruler ( you just need 2 marks on the ruler ) you can construct the cubic root of 2 and trisect any angle.

 

[jedishrfu]

Also origami folds can do it as well but that’s not a compass and straight edge.

http://www.cutoutfoldup.com/409-double-a-cube.php

 

[jedishrfu]

As @dagmar mentioned earlier a neusis construction could do it too but again it is not a straight edge and compass and so violates the spirit of the problem.

https://en.m.wikipedia.org/wiki/Neusis_construction

Here’s more history on the problem with additional references to search:

https://en.m.wikipedia.org/wiki/Doubling_the_cube

Having said this, why not try to define some three dimensional equivalent and then solve the problem. While it won’t be a solution in the classic sense, it could lead to some interesting and imaginative work. This is how new math or new inventions are often created.