Doubling a Cube: Can 3D Geometry Help?

In summary, in plane geometry, it is impossible to construct a line equal to the cubic 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. However, in solid geometry, using construction tools equivalent to a compass and straight-edge, it is possible to double a cube. This can be achieved through methods such as using a marked ruler or neusis constructions. While these methods may not fit the traditional definition of a compass and straight-edge, they still offer a solution to the problem and can lead to new discoveries in mathematics and technology.
  • #1
Thecla
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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)?
 
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  • #2
Thecla said:
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.
 
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  • #3
Thecla said:
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.
 
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  • #5
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.
 
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Related to Doubling a Cube: Can 3D Geometry Help?

1. What is the concept of "doubling a cube"?

"Doubling a cube" is a mathematical problem that involves constructing a cube with double the volume of a given cube using only a straightedge and compass. This problem has been studied for thousands of years and has been proven to be impossible using only these tools.

2. Why is it impossible to double a cube using only a straightedge and compass?

This problem is impossible because the cube root of 2, which is the number needed to double the volume of a cube, is not a constructible number. This means it cannot be created using only a straightedge and compass.

3. How does 3D geometry relate to the problem of doubling a cube?

3D geometry plays a crucial role in understanding the problem of doubling a cube. The concept of volume and the relationship between side lengths, surface area, and volume of a cube are all important aspects of 3D geometry that are relevant to this problem.

4. Are there any other methods or tools that can be used to double a cube?

While it is impossible to double a cube using only a straightedge and compass, there are other methods and tools that can be used to solve this problem. These include higher-level mathematical concepts such as calculus and algebraic geometry, as well as modern technology like computers and 3D printing.

5. Why is the problem of doubling a cube still studied today?

The problem of doubling a cube may seem like a simple mathematical puzzle, but it has significant implications in other areas of mathematics and even in real-world applications. It is also a useful exercise for developing problem-solving skills and critical thinking in mathematics.

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