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'Impossible' Solid's Volume (Posted on 2006-11-04) Difficulty: 4 of 5
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A solid whose plan view and front and side elevations resembled a multiplication symbol (x) bounded by a circle may be viewed here.

It is the third on that page.

The object can be created by imposing 3 cylinders on a cube in each of the x, y and z dimensions.

If the edges of the cube are of unit length, What is the volume of this object?

How mundane (as being simple) a solution can we get?

See The Solution Submitted by brianjn    
Rating: 4.0000 (3 votes)

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Starting... | Comment 1 of 11
Two ideas, nothing definite.

(1)  If the cube has unit length, then the cylindars all have unit diameter, or radius = 1/2 = r

In each plane, a relationship holds:
x^2 + y^2 <= r^2
y^2 + z^2 <= r^2
x^2 + z^2 <= r^2

I can imagine a computer solution in which a sector of the cube is meshed, and each point in the mesh is tested, then accepted or rejected as being part of the solid, subject to the above 3 equations.

(2)  Other thought is to first determine what percent of a cylindar's volume (r=1/2, h=1) is also included in a similar cylindar at 90 degrees.  Say it turns out to be 90%.  Since all 3 cylindars are orthogonal to each other I suspect it can then be deduced what the overlap is for all 3.

  Posted by Larry on 2006-11-04 14:36:36
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