Spherical Cube Perimeter

Geometry Level 5

Consider a sphere in the $xyz$ -coordinate system. Suppose there is a cube which circumscribes the sphere. We could represent each point on each of the cube's 12 edges in terms of spherical coordinates $(r, \theta, \phi)$ .

Suppose that, for every point on each of the 12 segments, we kept $\theta$ and $\phi$ the same and modified $r$ to make the point lie on the sphere.

What would be the ratio of the perimeter (combined length of all 12 segments) of the transformed cube to that of the original? Give your answer to 3 decimal places.

The answer is 0.615.

1 solution

Mark Hennings
Sep 13, 2017

If the sphere has radius $R$ , the cube has sides of length $2R$ . Two adjacent vertices of the cube and the centre of the cube (and the sphere) form an isosceles triangle of sides $R\sqrt{3},R\sqrt{3},2R$ , and so the angle at the centre is $\cos^{-1}\tfrac13$ . Thus the length of the curved segment on the sphere which corresponds to the side is $R\cos^{-1}\tfrac13$ . Thus the ratio of the curved length to the straight length is $\tfrac12\cos^{-1}\tfrac13$ , which is also the ratio of the perimeter of the spherical cube to that of the cube. The answer is $\tfrac12\cos^{-1}\tfrac13 = \boxed{0.6154797087}$ .

It should also be noted that the curved segment is indeed an arc of a circle (because the isosceles triangle defines a plane that cuts the sphere on a great circle). Had the arc been, for example, a parabola or something, the length wouldn't be that easy to compute.

Ivan Koswara - 3 years, 9 months ago

Maybe that will be the next one :)

Steven Chase - 3 years, 9 months ago

Of course, the curved segment is a segment of a great circle on the sphere. That is intrinsic to the problem. Every edge of the cube and the centre of the cube define a plane (passing through the centre of the sphere) in three-dimensional space. The process of mapping points on that edge to the sphere is simply radial projection onto the sphere in that plane. Thus the straight line that is the edge maps to a segment of a great circle...

Mark Hennings - 3 years, 9 months ago

Yes, that's true. Personally I think it should be at least mentioned ("the length of the curved segment, which is an arc of a great circle, ...") since I don't find it immediately obvious, but that's probably just me.

Ivan Koswara - 3 years, 8 months ago

@Ivan Koswara – Well, I did say, keep the angle parameters the same and adjust the radial parameter so that the point lies on the sphere. That should cover it. It's up to the problem solver to figure out the implications.

Steven Chase - 3 years, 8 months ago

@Steven Chase – Yes, that's what it says on the problem; that's precisely why I said it's not entirely obvious that it does make an arc of a great circle. The solver needs to figure it out.

Ivan Koswara - 3 years, 8 months ago

Niranjan Khanderia - 3 years, 8 months ago

Spherical Cube Perimeter

The answer is 0.615.

1 solution

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