Colossal Gardner, ch. 11


Spheres and Hyperspheres.
Gardner starts out with a simple definition of a circle – take a ruler with one end fixed in place and put a pen at the other end. Rotate the ruler around the fixed point and you’ll get a circle, defined in Cartesian coordinates as  x^2 + y^2 = r^2, where r is the radius. Let the ruler move in 3 dimensions and you get a sphere, as x^2 + y^2 + z^2 = r^2. Keep adding terms and you can go to as many dimensions as you want. The surface of the object has a dimensionality of n-1. That is, a circle’s surface is a 1D line, and a sphere’s surface is a 2D plane. In the late 19-century, many mathematicians and physicists felt that gravity and electromagnetism in our 3D universe could be transmitted over the surface of a large 4D hypersphere containing the universe.

Einstein himself suggested that within a hypersphere, a rocket moving on a straight line traveling far enough in one direction would end up reaching Earth again from the opposite direction. This concept of an enclosed warped space also shows up in the writings of George Gamow. It ties to the idea that space is warped, rather than flat. Since we’re within space as 3D beings, to us straight lines look straight because we’re tied to the same “surfaces” that the lines are painted on, much like how an ant on an inflated balloon is going to think that the line on the balloon is flat. Looking down on that 3D surface from the point of view of a 4D observer, it’d more obvious that things aren’t flat after all.

Circles and hyperspheres share other properties as well. n-spheres rotate around an n-2 space. Circles rotate around a point, spheres around an axis line, and 4D spheres around a plane. Cross sections are n-1: Cut a circle and you get two points; cut a sphere with a plane and you get a circle; cut a 4D sphere with a 3D plane and you get a sphere. Just like you can turn a thin rubber ring inside out, from 4D space you’d be able to turn a 3D sphere inside out.


(What’s the size of circle #4 in both figures?)

We then move to the maximum number of mutually touching n-spheres. On a plane, you can have no more than 4 circles touching together, either with all 4 being external, or 3 within the larger outer one. The formula is 2(a^2 + b^2 + c^2 + d^2) = (a + b + c + d)^2, where a, b, c and d are the bends (curvatures) of each circle, where the curvature is 1/radius. Generalized, the maximum number of mutually touching spheres is n+2, and n times the sum of the squares of the bends equals the square of the sum of the bends.

For packing problems for 3D balls, refer to last week’s blog entry. Generalized answers for N-spheres had not been found at the time of Gardner’s book. He does mention a paradox found by Leo Moser for 9-space, where a 9D sphere in the center of a sphere cluster actually sits outside the cluster while the 2^9 = 512 corners of the enclosing box still have room to hold 512 unit 9-spheres. (Unfortunately, I can’t find a copy of Moser’s paradox online.)

Puzzles:
For the above touching circles, if three of the radii have lengths of 1, 2 and 3 units, what’s the radius for circle four for both cases?

You have 3 perfectly spherical grapefruits resting on a counter, each touching the others. Under them you have a smaller orange also resting on the counter. The grapefruits have a radius of 3 inches. What’s the radius of the orange?

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