Say, that you have a N-dimensional hypercube with the edge of 1 (unit).

You can then, of course, put 2^N spheres inside, each with a diameter of 1/2.

For example, eight 50 cm diameter spheres, inside a 1 meter cube, fit nicely. As four circles each of 50 cm diameter fit well inside a unit square.

Now, you squeeze another sphere in the middle, touching every other of those 2^N spheres.

It has been recently noted, that the middle sphere can have a diameter bigger than 1!

Question No. 1: At which dimension X, does this (first) happen?

Question No. 2: At which dimension Y (if at any), the middle hypersphere has bigger hypervolume than that of its “encompassing” hypercube?

Question No. 3: How many 100 dimensional hyperspheres with the diameter of 2, can be squeezed (with no deformation) inside a 100 dimensional hypercube with the edge of 1?

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module DimVolume where

import Data.Number.BigFloat

type BigFloat50 = BigFloat Prec50

(^.) :: Epsilon a => BigFloat a -> BigFloat a -> BigFloat a

(^.) _ 0 = 1

(^.) x y = x * ( x ^. (y-1))

factorial :: (Eq n, Num n) => n -> n

factorial 0 = 1

factorial n = factorial (n – 1) * n

kFromDim :: (Epsilon n) => Integer -> BigFloat n

kFromDim dim

| even dim = fromInteger dim / 2

| otherwise = (fromInteger dim – 1) / 2

radiusFromDim :: (Epsilon n) => Integer -> BigFloat n

radiusFromDim dim = ((sqrt . fromInteger) dim – 1) / 4

volume :: (Epsilon n) => Integer -> BigFloat n

volume dim

| kEven = (pi^.k / factorial k) * (r^.(2*k))

| otherwise = ((2 * factorial k * (4 * pi) ^. k) / factorial (2 * k + 1)) * r^.(2*k + 1)

where

kEven = even dim

k = kFromDim dim

r = radiusFromDim dim

main :: IO ()

main = print (volume 1206 :: BigFloat50)

It’s some discussion about this here:

http://lesswrong.com/r/discussion/lw/ogr/open_thread_jan_16_jan_22_2016/dkt1