What’s a sphere?
Geometrically, a sphere is the locus of points equidistant from a given point (which we call the ‘center’. Algebraically, it is the solution set to the equation
where d is the number of dimensions the sphere lives in, and r is some positive constant which we call the radius.
In one dimension, a 0-sphere is just two points, one on each side of the ‘center’. In two dimensions we call a 1-sphere a circle. The words sphere itself is typically reserved for a 2-sphere. Higher dimensional (d-1)-spheres are typically just referred to as hyperspheres. Why d-1? Think of a circle in two dimensions… how many dimensions does the circle actually have? You can only slide along the circle either forward or backward, hence it is one dimensional. A moment thinking about a sphere in three dimensions will get you to the same conclusion
So, whats the surface area of one of these hyperspheres? We could do some nasty area element calculation from the algebraic constrain above, but that would make everyone cry (except Russians…. Russians do not need fancy techniques when they can do something with brute force). Instead, let’s integrate an isotropic multivariate Gaussian with vanishing covariance two ways. A what now? Consider the integral:
where the covariance matrix is diagonal with equal entries (σ²). Because of the pleasant structure of the covariance, we can break this integral up into d Gaussians and integrate:
Yay! Well… not yet, stop being so excited.
To connect this to hyperspheres we’re going to change to polar coordinates. The meassure will then be over all radii and solid angles:
where we have used the definition of the Euler Gamma Function. Try doing the integral yourself! We can now equate these two results and solve for the total solid angle of a hypersphere in d dimensions, and then get the surface area of a hypersphere:
We can plot this as a function of dimension (because of our know how with the Gamma function, we can even look at fractal dimensions…):
The function maximizes at d=7.25… dimensions. That’s weird. WHAT DOES IT MEAN?!?!?!?!?
Well, the total solid angle of a space can be thought of mechanically as how much you would have to move your head around to survey everything in every direction. Clearly in 1d you just have to look ahead of you and behind you. In 2d you have to pull an Exorcist move, and spin your head 360 degrees. In 3d you have to add to that and look above and below you. As you keep increasing the number of dimensions, the amount of neck movement keeps getting worse. Until you hit 7 dimensions. Then as you increase the number of dimensions, you realize you have to move your head less and less. In fact, by the time you’re in 18 dimensions, you have to move your head less than in 1 dimension, to see everything. Higher dimensions are weird.
Ok, what about the volume of hyperspheres? That’s pretty easy. We just integrate from a radius of 0 to some final radius to get:
This too is interesting. For a fixed radius, The volume maximizes as d = 5.25… dimensions. Higher dimensional unit-spheres just don’t take up that much space. Finally we can also quickly calculate the surface area to volume ratio. This turns out to be just d/r. As expected, the larger hyperspheres get, the more volume to surface area they have.
This scaling is very important, and the main reason Life uses cells as its fundamental unit. The energy usage of a cell is proportional to its volume, whereas its energy absorption is proportional to its surface area. If this ratio of absorption to usage is greater than one, then the cell is making excess energy and can grow in size. Eventually, however, because this ratio is proportional to the surface area to volume ratio, it will become less than 1 as the radius increases, and the cell will not be able to support further growth. At this point it is more efficient to split into two cells and CELLS ARE EVERYWHERE.
Interestingly, due to the scaling with dimension, we would expect cells that evolve in universes with more spatial dimensions to be larger. Assuming energy usage and absorption don’t change in higher dimensions, we would not expect multi-cellular life to ever evolve in worlds where the number of spatial dimensions is incredibly high. Someone really should make a B-movie entitled: Attack of the Amoeba of a Billion Dimensions!!!!