I’ve always had a strong curiosity about how things work. My dad used to despair how I’d take things apart but rarely put them back together. My interest was inside — in understanding the mechanism. (The irony is that I began my corporate career arc as a hardware repair technician.)
My curiosity includes a love of discovery, especially unexpected ones, and extra especially ones I stumble on myself. It’s one thing to be taught a neat new thing, but a rare delight to figure it out for oneself. It’s like hitting a home run (or at least a base-clearing double).
Recently, I was delighted to discover something amazing about spheres.
I was reading Three Roads to Quantum Gravity (2001), by Lee Smolin, and he had a section on the holographic principle. Briefly, that’s the idea that the surface (or boundary) of a volume contains all possible information the volume can store. (See: BB #76: The Holographic Theory)
I’ve never been sure what to make of that idea, but I think it supervenes on the maximum possible density of a volume — which means we’re talking a black hole. Since there’s nothing inside a black hole other than the singularity, I think I can see how the surface might suffice to describe the entire volume (i.e. the singularity). If there is less than the max, then there is “extra” space on the surface to account for 3D distances.
The above from an argument involving a ray drawn from each Planck area on the surface — each possible information “pixel” — to the center. The rays, under the holographic principle, account for all possible information in the volume. But that suggests each ray can be only one bit of information. That only makes sense if the inside is empty except for the singularity.
It got me thinking about the volume versus the surface area of a sphere. It’s really hard to see how the surface can contain all necessary information about the volume. The former famously grows as the square of the radius while the latter grows as its cube. Those two things are not the same!
So I’d written out, and was staring at, the formulae for the volume and for the surface area of a sphere:
Looking at them together like that I noticed that the surface area is the derivative of the volume. “Well that’s odd,” I thought. A circle is a two-dimensional sphere, so I wondered if the same thing applied to circles.
I wrote out the formulae for a circle’s area and circumference:
And, whoa, again the boundary of the space is the derivative of the enclosed space. (Turns out this is well-known to mathematicians, of course, but it was a discovery to me.)
If you remember anything from calculus, you likely remember derivatives:
A derivative is a function that gives the rate of change of another function.
The canonical example is that velocity is the derivative of distance over time, and acceleration is the derivative of velocity over time. (Which means acceleration is the second-derivative of distance over time.) Another common example is that the slope of a hill is the derivative of height over distance.
When we say something like distance over time, we mean a function that, given a timespan, returns a distance traveled. Mathematically:
Given function f, its derivative f’ (sometimes called g) takes the same time span and returns the change in distance, or velocity:
Given function f, then function f’ is its first derivative, function f” is its second derivative, and so on, each being the rate of change of the previous function. Sometimes g and h, respectively, are used for the first and second derivatives.
All that matters here is this simple rule:
That is, given some function f with an exponential, its derivative multiplies the expression by the value of that exponential and then subtracts one from it. Here’s an example:
Applying this rule to the volume equation for spheres — taking its derivative — gives the equation for surface area. That discovery surprised me; it’s not something I recall any math teacher or text mentioning.
Put graphically, given some curve, its derivative is another curve showing the slope of the first curve. (See my Solar Derivative post for details.) That means a graph of the sphere’s surface area (over radius) is also a graph of the slope of the matching volume graph.
This applies to a circle’s area and parameter, too, so there is some universality. It’s a matter of interiors and boundaries; the latter somehow being the rate of change of the former.
The next obvious step, if it works with circles and spheres, does it also work with squares and cubes (and tesseracts)? Both circles and squares share a notion of being very symmetrical about the origin.
At first it looked like it was off by a factor of two. Here are the usual formulae for the area and parameter of a circle:
And for the volume and surface area of a cube:
For the square, we want the derivative of s2, which is 2s, not 4s for parameter length. Unfortunately, 4s is the correct formula.
For the cube we’d likewise want 3s2, not 6s2, but again the latter is the correct formula. Both of these are off by a factor of two.
Upon digging into this boundary-is-the-derivative-of-the-interior thing, I read that using an equivalent to radius makes it work. Normally (as seen above), the formulae use the length of the side. That would be like using the diameter of a circle or sphere. The equivalent to radius would be using d=s/2 — half the side length.
Now the (formula for the) boundary is the derivative of the (formula for the) interior:
Both with the square (above) and with the cube:
This extends to the tesseract:
And beyond. The sphere, likewise, extends to higher dimensions. For example, here are the formulae for a 4D sphere:
As a way to test your understanding, here are just the volume formulae for 5D and 6D spheres. See if you can figure the surface area formulae by doing their derivatives:
Just use the simple rule from above.
Looking at other shapes, I realized spheres and cubes are special cases. In general this relationship doesn’t apply. A couple of cases come close or can be made to work with a concession.
For instance, the formulae for the volume and surface area of a cylinder are:
The first term is the derivative, but that second term is extra — its due to the surface area of the end caps. If the cylinder is open, however (no end caps), then the surface area of the tube is indeed the derivative of the volume it could contain.
This also works for a torus if R, the major axis, is taken as a constant:
And that’s about it. The trick apparently requires symmetry about the axes to work.
For instance, it doesn’t work with cones:
Which are pretty simple shapes that do have symmetry along their length, but their surface area definitely isn’t the derivative of the volume. The derivative of that would be:
Which is rather different from the real formula. (It also assumes the height, h, is a constant.)
The ellipsoid seemed like a symmetrical shape, almost spherical-ish, but the surface area definitely isn’t the derivative of the volume:
(Yikes!) The variables a, b, and c, are the three radii. What’s interesting is that if a=b=c, the ellipsoid reduces to a sphere, and then the surface area should be the derivative of the volume.
Apparently the ellipticity is the issue. Although the ellipse has a (very simple) formula for area:
There isn’t one, certainly not a derivative, for the parameter. (Calculating that apparently requires doing an interval.)
The obvious question is: What’s going on here?
Actually, it does make sense.
Imagine you have a ball with radius r. It has a current volume, given by:
Now imagine we apply a thin coat of paint to the ball. Since the paint coat is thin, the amount of paint is essentially the surface area of the ball. Which is given by:
The new coat of paint increases the ball’s volume just slightly. The new volume is the old volume plus the paint — which we know is essentially the surface area. (As paint coat thickness approaches zero, the amount approaches the surface area exactly.)
So the growth rate of the ball is its current surface area. As the ball gets larger, the grow rate, of course, increases.
This logic works for spheres of all dimensions (including circles), and it also works for cubes of all dimensions (including squares). It also works with open cylinders and fix-radius torii, but so far that’s about it.
The more interesting question, I think, is why it fails in other shapes.
I took a look at the Platonic solids, and they don’t appear to follow the pattern. The math for surface area and volume is a bit more involved, but at first blush I’m not seeing the former as the derivative of the latter.
But cubes are one of the Platonic solids, so the math works for at least one of them. Perhaps it can be untangled to work with the others. On the other hand, the sphere being a special case of the ellipsoid (and circles being a special case of ellipses) didn’t seem to help with the elliptical case.
Obviously it requires a very simple relationship between the boundary and the interior. Spheres, cubes, tubes, and torii, are all very simple shapes. The key may be the number of parameters necessary to specify them. Spheres and cubes only need one, tubes and torii only need two. (But the Platonic solids all only need one, so maybe parameters isn’t key?)
All-in-all, an interesting problem to ponder. For me, in stumbling over it, it echoed the most common exclamation in science: “Huh! That’s weird!”
That has been the beginning of a whole lot of science!
Stay curious, my friends! Go forth and spread beauty and light.