I don’t know about the rest of you, but I find mathematics and physics a fascinating subject. Such clockwork perfection. Probably why I still believe in a Creator (of one form or another).
I digress though. My question is purely a physics one. Finding the area of a two-dimensional object, like a circle for example, x[SUP]2[/SUP]+y[SUP]2[/SUP]=r[SUP]2[/SUP] is easy. I don’t know the formula off-hand. But I know I’ve probably seen it many times. Three-dimensional objects have volume. And again, it is fairly straightforward.
But what happens when you get to 4 dimensions? (And I might as well ask, 5 or 6 for that matter?) What is it called? And what is the formula for a, for example, 4th-dimensional hypersphere and hypercube?
I guess I’d call it hypervolume and it is defined pretty much the same as in lower dimensions. The hypervolume (HV) of a hypercube is the 4th power of the length of a side. For any other figure, find a collection of cubes that cover it and that gives a larger HV and a disjoint collection of cubes whose union fits inside it. The sum of the HVs of the first gives a number and the sum of the HVs of the second collection gives a (provably) smaller number. Do this for all collections of the first sort and look at the greatest lower bound of them (called the outer measure) and similarly define the inner measure as the least upper bound of the numbers over all possible sorts of the second sort. If the inner and outer measures agree (which they will for any figure of the kind you will readily think of–spheres, polytopes, and the like) that number is called the measure or hypervolume of the shape.
A two dimensional circle has a two dimensional interior (area) and a one dimensional exterior (circumference).
A three dimensional sphere has a three dimensional interior (volume) and a two dimensional surface (surface area).
A four dimensional hypersphere has a four dimensional interior (hypervolume), and a three dimensional surface.
An interesting topological fact is that in four dimensions, a one-dimensional curve cannot form a knot, like it can in three dimensions. However, a two dimensional surface can be knotted in four dimensions.
No, just as the area of a 2-d square is L[sup]2[/sup] and the volume of a 3-d cube is L[sup]3[/sup], the hypervolume of a 4-d tesseract would be L[sup]4[/sup].
It doesn’t go into how we find formulas for hypervolume, but it transforms a probability question involving a dart board into a hypergeometry problem and gets into the fascinating way inscribed n-spheres take up a decreasing portion of their corresponding n-cubes.
One tidbit I remember off the top of my head: Just as the formulas for the interior of a 2-d circle and 3-d cube have a factor of pi, for 4 and 5 dimensions, they have a factor of pi squared, and for 6 and 7, it’s pi cubed, and so on.
Note that if you fix a radius, let’s say r, but allow the number of dimensions n to increase, the volume of the n-sphere will actually tend to zero! The volume of a cube is (proportional to) r[sup]n[/sup] of course.
This is fairly easy to see from the 2d to 3d transformation.
The round shape that takes up the same proportion of a cube that a circle takes up of a square is a cylinder. You just extend the 2d shape into the 3rd dimension. But a sphere lops off all the “corners” of the cylinder. Higher dimensional spheres do the same thing (but are much harder to visualize). You can sort of imagine a 3d sphere extended linearly into 4 dimensions as a hyper-cylinder, with 3 circular dimensions and one linear one, then cutting off the extra bits to make a 4-sphere.
I was just reading (in a book by John Allen Paulos) an argument that most people are very strange, in some way (“people who, at least along some measurable dimensions, are statistically way off the charts”).
He imagines a multi-dimensional hypercube made up of many different dimensions along which people can be measured, and calls the middle 90% of each side the “normal” side, and the part at each end—the half an inch at either end of a 10-inch-long line—the extreme. As the number of dimensions goes up, the percentage of the hypercube that is completely “normal”—not extreme on any of its dimensions—goes way down. For 100 dimensions, the interior—the part that isn’t within half an inch of the edge–makes up about.0027 percent of the whole.
Thus, “most people live on the extreme, abnormal edges of the human multi-dimensional hypercube.”
Is two square inches greater or less than one cubic inch? The question is unanswerable, unless you say that any finite number of square inches is less than any nonzero number of cubic inches.
Sure, I didn’t realize your quibble was with the concept of comparing n-volumes for different n’s since you cleverly hid it by using the definition of n-volume.
Gotta be really skeptical :dubious: of any Creator who, unsatisfied with merely creating not only irrational but transcendental numbers, then went on to make some of those (like e and pi) so important. What was He thinking?
It’s always seemed to me that it’s the other way around: God made the real numbers, but integers exist only in the minds of humans. Sure, one sheep, two sheep, three sheep, but a big sheep can produce more wool and mutton than a small sheep, and inches of wool and pounds of mutton can take on any value.
