Follow my logic. A point has no dimension. A line only has length. I don’t know if it applies to this. But I know the circumference of a circle is 2πr. Next comes area. The area of a circle, is πr^2. Then next comes volume. The volume of a sphere is (4/3)πr^3.
Anyways, this is all following a predictable pattern. With the addition of each dimension (my example is the circle/sphere), comes a new measure: length, area, volume… My question is: what comes after volume? What does this lead to with the fourth dimension (which I believe is time)? What is it called, and what does it measure?
Also interesting for this is the subject of calculus. It has been a while since I took calculus. And I didn’t do too well in my class when I took it (which is why I avoid bringing it up even). But there is another pattern developing here. I could be wrong, but I believe each next “measure” is in fact the integral of the last one. Am I right? And what then is the integral of (4/3)πr^3 and what does it mean?
The volume of a four dimension hypersphere is π^2*R^4/2. The volume gets a new factor of square root of π every time you add a dimension, but there is a square root of π in the denominator in every odd number of dimensions, so you only see integral powers of π.
Yes, the formual does come from an integral, but it is not the integral of the formula for the volume in the previous dimension. You can read all about it at:
If time is a dimension, then duration would be next after volume. A sphere that exists for a week is seven times “bigger” than a sphere that exists for a day.
Generally you can stick on the prefix “hyper” to make 4-dimensional (or sometimes higher) analogs of 3-dimensional things, or you can stick the dimensionality of the thing as a prefix, or you can use the word “glome”.
The only thing that’s tricky here is that, technically, circles and spheres are surfaces, not the things bounded by those surfaces.
A 1-sphere is a circle, the set of all points in a plane the same distance from the center. It has a circumference of 2πr, and it bounds a 2-ball with an area of πr^2.
A 2-sphere is the familiar sphere, the set of all points in a 3-D space the same distance from the center. It has a surface area of 4πr^2, and it bounds a 3-ball with a volume of 4/3πr^3.
A 3-sphere is a hypersphere, or glome, the set of all points in a 4-D space the same distance from the center. It has a surface volume (aka surface hyperarea) of 2π^2r^3, and it bounds a 4-ball with a hypervolume of 1/2π^2r^4.
As for what it measures, that depends on the space you draw it in. You can draw a sphere in any 3 dimensions, even abstract ones, and it works the same way as a sphere in 3-dimensional physical space. So, the same is true with a hypersphere. In fact, half the time, when you deal with a 3-sphere, you’re defining it over 2 complex dimensions or 1 quaternion dimension rather than 4 real dimensions.
But forget about that. If you draw a hypersphere in the 4-dimensional spacetime you remember from school, what does it mean? Think about what you’re drawing. It’s a sphere that starts as a point, expands (at the speed of light) and then contracts evenly, then vanishes again as a point.
It’s pretty hard to visualize 4 dimensions, so let’s drop left-right and think of a 3-dimensional spacetime, where time is drawn from left to right. What is a sphere in that case? It’s a circle that expands from nothing, then shrinks back to nothing, as you move from left to right. Its volume is a nice sum of the total area bounded by the circle over time. (You see where the integrals come into this?)
In the same way, the hypervolume of your hypersphere is a nice sum of the total volume of that growing-and-contracting sphere over time.
You probably noticed one problem here: The volume in your 3D spacetime has to be multiplying two lengths and one timespan, so you end up with m^2s, rather than the familiar m^3. And likewise, the hypervolume is going to end up with m^3s. Well, the whole point of spacetime is that space and time are interchangeable. (If you use Planck units instead of SI units, they even use the same unit; otherwise, you have to multiply by c to get from m^2s to m^3.) But sometimes you don’t want to interchange them. That “area bounded over time” interpretation or “volume bounded over time” makes sense in the units you got, right?
(I cheated a bit here, since I’m using a spacetime with a +++ or ++++ metric, which people don’t generally do. But don’t worry about that, unless you want to toss in some constants of -1 and i.)
Almost but, not quite. 4/3πr^3, a sphere’s volume, is the integral of 4πr^2, which is a sphere’s surface area, not a circle’s area. But you can see that it’s only different by a factor of 4.
In the same way, 1/2π^2r^4, a hypersphere’s hypervolume, is the integral of 2π^2r^3, a hypersphere’s surface volume, not a sphere’s volume, but it’s only off by a constant of 2/3π.
No, it’s volume * duration that’s next after volume, not duration itself (which is only 1-dimensional).
Think of it this way: next after width is width * height, aka area; next after area is area * depth = volume; next after volume is volume * duration = hypervolume.
I’ve never heard a distinct name for this property; it’s apparently just called ‘hyper-volume’.
Only if you define it to be: There’s nothing (mathematically) inherent in the fourth dimension that makes it have to be time. It could be just another spatial dimension, like the three you’re used to.
If it’s a spatial dimension, it measures distance, just like the other three.
There’s a “Minkowski Volume” defined in Minkowski spaces, so I guess that would be the name of a 4 dimensional volume with three spatial and one temporal dimension.
Yeah, I explained in my previous post (the long, rambling one above) that you can use any 4 dimensions you want, even abstract ones.
But I was pretty sure Little Nemo was talking about 4D spacetime, not just any 4 dimensions (otherwise, why bring up duration in the first place?), so I didn’t want to get into it again.
Sorry, you don’t get to define new terms until you publish.
More seriously, reusing everyday words with new definitions like this can be misleading and confusing. Think about it: if you were reading a pop-science book, and it mentioned the “existence of your world tube”, would you have any idea it actually meant the 4-volume, or would you just think it was talking about the property of it being there?
Then again, avoiding doing so can be just as confusing. I think some branches of math use “4-sphere” where others (and physicists, and me) use “3-sphere”, and “glome” is just a silly word that doesn’t make sense even if you’re a Latin scholar, and just avoiding the issue by talking about an x^2+y^2+z^2+w^2=r^2 surface without a name at all is even worse. So maybe there’s no good answer.
Or maybe we need to create a whole new occupation to come up with names for concepts from math and science. If nothing else, it would force the science geeks to talk to all those communications majors, which might get them a few dates.
In general, the n-volumes of an n-dimensional affine space with underlying vector space V can be identified with the n-ary antisymmetric tensor power of V. [Thus the relationship between determinants and volumes]. This doesn’t depend on any inner product structure associated to V (in other words, it doesn’t depend on a metric signature), and so I would think the 4-volume of Minkowski space worked just the same as that of 4-dimensional Euclidean space. But perhaps there’s also another volume concept worth studying in that context for some reason.
