The n-Sphere

My research involves a field called dimensionality reduction. As such, I do a lot of thinking about dimensions and, unfortunately, have to think about dimensions greater than three. If you have never thought of a cube in 4 dimensions, try it now. Go ahead. I’ll wait.

That wasn’t much fun, was it? You weren’t very successful, were you? It’s ok. I never get far either.

Today I want to bring a crazy quirk about dimensions to your attention. We are going to look at the volume of the n-sphere, which is just a fancy term for an ordinary sphere in different dimensions (n+1 dimensions, to be precise). Now, the volume of the (n-1)-sphere is given by a messy equation,


Rather than explain the gamma function, I can use the helpful table given below to explain my quirk (thanks, Wikipedia!). Here are the volumes for a sphere of radius R in dimensions 0 through 9. (Note that, when we say “volume,” we refer to how much “stuff” we can fit inside a sphere in a given dimension. For example, the volume of a line segment is just its length.)


Notice that the volume scales with the radius raised to the power of the dimension? Notice how the constant that multiplies the radius peaks between 5 and 6 (actually, it peaks at n = 5.2569464) dimensions? Does this seem weird to you?

Let me explain in words. This says that if you take a sphere of radius 1 and compute it’s volume, the volume will increase as you add dimensions UNTIL you get to about 6 dimensions, whereupon it reverses and heads back to zero! A sphere of radius 1 has a bigger volume in 3D than it does in 1 million-D.

What about the volume of a sphere with radius 2? Before, with R=1, the R^D portion of the volume equation did not contribute (recall that one raised to any real exponent is still one). Now, if R=2, the volume increases with increasing dimension (edit: provided the gamma in the denominator does not trounce the numerator). A sphere of radius 2 has a much much much smaller volume in 3D than it does in 20D!

This is crazy when you think about it. Adding dimensions (once you get past 5) to a sphere of radius one, or less than one, actually makes its volume smaller. Adding dimensions, up to a point, to a sphere of radius greater than one makes its volume larger. This is not intuitive to our 3D minds. We think you can fit more “stuff” into a 3D sphere than into a 2D circle, provided the radius is the same. The math says this idea does not generalize. Adding more dimensions to the unit sphere makes its volume smaller!


~ by wcuk on March 13, 2008.

18 Responses to “The n-Sphere”

  1. this is the awesomest post of all time. for a second there, i thought i was reading off wolfram’s website and not wcuk. awesome. just awesome.

  2. p.s. so which is larger, a third dimensional sphere of radius 1 yard, or a millionth dimensional sphere of 3 feet?

  3. Mike hit one of my two responses on his second comment. This is a numbers game where the playing field changes with units. Speaking of units, comparing 3 feet to xxx feet^9th is like comparing apples to Chicago.

    However, I concede that I don’t know anything about how this would apply to the world of dimensionality reduction. Perhaps this is, in fact, an astonishing observation that I am failing to grasp.

  4. Gamma(100) ~ 10^155, Gamma(500,001) ~ Inf. anything^500000 ~ Inf. 4pi/3 ~ 4.188

    Gamma beats the pants off of the exponential in a race, so I’d have to call pi^500000/Gamma(500001) ~ 0

    1 yard sphere in 3D FTW!

  5. Charlie, you are correct. Our volume here has different units. Perhaps it’s better to ask how much “N-dimensional water” could we pour into an “N-dimensional shape”?

    Also, I have entered contract talks with Gatorade to market N-dimensional water as a performance drink. I have the patent, so don’t try anything funny.

  6. zomg charlie i was going to post a “p.p.s.” offering a cookie to anyone who could answer my question using fruit (since i would formulate the answer myself in terms of apples and oranges). your comparison is better, you win TWO cookies (and a rematch in SSBB).

    it’s like comparing torque to work. both can be expressed in identical units but neither can be directly compared (can the MechE/Physics folks confirm this for me?)

    my brain hurts now.

  7. Mike,

    Don and I had this discussion in a car once. Though Torque can be expressed in the same units as work (Newton-meters, for example), Torque can not be directly compared to work. The reason that they are in the same units, is because the ‘distance’ portion of the rotational work = force x distance equation is radians, which is dimensionless. Work = Torque x radians.


    N-dimensional water: fueling needs you didn’t know your body had.
    I guess that the difficulty in blogging something scientific is in the lack of an introduction sufficient for your readership.

  8. This *is* interesting.

    Now at the time of the “Big Bang” the universe had >6 dimensions according to various models of unified field theory (8-12). Is the reduction in dimensionality due to symmetry breaking what caused inflation? If so, it certainly clarifies things though only up to the weirdness of the math.

  9. Science is rife (or rather, “Reif”) with n-spheres, as our teacher once said.

  10. I don’t want to alarm anyone, but I dream in 4 dimensions (not including time). Also, I once wrote a C++ program for 4-D checkers, tic-tac-toe, and chess (the last one was really confusing to play).

  11. Jeff- reduction in dimensionality didn’t cause inflation, it was the expansion of the money supply at a rate greater than that of GDP growth. Sheesh. That’s Econ 101!

    I kid, I kid!

  12. @theJenkster


    Only getting back here later on. Forgot to click follow-up I guess.

  13. I’m not quite sure I understand this. I can’t see why it is right to say that the volume of a 10 dimensional sphere is smaller than a 3 dimensional one. Volume in 2 dimensions is what we would call area, and this is measured in completely different units to volume in 3 dimensions, so even if a 3 dimensional shape had more cm^3 in it than there were cm^2 in a 2 dimensional shape, it would still presumably be meaningless to say that the 2 dimensional shape had a “larger” volume. Take an example of a square length 1 cm, area 1cm^2, I could choose to measure it in meters and call the length 0.01 and the area 0.0001. If I now work out the volume of the cube with the same lengths, it would have either 1cm^3 volume or 0.000001m^3 volume. If I measured in meters the “volume” appears to be less in the “cube” but if I measured in centimeters then the volume appears to be the same, even though the shapes are identical. Surely this shows that it is meaningless to talk of one shape having more “volume” than another, when one is in higher dimensions than the other.

  14. Toby, you’re getting hung up on units. In Mathematics, units don’t matter.

  15. Toby: It’s true that any given volume has fewer cubic meters than cubic centimeters, but a cubic meter compensates by being *a lot larger* than a cubic centimeter! So it all makes sense.

  16. Mike: That’s the point I’m making really, the volume appears to go up or down in different dimensions depending on the units you use, even though, as theJenksster pointed out, units don’t matter. I was using those units as an example to show that it is meaningless to say that a four dimensional shape has more volume than a 3 dimensional shape, just as it is meaningless to say that a cube has more volume than a square, area and volume are completely different quantities, as are 3 dimensional volume and 4 dimensional volume.

  17. You’re right, physically speaking, about the different units. I agree it doesn’t make any physical sense to setup an inequality between a square centimeter and a cubic centimeter.

    Mathematically, we are focusing on the coefficient of V_n in the original post, and it’s certainly not meaningless to compare *them*, since they are just real numbers. The troubling fact is that the coefficients increase to a peak and then decrease — this just doesn’t agree with our intuition. But it doesn’t have to…

    I think we’re on the same page.

  18. How do u calculate the volume of the intersection of two sphere in higher dimensions D>3?

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