See "The Planiverse" by A. K. Dewdney on my Book List.
N-Dimensional Hyperspheres and Hyperballs
The volume of an n-dimensional hypercube is simply sn where s is the length of a side.
What is the volume of an n-dimensional hyperball?
First, let's define some familar geometric shapes.
- A circle is the 1-dimensional rim of 2-dimensional disk.
- A disk is the 2-dimensional area filling a circle.
- A sphere is the 2-dimensional surface of a 3-dimensional ball.
- A ball is the 3-dimensional volume filling a sphere.
In higher dimensions we use the terms hypersphere and hyperball, or, to be more specific, an n-sphere or n-ball.
- An n-sphere is a hypersphere with n dimensions.
- An n-ball is a hyperball with n dimensions.
We can redefine the familar shapes using our new terms.
- A circle is a 1-sphere.
- A disk is a 2-ball.
- A sphere is a 2-sphere.
- A ball is a 3-ball.
The n-content is the n-dimensional "area" or "volume" of a geometric shape. For example:
- The 1-content of a circle is its circumference.
- The 2-content of a disk is its area.
- The 2-content of a sphere (2-sphere) is its surface area.
- The 3-content of a ball (3-ball) is its volume.
- The 3-content of a 3-sphere (hypersphere) is its hyper-surface-area.
- The 4-content of a 4-ball (hyperball) is its hyper-volume.
Here is a table showing, for different dimensions, the n-content ("volume") of hyperballs and the boundary (n-1)-content ("surface area") of their corresponding hyperspheres:
|Dimension (n)||Full Shape||Full n-Content
|Boundary Shape||Boundary (n-1)-Content
|2||disk (2-ball)||π r2||circle (1-sphere)||2π r|
|3||ball (3-ball)||(4/3)π r3||sphere (2-sphere)||4π r2|
Isn't it strange that the power of π increases by one only when the dimension increases by two?
In general, the n-content ("volume") of an n-dimensional hyperball is:
|if n is even:||(1/(n/2)!)πn/2rn|
|if n is odd:||(2n((n-1)/2)!/n!)π(n-1)/2rn|
where n! = n(n-1)(n-2)... (factorial) and n!! = n(n-2)(n-4)... (double factorial).
In general, the boundary "surface area" ((n-1)-content) of an n-dimensional hyperball is the "volume" (n-content) multiplied by (n/r).
Using differential calculus, you can find the "surface area" ((n-1)-content) of an n-ball by differentiating its "volume" (n-content) with respect to the radius, r. This is a fun exercise for calculus beginners.
Using integral calculus, you can derive the formulas for n-dimensional balls based on the formulas for (n-1)-dimensional balls. The "volume" of an n-dimensional ball can be found by integrating the "surface area" of (n-1)-dimensional spherical shells from 0 to r, like the layers of an onion.