Of course a 3-dimensional thing is bigger than a 2-dimensional thing. Just like a beach ball is bigger than a circle cut from paper. And it’s equally obvious that a 4-dimensional thing would be bigger than a 3-dimensional thing.
Or … is anything obvious? Following is a lesson in multi-dimensional reasoning.
A 2-dimensional “ball” is a circle. Two-dimensional points with two coordinates comprise the 2-ball. Any pair of rectangular coordinates on the circle satisfy:*
.
For example the pair (√½, √½) is on the circle. And the pair (√¼, √¾) is also on it. But (½, ½) isn’t.
A 3-ball is made up of the 3-points whose three rect-coords satisfy
.
That’s just a normal sphere from real life.
And a 4-D ball would consist of any array of four values whose Pythagorean sum** is 1:
A 5-ball is likewise the set of all 5-points with Pythagorean norm 1. It’s bigger than a 4-ball, of course.
A centered unit 6-sphere is the set of all 6-points with norm 1. It’s smaller than a unit 5-sphere.
- You: What?
- Me: Don’t ask what, you heard me.
A 6-dimensional ball is smaller than a 5-dimensional ball.
The 7-ball is smaller still, and higher dimensions keep getting smaller in volume.
True story. I am not making this up.
If you don’t believe me, just do the sextuple integral. The volume of a 6-sphere is 5.6771 and the volume of a 7-sphere is 4.7247. A 13-ball is less than 1 unit volume.
* I’m talking about a “unit circle” with radius 1, but that could be radius 1 mile or radius 1 nanometer. Or, like, whatever.
** Pythagorean sum? I’m being sly. Hinting at future posts about
- L_p norms
- √2
- Pythagoras in higher dimensions
- statistical variance as a Euclidean phenomenon.
- the logic of multiple dimensions
