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