The N-sphere

Aug 26, 2017

It's very common for high-school students to learn the geometry of a circle.
Trigonometric functions (sin(), cos(), etc.) are required knowledge for
high-school graduates, and they have a close relationship to circles. This means
that most students have seen the equation for a unit circle:

y = \sqrt{1 - x^2}

Or put in a more general form:

x^2 + y^2 = 1

Not as many have taken calculus, but for those that have they've probably also
seen the equation for a unit sphere:

x^2 + y^2 + z^2 = 1

Once you've seen both equations, it's easy to notice the pattern - every dimensional variable is squared, and those squares are summed to equal 1. This means that a hyper-sphere, regardless of how hard is may be to visualize, is no more mathematically complicated than the sum of 4 squares. In fact, any n-dimensional sphere (or n-sphere as they are known) can be calculated as

{x_1}^2 + {x_2}^2 + ... + {x_n}^2 = 1
\sum_{i=1}^n {x_i}^2 = 1

In summation-notation

The n-sphere isn't something most people were taught about. That's probably because it's not something that you'd run into unless you either learn about it yourself or you've take a few different calculus courses.

But there is a simpler form that could be discussed earlier, perhaps as a way to introduce students to multi-variate calculus early on. In the summation form of the n-sphere shown above, i can be less than two. You can have a 1-dimensional sphere.

x^2 = 1

Also known as a 0-sphere

The above equation does not produce a shape - since the drawing space for a 1-D function is just a single line. Rather, the 0-sphere looks like two points. For the unit 0-sphere those points are 1 and -1 (as both equal 1 when squared).

The red dots above are the 0-sphere.

The 0-sphere might seem useless, but it actually shows up much more often than you might think. When two spheres intersect, the shape on their surfaces that intersects each other is a circle. And when two circles (A and B) intersect, their intersection is a 0-sphere C.

References

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