# A Geometry Thing That’s Left Me Unsettled

I came across a little geometry thing that left me unsettled, even as I have to admit it’s correct. Start with a two-dimensional space, or as the hew-mons call it, a plane. Draw a square with sides of length two and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1), (+1, -1), (-1, +1), and (-1, -1). Around each of these corners draw a circle of radius 1.

There is some largest circle that you can draw, centered on the origin, the dead center of the square, with Cartesian coordinates (0, 0), and that just touches all of the corners’ circles. It has a radius of a little under 0.414.

Now think of the three-dimensional analog. Three-dimensional space. Draw a box with sides all of length two and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1, +1), (+1, +1, -1), (+1, -1, +1), (+1, -1, -1), (-1, +1, +1), (-1, +1, -1), (-1, -1, +1), and (-1, -1, -1). Around each of these eight corners draw a circle of radius 1.

There is some largest sphere that you can draw, centered on the origin, the point with Cartesian coordinates (0, 0, 0), that just touches all of the corners’ circles. It has a radius of a little under 0.732.

Think of the four-dimensional analog. This is harder to sketch. But a four-dimensional hypercube, with each side of length 2 and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1, +1, +1), (+1, +1, +1, -1), (+1, +1, -1, +1), (+1, +1, -1, -1), and you know what? Will you let me pretend we listed all sixteen corners? Thanks. Around each of these corners draw a circle of radius 1.

There is some largest hypersphere you can draw, centered on the origin, the point with Cartesian coordinates (0, 0, 0, 0), and that just touches all of these corners’ circles. It has a radius of 1.

Keep going. Five-dimensional space, with corners like (+1, +1, +1, +1, +1). Six-dimensional space, with corners like (+1, +1, +1, +1, +1, +1). Seven-dimensional space. And so on.

Eventually, the space is vast enough that the radius of this largest-touching hypersphere is bigger than 2. That is, reaching out more than twice as far as the original box goes, this even though the corner hyperspheres line the edges of the box, and touch one another along those edges.

Non-Euclidean geometry has the reputation of holding deep, inscrutable mysteries. To say something is a non-Euclidean space, outside of a mathematical context, is to designate it as a place immune to reason and beyond human comprehension. This is not such a case. This is a completely Euclidean space; it’s just got a lot of dimensions to it. Strange things will happen.

Another weird, but to me not so unsettling matter, concerns the surface (or hypersurface) area and the volume of these spheres. The circumference of a unit circle is, famously, 2π. The area of a unit sphere is 4π. For a four-dimensional hypersphere the surface area is a bit bigger yet. And bigger again for five and six and seven dimensions. But at eight dimensions the surface area starts shrinking again, and it never grows again. Have a great enough number of dimensions and the unit hypersphere has almost zero surface area. The volume of a unit circle is π. Of a unit sphere, $\frac43 \pi$. For a four-dimensional hypersphere, $\frac12 \pi^2$. For a five-dimensional hypersphere, $\frac{8}{15}\pi^2$. It is never so large again; for six or more dimensions the volume starts to shrink again. As the number of dimensions of space grows, the volume of the unit hypersphere dwindles to zero.

You know, that’s unsettling me more now that I’m paying attention to it.

## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

## 4 thoughts on “A Geometry Thing That’s Left Me Unsettled”

1. As unsettling as the Dirac delta function having unit area😅

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1. Yes, that’s another good and unsettling piece.

Now I’m curious whether there’s a specific link between unit hyperspheres and the Dirac delta function. Probably is, if you dig around and manipulate things enough.

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1. Thanks kindly! I’m near the end of my post-a-day energy, but do certainly plan to do another sequence like this next year.

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