# Is the surface of a sphere a two- or three-dimensional space?

On the last page of last month’s newsletter, my company published the following brain teaser:

My answer was a circle:

But I’ve been informed my answer is incorrect. The correct answer is the surface of a sphere.

Huh?? Isn’t the surface of a sphere a three-dimensional space? I am not – by any means – an expert on this, but my first hunch is that the surface of a sphere is a two-dimensional surface that exists in three dimensional space…but the surface itself only has two dimensions.

The surface of a sphere exists in three-dimensional space, and if you don’t have a three-dimensional space you can’t have a full sphere.

But every point on the surface of a sphere can be designated uniquely with only two coordinates. That is, if you stipulate that you are on the surface of a sphere, you only need two bits of information to tell where you are (Latitude and longitude on the Earth or a Globe; or theta and phi in spherical coordinates, or some other such scheme).

So both answers are correct, but they rely on different assumptions.

The surface is two-dimensional, it doesn’t contain the space within it. Similar to your answer about a circle, except the line defining the perimeter of a circle is one dimensional, not two. The definitions here are confusing because both circles and spheres are defined by a point and a radius, and typically are considered to define the space within and without them. By referencing the surface of the sphere, or the perimeter of the circle you remove a dimension.

Look up “manifold”. The surface of a sphere is a 2-D manifold curved in 3-D space.

Any point on a sphere’s surface can be represented by three numbers - 1> the radius of the sphere 2> the angle the point makes with the vertical plane 3> the angle the point makes with the horizontal plane.

If the radius of the sphere is fixed (like the earth) then it becomes 2 dimensional - you only need the last 2 numbers to locate the point.

Its the same way as a circle, if the radius is fixed, then all you need is one number (the angle ) to locate the point on the circle and it becomes one dimensional.

But the surface of a sphere moves in all three planes (x,y and z). If it only moved in two of them you’d have a disk (or, rather, a projection of a sphere into 2D).

Um, don’t you need more than just latitude and longitude? I mean, what/where is zero degrees? Don’t you have to arbitrarily define what/where zero degrees is? And if so, isn’t that… an edge?

The definition of the surface is based on 3 dimensions, but the surface itself is infinitely thin. In the same manner the perimeter of a circle is one dimensional. Spheres as a whole are 3 dimensional, defining all the points within (and without) a radius from a point. The 3rd dimension there is defined by a line that intersects a point defining the sphere in 3 dimensions. That line is a dimension that isn’t part of the surface.

ETA: It does seem kind of weird from a common sense view. Consider though a form of non-Euclidean geometry where concentric circles that aren’t defined from the same point can be considered to be parallel lines that are not equidistant. The circles there are one dimensional lines, not counting the enclosing space between them.

You could extend that sort of argument to a plane, as well – don’t I need to know where the origin and orientation of my coordinate system is in an infinite plane? I can’t tell where anything is unless I know where “zero” is and which way the “x” axis runs.

That’s quite true, but it’s assumed that you do know, and that once you do you can uniquely specify any position on the plane with a pair of numbers (x and y in cartesian coordinates, or r and theta in circular ones, or other combinations in a slew of other, more eccentric coordinate systems).

Just so, it’s assumed we know where the “north pole” and “Greenwich Meridian” of a sphere are. Or the 0 degree phi and zero degree theta of spherical coordinates or…, well, there are lots of books to look up the other arcane two-D-on-a-spherical-surface-coordinate systems.

I find your last statement odd, though. Zero is an edge? At best it’s a point, but I suspect that’s why you put a question mark after that.

More precisely, the zero degree line is a boundary, I would think. If so, then the answer can’t be the surface of a sphere?

Oops, just saw your response, CM.

The “surface of a sphere” as the OP imagines it is a two dimensional manifold (the surface) embedded in a three dimensional manifold (the space the sphere is in). Basically, there are some properties of the “sphere” concept that depend on how it is embedded in the larger space, but there are also a whole bunch of properties of the sphere that don’t depend on the embedding at all.

This is related to the perennial question about General Relativity: “If space is curved, does that mean that there’s some bigger, higher-dimensional space that space is curved into?”, to which the answer is “maybe, maybe not – the equations of GR depend only on those properties of space that aren’t dependent on how space is or is not embedded in some hypothetical bigger manifold, so even if there was a bigger space to be seen, we couldn’t observe it”.

There’s no line on the surface of a sphere. It’s an area, so the surface has two dimensions. Which may be the best way to explain it, a line segment has length, that’s one dimension, and the same applies to the perimeter of a circle. A sphere has three dimensions, but it’s surface only has two.

Nitpick : Latitude, Longitude and Altitude/Depth are needed to specify a place on earth. Sometimes Altitude is just assumed to be the surface but it need not be so.

A ship and submarine could have the same Lat/Long - and yet be far away.

That’s certainly not an edge of the sphere. What it is is an interesting part of the coordinate chart you use you describe the sphere. If we picked another great circle to be zero degrees, or even another form of coordinate system entirely, you wouldn’t have done anything important to the sphere itself, just to your description of it.

Yes, but here you’ve strayed well beyond the OP’s constraint of the surface of a sphere.

Since you can define a one to one relationship between all of R2 and the real line, can you claim a sphere is 1 dimensional?

THere are an infinite number of finite two dinensional surfaces without edges/boundaries, aren’t there? Not just spheres, but all manner of ovoids and tori, correct? A Klein bottle would seem to qualify, as well.

And that’s before we get to the innumberable irregular 2D surfaces without edges. Heck, the surface of a human body (put on a bodysuit for an easier visual) would meet the condition.

No, because (speaking colloquially) you can’t do so in such a way that points on the sphere that are close to one another always have (single) coordinates that are close together. This issue is discussed in the thread What is a “Dimension”?

If you’re looking at topological equivalence, there really aren’t that many possible edgeless surfaces. These are categorized by whether they are orientable or not, and a non-negative integer called the “genus” of the surface.

[ul]
[li]The sphere is an orientable manifold of genus 0.[/li][li]The torus is an orientable manifold of genus 1[/li][li]The Klein bottle is a non-orientable manifold of genus 2.[/li][/ul]

Basically the orientable surfaces of genus n, where n>= 0, and the non-orientable surfaces of genus k where k >= 1 comprise all there there is.