# Is the surface of a sphere a two- or three-dimensional space? **URL:** \ **Category:** Factual Questions\ **Created:** [January 9, 2015, 1:54pm UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390 "2015-01-09T13:54:48Z")\ **Posts on this page:** 12\ **Page:** 2
**Author:** ![Snarky\_Kong](https://avatars.discourse-cdn.com/v4/letter/s/a183cd/32.png) [@Snarky\_Kong](https://boards.straightdope.com/u/Snarky_Kong)\ **Post date:** [January 9, 2015, 5:20pm UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/21 "2015-01-09T17:20:44Z")
> [@Thudlow\_Boink](#): > > 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”?](http://boards.straightdope.com/sdmb/showthread.php?t=617954) Ah. Thank you. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 9, 2015, 5:38pm UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/22 "2015-01-09T17:38:42Z")
> [@leahcim](#): > > 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. I find it amusing that you can (correctly and meaningfully) say “there really aren’t that many” about something of which there are infinitely many (one for each integer n \>=0 and each k \>= 1). 🙂 ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [January 9, 2015, 6:17pm UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/23 "2015-01-09T18:17:46Z")
> [@Thudlow\_Boink](#): > > I find it amusing that you can (correctly and meaningfully) say “there really aren’t that many” about something of which there are infinitely many (one for each integer n \>=0 and each k \>= 1). 🙂 The number infinite, but it isn’t one of the _big_ infinities. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 9, 2015, 9:40pm UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/24 "2015-01-09T21:40:55Z")
Isn’t a Klein bottle of the same genus as a torus? ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 9, 2015, 9:52pm UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/25 "2015-01-09T21:52:52Z")
Technically speaking, there’s no such thing as the _surface of a sphere_, or at the least it’s redundant, because a sphere is _defined_ as the set of points at some distance from a point and excludes the interior points. A _ball_ however includes the interior points. The surface of a 3D ball is a sphere (which is a 2D surface). Obviously these are the mathematical definitions and they don’t quite match the common English definition. Just something to watch out for. ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [January 9, 2015, 10:29pm UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/26 "2015-01-09T22:29:27Z")
> [@Chronos](#): > > Isn’t a Klein bottle of the same genus as a torus? I think not. Torus is (orientable) genus 1, Klein bottle is (non-orientable) genus 2. A [cross-cap](https://en.wikipedia.org/wiki/Cross-cap) is non-orientable genus 1. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [January 10, 2015, 1:43am UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/27 "2015-01-10T01:43:04Z")
Let me point out that the interior of a flat disk (all the points not on the boundary) also constitutes a finite 2 dimensional surface without a boundary. Oh, you meant a compact surface? Then Leahcim is correct, of course. No, I am not going to attempt to define compact. Okay I lied. Although this is not the general definition, a compact surface is characterized by the fact that every infinite sequence of points contains a convergent subsequence. On the interior of the unit disk, the points whose distance to the origin is less than 1, the sequence of points (0,0), (1/2,0), (2/3,0), (3/4,0), … does not converge since its limit, (1,0) is not in the interior of the disk. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [January 10, 2015, 8:32am UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/28 "2015-01-10T08:32:30Z")
So, an ordinary sphere is _two-dimensional_. There is such a thing as a “_three-dimensional_ sphere”, but it is NOT the three-dimensional “ball.” A good overview of such things for the layman is _The Poincaré Conjecture_ by Donal O’Shea. Nitpick: While local regions of 2-D manifolds have two coordinates, I don’t think good _global_ coordinate systems exist in general. The torus may have a better such system (latitude and longitude both circular 0→2pi) than the sphere has, but can anything nice be conjured up for a two-holed donut? ---
**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\ **Post date:** [January 10, 2015, 4:06pm UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/29 "2015-01-10T16:06:53Z")
> [@septimus](#): > > Nitpick: While local regions of 2-D manifolds have two coordinates, I don’t think good _global_ coordinate systems exist in general. The torus may have a better such system (latitude and longitude both circular 0→2pi) than the sphere has, but can anything nice be conjured up for a two-holed donut? Every closed surface can be described by [taking a polygon and identifying edges](http://en.wikipedia.org/wiki/Fundamental_polygon). For example, a two-holed donut can be gotten from an octagon. So, put a nice set of coordinates on the octagon and that gives you global coordinates on the two-holed donut. Of course, there will be singularities where you identified edges, but that’s true for latitude and longitude at the poles of the sphere, too. ---
**Author:** ![MsKaren](https://avatars.discourse-cdn.com/v4/letter/m/a183cd/32.png) [@MsKaren](https://boards.straightdope.com/u/MsKaren)\ **Post date:** [January 10, 2015, 4:37pm UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/30 "2015-01-10T16:37:42Z")
In a simpler common sense approach, the surface of a sphere would be measured in square units like square inches. That makes it 2 dimensional. In the same way, its volume would be cubic units like cubic inches therefore 3-D, but a circumference would be just inches so it is 1 dimensional. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 11, 2015, 1:51am UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/31 "2015-01-11T01:51:58Z")
> [@Chronos](#): > > Isn’t a Klein bottle of the same genus as a torus? You are perhaps thinking of the fact that they both have the same Euler characteristic (specifically, zero). But genus is different from Euler characteristic. Genus is sort of an odd concept, but basically, the genus of an orientable surface is how many toruses are joined together to make it (every orientable surface being decomposable in this manner, and toruses not being any further decomposable), and the genus of a non-orientable surface is how many projective planes are joined together to make it (every non-orientable surface being decomposable in this manner, and projective planes not being any further decomposable). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 11, 2015, 2:00am UTC](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390/32 "2015-01-11T02:00:15Z")
> [@Last edited by Indistinguishable; Today at 05:54 PM..](#): > > Reason: By “surface”, I mean throughout “compact connected surface with no boundary” And by “joined together”, I mean via the “connected sum”… [Previous page](https://boards.straightdope.com/t/is-the-surface-of-a-sphere-a-two-or-three-dimensional-space/709390.md?page=1)