# Geometry: point, line, plane--then what? **URL:** \ **Category:** Factual Questions\ **Created:** [March 30, 2003, 1:06pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210 "2003-03-30T13:06:26Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\ **Post date:** [March 30, 2003, 1:06pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/1 "2003-03-30T13:06:26Z")
In Euclidean geometry, there’s the point (dimensionless), line (one dimension), plane (two dimensions). What is the analogous three-dimensional construct once you start talking about more than three dimensions? ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [March 30, 2003, 1:16pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/2 "2003-03-30T13:16:14Z")
Point, line, polygon, polytope. ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [March 30, 2003, 1:23pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/3 "2003-03-30T13:23:42Z")
Or you can say for example 3-gon, 4-gon,… n-gon. Or 3-tope, 4-tope,… n-tope. ---
**Author:** ![hawthorne](https://avatars.discourse-cdn.com/v4/letter/h/c89c15/32.png) [@hawthorne](https://boards.straightdope.com/u/hawthorne)\ **Post date:** [March 30, 2003, 1:25pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/4 "2003-03-30T13:25:11Z")
hyperplane ---
**Author:** ![heresiarch](https://avatars.discourse-cdn.com/v4/letter/h/278dde/32.png) [@heresiarch](https://boards.straightdope.com/u/heresiarch)\ **Post date:** [March 30, 2003, 1:26pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/5 "2003-03-30T13:26:29Z")
> [@](#): > > \*Originally posted by CookingWithGas \* > \*\*In Euclidean geometry, there’s the point (dimensionless), line (one dimension), plane (two dimensions)… \*\* space (three dimensions). You could call it “3-space” to generalize it. Then higher dimensions would be 4-space, 5-space, … n-space. I’m not saying that’s standard terminology. I’m not a mathematician. I’m also not sure how well Euclidean geometry corresponds to the real universe for any number of dimensions greater than 3. If someone would like to address that, I’d be very interested. ---
**Author:** ![Keeve](https://avatars.discourse-cdn.com/v4/letter/k/f07891/32.png) [@Keeve](https://boards.straightdope.com/u/Keeve)\ **Post date:** [March 30, 2003, 1:35pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/6 "2003-03-30T13:35:06Z")
I disagree with Desmostylus, because the OP is asking for names of the place in which a something could exist, not the names of the body itself. In other words, a line is an area which can contain any number of points, line segments, and rays. And a plane is a surface which can contain any number of polygons, circles, curves, lines, points, and such. The OP’s specification of “plane (two dimensions)” shows that the three-dimensional item of the list would not be cube, sphere, or any other such body, but the name of the place in which such a body might be. ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [March 30, 2003, 1:37pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/7 "2003-03-30T13:37:53Z")
I misread the OP. Ignore my earlier stuff. ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [March 30, 2003, 2:21pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/8 "2003-03-30T14:21:31Z")
I’m not a mathematician yet, but I’m doing a maths degree. Either hyperplane, 3-plane or 3-space would be correct. Sometimes you’d use ‘volume.’ (All n-spaces are all considered hyperplanes, so you’d only use hyperplane if you were in 4-d space.) ---
**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:** [March 30, 2003, 4:07pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/9 "2003-03-30T16:07:16Z")
The standard usage is that a hyperplane is an (n-1)-dimensional flat subspace of an n-dimensional space. What seems to be meant here is what is the name of the space after point, line, plane. The correct term is 3-dimensional space, usually shortened to 3-space. ---
**Author:** ![Trigonal\_Planar](https://avatars.discourse-cdn.com/v4/letter/t/0ea827/32.png) [@Trigonal\_Planar](https://boards.straightdope.com/u/Trigonal_Planar)\ **Post date:** [March 30, 2003, 5:49pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/10 "2003-03-30T17:49:13Z")
I thought after plane, geometry generally broke into non-euclidean geometry, conics? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 30, 2003, 5:53pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/11 "2003-03-30T17:53:23Z")
What **Shade** and **Hari Seldon** said is correct. n-space is the term I would use, with a line being a 1-space, a plane a 2-space, and so on. I guess a point would be a 0-space, but I’ve never seen that used. ---
**Author:** ![KarmaComa](https://avatars.discourse-cdn.com/v4/letter/k/e36b37/32.png) [@KarmaComa](https://boards.straightdope.com/u/KarmaComa)\ **Post date:** [March 30, 2003, 6:20pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/12 "2003-03-30T18:20:34Z")
Word. n-dimensional subspace. ---
**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:** [March 30, 2003, 6:59pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/13 "2003-03-30T18:59:51Z")
> [@](#): > > I thought after plane, geometry generally broke into non-euclidean geometry, conics? Yes and no. You can have Euclidean geometry of any number of dimensions. What we usually work with is 3-d Euclidean geometry, already somewhat above planes, and you can, in principle, discuss Euclidean geometry with any number of dimensions (although physical examples start getting scarce for more than three dimensions). You can also depart from plane Euclidean geometry by adding curvature. In a curved two-dimensional geometry, for instance, the Pythagorean Theorem no longer holds. And you can also depart from Euclid by introducing timelike dimensions, which behave a little differently from spacelike dimensions. You actually get all three of these departures in relativity. And back to the original topic, in relativity we generally refer to a 3-dimensional unbounded structure as a hypersurface. “Space” usually means “spacetime” to us, and “hyperplane” would imply a flat surface. On the other hand, a hypersurface is allowed to be closed, which a hyperplane (I presume) wouldn’t be, so maybe it’s not the right term for what the OP is asking. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [March 30, 2003, 8:33pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/14 "2003-03-30T20:33:04Z")
Another term is “3-manifold”. This can be used to describe both Euclidean and non-Euclidean spaces. It sounds like “manifold” might mean the same thing as the term “hypersurface” that **Chronos** mentioned. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 30, 2003, 9:23pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/15 "2003-03-30T21:23:24Z")
> [@](#): > > \*Originally posted by Chronos \* > **In a curved two-dimensional geometry, for instance, the Pythagorean Theorem no longer holds.** Interestingly enough, there are theorems in the hyperbolic and spherical planes that my geometry text (Brannan, Esplen, & Gray) identify as the Pythagorean theorem for that geometry. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [March 30, 2003, 10:39pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/16 "2003-03-30T22:39:25Z")
> [@](#): > > \*Originally posted by Tyrrell McAllister \* > \*\*Another term is “3-manifold”. This can be used to describe both Euclidean and non-Euclidean spaces. It sounds like “manifold” might mean the same thing as the term “hypersurface” that **Chronos** mentioned. \*\* A hyperplane or n-space is a special case of a manifold, as is a hypersurface. > [@](#): > > _Originally posted by ultrafilter_ > \*\* Interestingly enough, there are theorems in the hyperbolic and spherical planes that my geometry text (Brannan, Esplen, & Gray) identify as the Pythagorean theorem for that geometry.\*\* There is also a law of cosines and a law of sines for both spherical and hyperbolic geometry, analogous to the plane version in each case. The laws in the Euclidean plane are actually limiting cases of the spherical and hyperbolic versions. However, spherical and hyperbolic geometry are two very specific curved spaces; in an arbitrarily curved space there may be no analogue to the Pythagorean theorem. ---
**Author:** ![raisinbread](https://avatars.discourse-cdn.com/v4/letter/r/8491ac/32.png) [@raisinbread](https://boards.straightdope.com/u/raisinbread)\ **Post date:** [March 30, 2003, 11:02pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/17 "2003-03-30T23:02:34Z")
Polytope? Then what’s a polychoron? I always figured the polychoron was the next dimensional construct. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [March 30, 2003, 11:06pm UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/18 "2003-03-30T23:06:49Z")
Isn’t the next one up from _plane_ simply called _solid_? ---
**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:** [March 31, 2003, 2:46am UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/19 "2003-03-31T02:46:13Z")
No, “solid” is the next up from “figure”. A figure lies in a plane; a solid lies in a space. And I’m aware that there are curved-space theorems analogous to the Pythagorean; I just meant that a[sup]2[/sup] + b[sup]2[/sup] = c[sup]2[/sup] isn’t true any more. And unlike a hypersurface, a manifold can have an edge, so long as the edge isn’t actually part of the manifold. For instance, the interval (0,1) (all numbers greater than zero but less than one) is a manifold, but the interval [0,1] (all numbers greater than or equal to zero, but less than or equal to one) is not. ---
**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:** [March 31, 2003, 3:40am UTC](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210/20 "2003-03-31T03:40:06Z")
A “me too” for _n_-space. I’m chiming in here to nitpick what Chronos said: A manifold _is_ allowed to contain an “edge”, usually called its boundary. So, [0,1] is a valid manifold-with-boundary, while (0,1) is a manifold with empty boundary. (In fact, for my purposes, (0,1) is indistinguishable from the whole real line, but then again I’m a topologist, not a geometer.) [Next page](https://boards.straightdope.com/t/geometry-point-line-plane-then-what/165210.md?page=2)