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?
Point, line, polygon, polytope.
Or you can say for example 3-gon, 4-gon,… n-gon.
Or 3-tope, 4-tope,… n-tope.
hyperplane
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.
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.
I misread the OP. Ignore my earlier stuff.
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.)
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.
I thought after plane, geometry generally broke into non-euclidean geometry, conics?
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.
Word. n-dimensional subspace.
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.
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.
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.
A hyperplane or n-space is a special case of a manifold, as is a hypersurface.
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.
Polytope? Then what’s a polychoron? I always figured the polychoron was the next dimensional construct.
Isn’t the next one up from plane simply called solid?
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.
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.)