# Hyperplanes as seen in lower dimensions **URL:** \ **Category:** Factual Questions\ **Created:** [November 15, 2011, 6:50am UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988 "2011-11-15T06:50:56Z")\ **Posts on this page:** 14\ **Page:** 1
**Author:** ![Swords\_to\_Plowshares](https://avatars.discourse-cdn.com/v4/letter/s/b38774/32.png) [@Swords\_to\_Plowshares](https://boards.straightdope.com/u/Swords_to_Plowshares)\ **Post date:** [November 15, 2011, 6:50am UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/1 "2011-11-15T06:50:56Z")
Hopefully some math wizards can help me with this. A 3-dimensional hyperplane (aka a “plane”) looks like a line in 2 dimensions, unless it’s parallel to the 2D plane, in which case it’s different, I guess. What does a 4-dimensional hyperplane look like in 2 dimensions? Can it look like any 2-D shape or configuration? What about a 5-D hyperplane in 3 dimensions? Going even one step further, what would a 2\*d dimensional hyperplane look like in d-dimensional space? Is there any special relationship here? ---
**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:** [November 15, 2011, 6:57am UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/2 "2011-11-15T06:57:49Z")
The intersection of an N-plane and an M-plane can be a plane of any dimension up to whichever of N and M is lower. So, for instance, a 4-D (or 5-D or 6-D or…) plane intersecting with our own space (a 3-plane) could be a point, a line, a (2-D) plane, or fill the entire space. You can’t get any interesting shapes, unless the planes you’re intersecting already have interesting shapes. ---
**Author:** ![Swords\_to\_Plowshares](https://avatars.discourse-cdn.com/v4/letter/s/b38774/32.png) [@Swords\_to\_Plowshares](https://boards.straightdope.com/u/Swords_to_Plowshares)\ **Post date:** [November 15, 2011, 7:00am UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/3 "2011-11-15T07:00:58Z")
What about the intersection of a shape in a space with a higher dimensional hyperplane? Say, a 4-d hyperplane intersecting with a 2-d circle, or a N-d hyperplane intersecting with a 3-d sphere? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 15, 2011, 7:48am UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/4 "2011-11-15T07:48:06Z")
A hyperplane intersecting with an N-d shape can only produce either the entirety of the shape, or the portion of the shape obtained by intersecting it with a plane of dimension less than N. So, the possibilities are: the whole shape, an (N-1)-dimensional slice of the shape, an (N-2)-dimensional slice of the shape, …, a 1-dimensional slice of the shape, a point, and nothing at all. So a 4-d hyperplane intersecting with a 2-d circle can only produce the same shapes as a 2-d hyperplane intersecting with a 2-d circle can: the whole circle, or a pair of points, or a single point, or nothing. [Or, if you mean the whole disk inbetween the circular boundary: the whole disk, or a line segment, or a single point, or nothing]. And an N-d hyperplane intersecting with a 3-d sphere can only produce the same shapes as a 3-d plane intersecting with a 3-d sphere can: the entirety of the sphere, or a 2-d slice of it (which will be an ellipse), or a 1-d slice of it (which will be a pair of points [and the line segment inbetween them, if you include the whole ball and not just its surface sphere]), or a single point, or nothing at all. ---
**Author:** ![cmyk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cmyk/32/3353_2.png) [@cmyk](https://boards.straightdope.com/u/cmyk)\ **Post date:** [November 15, 2011, 7:48am UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/5 "2011-11-15T07:48:30Z")
It’ll either annihilate the universe, or create a unicorn. You can never tell what you’ll get. Couldn’t you perform a boolean intersecting operation of a 4D plane and a 3D sphere or 2D disc? You’d end up with some interesting topology and shapes, but it’d be arbitrary as it depends on the arrangement / orientation and intersection of the hyperplane relative to the sphere/disc. Of course a plane has no volume, but a hypercube and a sphere intersection could create a shell that would morph in weird ways as you rotated the hypercube in 4 dimensions. How difficult would this be to model algebraically, and graph? disclaimer: IANAMathematician ---
**Author:** ![Swords\_to\_Plowshares](https://avatars.discourse-cdn.com/v4/letter/s/b38774/32.png) [@Swords\_to\_Plowshares](https://boards.straightdope.com/u/Swords_to_Plowshares)\ **Post date:** [November 15, 2011, 4:06pm UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/6 "2011-11-15T16:06:09Z")
I’m not sure if I’m getting quite the result I was hoping for. So the hyperplane can intersect the shape, which is great. But would it also be covering area outside the shape as well? Basically I was hoping to prove that a hyperplane going down to lower dimensions can equal a shape in that lower dimension. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 15, 2011, 5:27pm UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/7 "2011-11-15T17:27:38Z")
> [@Swords\_to\_Plowshares](#): > > I’m not sure if I’m getting quite the result I was hoping for. So the hyperplane can intersect the shape, which is great. But would it also be covering area outside the shape as well? > > Basically I was hoping to prove that a hyperplane going down to lower dimensions can equal a shape in that lower dimension. In general, it can’t. The intersection of two affine spaces has to be an affine space, which means that you’re going to get either a point, a line, a plane, or some higher dimensional equivalent. You’re not going to get circles, spheres or anything else interesting. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 15, 2011, 7:18pm UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/8 "2011-11-15T19:18:48Z")
In other words: The intersection of two (or more) hyperplanes is always a hyperplane, of some dimension or another. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 15, 2011, 9:38pm UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/9 "2011-11-15T21:38:29Z")
> [@Indistinguishable](#): > > In other words: The intersection of two (or more) hyperplanes is always a hyperplane, of some dimension or another. As a lay answer, this is probably the clearest, but it’s no good mathematically. A hyperplane in a vector space V is defined to be an affine space of codimension one, which means that a hyperplane in V is not going to be a hyperplane in any larger vector space containing V. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 15, 2011, 11:24pm UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/10 "2011-11-15T23:24:47Z")
Ah, like **Chronos** , I was taking “hyperplane” simply to mean “affine space”, with no need for consideration of any ambient space in which it may be embedded. But re-reading the OP, this isn’t the way the terminology is being used (e.g., the OP uses “3-d hyperplane” to refer to a standard plane with 2 intrinsic dimensions), and you are correct as to how it is being used instead. Still, I wouldn’t say it’s “no good mathematically”. It’s just a matter of terminology to make it fine. The intersection of two (or more) affine spaces is always an affine space, of some dimension or another. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 15, 2011, 11:33pm UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/11 "2011-11-15T23:33:27Z")
> [@Indistinguishable](#): > > Ah, like **Chronos** , I was taking “hyperplane” simply to mean “affine space”, with no need for consideration of any ambient space in which it may be embedded. Or rather, I was taking “hyperplane” simply to mean “affine subspace”, with no need for consideration of the specific _dimension_ of the ambient space in which it may be embedded. Of course, for determining intersections, the actual embeddings into a common superspace matter a great deal. Anyway… _terminology!_ ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 15, 2011, 11:40pm UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/12 "2011-11-15T23:40:28Z")
> [@Indistinguishable](#): > > It’s just a matter of terminology to make it fine. The intersection of two (or more) affine spaces is always an affine space, of some dimension or another. Oh, right, I forgot, that’s what **ultrafilter** said in the first place. Heh. :smack: ---
**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:** [November 16, 2011, 12:09am UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/13 "2011-11-16T00:09:19Z")
> [@](#): > > And an N-d hyperplane intersecting with a 3-d sphere can only produce the same shapes as a 3-d plane intersecting with a 3-d sphere can: the entirety of the sphere, or a 2-d slice of it (which will be an ellipse)… While this is true, wouldn’t it have been better to be a bit more specific here and just say “circle” instead of “ellipse”? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 16, 2011, 12:10am UTC](https://boards.straightdope.com/t/hyperplanes-as-seen-in-lower-dimensions/602988/14 "2011-11-16T00:10:32Z")
Heh, good point; I wasn’t thinking enough. Another :smack: then… In short: the intersection of an affine subspace with a hyper-sphere/-ball is just another hyper-sphere/-ball, of some dimension or another (no larger than either of them).