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?
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.
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?
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.
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?
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.
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.
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.
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.
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).