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Say you have defined a 3D space, which consists of 0D points. What is it called when you reinterpret it as a 1D space, in which each "point" is a 2D subspace of the original 3D space?

fumigail
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  • How would you define scalar multiples of a subspace? – hardmath Nov 23 '14 at 21:01
  • They don't necessarily need to be scalar multiples, you might be slicing $R^3$ along one axis. Represent the 3D space as a function of 3 arguments that returns a point, then turn it into a function of 1 argument that returns a function of 2 arguments that returns a point. – fumigail Nov 23 '14 at 21:05

1 Answers1

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The set of all two-dimensional subspaces of real three dimensional space is called the Grassmannian $\operatorname{Gr}(2,\mathbb{R}^3)$. It is actually a two-dimensional manifold homeomorphic to the real projective plane. It can be constructed by taking all unit vectors (i.e. points on the sphere centered at the origin) and considering them as the normal vectors for planes. This suggests that the space would be homeomorphic to a sphere, but the vectors $v$ and $-v$ give you the same plane, so it's the sphere with antipodal points identified. It is difficult to visualize because it cannot be embedded in three dimensional space.

Matt Samuel
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  • This answers my question but makes me realize I asked the wrong question.. What I'm looking for is a term for when I turn, say, turn a cube into a line whose points consist of planes (or even a plane whose points consist of lines). But I now realize that these planes aren't really subspaces, because I would need to redefine addition as only involving the last 2 dimensions to make them closed. – fumigail Nov 23 '14 at 21:39
  • The construction you are looking for in that case is the Cartesian product. If $L$ is a line segment and $S$ is a square, then the Cartesian product is the cube $L\times S$. It is what you get by having one square for every point in the line segment. – Matt Samuel Nov 23 '14 at 21:43
  • I don't know that Cartesian product fits here, because my points might be distinguishable, so every planar "slice" of the cube could be unique, and I wouldn't be able to reconstruct the original cube from a product of a line and a square. Rather, I need to re-parameterize the cube, so that the constituent squares are hung on the line like beads on a necklace. – fumigail Nov 23 '14 at 21:51
  • That still sounds like the Cartesian product to me, but I may just be misunderstanding. – Matt Samuel Nov 23 '14 at 21:55
  • The misunderstanding is likely mine, as I keep trying and failing to ask the right question. Here was an earlier attempt, perhaps if you have time you could read it and take a guess at what I'm looking for? I need to describe $g$: http://math.stackexchange.com/questions/1035397/what-is-the-name-of-this-operation – fumigail Nov 23 '14 at 22:00
  • Currying does seem like the correct answer. What you have is an indexed sequence of functions that really just reinterprets a single function. Note that any function is a subset of the Cartesian product of its domain and its range. Your function $h$ is a subset of $(A\times B\times C)\times T$. You're picking an element $a$ of $A$ and interpreting the subset contained in $({a}\times B\times C)\times T$ as a new function $B\times C\to T$, which is a subset of $(B\times C)\times T$. You consider the function $g$ as a subset of $A\times ((B\times C)\times T)$. – Matt Samuel Nov 23 '14 at 22:06
  • So, would you say $g$ represents the currying itself? – fumigail Nov 23 '14 at 22:08
  • Yes. See http://en.wikipedia.org/wiki/Currying#Motivation – Matt Samuel Nov 23 '14 at 22:09
  • Yes, I think I understand now. I kept thinking of currying as producing a new function by binding some concrete arguments, but thats actually partial application. – fumigail Nov 23 '14 at 22:13