[math-fun] Puzzle: Skew planes in 4-space?
Fred's query reminds me of the time in 1994 that I gave a lecture -- about 4-dimensional space -- to the Hampshire College summer math program for mathematically talented high school students. Some brilliant youngster asked me whether there could be skew (2D) planes in 4-space. (Skew meaning neither parallel nor intersecting.) I thought about it for a few seconds . . . and then a few more seconds . . . and had to admit I didn't know. But I went home and soon figured it out. Can you? --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx
Consideration of vector bases easily establishes that in projective space of dimension n-1, subspaces of dimensions k-1 and m-1 must meet in a common subspace of dimension at least k+m-n-1, when this is non-negative. Here n = 5, k = m = 3, so there must be a point in common somewhere. But it depends a little on what you mean by "parallel": distinct planes in 4-space might meet in a line at infinity, only a point at infinity, or in a finite line or finite point. In the second case, they're only "semi-parallel" --- but that would hardly qualify as "skew" to your student, I don't suppose. WFL On 3/22/11, Dan Asimov <dasimov@earthlink.net> wrote:
Fred's query reminds me of the time in 1994 that I gave a lecture -- about 4-dimensional space -- to the Hampshire College summer math program for mathematically talented high school students.
Some brilliant youngster asked me whether there could be skew (2D) planes in 4-space. (Skew meaning neither parallel nor intersecting.) I thought about it for a few seconds . . . and then a few more seconds . . . and had to admit I didn't know.
But I went home and soon figured it out. Can you?
--Dan
________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx
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On 3/22/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
Consideration of vector bases easily establishes that in projective space of dimension n-1, subspaces of dimensions k-1 and m-1 must meet in a common subspace of dimension at least k+m-n-1, when this is non-negative. ...
Just in case anybody here might be interested, I posted the following on the geometric_algebra list: principal angles were discussed here first. A recent enquiry on the math-fun list concerning a formula for the distance between two lines in Euclidean 3-space set me thinking about the more general expression for distance d between (k-1)-, (l-1)-subspaces X,Y in Euclidean (m-1)-space ["l" is "ell"]. The appropriate algebra is Cl(m-1,0,1), with vectors representing (reflections in) hyperplanes: it's time this had a name, and I propose calling it "EGA" in future. I conjecture that d^2 = ||<XºY>_{m-k-l}|| / ||<X•Y>_{k+l-2}|| where X•Y denotes Clifford product, XºY == (X*•Y*)* dual product, ||X|| == (X~)•X (scalar) magnitude, <X>_k the k-grator (grade-k part). I've verified this for all 2-space and 3-space cases, but currently have not the faintest idea how to prove it in general. It eventually dawned on me that there is a strong connection between this and the topic of principal angles, which was discussed on this list earlier (thread beginning 5th Jan 2010). There I defined a "grade expansion polynomial" whose coefficients are the magnitudes of the k-grators of X•Y, and whose roots yield the angles between the subspaces. In the case of distance, the associated root is infinite; however, the expression above shows that it is still possible to extract the actual value from the product! Fred Lunnon
Hmm. EGA makes me think of Grothendieck. see: http://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C... On Tue, Mar 22, 2011 at 6:26 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
[.... snip....]
The appropriate algebra is Cl(m-1,0,1), with vectors representing (reflections in) hyperplanes: it's time this had a name, and I propose calling it "EGA" in future.
[... snip ...]
Urggghhh .... schemes ... one more thing I've given up hope of ever understanding. Maybe I'm going to have to change that acronym --- wouldn't want anybody confusing the two! WFL On 3/23/11, James Buddenhagen <jbuddenh@gmail.com> wrote:
Hmm. EGA makes me think of Grothendieck. see: http://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C...
On Tue, Mar 22, 2011 at 6:26 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
[.... snip....]
The appropriate algebra is Cl(m-1,0,1), with vectors representing (reflections in) hyperplanes: it's time this had a name, and I propose calling it "EGA" in future.
[... snip ...]
participants (3)
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Dan Asimov -
Fred lunnon -
James Buddenhagen