Yes, my solution obviously only works for n >= 3. Tom
A pox on brilliant youngsters. Here's a case I can solve: In 2-space you can have one skew point.
Steve Gray
On 3/21/2011 6:45 PM, Tom Karzes wrote:
I'd say yes. In general, you can have two skew (n-2)-dimensional "planes" in n-space. In 3-space you can have two skew lines, in 4-space you can have two skew planes, etc.
For lines in 3-space, coordinates (x1, x2, x3):
Line 1: x1 = 0, x2 = 0, x3 = free Line 2: x1 = 1, x2 = free, x3 = 0
For planes in 4-space, coordinates (x1, x2, x3, x4):
Plane 1: x1 = 0, x2 = 0, x3 = free, x4 = free Plane 2: x1 = 1, x2 = free, x3 = 0, x4 = free
For higher dimensions, just keep adding more xi = free instances as needed. The x1 restriction prevents them from intersecting, and the x2/x3 restrictions prevent them from being parallel.
Tom
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
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