Skew Flats in Higher Dimensions
In higher dimensional space, a flat of dimension k is referred to as a k-flat. Thus, a line may also be called a 1-flat.
Generalizing the concept of skew lines to d-dimensional space, an i-flat and a j-flat may be skew if i + j < d. As with lines in 3-space, skew flats are those that are neither parallel nor intersect.
In affine d-space, two flats of any dimension may be parallel. However, in projective space, parallelism does not exist; two flats must either intersect or be skew. Let I be the set of points on an i-flat, and let J be the set of points on a j-flat. In projective d-space, if i + j ≥ d then the intersection of I and J must contain a (i+j−d)-flat. (A 0-flat is a point.)
In either geometry, if I and J intersect at a k-flat, for k ≥ 0, then the points of I ∪ J determine a (i+j−k)-flat.
Read more about this topic: Skew Lines
Famous quotes containing the words flats, higher and/or dimensions:
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—Sir John Betjeman (1906–1984)
“Wherever a man separates from the multitude, and goes his own way in this mood, there indeed is a fork in the road, though ordinary travelers may see only a gap in the paling. His solitary path across lots will turn out the higher way of the two.”
—Henry David Thoreau (1817–1862)
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—Doris Lessing (b. 1919)