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:
“I have a Vision of the Future, chum.
The workers’ flats in fields of soya beans
Tower up like silver pencils, score on score.”
—Sir John Betjeman (1906–1984)
“Do they merit vitriol, even a drop of it? Yes, because they corrupt the young, persuading them that the mature world, which produced Beethoven and Schweitzer, sets an even higher value on the transient anodynes of youth than does youth itself.... They are the Hollow Men. They are electronic lice.”
—Anthony Burgess (b. 1917)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)