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)
“Man is a stream whose source is hidden. Our being is descending into us from we know not whence. The most exact calculator has no prescience that somewhat incalculable may not balk the very next moment. I am constrained every moment to acknowledge a higher origin for events than the will I call mine.”
—Ralph Waldo Emerson (1803–1882)
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)