Examples of Higher Rank
As stated above, this concept of an abstract polytope is very general, and includes:
- Apeirotopes, i.e. infinite polytopes or tessellations (tilings)
- Decompositions of other manifolds such as the torus or real projective plane
- Many other objects, such as the 11-cell and the 57-cell, that don't fit well into "normal" geometric spaces.
In general, the set of j-faces (−1 ≤ j ≤ n) of a traditional n-polytope form an abstract n-polytope.
Read more about this topic: Abstract Polytope
Famous quotes containing the words examples of, examples, higher and/or rank:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“There are no lower or higher or median moralities. There is only one morality, and it is precisely the one that was given to us during the time of Jesus Christ and that stops me, you and Barantsevich from stealing, offending others, lying etc.”
—Anton Pavlovich Chekhov (1860–1904)
“If, in looking at the lives of princes, courtiers, men of rank and fashion, we must perforce depict them as idle, profligate, and criminal, we must make allowances for the rich men’s failings, and recollect that we, too, were very likely indolent and voluptuous, had we no motive for work, a mortal’s natural taste for pleasure, and the daily temptation of a large income. What could a great peer, with a great castle and park, and a great fortune, do but be splendid and idle?”
—William Makepeace Thackeray (1811–1863)