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, higher and/or rank:
“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)
“In the mountains of truth you will never climb in vain: either you will already get further up today or you will exercise your strength so that you can climb higher tomorrow.”
—Friedrich Nietzsche (1844–1900)
“West of this place, down in the neighbor bottom,
The rank of osiers by the murmuring stream
Left on your right hand brings you to the place.”
—William Shakespeare (1564–1616)