Discrete Examples
Three of the six regular 4-polytopes – 8-cell (tesseract), 24-cell, and 120-cell – can each be partitioned into disjoint great circle (regular polygon) rings of cells forming discrete Hopf fibrations of these polytopes. The tesseract partitions into two interlocking rings of four cubes each. The 24-cell partitions into four rings of six octahedrons each. The 120-cell partitions into twelve rings of ten dodecahedra each.
The 600-cell also partitions into 20 rings of 30 tetrahedra each in a very interesting, quasi-regular chain called the Boerdijk–Coxeter helix. When superimposed onto the 3-sphere curvature it becomes regular.
Read more about this topic: Hopf Fibration
Famous quotes containing the words discrete and/or examples:
“One can describe a landscape in many different words and sentences, but one would not normally cut up a picture of a landscape and rearrange it in different patterns in order to describe it in different ways. Because a photograph is not composed of discrete units strung out in a linear row of meaningful pieces, we do not understand it by looking at one element after another in a set sequence. The photograph is understood in one act of seeing; it is perceived in a gestalt.”
—Joshua Meyrowitz, U.S. educator, media critic. “The Blurring of Public and Private Behaviors,” No Sense of Place: The Impact of Electronic Media on Social Behavior, Oxford University Press (1985)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)