Definition
An n-dimensional building X is an abstract simplicial complex which is a union of subcomplexes A called apartments such that
- every k-simplex of X is within at least three n-simplices if k < n;
- any (n – 1 )-simplex in an apartment A lies in exactly two adjacent n-simplices of A and the graph of adjacent n-simplices is connected;
- any two simplices in X lie in some common apartment A;
- if two simplices both lie in apartments A and A ', then there is a simplicial isomorphism of A onto A ' fixing the vertices of the two simplices.
An n-simplex in A is called a chamber (originally chambre, i.e. room in French).
The rank of the building is defined to be n + 1.
Read more about this topic: Building (mathematics)
Famous quotes containing the word definition:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)