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Simplicial complex
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Tue Aug 15 2000 at 1:23:44
Mathematical structure, an assemblage of a number of
simplex
es.
No, "simplicial complex" is not an
oxymoron
.
Simplicial complexes, often called just "complexes", are used in
Homotopy
theory but are also very useful in modeling spatial objects in various computer graphics-related disciplines.
Each complex has its own dimension, (let's call it
k
), such that
k
>= 0, and contains one or more
k
-simplexes, none of which have points in common, except that two simplexes may share a face. Each complex is constructed with reference to a Euclidean Space whose dimension >=
k
.
As is pointed out in the node on
simplexes
, each
k
-simplex has (
k
+1) "faces" forming its outer boundary. Each face is a (
k
-1)-simplex.
So we can
recursive
ly take the
k
-simplexes, all the (
k
-1)-simplexes that are faces of the
k
-simplexes, all the (
k
-2)-simplexes which are faces of the faces, and so on, until we have 0-simplexes, which are all of the simplexes'
vertices
. The assemblage of all these simplexes is a simplicial complex.
For computer graphics
cognoscenti
, a
wire frame
is a 2-complex constructed in 3-dimensional space.
A simplicial complex is
not
a
topological space
(although it can serve as a
base
for the
discrete topology
of the set of its constituent simplexes). However, you can take the space the complex is constructed in, and restrict it to the set of all the points in any of a complex's simplexes. The resulting subspace is called the
polyhedron
of that complex.
(
thing
) by
artemis entreri
(10.9 mon)
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Sun Feb 11 2001 at 5:48:34
(a) a
family
of
finite sets
, closed under
subsets
.
(b) a collection of
simplices
meeting
face-to-face
.
The first type of
complex
can be made into the second, by
embedding
the members of the
sets
in
general position
in some
space
of suffiently high
dimension
, and forming
simplices
by taking the
convex hull
of each
set
. The
sets
of a
simplicial complex
are also known as
faces
.
--back to
combinatorics
--
printable version
chaos
Simplex
Combinatorics
Symplectic Topology
Polygonal modeling
Triangulation
Topological Space
Covering
Homology groups
polyhedron
wire frame
complex
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