Graphics Reference
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Figure 7.1:
Examples of simplices
0
,
1
,
2
and
3
[
24
].
that meet only along a common face and their faces of any dimension. A concrete example of
a simplicial complex is given by triangulated surfaces, where the vertices, edges and faces of the
triangulation are
0
-,
1
- and
2
-simplices, respectively. e
dimension
of a simplicial complex is the
maximum dimension of its simplices.
A
subcomplex
of a complex
K
is a simplicial complex whose set of simplices is a subset of the
set of simplices of
K
. Particular instances of subcomplexes are given by the star and the link of a
simplex. Given a simplex
, the
star
of
is the union of all the simplices containing
. e
link
of
consists of all the faces of simplices in the star of
that do not intersect
. e concepts of
star and link are illustrated in Figure
7.2
for the case of a
0
-simplex. Other useful subcomplexes
of a complex
K
are its skeletons: for
0r
dim
.K/
, the
r
-
skeleton
of
K
is the complex of all
the simplices of
K
whose dimension is not greater than
r
.
(a)
(b)
(c)
Figure 7.2:
(a) A
0
-simplex, (b) its star and (c) its link [
24
].
Note that it is also possible to define an
abstract simplicial complex
without using any ge-
ometry, as a collection
, so is every
non-empty subset of
A
. Simplicial complexes can be seen as the geometric realization of abstract
simplicial complexes.
For more details on simplicial complexes refer to [
149
].
A
of finite non-empty sets such that if
A
is any element of
A
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