 | simplex |
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Name
simplex — A single top-dimensional simplex in a
dim-dimensional triangulation
Synopsis
| Content Model | |
simplex ::= (#PCDATA) | |
| Attributes | |
| Name | Type |
desc | CDATA |
Description
A simplex element stores a single top-dimensional simplex in a
dim-dimensional triangulation, along with its
associated facet gluings, for dimensions
5 ≤ dim ≤ 15.
The character data of this XML element should be a
whitespace-separated list of dim+1
integer pairs, representing the gluings of facets
0, 1, ..., dim of this simplex. Note that
facet i is always opposite vertex
i in a top-dimensional simplex.
For each pair, the first integer represents the other top-dimensional
simplex to which the facet is glued (note that simplices in a triangulation
are numbered 0, 1, 2, etc.). The second integer represents the
permutation of vertices from this simplex to the other simplex
describing precisely how this gluing takes place. This permutation
will take the current facet number of this simplex to the corresponding
facet number of the adjacent simplex, and the other
dim vertex numbers of this simplex
to the corresponding dim vertex numbers on
the adjacent simplex to which they are identified by this gluing.
A permutation is represented as an integer as follows.
If the permutation maps 0, 1, ..., dim to
a0, a1, ...,
aDim respectively (where
a0, a1, ...,
aDim are 0, 1, ..., dim
in some order), then the corresponding integer is
(a0 +
B a1 +
B2 a2 +
... +
Bdim aDim).
Here the “base” B is
B = 8 for dimensions
dim = 5..7, and
B = 16 for dimensions
dim = 8..15.
As an example, the identity permutation for dimension
dim = 8 which maps (0, 1, ..., 8) to
(0, 1, ..., 8) is represented by the integer
(0 + 1×16 + 2×256 + ... + 8×4294967296),
which is 36344967696.
If a facet is a boundary facet (i.e., it is not glued to anything), the two corresponding integers stored in the XML character data should be -1 and -1.
Warning
For dimensions dim = 5..7,
permutations are represented by 32-bit integers.
In contrast, for dimensions dim = 8..15,
permutations are represented by 64-bit integers.
If you are writing code that reads Regina data files, you may need
to use a large integer type
(such as long long in C++) accordingly.
Attributes
descA human-readable description of the role that this top-dimensional simplex plays in the overall triangulation.
Example
The following XML snippet represents simplex number 0 in a 6-dimensional triangulation. Facet 0 of this simplex is glued to facet 6 of simplex number 1, with a gluing permutation that maps (0,1,2,3,4,5,6) to (6,0,1,2,3,4,5). Facet 6 of this simplex is glued to facet 0 of simplex number 1, with a gluing permutation that maps (0,1,2,3,4,5,6) to (1,2,3,4,5,6,0). The remaining facets 1,...,5 of this simplex are all boundary facets.
<simplex> 1 14038087 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 2054353 </simplex>
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