Download - Operations on Maps
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Operations on Maps
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• There are three fixed point free involutions defined on M: 0,1,2.
• Axioms for maps: • A1: < 0,1,2> acts
transitivley.• A2: 02 = 20 is
fixedpoint free involution.• There are four flags per
edge: , 0(), 2(), 2( 0()) = 0( 2()).
Maps and Flags
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Flag Systems are General
• One may use flag systems to describe nonorientable surfaces such as Möbius bands or even complexes that are not surfaces, such as books!
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Dual Du
• Dual Du interchanges the role of vertices and faces and keeps the role of edges.
• For instance the dual of a cube is octahedron.
Du
v e
f
v e
f
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Dual Du - continued
• Only the labelings on the flags are changed.
• The exact definition is given by the matrix on the left.
f
e
v
v
e
f
001
010
100
Du
v e
f
v e
f
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Truncation Tru
• Truncation Tru chops away each vertex and replaces it by a polygon.
• For instance the eitght corners of a cube are replaced by triangles. Former 4-gons transform into 8-gons.
v e
f
Tru
v e
f
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Truncation Tru - continued
• Each flag is replaced by three flags.
• The exact definition is given by the three matrices on the left.
v e
f
Tru
v e
f
f
e
v
f
e
ev
100
010
02/12/12/)(
f
e
v
f
fv
ev
100
2/102/1
02/12/1
2/)(
2/)(
f
e
v
v
fv
ev
001
2/102/1
02/12/1
2/)(
2/)(
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Medial Me
• Medial Me chops away each vertex and replaces it by a polygon but it does it in such a way that no original edges are left.
• The resulting map is fourvalent and has bipartite dual.
Me
v e
f
v e
f
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Medial Me - continued
• Each flag is replaced by two flags.
• The exact definition is given by the two matrices on the left.
f
e
v
f
fv
e
100
2/102/1
010
2/)(
f
e
v
v
fv
e
001
2/102/1
010
2/)(
Me
v e
f
v e
f
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Composite Transformations
• Obviously we may combine two or more transformations into a composite transformation. If S and T are two transformations then S o T () = S(T()).
• Here are some examples:
Du Me Tru
Du Id An Su2 o Du
Me Me Me o Me Me o Tru
Tru Le Tru o Me Tru o Tru
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Rules for Composite Operations
• Rule: Let M1, M2, ...be matrices defining
transformation T and let N1, N2, ... be
matrices that define S. Then the composite transformation T o S is defined by the set of all pairwise matrix products M1N1, M1N2, ..., M2N1, M2N2, ...
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Twodimensional subdivision Su2
• As we defined earlier Su2 = Du o Tru o Du
• It is interesting that many early gothic blueprints of churches contain transformation Su2 on the infinite square grid.
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• Go = Du o Me o Tru• The resulting graph is
bipartite with quadrilateral faces.
• This transformation can be found on the ceilings of various late gothic churches in Slovenia.
• Note that there are 6 matrices needed in order to define Go.
The Gothic Transformation
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Two Examples
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Strawberry Fields
Slika 20. Operacija Go nad šestkotniki na stropu neke angleške hiše iz 18. stoletja.Gothic transformation over hexagons on a ceiling of an 18 century mansion in England.
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The Gothic Cube
• The results of Go on the cube are visible on the left.
• We can apply it to any tiling or polyhedron.
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Onedimensional subdivision Su1
• Onedimensional subdivision Su1 inserts a vertex in the midpoint of each edge.
• The resulting map is bipartite.
Su1
v e
f
v e
f
f
e
v
f
ev
v
100
02/12/1
001
2/)(
f
e
v
f
ev
e
100
02/12/1
010
2/)(
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More Composite Transformations
• We extend our table of composite operations
Du Me Tru Su1 Su2 BS
Du Id An Su2 Du o Su1 Du o Su2 Co
Me Me Me o Me Me o Tru Me o Tru Me o Tru Me o BS
Tru Le Tru o Me Tru o Tru Tru o Tru Tru o Tru Tru o BS
Su1
Su1 o Du
Su1 o Me
Su1 o Tru
Su1 o Su1
Su1 o Su2
Su1 o BS
Su2
Su2 o Du
Su2 o Me
Su2 o Tru
BS Su2 o Su2
Su2 o BS
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Representations of flag systems
• Let be a flag system and let :V ! V be a vertex representation. We can extend the representation in the following way. – For each element e from E or (F)
• (e) = apex{(v)| v ~ e}.
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A Local Example
• The pattern on the left can be obtained from a usual hexagonal tiling:
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MetaSeattle = {
{Metaef,Metavf2,Metaf},
{Metaef,Metae2f,Metavf},
{Metae,Metave,Metavf},
{Metav,Metave,Metavf},
{Metae,Metae2f,Metavf},
{Metaef,Metavf2,Metavf}
};
The Seattle Transformation
Seattle[m_SurfaceMap] := TransformS[MetaSeattle,m];
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Seattle on Penrose Tiles
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Matrices and representations
• Let be a flag system with representation and let T be a transformation (defined by some set of matrices).
• R can be extended to a representation of T() as follows:
• The interpretation r on T() is determined in three steps:– Using matrices we get the first representation.– We keep only the vertex part– We extend it by the apex construction to the final
representation.
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The Möbius-Kantor graph
• Here is the generalized Petersen graph G(8,3), also known as the Möbius-Kantor graph. It is the Levi graph of the Möbius-Kantor configuration, the only (83) configuration.
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The Möbius-Kantor graph, Map M on the surface of genus 2.
• The Möbius-Kantor graph gives rise to the only cubic regular map M of genus 2 (of type {3,8}) . The faces are octagons.
• We are showing the Figure 3.6c of Coxeter and Moser.
• The fundamental polygon is abcda-1 b-1 c-1 d-1
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Co(M) = Du(BS(M)) = Du(Su2(Su1(M))).
• The skeleton of Co(M) is a trivalent graph on 96 vertices. It is the Cayley graph for the group
• <x,y,z| x2 = y2 = z2 = (xz)2 = (yz)3= (xz)4=1>
• This is Tucker’s group, the ONLY group of genus 2.
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My Project at Colgate
• I am working with a sculptor, two arts students and math students to build a model of this Cayley graph on a double torus.