Operations on Maps

• 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

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!

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

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

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v e

f

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

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

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f

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2/102/1

02/12/1

2/)(

2/)(

f

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001

2/102/1

02/12/1

2/)(

2/)(

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

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

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fv

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001

2/102/1

010

2/)(

Me

v e

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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

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, ...

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.

• 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

Two Examples

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.

The Gothic Cube

• The results of Go on the cube are visible on the left.

• We can apply it to any tiling or polyhedron.

Onedimensional subdivision Su1

• Onedimensional subdivision Su1 inserts a vertex in the midpoint of each edge.

• The resulting map is bipartite.

Su1

v e

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v e

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02/12/1

001

2/)(

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2/)(

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

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}.

A Local Example

• The pattern on the left can be obtained from a usual hexagonal tiling:

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];

Seattle on Penrose Tiles

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.

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.

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

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.

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.