Operations on Maps
Maps and Flags
t2(f)
f
t1(f)
t2t0(f)
t0(f)
• There are three fixed
point free involutions
defined on F M: t0,t1,t2.
• Axioms for maps:
• A1: < t0,t1,t2> acts
transitivley.
• A2: t0t2 = t2t0 is
fixedpoint free involution.
• There are four flags per
edge: f, t0(f), t2(f), t2(
t0(f)) = t0( t2(f)).
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
f
f
Du
v
e
v
e
• Dual Du interchanges
the role of vertices
and faces and keeps
the role of edges.
• For instance the dual
of a cube is
octahedron.
Dual Du - continued
f
f
Du
v
v
e
 f  0 0 1  v 
 e   0 1 0  e 
  
 
 v  1 0 0  f 
e
• Only the labelings on
the flags are
changed.
• The exact definition is
given by the matrix on
the left.
Truncation Tru
f
f
Tru
v
e
v
e
• 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 4gons transform into 8gons.
Truncation Tru - continued
f
f
Tru
v
e
v
 (v  e) / 2  1 / 2 1 / 2 0   v 
(v  f ) / 2  1 / 2 0 1 / 2  e 

 
 
v
0
0   f 

  1
 (v  e) / 2  1 / 2 1 / 2 0   v 
(v  f ) / 2  1 / 2 0 1 / 2  e 

 
 
f
0
1   f 

  0
(v  e) / 2 1 / 2 1 / 2 0  v 

 0
e
1 0  e 

 
 
f
0 1  f 

  0
e
• Each flag is replaced
by three flags.
• The exact definition is
given by the three
matrices on the left.
Medial Me
f
f
Me
v
e
v
e
• 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.
Medial Me - continued
f
f
Me
v
e
v
e

  0 1 0  v 
(v  f ) / 2  1 / 2 0 1 / 2  e 

 
 
v

  1 0 0   f 
e

  0 1 0  v 
(v  f ) / 2  1 / 2 0 1 / 2  e 

 
 
f

  0 0 1   f 
e
• Each flag is replaced
by two flags.
• The exact definition is
given by the two
matrices on the left.
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 (F) = S(T(F)).
• Here are some examples:
Du
Me
Tru
Du
Id
Me
Le
Me
An
Me o Me
Tru o Me
Tru
Su2 o Du
Me o Tru
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.
The Gothic Transformation
• 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.
Two Examples
Strawberry Fields
Gothic transformation over hexagons on a ceiling of an 18
century mansion in England.
Slika 20. Operacija Go nad šestkotniki na stropu neke angleške hiše iz 18. stoletja.
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
f
f
Su1
v
e
v
e
v
0 0  v 

  1
(v  e) / 2  1 / 2 1 / 2 0  e 

 
 
f
0 1  f 

  0
e
1 0  v 

  0
(v  e) / 2  1 / 2 1 / 2 0  e 

 
 
f
0 1  f 

  0
• Onedimensional
subdivision Su1
inserts a vertex in the
midpoint of each
edge.
• The resulting map is
bipartite.
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
Su
1
Su1 o
Du
Su1 o
Me
Su1 o Tru Su1 o
Su1
Su1 o
Su2
Su1 o BS
Su
2
Su2 o
Du
Su2 o
Me
Su2 o Tru BS
Su2 o
Su2
Su2 o BS
Representations of flag systems
• Let F be a flag system and let r:VF ! V be
a vertex representation. We can extend
the representation in the following way.
– For each element e from E or (F)
• r(e) = apex{r(v)| v ~ e}.
A Local Example
• The pattern on the left
can be obtained from
a usual hexagonal
tiling:
The Seattle Transformation
MetaSeattle = {
{Metaef,Metavf2,Metaf},
{Metaef,Metae2f,Metavf},
{Metae,Metave,Metavf},
{Metav,Metave,Metavf},
{Metae,Metae2f,Metavf},
{Metaef,Metavf2,Metavf}
};
Seattle[m_SurfaceMap] :=
TransformS[MetaSeattle,m];
Seattle on Penrose Tiles
Matrices and representations
• Let F be a flag system with representation r and let T be
a transformation (defined by some set of matrices).
• R can be extended to a representation of T(F) as
follows:
• The interpretation r on T(F) 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 MöbiusKantor 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.