We shall here consider one of the -configurations in some detail, referring the reader to [96]
for a discussion of the other cases. The configuration we will treat is the well known Desargues
configuration, displayed in Figure 54
. The Desargues configuration is associated with the 17th century
French mathematician Gérard Desargues to illustrate the following “Desargues theorem” (adapted
from [145]):
Let the three lines defined by and
be concurrent, i.e.,
be intersecting at one point, say
. Then the three intersection points
and
are colinear.
Another way to say this is that the two triangles and
in Figure 54
are in
perspective from the point
and in perspective from the line
.
As we will see, a new fascinating feature emerges for this case, namely that the Dynkin diagram dual to
this configuration also corresponds in itself to a geometric configuration. In fact, the Dynkin diagram dual
to the Desargues configuration turns out to be the famous Petersen graph, denoted , which is
displayed in Figure 55
.
To construct the Dynkin diagram we first observe that each line in the configuration is
disconnected from three other lines, e.g., have no nodes in common with the lines
,
,
. This implies that all nodes in the Dynkin diagram will be
connected to three other nodes. Proceeding as in Section 10.2.2 leads to the Dynkin diagram
in Figure 55
, which we identify as the Petersen graph. The corresponding Cartan matrix is
Because the algebra is Lorentzian (with a metric that coincides with the metric induced from the
embedding in ), it does not need to be enlarged by any further generator to be compatible with the
Hamiltonian constraint.
It is interesting to examine the symmetries of the various embeddings of the Petersen graph in the
plane and the connection to the Desargues configurations. The embedding in Figure 55 clearly
exhibits a
-symmetry, while the Desargues configuration in Figure 54
has only a
-symmetry. Moreover, the embedding of the Petersen graph shown in Figure 56
reveals yet
another symmetry, namely an
permutation symmetry about the central point, labeled
“10”. In fact, the external automorphism group of the Petersen graph is
, so what we see
in the various embeddings are simply subgroups of
made manifest. It is not clear how
these symmetries are realized in the Desargues configuration that seems to exhibit much less
symmetry.
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