The Mathematics behind Escher prints: from symmetry to groups

Summer Institute for Mathematics at the University of Washington 2006
Julia Pevtsova


In this class we shall make a transition from a very geometric notion of symmetry of two dimensional objects and wall patterns to a very algebraic notion of a group. The main theme can be expressed with a quote from Sophie Germain:

L'algèbre n'est qu'une géométrie écrite;
la géométrie n'est qu'une algèbre figurée.

(Algebra is nothing but written geometry;
Geometry is nothing but pictured algebra.)

We shall touch upon a curious similarity between mathematicians and biologists: namely, their urge to classify things. In particular, we shall discuss why an algebraist may consider a wall pattern made of Escher's Clowns to be the "same" as the one made of simple triangles.

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