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Abstract geometry and the objects of the material world
The topology of concrete objects
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angle angle-sum axis boundary braid branch-points bridge called centre characteristic circle closed path colours are sufficient connectivity Consider corresponding crossings curve cylinder deform denote described diagonal end-lines end-points Euler's theorem example faces figure five colours four four-colour problem genus geometry heptahedron holes image plane infinite intersect invariant joined Klein bottle knight's tour line-complex line-segments line-system loop lower sheet mathematical maze mesh Mobius band Mobius strip n-gon neighbouring number of edges number of points number of vertices obtain octahedron one-sided surface opposite orientation pieces point at infinity polygonal polyhedra polyhedron position possible projective plane proof proved pulled reader rectangle rectangular regions regressive cut regular graph removed represented Riemann surface rope rotation round shown in Fig shows sides simple solid sphere square structure Suppose symmetric tetrahedron theory thread topologically equivalent torus tree trefoil knot triangle triangular twisted unicursal upper sheet vertex weft z-plane