automorphisms

• “The basis elements of the automorphisms are e_a = ?”

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• “In his hands, it turned from the study of substitutions in formulas and of functions invariant under such substitutions into the study of field extensions and corresponding automorphisms (Stein 1988, earlier Mehrtens 1979b).”

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• ““NFU can be understood as the theory of certain nonstandard models of Zermelo-style set theory with automorphisms””

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• “It is again a relative concept and, again, a group of automorphisms of the space is involved.”

  SYMMETRY AND ASYMMETRY

• “From this approach, an “obvious” asymmetry is frequently a regular symmetry with respect to a group of automorphisms that are not strict orthogonal trans - formations but obvious distortions of such.”

  SYMMETRY AND ASYMMETRY

• “By this is meant that any point P of the space can be carried into any other point Q by at least one of the automorphisms.”

  SYMMETRY AND ASYMMETRY

• “In order to arrive at a general mathe - matically oriented notion of symmetry, it is also neces - sary to consider not only single automorphisms but certain assemblages of automorphisms called groups”

  SYMMETRY AND ASYMMETRY

• “The symmetry of a figure is “interesting,” or “mean - ingful,” or “relevant,” mathematically or aesthetically, if the automorphisms of the underlying group are”

  SYMMETRY AND ASYMMETRY

• “Now, the mathematical notion of symmetry demands that there shall be given some group of automorphisms.”

  SYMMETRY AND ASYMMETRY

• “A figure is not just symmetric tout court, but it is symmetric relative to a given group of automorphisms, which, in a logical sense, has to be given first.”

  SYMMETRY AND ASYMMETRY

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