Star gazing in affine planes
Degree GrantorUniversity of Canterbury
Degree NameResearch report
Even in the study of modern geometry it is as well to remember that geometry has its origin in the measurement of the earth, that is to say surveying, and that figures and diagrams are the very heart of the subject both for the transmission and preservation of information and for the development of new ideas and methods. Therefore, while accepting that a diagram in itself is not a proof, one should not be reluctant to introduce them into one's formal presentations. After all, if a figure has been helpful to you in composing your thoughts why selfishly deny that figure to others? In teaching a course on combinatorics I have found students doubting the existence of a finite projective plane geometry with thirteen points on the grounds that they could not draw it (with 'straight' lines) on paper although they had tried to do so. Such a lack of appreciation of the spirit of the subject is but a consequence of the elements of formal geometry no longer being taught in undergraduate courses. Yet these students were demanding the best proof of existence, namely, production of the object described. It seems to me that finite projective planes are not good objects to draw convincingly but affine planes are, in that lines which are technically parallel can be made visibly parallel at the cost of drawing them piecewise in straight segments. It may have been serendipity that the first case I tried worked out so nicely and it may have been just coincidence that the resulting figures were enough to activate an interest in finite inversive planes, their construction from finite affine planes and their interrelationship with t-designs, but I now shamelessly present a fig.- laden dissertation on finite affine planes and circle geometries.
SubjectsField of Research::01 - Mathematical Sciences::0101 - Pure Mathematics::010102 - Algebraic and Differential Geometry
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