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MAED 5211 Classical and Modern Geometry - Homework 1: Incidence Geometry and Projective Geometry
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MAED 5211 Classical and Modern Geometry
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Homework 1
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1. Let ? be a model of incidence geometry.
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1. (a) Four distinct points, no three of which are collinear, are said to form a quadrangle. Suppose every line in ? has at least three distinct points lying on it. Prove that a quadrangle exists in ?.
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2. (b) Four distinct lines, no three of which are concurrent, are said to form a quadrilateral. Suppose ? is a model of projective geometry. Using (a), or otherwise, prove that a quadrilateral exists in ?.
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2. Construct a model of incidence geometry containing 5 points, which is not affine, not hyperbolic, and not elliptic.
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3. Let S be the following statement in the language of incidence geometry: “If ? and ? are any two distinct lines, then there exists a point ? that does not lie either ? or ?.”
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(a) Show that S cannot be proved from the axioms of incidence geometry.
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(b) Show that S holds in every projective geometry. (Note: (a) and (b) implies that S is independent of incidence axioms.
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(c) Use (b) to show that in any model of finite projective geometry i.e. a projective model with finite number of points, all the lines have the same number of points lying on them.
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(Hint: Given any two lines ? and ?, define a function ?: {?} → {?} such that ? is bijective, which implies that the number of points on ? equals the number of points on ?.)
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