By Luigi A. Rosati

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B) The EEP is a subplane of the Extended Plane C2 . X 2. Let ABCD be a quadrangle in the EEP. Let X = AB ∩ CD, Y = BD ∩ CA, Z = AD ∩ BC. The triangle XY Z is called the diagonal triangle. Draw the dual conﬁguration (the quadrilateral and the diagonal trilateral). A D 3. Assume that Desargues’ Theorem is valid in the Y EEP. (a) State the dual of Desargues’ Theorem. C (b) Draw the dual Desargues’ conﬁguration. B (c) State the converse of Desargues’ Theorem. (d) Compare your statements in (a) and (c).

10. Count all the incidences in S4 , a four-dimensional projective space of order q. That is, determine the number of (a) points and hyperplanes in S4 ; (b) lines and planes in S4 ; (c) points in each of a line, a plane and a hyperplane; (d) lines in each of a plane and a hyperplane; (e) lines through a point; (f) planes in a hyperplane; 36 Introduction to axiomatic geometry (g) planes through a point; (h) planes containing a line; (i) hyperplanes through each of a point, a line, and a plane.

The claim is that they are consistent. 2. Fano’s Postulate is required to prevent trivialities. 26 Introduction to axiomatic geometry 3. Axiom A8 is called the dimension axiom; it is usually given as: for all subspaces U , V of the Sr , dim(U ⊕ V ) = dim U + dim V − dim(U ∩ V ). 8 In an r-dimensional projective space Sr : • A one-dimensional subspace S1 is called a line. • A two-dimensional subspace S2 is called a plane. • A three-dimensional subspace S3 is called a solid. • An (r − 1)-dimensional subspace Sr−1 is called a hyperplane of the Sr .