## The Sensual (Quadratic) FormThe distinguished mathematician John Conway presents quadratic forms in a pictorial way that enables the reader to understand them mathematically without proving theorems in the traditional fashion. One learns to sense their properties. In his customary enthusiastic style, Conway uses his theme to cast light on all manner of mathematical topics from algebra, number theory and geometry, including many new ideas and features. |

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### Contents

PSL2Z and Farey Fractions | 27 |

THE SECOND LECTURE | 35 |

THE THIRD LECTURE | 61 |

THE FOURTH LECTURE | 91 |

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### Common terms and phrases

3-dimensional arithmetic audible base binary called character colors congruences conorms consider consists contain continue corresponding defined describe determinant diagonal dimensions discussion e₁ edge entries equal equivalent example fact figure finite follows form f formula four fourth function Gauss mean genus give given global relation glue vectors integers invariants labeled lattice least lecture Lemma length Mathematical matrix modulo multiple namely negative nonzero norm obtain obtuse superbase orthogonal p-adic p-adic symbol parameters particular plane players points positive definite possible powers prime problem proof prove quadratic form rational represents respect result river root lattices rule shape signature similar simple smallest space spinor squares sum of three superbase suppose Theorem theory third topograph trivial turns unimodular usually values Voronoi cell