This book, a compilation of the lectures Conway presented at the Joint Meetings of the American Mathematical Society and Mathematical Association of America in 1991, is a bit overstated in the title. In particular, the inclusion of the word "sensual." While some mathematicians no doubt get a significant buzz over the quadratic form, (polynomials in several variables where every term has degree two), I doubt if it reaches the point of sexual stimulation. In the first lecture he considers only equations of the form a*x^2 + b*x*y + c*y^2, where a, b and c are integers. While at first glance this may appear to be a severe restriction, that is not the case. For example, by rewriting the quadratic into the matrix form,

| a b |
( x y ) | | (x y )
| c d |

it is possible to work with matrices, determinants, bases, superbases, equivalence of forms, primitive vectors, and norms of the vectors. Tree diagrams are used to represent the bases of the forms, this leads to a large number of successive descriptive operations. By the time you reach the end of the book, Conway has gone through several of the human senses, considering the following questions.

*) Can you see the values of 3x^2 + 6xy - 5y^2?
*) Can you hear the shape of a lattice?
*) ... and can you feel its form?

The title of the fourth lecture is "The Primary Fragrances" so he "covers" four of the five senses.
I found the difficulty to be rather high, there were many times when I had to slow way down and process the material in small chunks. One of the primary reasons for this is that Conway introduces a lot of notations and terminology and it was not always immediately clear to me what he meant. However, I did manage to understand nearly all of it, which made reading it very worthwhile. Once I finished, I found myself surprised that such a "simple" equation led me down so many different mathematical paths.

Published in Journal of Recreational Mathematics, reprinted with permission

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