Addition theorems; the addition theorems of group theory and by Henry B. Mann.

By Henry B. Mann.

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1 -7 --6 show that if the coefficients a, h, b, in the +- form ax 2 + hxy + by 2 are integers, then the river is necessarily periodic in this way. The a h b determinant of this form is d = ab - (~h)2. If we suppose that the base corresponds to an edge of the river, then exactly one of a and b is negative, so ab is negative, and Idl = (~h)2 + labl. Hence I~hl < y'idj, and labl = Idl - (~h)2, so there are only finitely many values for a, b, and h. Now a, b, and h together determine all the other labels, and there are only fmitely many possible triples (a, b, h) for river edges.

__ ..... , \ I \ I \1 " ". __ /,-,'~/;,-' Y I \ __ ..... ,.. ,. __ - .... \ __ ..... , \ I \1 " ". __ .... -,\~/ Y __ ..... \ , I .... __ ..... - .... \ Y 27 28 THE SENSUAL (quadratic) FORM The picture shows H divided into fundamental regions for the group PSL 2 (Z) = r. The solid edges form a tree with three edges per vertex whose nodes and edges correspond to the superbases and bases for Z+. Each of the regions of our topograph consists of "fans" of fundamental regions. / / 1 1 I - _.... - - y ....

Hence I~hl < y'idj, and labl = Idl - (~h)2, so there are only finitely many values for a, b, and h. Now a, b, and h together determine all the other labels, and there are only fmitely many possible triples (a, b, h) for river edges. So some two edges of the river must be surrounded in the same way, and the river must be periodic (since the values of a, b, and h at any edge determine the entire topography). Now, can we really see the values of /(x, y) = 3x 2 + 6xy - 5y 2? For this form, we have /(1,0) = 3, /(0,1) = -5, and /(1, 1) = 4.

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