  A sequence $a_0<a_1<\ldots<a_n$ of nonnegative integers is called a Sidon
sequence if the sums of pairs $a_i+a_j$ are all different. In this paper we
construct CAT(0) groups and spaces from Sidon sequences. The arithmetic
condition of Sidon is shown to be equivalent to nonpositive curvature, and the
number of ways to represent an integer as an alternating sum of triples
$a_i-a_j+a_k$ of integers from the Sidon sequence, is shown to determine the
structure of the space of embedded flat planes in the associated CAT(0)
complex.
