  The Ulam sequence is given by $a_1 =1, a_2 = 2$, and then, for $n \geq 3$,
the element $a_n$ is defined as the smallest integer that can be written as the
sum of two distinct earlier elements in a unique way. This gives the sequence
$1, 2, 3, 4, 6, 8, 11, 13, 16, \dots$, which has a mysterious quasi-periodic
behavior that is not understood. Ulam's definition naturally extends to higher
dimensions: for a set of initial vectors $\left\{v_1, \dots, v_k\right\}
\subset \mathbb{R}^n$, we define a sequence by repeatedly adding the smallest
elements that can be uniquely written as the sum of two distinct vectors
already in the set. The resulting sets have very rich structure that turns out
to be universal for many commuting binary operations. We give examples of
different types of behavior, prove several universality results, and describe
new unexplained phenomena.
