We construct explicitly, using the generalized Weierstrass representation, a complete embedded minimal surface $M_{k,\theta}$ invariant under a rotation of order $k+1$ and a screw motion of angle $2\theta$ about the same axis, where $k \gt 0$ is any integer and $\theta$ is any angle with $|\theta| \lt \pi/(k+1)$. The existence of such surfaces was proved in [Callahan et al. 1990], but no practical procedure for constructing them was given there.

We also show that the same problem for $\theta=\pm\pi/(k+1)$ does not have a solution enjoying reflective symmetry; the question of the existence of a solution without such symmetry is left open.

We study two families of excitable cellular automata known as the Greenberg-Hastings model and the cyclic cellular automaton. Each family consists of local deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by the range $\rho$ of interaction, $\ell^p$ shape of its neighbor set, threshold $\theta$ for contact updating, and number $\kappa$ of possible states per site. These models are mathematically tractable prototypes for the spatially distributed periodic wave activity of so-called excitable media observed in diverse disciplines of experimental science.

Fisch, Gravner and Griffeath studied experimentally the ergodic behavior of these models on $\Bbb Z^2$, started from random initial states. Among other phenomena, they noted the emergence of asymptotic phase diagrams (and dynamics on $\Bbb R^2$) in the threshold-range scaling limit as $\rho,\theta\to \infty$ with $\theta/\rho^2$ constant.

Here we present several rigorous results and some experimental findings concerning various phase transitions in the asymptotic diagrams. Our efforts focus on evaluating bend$(p)$, the limiting threshold cutoff for existence of the spirals that characterize many excitable media. Our main results are formulated in terms of spo$(p)$, the cutoff for existence of stable periodic objects that arise as spiral cores. Some subtle consequences of anisotropic neighbor sets $(p\neq2)$ are also discussed; the case of box neighborhoods $(p=\infty)$ is examined in detail.

When a quadrilateral undergoes a certain infinite folding process, an intricate curve is traced out by its transformed vertices. The curve possesses attractive symmetry properties. We study the folding as a dynamical system in $\C^4$, which is partially described by a billiard-like dynamical system on the bounded component of an elliptic curve associated with the squared diagonal lengths.

The radical $\mathrm{rad}\,n$ of an integer $n\ne0$ is the product of the primes dividing $n$. The $abc$-conjecture and the Szpiro conjecture imply that, for any positive relatively prime integers $a$, $b$, and $c$ such that $a+b=c$, the expressions $$ {\log c\over \log \mathrm{rad}(abc)} \quad\hbox{and}\quad {\log abc \over \log \mathrm{rad}(abc)} $$ are bounded. We give an algorithm for finding triples $(a,b,c)$ for which these ratios are high with respect to their conjectured asymptotic values. The algorithm is based on approximation methods for solving the equation $Ax^n-By^n=Cz$ in integers $x$, $y$, and $z$ with small $|z|$.

Additionally, we employ these triples to obtain semistable elliptic curves over $\mathbf{Q}$ with high Szpiro ratio $$ \sigma={\log|\Delta|\over\log N}, $$ where $\Delta$ is the discriminant and $N$ is the conductor.

The totally real algebraic number field $F$ of degree 6 with Galois group $A_6$ and minimum discriminant is determined. It is unique up to isomorphy, and is generated by a root of the polynomial $t^6 - 24 t^4 + 21 t^2 + 9 t + 1$ over the rationals. We also give an integral basis and list the fundamental units and class number of $F$.