Matpic
mathematics, in pictures
The collection

Every essay — page 4

Page 4 of 4, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Ladders Concepts Search

Dynamics

One rule, applied over and over, and what the sequence does in the end.

0123456740 steps of a rotation by √2 − 1 of a turnthe gaps between neighbouring points take 3 distinct values — never more than three, at any number of steps

The orbit that must come back

A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

6 figures the Lorenz system at ρ = 28, projected onto the x–z planepast the critical ρ of 24.74: two fixed points, neither attracting, and a trajectory that settles on neither

Two lobes and no cycle

Three equations, three variables, and a trajectory that never crosses itself, never repeats, and never leaves a region of zero volume. The set it settles onto is not a point, not a loop, and not a surface.

6 figures 9232127 reaches 1 in 111 steps, after climbing to 9232the vertical scale is logarithmic, because the climb is the interesting part and it is steep

The question nobody can answer

Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.

6 figures