Whole Number Dynamics I

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

Whole Number Dynamics II

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

Whole Number Dynamics III

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

Spiroflowers

Age 16 to 18 Challenge Level:

A spirolateral is a continuous path drawn by repeating a sequence of line segments of lengths $a_1, a_2, a_3, ... a_n$ with a given angle of turn between each line segment and the next one. (Alternatively the path can be considered as a repeated sequence of 'bound' vectors: $\overrightarrow {P_1P}_2, \overrightarrow{P_2P}_3,... \overrightarrow{P_n P}_{n+1}$, each vector starting at the endpoint of the previous vector.)
In the first diagram the lengths of the line segments are equal and the angles of turn vary periodically in sequences of length 3. In the second diagram the lengths of the line segments vary periodically in sequences of length 5 and the angles of turn are equal. In the third diagram both the lengths and the angles vary.

Investigate these patterns, give sequences of instructions which would produce similar paths and explain why in each case the spirolateral paths are closed producing a cyclic pattern when the sequence is repeated infinitely often.

 Why does the spirolateral in this diagram continue indefinitely, shooting off to infinity if the sequence is repeated infinitely often?