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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.

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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.

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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.


Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The spiropath contains a pattern which is repeated over and over again with different starting points and in different initial directions. Imagine replacing each pattern by a single line segment joining the starting point and end point of that pattern (or equivalently by the single vector which is the sum of all the vectors forming a single example of the pattern). Then the repeated pattern reduces to repeated vectors of the same length end to end with a given angle of turn between the vectors, so the question of whether the path will close up reduces to considering the result of the Spirostars problem .

The diagrams given contain a small triangle which is the Logo 'turtle'. As the path is drawn by the Logo software you can see the turtle moving around. It can be hidden but we have chosen to show it to indicate astarting point and initial direction for the spiropath.

If we define a 'turtle' as a point and a direction, we can use the notation $(x, y, \theta)$ or $(z, \theta)$ or $ze^{i\theta}$. The motif which forms the pattern in the spiropath is repeated but from a different starting point each time and, in general, with a different initial direction. Each repetition of the set of instructions which draws the motif has the effect of mapping turtles to turtles where the turtle gives the initial point and the initial direction. These motifs have the same form but different starting points and initial directions. When the motif is repeated over and over again it may return to the same initial 'turtle' and repeat a cyclic pattern as in the first three examples, or it may never return to a previous starting point and path may go on to infinity.