The Phi line Graph

If we draw the line y = Phi x on a graph, (ie a line whose gradient is Phi) then we can see the Rabbit sequence directly.

Where the Phi line crosses a horizontal grid line (y=1, y=2, etc) we write 1 by it on the line and where the Phi line crosses a vertical grid line (x=1, x=2, etc) we record a 0.

Now as we travel along the Phi line from the origin, we meet a sequence of 1s and 0s - the Rabbit sequence again!

1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ...

The following sections explore this relationship using functions such as "the next integer below" (the floor function) and "the next integer above" (the ceiling function) which will tell us which grid-line we have just crossed.

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