Pattern of Buds on the main Cardioid and “Mandelbrotian” Strings
The Mandelbrot Set, Showing the Main Cardioid and Buds, with Some Bud Periods (Devaney)
Notice the following pattern of the Buds attached to the main cardioid:
Every adjacent pair of Buds has a third Bud between them, whose Period is the sum of theirs.
Imagine walking around the periphery of the cardioid, starting at its cusp, and proceeding CCW, noting only the largest Buds’ periods. That would generate strings of numbers like:
[323]
or
[4352534]
etc.
Now we have strings of integers.
In this new domain, we can construct an algorithm that successively generates such strings, by saying “Between every neighboring pair of integers, insert a third that is their sum.”.
For instance, from [1 1], we would get [1 2 1] – and then [1 3 2 3 1]. Note that the latter is just the first of the examples above with “1” tacked onto either end.
Without explaining why, exactly (We could say “Consider the cardioid to give us the starting series, “[1 1]”.), we do that just so our sequences of integers match (ignoring the “1”s) the Buds’ Periods.
It turns out that this – now a “Pure Math” analogue- algorithm generates the Periods of all of the Buds around the periphery of the cardioid – and so, in effect all of the attached Buds – each with its correct Period!
It turns out that the latter, now a “Pure Math” analogue, can be used to generate the Periods of all of the Buds around the periphery of the cardioid!
- Every adjacent pair of Buds has a third Bud in-between them, whose Period is the sum of theirs.
- If we consider the related structure of strings of integers and make a generative, rather than just descriptive statement, we get;
- Between every neighboring pair of integers, insert a third that is their sum.
It turns out that 2) generates all of the attached Buds – each with its correct Period!
(Note that the coordinates of the Buds are not produced – just their relative positions.)
Generating any “Layers” of Buds ….
I have written a Python program to generate any ~layer~ of these Buds, produced by repeatedly applying this algorithm.
Starting from [1 1], representing the cardioid, we get, successively:
[1 2 1]
[1 3 2 3 1]
[1 4 3 5 2 5 3 4 1]
and so on.
Remembering the connection to Buds, the second of these picks out one Bud 3, Bud 2 and the other Bud 3.
And so on.
Continuing the applications of the algorithm generates – in effect – successively “deeper” traverses around the periphery, visiting successively smaller and smaller – or higher and higher periods of buds. This can be continued forever!
I have written a Python program to graph these series for any of these traverses. In these plots, we can see the contributions of buds (of period) 2, 3, 4, …”
In the following graph of Buds’ Periods, the Y-axis shows the Periods; the X-axis values are only for reference from the text below:
(Ignoring the decimals…)
At the x-value for:
2. is the cardioid
3. is bud 2
4. is bud 3
and so on.
Here’s traverse 20:
Again, the vertical lines show the Periods.)
Note:
- The lines of height 1, for the cardioid, are at the extreme ends of the graph.
- The center vertical line represents Bud 2.
- The vertical lines representing the lowest-Period Buds are spread out similarly, by the insertion of the lines for the intermediate-Period Buds.
Here’s 64, in a different style of plot:
(with the biggest buds in the center)
Note: The X-Axis values meaning (the step in the traverse of the cardioid’s periphery at which each Bud is encountered) is not particularly important.