Karl Palmen's Seed for a Fractal Calendar


Sent: 28 August 1998
To: CALNDR-L

...reminded me of a fractal calendar I discovered whose mean period is the golden ratio of 1.618... .

It can be defined as follows:
Each period has either 1 or 2 days.
These can be grouped into building blocks of 2,1 and 2,1,2.
The pattern of these building blocks is exactly the same as the pattern of 1's and 2's respectively.

This gives rise to: (with building blocks identified below their 1)

2 1 2 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 .....
  2     1   2     2     1   2     1   2     2     1   2     2     1
        2               1         2               2               1
                        2                                         1
                                                                  2

each number below a 1 is the number of 2s that follow it before the next 1.

I believe it has some interesting properties:
1) the Nth period of length 1 comes N periods after the Nth period of length 2.
2) The Nth period has 1 day if and only if the Nth day does not begin a period.
3) it can be constructed as follows:
s0 = 1
s1 = 2
s2 = s1,s0 = 2 1
s3 = s2,s1 = 2 1 2
s4 = s3,s2 = 2 1 2 2 1
s5 = s4,s3 = 2 1 2 2 1 2 1 2
s6 = s5,s4 = 2 1 2 2 1 2 1 2 2 1 2 2 1
etc.

It also shows how calendars can be constructed by cyclically interrupting cycles and then cyclically interrupting the cycles of interruption several times over.

Karl Palmen


Friday Nine of Hearts

Saturnight 1 Month 6 Yerm 2


Mario Hilgemeier replies:

Since the start is always the same, it is easy to continue, once started. Clifford Pickover gives a similar L-system-like sequence with the generation rule "followed with the inverse":

take 0 as the inverse of 1, then the generations look like
0
01
0110
01101001
0110100110010110

This makes an interesting broken rhythm, when turned it into audio. some drummers really use 1001_0110 (0 low tone, 1 high tone), where the interesting thing is the accent shift to 0101_1010 (the "_" is not really played, it's there to indicate perception).

Karl Palmen replies:
As a calendar it does not have optimum short term accuracy. Only 01010101 or 101010 can achieve that for 0.5 . I believe my golden ratio calendar has optimum short term accuracy. Indeed the period of day N may be N-1 divided by the golden ratio rounded down with one added.

Analysis of Metonic Cycle:

13 12 12 13 12 12 13 12 13 12 12 13 12 12 13 12 13 12 12
 3        3        2     3        3        2     3
                   3                       4

The above emails have been subject to editing - refer to the CALNDR-L archive for full text


Karl Palmen's Calendars

Calendrics

Essays on mathematical themes


Copyright 1998 by the authors
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