"It is truly remarkable that the human mind has created

powerful new ways to represent and explore events in the distant past.

May we continue to do so." (Lance Latham)

A fractal , as defined by Benoit Mandelbrot (who coined the term) is a geometric object that is self-similar on many scales (like a cauliflower or coastline). A non-integer dimension can be attributed to a fractal (see also Fractal FAQ). There is an infinite number of fractals.

The fractal dimension D of a geometric object can be computed by

D = ln(N) / ln(r)

where

N is the number of copies of the original being made

r is the shrinking factor for each copy

An example for this is the Koch curve. The original is a simple line of finite length.
Then, in the first iteration, it is copied upon itself four times, N=4. The first and last third remain the same, the middle third is replace by two "thirds" with an 120 angle between them, thus forming a triangular "dent". Since each of the copies is a third of the length of the original, r=3.

Therefore D = ln(4) / ln(3) = ln(1.38629) / ln(1.09861) = 1.26186...

On each iteration, the line grows in length by 33.3 %. In other words, the mathematical Koch curve has infinite length but finite dimension greater than 1 (which lines normally have).

A fractal calendar could be based on such a generator pattern for self-similarity, starting or ending at a certain date. Terence McKenna's Timewave is an example for this: an eschatological calendar (fixed end). It assigns a rational "novelty" value to each second, zero being maximal novelty. The fractal dimension of this calendar is 1.4308... = ln(384) / ln(64) (lunar year and number of I Ching hexagrams) . It is interesting if you employ the ten fingers and five Chinese elements you get ln(10) / ln(5) = 1.43068..., only 0.011 % percent off. There are many possibilities how that fractal might look.

The division of a given timespan in parts and the division of each of the parts in the same way (the generator pattern) could also be seen as fractal, even without assigning a number like the Timewave.

Example A - **equal parts**:
Seven Space calendar
- division into seven equal parts on all scales.

Example B - **different parts**: division into three parts of 4/12, 5/12, and 3/12 length. (Not a very practical calendar)

Let's start with a tropical year (in 1994-1998, the tropical year was 365.242190 days) and divide it. According to the rule we get
121.747397, 152.184246, and 91.3105475 days. Now watch the scaling in the next subdivisions of the timespans

365.242190 (4/12) 121.747397 (4/12) 40.5824656 (5/12) 50.7280819 (3/12) 30.4368492 (5/12) 152.184246 (4/12) 50.7280819 (5/12) 63.4101024 (3/12) 38.0460615 (3/12) 91.3105475 (4/12) 30.4368492 (5/12) 38.0460615 (3/12) 22.8276369Three further subdivisions, and we get timespans from .356682 to 4.58696 days. The resulting timespans are of uneven length, but some lengths recur (which stems from the permutation of factors). This system can also be used for creating year-dozens (12 years 4-5-3), dozen-dozens (= 1.44 century), and so forth. A dozen-dozen is built up as 4-5-3 dozens, or [(4-5-3)(4-5-3)(4-5-3)(4-5-3) - (4-5-3)(4-5-3)(4-5-3)(4-5-3)(4-5-3) - (4-5-3)(4-5-3)(4-5-3)]. This is similar to L-systems, string-rewriting algorithms for the generation of fractals, especially plant forms.

Example A and B have both a fractal dimension of 1, since every day is part of the calendar. For the Seven Space calendar D = ln(7) / ln(7) and for example B we have D = ln(12/ ln(12).

On the other hand, if a calendar uses gaps, i.e. parts of the generator pattern are left permanently empty, it will have a fractal dimension between 0 and 1.

As an example for this, consider a calendar which is made of a self-similar seven-part cycle, just like the Seven Space calendar. But in this example, suppose every sixth day does not count as time - it is "time out" for the calendar, a gap.

The fractal dimension is D = ln(6) / ln(7) = 0.92078. Since every cycle of the calendar has a gap of 1/7, the amount of "real" time that is not "time out" gets less and less (by a factor of 0.85714 with each iteration), so that in the limit case all is "time out" plus some infinitesimally short "intervals", or rather: points of "real" time - a fractal dust.

Fractal dusts do not seem very useful as calendars need to assign a name or value to each nanosecond, even "time out" is a name. Calendar-making in its essence is just this: the creation or discernment of differences in time. We have seen above in examples A and B that the fractal dimension remains 1, if no or the same value is assigned to all time units. But days are made different by calendars, and this means, that an extra dimension, a value, is added to time, and this dimension is not one of the three space dimensions.

The prototypical timewave calendar does assign values. And we do this, too, when treating a Sunday differently than a Monday. It is just rather arbitrary to assign number values to the weekdays. But there is no other way if we want to compute the fractal dimension D.

Lets state that calendar time is fractal, but it is often difficult to know its exact fractal dimension. This depends on the calendar system you use. If you add 3 numerical values to each time instant, you can easily arrive a D between 3 and 4. And if all seconds are the same in your calendar, time will be a flat, one-dimensional bore.

The fractal qualities of my All-Cycle Life Calendar and my Freetime Calendar have still to be worked out.

Introduction (previous page)

Fractal Time (next page - link presently defunct)

Contents of "On Fractal Time"

Other Calendrics at this site.

Essays on mathematical themes.