I will start by stating I was not an ‘A’ student in geometry in high school. In fact, I was doing good if I managed a ‘C’ in geometry. Now, since then I have somewhat redeemed myself by dabbling in more advanced math concepts such a trigonometry that culiminated in a math test where I received an ‘A’ despite my apprehension on my mathmatical accumen. I will also add that I have always been a history nerd and so it is somewhat surprising (at least to me) that I have taken an interest in the subject of fractal geometry. Around the same time I began reading up on fractal geometry I took a deeper look at the concept of decision cycles and how the two are related. What is fractal geometry and what does it have to do with decision cycles?

Fractal geometry is a mathematical toolkit to manage the irregularities of an object by spotting the regularity within. One could think of fractal geometry as structural forms derived from chaotic processes (Peitgen,1992, viii). Credited to Benoit Mandelbrot, he stated chaotic systems start with a specific point and through a repeating process reaches an unpredictable end state. Balance a quarter on its end and every time it falls it will appear to do so unpredictability, but the direction of fall will be dependent on the initial point. Fractals scale, branches of trees appear to be trees, beaches become coastlines. There is a certain symmetry to a fractal where the parts echo the whole with precise measurability (Mandelbrot, 2006, 125, 127). Fractals can be self-similar scaling up or down or multi-fractal scaling different ways at different points.

Cauliflower as a fractalCauliflower as fractal

This concept is the self-similarity dimension, where an object is self-similar if it can be broken down into random pieces, each being a small replica of the entire structure (like the cauliflower above). I should note that something can be self-similar and not be a fractal, such as a straight line (line segment) or a cube. I should also note that self-similarity dimension is a special form of fractal dimension and that there are many different notions of dimension (it all gets rather confusing and requires caffeine to navigate). Fractals can be dimensional. Dimensions are a tool to measure how irregular an object is and dependent on the observer's point of view (Mandelbrot, 129-130). Dimensions are where things get interesting…

Rod Sterling takes watchers into other dimensions and so has mathematicians by defining dimensions zero thru four (I am excluding string theory and referencing “older” definitions to make my point - also because I think Poincaré is underrated). Zero for a disconnected point (no line segments) or dots, one for a line segment, two for a square, three for a cube, and four for time (Mandelbrot, 129; Poincaré, 2001, 57-58). A series of disconnected points has zero dimensions. A three-dimensional space is the space we live in, and the fourth dimension of time is psychological (qualitative) and physical (quantitative) (Poincaré, 211).

Psychological (qualitative) time is a matter of perception. You see event B, which can only occur after event A, but you did not see event A. You still perceive event A as occurring prior to event B. Physical (quantitative) time is a matter of measurement. Did the light from a distance star originate before or after the Magna Carta was signed? Which came first is the qualitative problem, as these events could have happened simultaneously. This means event A and event B could be of sequence AB or BA. The solution is to use the quantitative aspect to measure the light velocity from the star to find the actual time the light originated. This will then confirm the order of event A and event B answering if the Magna Carta was signed before or after the light originated.

Returning to fractals and self-similarity, let us think of fractals as a black and white object. A multi-fractal would also be a black and white object but with the addition of shades of gray. A multi-fractal has more than one scaling ratio in the object. So, some parts of the object will shrink or expand at different rates – this is our shades of gray (Mandelbrot, 208). The ‘multi’ in multi-fractals represents variability, multi-fractals contain (or capture) variations. Multi-fractals are more representative of the systems we exist around such as financial markets or the dispersion of water in the ground or why every backyard in America is not filled with (an equal amount) gold ore.

Mandelbrot discussed two other concepts worth looking at related to multi-fractals. The first is the Noah Effect, the wild price swings (crashes, bubbles) we see in financial markets, represent abrupt change like the Great Flood (Mandelbrot, 200). The second, is the Joseph Effect, the long-range memory or dependence, anchoring today’s price to yesterday’s (Mandelbrot, 201). Mandelbrot derived the name from the biblical account of seven years of famine followed by seven years of prosperity in Egypt. News of high inflation has continued to echo across the months, framing personal financial decisions. The mood is pessimistic, the perception is everything is more expensive and will be more so tomorrow. The Noah Effect and the Joseph Effect illustrate the variability in multi-fractals.

Applying the four dimensions, multi-fractals are used to model asset returns by manipulating the dimension of time. This is done by shortening or lengthening the time axis, compressing or stretching time. If we take a rectangle and divide it into two halves, then we cut the two halves into two more halves. The quarters we created are divided in half yet again and the cycle continues until the rectangle no longer looks like a rectangle but rather a jagged, irregular series of peak and valleys much like a stock chart (Mandelbrot, 215). This mathematical process is called the multiplicative cascade and it is how Mandelbrot described the time-deformation process in multi-fractal time.

Irregular spaced segmentsirregular spaced segments

Time-deformation gets us time that is bunched into irregular spaced segments. Physical time is a quantitative measurement of time, such as measuring the velocity of light. This implies a constant unit of measure (even if that unit of measure is an approximation because we lack the ability to define it precisely). Psychological time is qualitative and subject to our perceptions (there is also an inherent understanding of the natural order of things, so perception is not the definitive rule).

This influence could be emotional such as watching the one-minute chart of a volatile stock, shaping our perception of reality as it relates to the chart. The influence on emotions and ultimately our perceptions will influence the decisions we make and the time-scales we make them. I cannot overstate this point - time-deformation messes with our perceptions in far reaching ways with important ramifications in our lives. Every every reaction or movement is a result of time-deformation creating a perception filter that drives a decision cycle leading to an event. Variations in perception filters to time-deformation in trading cycles will create different decision cycles that will influence markets. A day trader’s perceptions about the time-deformation events that they see will drive a different decision cycle aiming for different outcomes than that of a swing-trader or boglehead trader. Friction is generated when these cycle outcomes collide further influencing the perceptions.

As time-deformation occurs, it messes with our perceptions.

Looking more closely at decision cycles, the bunching of time into irregular spaced segments creates and influences decision cycles. Cycles for traders to make decisions on the buying and selling of securities. We can see this multi-fractal time as a multi-fractal decision framework that compresses or stretches the time dimension in decision cycles. Decision cycles are important, and not just for traders, but for a multitude of things like aerial combat.

Colonel Boyd was an instructor at the U.S. Air Force Weapons School in the 1950’s (when it was simply the Fighter Weapons School). Boyd spent considerable time thinking about fighter energy maneuverability – essentially a quantitative model for aircraft performance. He compared turn radius, turn rate, and G in aircraft maneuverability as it related to air combat (Boyd, 1976, 8). Boyd defined maneuverability as being able to change speed, altitude and direction in any combination (Boyd, 6). The goal was to do so faster than an opponent in a dogfight, but changing altitude, direction and putting G onto an aircraft takes energy, the loss of which could determine the fight. What was needed was an aircraft that had a higher energy rate for any combination of turn rate/radius. A fighter that could gain and lose energy faster than an opponent and at the same time out-turn them (Boyd, 18). This introduced the concept of fast transients - operating at a faster tempo.

A fast transient aircraft can control engagements by forcing the overshoot of an attacker or stay inside (the turn circle of) a hard turning defender. A fast transient aircraft operates at a faster tempo than the opponent and thus operating inside their time scales. Boyd stating that in doing so, the fighter would appear ambiguous to the opponent, creating confusion and disorder (Boyd, 19). Just like executing a fake out in football to where the opponent shifts the wrong direction long enough to create a gap to exploit for extra yardage or a touchdown.

The footballer is able to fake out the opponent because they have generated rapid change, this causes the opponent to be unable to adapt fast enough to the change. In essence, the footballer is operating at faster time scales and thus faster decision cycles than the opponent. The opponent observes the footballer’s movements, and these observations are distorted leading to an uncertainty in their orientation (of the situation), the opponent decides (slower than the footballer) and reacts incorrectly. The more I think on this concept, the more it drives me to look into the concepts of tempo and rhythm.

In a way, the footballer has created time-deformation.

Tempo originates from a fencing term meaning the timing of an attack (Lindberg, Stevenson, 1787). Tempo could be thought of as rhythm, but that would be an oversimplification. Rhythm is a recuring sequenece of actions. Tempo then is the timing of those recurring actions.

Miyamoto Musashi wrote about rhythm and its core role in sword fighting (the 16th century equivalent to light saber duels). He spoke of understanding rhythm and understanding the rhythm of the opponent in such a way as to not be drawn into their rhythm (Musashi, 1992, 25). Musashi believed victory belonged to the warrior who understood the opponent’s rhythm and who could attack with a rhythm the opponent would not expect. Essentially operating at a different rhythm, faster rhythm – faster time scales, faster decision cycles. I should expand on this for a moment.

I have read a lot of Musashi. I have read his Book of Five Rings multiple times, I have three different translations of the text. I have also taken the time to read some of his other texts such as “Heidōkyō” or “Heihō-Kakitsuke”. I read Eiji Yoshikawa’s epic novel, Musashi, and watched The Samurai Trilogy starring Toshiro Mifune in the role of Musashi. While these are both works of fiction I believe Yoshikawa as writer and Toshiro as actor - channeled the essense of Musashi. Musashi was a synthesis of different arts, perfectionist in his technique, he sought out opponents to practice and refine and learn. A common theme in his writings is a focus on cadence or rhythm - rhythm of one’s movements, rhythm of the opponent’s movements and ultimately an ability to exploit rhythm toward the advantage in duels. Musashi’s teachings are specific to his Hyoho Niten Ichi Ryu and may seem slightly ambiguous in meaning and searching for applicability outside of this school. Colonel Boyd’s research was specific to aerial combat and fighter aircraft, technical and specific in a narrow band of application. Yet, Musashi and Boyd were very similar to each to other when drilling down the essense of what they both were trying to say. I believe that as such, both have applicability outside their original avenues of application.

Compare the two statements below, the first is a synthesis of Boyd and the second of Musashi:

The suppression or distortion of the opponent’s observations, change direction and speed faster, leaving them in an uncertain, ambiguous orientation, driving them to make slow decisions, and thus under or overreact.

Keep the opponent off their footing, leading to disorientation, take initiative, perform unexpected actions, leaving them in fear and confusion, breaking their rhythm, now out of sync, they are making slower decisions, where one can strike before the opponent can react.

Both concepts possess a self-similarity almost like a self-similar fractal. Once you start looking for self-similarity you start to see it in unusual places. It is almost as if everything was programmed by design to be that way; however, the immediate point is that Musashi and Boyd share a common framework when thinking and acting in contests where perceptions influence decision cycles (to be read as ‘all contests’). While I think Mandelbrot was speaking of time-deformation as it relates to the modeling of asset returns with the shortening or lengthening of the time axis, I think Musashi and Boyd, indirectly (or unintentionally) were speaking of creating time-deformations through one’s movements and the rhythm of that movement in relation to an opponent.

The concept of multi-fractals, where an object has more than one scaling ratio, creates variability. This variability leads to time-deformation in the fourth dimension when working with multi-fractals that have a time axis. That time-deformation influences the psychological time component leading to perception filters that create distortions in time influencing decision cycles. This connection from fractals, to the fourth dimension and ultimately to decision cycles is a connection George likely would have made if not for the Portuguese waitress.

George is holding a copy of *Chaos and Fractals*George is reading ‘Chaos and Fractals’

Where do we go from here? Clearly, George would have realized that if he was ever in a lightsaber duel with a twelve-year-old that operating at a faster tempo would yield a decision cycle superior to his opponent. Yet, George would have bored his friends by explaining the deeper lesson of the importance of guarding against things that mess with our perceptions.

Still where do we go from here? This concept interests and troubles me deeply. I think it is the key, but as of yet have not been able to articulate it (thus, the effort to write about this concept). This research into fractals, self-similarity has led me into thinking about symmetry and asymmetry which I will explore in Part 2.


Ackerman, Andy, dir. 1996. Seinfeld, Season 8, episode 9, “The Abstinence,” Aired November 21, 1996, NBC video.

Boyd, Colonel John. 1976 “New Conception for Air to Air Combat.”

Jürgens, Hartmut, Peitgen, Heinz-Otto, and Saupe, Dietmar. 1992. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag.

Lindberg, Christine, Stevenson, Angus, ed. 2010. The New Oxford American Dictionary, 3rd ed. New York: Oxford University Press.

Mandelbrot, Benoit. 2006. The (Mis) Behavior of Markets. New York: Basic Books.

Musashi, Miyamoto. 1992. The Book of Five Rings, trans. Bradford J. Brown, Yuko Kashiwagi, William H. Barrett, and Eisuke Sasagawa. New York: Bantam Books.

Poincaré, Henri. 2001. The Value of Science: Essential Writings of Henri Poincaré. Edited by Stephen Jay Gould. New York: The Modern Library.


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