Circle Maps / Process Geometry:

event symmetry with time-achronicity

a synopsis by Christopher Walsh


Introduction   |   Developmental Theory

Contemporary Research   |   Mystical Origins

Technical Remarks   |   Further developments

Circle Map Sine Wave / Process Geometry, video


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Circle Maps and the general forms of Process Geometry are tools for studying the symmetry of processes, by converting magnitudes in time-series into angles of chords within fixed or variable radius polar coordinates.

A moment is a point on the circumference, and consecutive events are represented by a chord between the given points. A process is represented by the aggregate geometry of chords. An event within a process is a continuous envelope of intersecting edges, and more generally, clustered intersections. The complete graph reveals symmetry and rhythm of events.

It is like an oscilloscope, but instead of phase and frequency, the process geometry maps changes in magnitude as angles for a given window of time.

  1. The same magnitudes of change at different points in a sequence will result in symmetrical angles, separated by degrees proportional to their absolute magnitudes within the normalized range of the Circle Map function: Signal(min,max)→Circle(0,360).
    1. Premise: Rotation of a Circle Map is equivalent to Cartesian addition, f(x)= signal+c.
      1. The distances between successive magnitudes of the signal are independent of their absolute magnitudes, as in Euclidean distance, (y.2-y.1) = d.
      2. Therefore, if we add some constant to the signal, the set of distances between heights of the sampled histogram will remain unchanged.
      3. Alternatively, since each point on the circle is a monotonically increasing degree up to the modulus, addition moves every point by some constant in the same direction as the increase of degrees.
    2. Implication: Equal changes of magnitude in a sequence will map to symmetrical angles, separated in degrees by their absolute magnitudes, according to the map Signal(min,max)→Circle(0,360). This demonstrates the basis for interpreting the process structure of a signal.
  2. Evidently, the map is achronical, or eternal, since we can consider an infinite duration without effecting the geometry of the circle graph.
    1. Premise:
      1. Given a ‘window of time’, or a sequence of a fixed length, our concern is the measurement of the minimum and maximum magnitudes which occur in the window.
      2. For a random variable, the Signal(min,max) is independent of the length of the sequence.
    2. Exception:
      1. If the variable is convergent or divergent in proportion to the time interval, as in a population explosion, points on the circumference will become regionalized by the time of occurrence, which will skew angles.
        1. Since by definition, the magnitudes are time-dependent, time will become causally evident.
      2. In this case, we might choose to renormalize the signal:
        1. to a mean height, to focus on the local sequential changes.
        2. Or, superimpose sections of the sequence with similar relative ranges.
        3. finally, we could remap Signal→Sine(signal).
          1. Sine maps have a periodic, variable rate of change, which results in a particular geometric structure*
          2. This consistency is a medium for the signal; in effect, the Sine generates a field that is transformed by the signal, so that one can analyze the structural disturbances, as in optical diffraction.

Further, the many categories of sequence, whether it be linguistic, mathematical, or natural data sources, become interrelated by their common sequential structure. All processes can be understood in terms of a set of angular changes, which naturally fits within some fraction of degrees in the circle.

This means that we can measure the relative order of unordered sets, or subjectively ordered sets, such as King Wen’s arrangment of I Ching hexagrams, or dictionary lists of words. Wherein the relative behavior of sequences across phenomena can be visually compared and modulated geometrically. Here it is natural to use the terminology of a circularly oriented graph with a path. The selected order of the circumference vertices becomes a unit process, or the basis for measurement of a range of processes (which are subsets of the unit process).

Intuitively (not generally), I am representing the geometry of events in time. Think of a clock with it’s 12 hours, and when a certain event occurs during the day, perhaps whenever I open a door, I record the time. Later, for each recorded time in the series, I draw a point on the clock, and draw a line between each consecutive point. An angle is formed by the slope of consecutive lines, resulting in a geometry. For instance, if I marked five evenly spaced intervals during one day, it would approximate a pentagram. Since each day keeps winding around the same clock, I could mark points for weeks on end, and eventually a rhythmic structure would reveal itself through the angles formed by consecutive points.

With a circle map, you can watch numbers grow, and reach towards infinity by combining, multiplying. The interaction of prime and composite numbers oscillates between randomness and perfect order. In between the extremes is a dynamic interaction of symmetry that speaks to the complexities of life.

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The Developmental Basis of Process Geometry:

The circle is a means to understand time and process, because the world literally revolves; we experience each day as if it repeats a circular process, with more or less change. An impression of patterns emerges from our experience of interweaving of processes. These experiences lead to a circle-symbology, which is historically at the heart of our understanding of the world; i.e., the 12-hour clock, Western chromatic musical scales, and occult sigils such as the Pentagram, Enneagram, or the Star of David.

With a pragmatic philosophy, it is enlightening to mathematically generalize such mystical impressions; e.g., from repetition, to circularity, to tables of sines. Indeed, our vague impressions and mythic symbols can lead to quantitative models with expressive power. Symbols are persistent, and through empirical method, the symbol and its semantic domain can evolve into reality fitting explanations. For instance, the study of geometry leads to the consideration of time and perception, to the threshold of physics, as in the conclusion of Bernhard Riemann’s “Habilitation Dissertation“, (video summary):

The answer to these questions can only be reached by starting from the conception of phenomena which has hitherto been justified by experience, and which Newton assumed as a foundation, and by making in this conception the successive changes required by facts which it cannot explain. Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.

This leads us into the domain of another science, that of physics, into which the object of today’s proceedings does not allow us to enter.

In the case of Process Geometries, we ask: what remains fundamentally true of the mystic conceptions of the circular-geometric model of time processes? At first, the shift from specific geometries to general maps. Simplification follows: one can reduce the complexity of the considered processes, in order to develop analytical methods. Thirdly, we shift from time-as-circumference, to a general map of a parent set against a subset. Lastly, we remove considerations of the circle, and enter the domain of graph theory paths; wherein: optimal arrangements of the points can be sought algorithmically (power graphs* , and multi-dimensional scaling*).

But we can still draw from the mystic conception for research (or actively engage in mystical conception for the express purposes of research impetus (this can quickly shift into a pattern-matching game of mathematics with idealized rules)): 

Originally, I conceived of angle symmetry as a means to map states of consciousness during experimental drawing tests which utilize unicursal line. Since I can encode a continuous sequence of feelings as differentiations of angular direction in the plane, to varying degrees of magnitude. Each drawing becomes a geometric map of the process, in this sense. But here the line between mystical thought, art, and mathematical pattern matching starts to blend.

A divergent note regarding developmentalism: whether the evolution of impressions to symbology, to quantitative prediction is convergent across cultures, or whether the bifurcations of ontologies would lead to unique technologies, as concerns Michel Serres in “The Birth of Physics“. One thing is certain: increases in communication tend to merge ontologies, which also corresponds with scientific progress, as the human collective trades in the broad range of its experiences to solve and create. (See: “I, Pencil”, for an application of this idea in free market theory.)




Contemporary Research:




Mystical Origins


The broad notion of Process Geometry begins with Gurdjieff’s teachings of the ‘Fourth Way Enneagram’. He attempts to convey the notion of living symbolism, or symbolism in motion through a Circle Map of the digital sequence .142875…, derived from the base 10 equation 1/7. Specifically, he devised geometric dance compositions as means for living contemplation of a process (film excerpt from “Meetings with Remarkable Men”).

Time passes along the circle circumference. Each point in the sequence has ‘insight’ into the neighboring points connected by edges, forwards and backwards in time.

The base 10 counting process creates a group structure of thirds, 147, 258, 369. The {3,6,9} set governs the underlying geometry of the enneagram sequence: that a process has a beginning {3}, an ending {6}, and transcendence {9} (or octave).

Regarding our discussion of the Developmental Theory:*

The conceptual reduction from a dogmatic choice of geometry, to the consideration of geometries of process in general, naturally leads us to measure and map the geometry of physical processes. The underlying significance of the map remains, as well as the intuitive conception of eternity, or circular time. The interaction of consciousness and the symbol has attained an octave, nevertheless!

While it is tempting to consider one geometry over another as representing a general type of process, it is much more exciting to map diverse data sources, and empirically research their taxonomy.

Natal Chart geometries lead to the generalization of superimposed Circle Maps: where the structural differences of two signals are independently compared.

The categories of Circle Map dynamics in astrological natal charts are:

{ Sextile, conjunct, trine, square, and opposite }

Astrologers qualitatively compare the sets of chord angles that result from a function of dates of birth and astronomical phase. For instance, two natal charts that result in a conjunct set of angles would be complimentary.


Vortex Based Mathematics is a model of oscillation based on the finite group structures of multiplication modulo 9.  Base 10 integers are defined as energy levels, and multiplication is a motion of the energy in a torus geometry. The multiplication sequences are tiled in a rhombic grid, which is isomorphic to a torus. A negative-slope diagonal motion is an iteration of multiplications, while a positive-slope diagonal motion sums to the ‘Fibonacci Nexus’ sequence (source), resulting in a vortex motion of the torus. Viz., Fibonacci numbers divisible by 144 occur in intervals of 12. After division by 144, and the modulo 9 is taken, it forms the “nexus key”: 743883479256116529. In general, for k=1→∞, F(k*N) is divisible by F(N). It turns out, beginning with N=4, every N+(k*8) generates the Nexus. Lucas numbers have a similar rule for N+(k*4).

While the model of VBM lacks certain empirical tests to justify its energy hypothesis, as well as its choice of modulo 9, it does display an elegant weave of patterns. Extending this approach, I am able to implement an ‘applied number theory’, utilizing the idiosyncrasies of modular group structures: e.g.,

Here I am modulating one sequence with another, to show a binary switching effect:



Some Technical Discussion:

Calculating Maps:

The category of techniques for mapping magnitude to angle includes: modular arithmetic, by dividing the circle into N degrees for functions modulo N; scalar maps that renormalize the range of a signal between (0,1)=(0,360) degrees; and Sine functions, which bound the input of sine(t) to the range (0,1). (Note: the Sine() rate of change is not constant, and therefore has unique geometric properties in Circle Maps.*).

General Maps:

A set P that maps to the circumference set Q must be some subset of Q; however, Q does not need to be ordered: the relative order of sets can be measured empirically. Thus, other notable approaches include syntactic processes*, and more interestingly, conceptual processes utilizing ontology induction*.

Time and Sample:

(Time-window vectors)

(Sample-rate summations)

(entropy calculations for differential of sample-rate interval)

(Sampling frequency proportional to derivatives of smooth signal), (intervals create ‘derivative’ graph)

(Groups of higher order derivatives)

(Color injects time into the achronical map)


The modulus is an upper bound of a sequence. When the proportion of the modulus to the signal is one (M : S = 1), where S = max(signal)-min(signal), there is a one-to-one correspondence between points on the circumference and points in the signal.

However, when the modulus is less than the range of the signal, the remainder, R, = (signal.samples > S), are equivalent with existing points on the circumference. Therefore, the circle map becomes a probability map with probability of a point proportional to the ratio of the modulus to the signal.

N-Sphere Maps:

A Circle Map consists of the 1-Sphere, while general Process Geometry can involve N-Sphere rotation, whether a quantity of N signals are combined, or differentiated in higher dimensions of angular cycles.* These techniques define a spherical geometric space, which maintains the time-achronicity of the 1-sphere “circle map”.




Further developments:

Drafting plotter Fine Art prints: (commissions and selected prints are available*)

In these drafting plotter prints, a special dilution of ink lets pigment aggregate along the envelopes of curves. This presents the essential geometric features of a process:

Dimensional extensions based on N-Sphere rotation and projection:

The following sketches define a spherical geometric space for signals, which maintains the time-achronicity of the 1-sphere circle map. Each dimension of rotation could map a signal in the same manner as a 3-d attractor.

The next tweet links to the work of Ralph Abraham’s Visual Math Institute, for an example of a geometric attractor:

His work illustrates the measurement of geometric shape from a chaotic, continuous signal. The question is how to extract a discrete measurement with minimal entropy from such a signal source. In a general sense it is an ‘invariant probability measure’ of the time-evolution (David Ruelle, “Chaotic Evolution and Strange Attractors“, ch.7).