# Asymmetry, Symmetry and Beauty

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

_{1}, A

_{2}, …, A

_{N}, we construct the sequences (Euclidean vectors) y

_{i}of N successive members of a time series [y

_{i}= (A

_{i}, A

_{i+1}, A

_{i+2}, ..., A

_{t+N})] starting with each data point A

_{i}:. This is referred to as embedding in dimension N. Recurrence isometries are calculated by comparing the Euclidean norms of these vectors, and if the difference between them is less than a chosen cutoff radius (1%), a recurrence is plotted and counted. We have made calculations with many cutoff radii, anging from 0.1 to 50 % and the reults are similar. The quantification of recurrences at low and high embeddings allows one to consider both simple and complex patterns. Recurrence plots graph isometries as a function of time. Recurrences are colored according to their distance from red to violet. Both types of recurrence generate the same pattern in recurrence plots, but the number of isometries increases and the number of similarity recurrences decreases with the length of the vector. Isometry is the number of isometric recurrences as a percentage of the total number of possible recurrences in the sample (N x N / 2). The number of isometries (as % of the total number of possible recurrences) is computed for the original data and for a copy of the data randomized by shuffling. Novelty is defined as the increase in recurrence isometry produced by shuffling the data [29,30], and is quantified as the ratio of isometries after shuffling over isometries in the original data. Novelty is a key measure to distinguish creative from non-creative processes. Novelty is demonstrable for recurrence isometry but not for similarity recurrence. Embedding plots present the value of novelty computed with 2, 3, …, 100 embeddings.

_{i}-mean)

^{2}/N) of the series was recorded as a function of time. Series converging to an attractor display a decrease in S.D. with increasingly larger samples. In contrast, diversification is the increase in variance of a time series with an increase of the size of the sample (global diversification) or with increasing the embedding (local diversification): the SD is computed for sets (“embeddings”) of 2, 3, …, 100 consecutive terms of the time series, starting with each term in the series. The values obtained for each embedding are averaged for the entire series, and these averages are plotted as a function of the number of embeddings.

**H = s + d* log**

_{2}n## 3. Results

#### 3.1. Asymmetry and Symmetry

#### 3.2. Entropy

#### 3.3. Power Spectrum Analysis

#### 3.4. Biotic Patterns in Music and Poetry

## 4. Discussion

_{t+1}= A

_{t}+ k*t* sin(A

_{t}) and the diversifying equation A

_{t+1}= A

_{t}+ sin(k*t*A

_{t}) generate a sequence of patterns as time t increases: equilibrium, periodicities, chaos, bios and leaps (figure 2).

_{t}). These three elements correspond to the defining properties of the three mother structures of mathematics (Bourbaki): lattice asymmetry, group opposition, and topological continuous transformation of spatial form. These three elementary forms are also found by Piaget in fundamental cognitive operations, in the three dimensions of the Central Nervous System [4] and in many other physical, biological and human processes as illustrated by Table 4. Some of these may be mere analogies (as between the wing of a bird and the wing of an airplane) but we as hypotheses to be tested that many of them are homologies (as between the leg of a horse and the wing of a bird). We thus conceived the idea that these three forms – unidirectional flow of energy in asymmetric time (action), bipolar opposition such as the bidirectional communication of information by bidimensional and bipolar electromagnetic radiation, and tripolar, continuous transformation of tridimensional matter-- constitute basic principles of nature [4]. This is Process theory (Table 4).

**uni**directional gravitation,

**bi**polar and

**bi**dimensional electromagnetic force, and

**tri**-polar nuclear forces that unite 3 quarks to form protons and neutrons and thereby construct

**tri**dimensional atomic nuclei. More generally, action (energy * time) is asymmetric, information requires the distinction between two opposite entities, and matter is tridimensional.

## Acknowledgements

## References

- Anderson, P.W. More Is Different. Science
**1972**, 177, 393–396. [Google Scholar] [CrossRef] [PubMed] - Bourbaki, N. Éléments de mathématique; Actualités Scientifiques et Industrielles: Paris, France, 1952. [Google Scholar]
- Beth, E.W.; Piaget, J. Epistémologie mathématique et psychologie; Essai sur les relations entre la logique formelle et la pensée réelle: Paris, France, 1961. [Google Scholar]
- Sabelli, H. Union of Opposites: A Comprehensive Theory of Natural and Human Processes; Brunswick Publishing: Lawrenceville, VA, USA, 1989. [Google Scholar]
- Sabelli, H. Bios: A Study of Creation; World Scientific: Singapore, 2005. [Google Scholar]
- Sabelli, H.; Carlson-Sabelli, L. As simple as one, two, three. Arithmetic: a simple, powerful, natural and dynamic logic. In Sustainable Peace in the World System and the Next Evolution of Human Consciousness, Proceeding of International Systems Society 40th meeting; Hall, M.L.W., Ed.; Louisville, KY, USA, 1996; pp. 543–554. [Google Scholar]
- Sabelli, H.; Kovacevic, L. Quantum Bios and Biotic Complexity in the Distribution of Galaxies. Complexity
**2006**, 11, 14–25. [Google Scholar] [CrossRef] - Sabelli, H.; Kovacevic, L. Biotic Expansion of the Universe. In International Conference on Advances in Internet, Processing, Systems, and Interdisciplinary Research; Sveti Stefan, Montenegro, 2003; Electronic Publication IPSI-2003. [Google Scholar]
- Thomas, G.; Sabelli, H.; Kauffman, L.; Kovacevic, L. Biotic patterns in Schrödinger’s equation and the evolution of the universe. InterJournal
**2006**, 1787. [Google Scholar] - Sabelli, H.; Thomas, J.; Kovacevic, L.; Horan, D. Biotic Dynamics of Galactic Distribution, Gravitational Waves, and Quantum Processes. A Causal Theory of Cosmological Evolution. In Black Holes and Galaxy Formation; Wachter, A.D., Propst, R.J., Eds.; Nova Science Publishers: Hauppauge, NY, USA, 2009. [Google Scholar]
- Sabelli, H. Complex Biotic Patterns in LIGO Recordings point to the creativity of gravitational interactions. Complexity
**2010**, 15, 12–24. [Google Scholar] - Carlson-Sabelli, L.; Sabelli, H.; Zbilut, J.; Patel, M.; Messer, J.; Walthall, K.; Tom, C.; Fink, P.; Sugerman, A.; Zdanovics, O. How the heart informs about the brain. A process analysis of the electrocardiogram. In Cybernetics and Systems 94; Trappl, R., Ed.; World Scientific: Singapore, 1994. [Google Scholar]
- Carlson-Sabelli, L.; Sabelli, H.; Patel, M.; Messer, J.; Zbilut, J.; Sugerman, A.; Walthall, K.; Tom, C.; Zdanovics, O. Electropsychocardiography. Illustrating the Application of Process Methods and Chaos Theory to the Comprehensive Evaluation of Coronary Patients. Complex. Chaos Nurs.
**1995**, 2, 16–24. [Google Scholar] - Carlson-Sabelli, L.; Sabelli, H.; Messer, J.; Patel, M.; Sugerman, A.; Kauffman, L.; Walthall, K. Process method: Part I. An empirical measure of novelty differentiates creative organization from static order and chaos. In Systems thinking, globalization of knowledge, and communitarian ethics, Proceeding of International Systems Society; Rhee, Y.P., Bailey, K.D., Eds.; Kwanak Press: Seoul, Korea, 1997; pp. 1072–1090. [Google Scholar]
- Sabelli, H.; Carlson-Sabelli, L.; Patel, M.; Zbilut, J.; Messer, J.; Walthall, K. Psychocardiological portraits: A clinical application of process theory. In Chaos theory in Psychology; Abraham, F.D., Gilgen, A.R., Eds.; Greenwood Publishing Group: Westport, CT, USA, 1995; pp. 107–125. [Google Scholar]
- Sabelli, H.; Carlson-Sabelli, L.; Patel, M.; Sugerman, A. Dynamics and psychodynamics: Process Foundations of Psychology. J. Mind Behav.
**1997**, 18, 305–334. [Google Scholar] - Sabelli, H.; Messer, J.; Kovacevic, L.; Walthall, K. The biotic pattern of heartbeat intervals. Int. J. Cardiol.
**2010**, (in press). [Google Scholar] [CrossRef] - Sabelli, H.; Kovacevic, L. Biotic Complexity of Population Dynamics. Complexity
**2008**, 13, 47–55. [Google Scholar] [CrossRef] - Patel, M.; Sabelli, H. Autocorrelation and Frequency Analysis Differentiate Cardiac And Economic Bios From 1/F Noise. Kybernetes
**2003**, 32, 692–702. [Google Scholar] [CrossRef] - Sabelli, H. Bios, creative organization in economic, biological, and meteorological data. In International Conference on Advances in Internet, Processing, Systems, and Interdisciplinary Research; Sveti Stefan, Montenegro, 2003; Electronic Publication IPSI-2003. [Google Scholar]
- Sabelli, H.; Sugerman, A.; Kauffman, L.; Kovacevic, L.; Carlson-Sabelli, L.; Patel, M.; Messer, J.; Konecki, J.; Walthall, K.; Kane, K. Biotic Patterns in Biological, Economic and Physical Processes. J. Appl. Syst. Stud.
**2004**, 5, 14–26. [Google Scholar] - Sabelli, H. The Biotic Pattern of Prime Numbers. Cybern. Syst. J.
**2008**, (in press). [Google Scholar] - Levy, A.; Alden, D.; Levy, C. Biotic patterns in music. In Society for Chaos Theory in Psychology and Life Sciences Meeting, SCTPLS2006; Johns Hopkins University: Baltimore, MD, USA, 4-6 August 2006. [Google Scholar]
- Sabelli, H. Music, Poetry, Painting, and Bipolar Illness. Nonlinear. Dynam. Psychol. Life Sci.
**2010**, (in press). [Google Scholar] - Kauffman, L.; Sabelli, H. The Process equation. Cybern. Syst.
**1998**, 29, 345–362. [Google Scholar] - Sabelli, H. Complement plots: Analyzing opposites reveals Mandala-like patterns in human heartbeats. Int. J. Gen. Syst.
**2000**, 29, 799–830. [Google Scholar] [CrossRef] - Dekking, M.; Mendès-France, M. Uniform Distribution Modulo One. Journal für die reine und angewandte Mathematik
**1981**, 239, 149–153. [Google Scholar] - Sabelli, H.; Sugerman, A.; Kovacevic, L.; Kauffman, L.; Carlson-Sabelli, L.; Patel, M.; Konecki, J. Bios Data Analyzer. Nonlinear. Dynam. Psychol. Life Sci.
**2005**, 9, 505–538. [Google Scholar] - Sabelli, H. Novelty, a Measure of Creative Organization in Natural and Mathematical Time Series. Nonlinear. Dynam. Psychol. Life Sci.
**2001**, 5, 89–113. [Google Scholar] [CrossRef] - Sabelli, H.; Abouzeid, A. Definition and Empirical Characterization of Creative Processes. Nonlinear. Dynam. Psychol. Life Sci.
**2003**, 7, 35–47. [Google Scholar] [CrossRef] - Sabelli, H.; Patel, M.; Sugerman, A.; Kovacevic, L.; Kauffman, L. Process Entropy, a Multidimensional Measure of Diversity and Symmetry. http://creativebios.net/webjass/10Entropy.pdf.
- Sabelli, H.; Sugerman, A.; Carlson-Sabelli, L.; Patel, M.; Kauffman, L. Embedding Plots: A Tool to Measure Simplicity, Complexity and Creativity. J. Appl. Syst. Stud.
**2004**, 5, 159–201. [Google Scholar] - Sabelli, H.; Carlson-Sabelli, L. Biological Priority and Psychological Supremacy, a New Integrative Paradigm Derived from Process Theory. Am. J. Psychiatry
**1989**, 146, 1541–1551. [Google Scholar] - Torre, C. Chaos, Triadic Theory of Psychological Competence in the Academic Setting. In Chaos Theory in Psychology; Gilgen, A., Abraham, F., Eds.; Praeger/Greenwood Publishing: Westport, CT, USA, 1995; pp. 279–294. [Google Scholar]
- Carlson-Sabelli, L.; Sabelli, H. Phase plane of opposites: A Method to study change in complex processes, and its application to sociodynamics and psychotherapy. Social Dynam.
**1992**, 3, 1–6. [Google Scholar] - Sabelli, H. Biothermodynamics. Open Cybern. Syst. J.
**2009**, (in press). [Google Scholar] - Schrödinger, E. What is Life? The Physical Aspect of the Living Cell; The Macmillan Company: New York, NY, USA, 1945. [Google Scholar]
- Prigogine, I. From Being to Becoming: The New Science of Connectedness; Doubleday: New York, NY, USA, 1987. [Google Scholar]
- Prigogine, I. The End of Certainty; The Free Press: New York, NY, USA, 1997. [Google Scholar]
- Cohen, J.; Steward, I. The Collapse of Chaos; Penguin: New York, NY, USA, 1994. [Google Scholar]
- Leyton, M. Symmetry, Causality, Mind; MIT Press: Cambridge, UK, 1992. [Google Scholar]
- Petitjean, M. Order, entropy and symmetry: An awkward relation? Symmetry Cult. Sci.
**2005**, 16, 5–6. [Google Scholar] - Jakulin, A. Symmetry and information theory. Symmetry Cult. Sci.
**2005**, 16, 7–26. [Google Scholar] - Matsuno, K. Symmetry and Information: Symmetry as an Emergent Property of Information. Symmetry Cult. Sci.
**2005**, 16, 27–36. [Google Scholar] - Smith, A. A hierarchical perspective. Symmetry Cult. Sci.
**2005**, 16, 37–46. [Google Scholar] - Salthe, N. Asymmetry and self-organization. Symmetry Cult. Sci.
**2005**, 16, 71–90. [Google Scholar] - Darvas, G. Order, entropy and symmetry. Symmetry Cult. Sci.
**2005**, 16, 91–108. [Google Scholar] - Lin, S.K. The Nature of the Chemical Process. 1. Symmetry Evolution—Revised Information Theory, Similarity Principle and Ugly Symmetry. Int. J. Mol. Sci.
**2001**, 2, 10–39. [Google Scholar] [CrossRef]

**Figure 1.**Step pyramid archetype. Top: early pyramids in Egypt, Middle East, and Mexico. Bottom: Diagram illustrating how it captures asymmetry, symmetry of opposites, and transformation from simple and larger to a smaller top.

**Figure 2.**Process equation. The sequence of patterns in steps of increasing complexity in the time series generated by the equation as g increases (logarithmic scale).

**Figure 3.**

**Entropy-bin plot.**Plot of informational entropy as defined by Shannon. It shows the calculation of the degree of symmetry (entropy at 2 bins = 1 for perfect symmetry and less than 1 for asymmetry) and diversity (slope). Many natural and human processes are highly asymmetric. Biotic series and random walks have a small degree of asymmetry, while chaos, random, and periodic series are symmetric.

**Figure 4.**Histogram showing asymmetry in the statistical distribution of letters in poetry (Hugo’s A Sunset and Wordsworth’s Lines Written as a School Exercise at Hawkshead) and musical notes (Georges Bizet’s “Habanera” from Carmen and Bach’s Prelude). Units: the number of occurences are plotted in the y-axis.

**Figure 5.**Pareto histogram and logarithmic trendline (except for random walk, where a linear trendline fits).

**Figure 8.**Complement plots (left) and trigonometric walks (right) of musical compositions and of poems.

**Figure 9.**Entropy of musical compositions and of literary texts as a function of the number of bins. The musical compositions analyzed include: Eduardo di Capua’s O Sole Mio, Georges Bizet’s Habanera from Carmen, Bach’s Crab Canon, Mozart’s Fur Elise, the American Spiritual Swing Low Sweet Chariot, and Chopin’s Ballade no. 4. The poems are: Wordsworth’s Lines Written as a School Exercise at Hawkshead, Pushkin’s Eugene Onegin, Ginsberg’s Howl, Hugo’s A Sunset, Mandelstam’s Tristia, and Goethe’s To Luna.

**Figure 10.**Quantification of isometry as a function of the number of embeddings in musical compositions and literary texts as compared with their randomized copy (blue line).

**Figure 11.**Recurrence plots of musical compositions show temporal complexity. The quantification of isometries shows novelty and the quantification of consecutive isometries supports non-random causation.

**Figure 12.**Series of differences between consecutive terms Pattern in the recurrence plots and the quantification of isometry and of consecutive isometries in the show non-random causation.

Series | Max | Min | Middle | Mean | Median | Mean - median | Middle - median | Middle– mean | Cosine/sine average |
---|---|---|---|---|---|---|---|---|---|

Random | 9.97 | 0.02 | 5.00 | 4.91 | 4.94 | -0.03 | 0.039 | 0.065 | 4.91 |

Random Walk | 5.97 | -5.00 | 5.49 | 0.33 | 0.16 | 0.16 | 5.32 | 5.159 | 0.33 |

Chaos | 5.79 | 0.49 | 2.65 | 3.16 | 3.10 | 0.06 | -0.45 | -0.50 | 3.16 |

Bios | 17 | -43.87 | 30.43 | -14.02 | -13.02 | -1.00 | 43.45 | 44.45 | -14.02 |

Prime Numbers | 7919 | 2 | 3959 | 3683 | 3576 | 106.91 | 382.5 | 275.59 | -0.004 |

Music | |||||||||

Bach: Prelude | 63 | 16 | 23.5 | 40.52 | 40 | 0.522 | -16.5 | -17.02 | 40.52 |

Bach: Fantasia | 59 | 11 | 24 | 39.2 | 41 | -1.80 | -17 | -15.20 | 39.20 |

Freeman: Affair in San Miguel | 59 | 8 | 25.5 | 36.33 | 42 | -5.67 | -16.5 | -10.83 | 36.33 |

Duke: April in Paris | 59 | 3 | 28 | 37.47 | 36.5 | 0.97 | -8.5 | -9.47 | 37.47 |

Poetry | |||||||||

Whitman: A Sight in Camp | 121 | 32 | 44.5 | 92.3 | 104 | -11.70 | -59.5 | -47.80 | 92.30 |

Rumi: Descent | 122 | 32 | 45 | 90.59 | 104 | -13.41 | -59 | -45.59 | 90.59 |

Neruda: Puedo escribir | 122 | 32 | 45 | 92.16 | 105 | -12.84 | -60 | -47.16 | 92.16 |

Fierro: Aquí me pongo a cantar | 243 | 32 | 105.5 | 96.12 | 105 | -8.88 | 0.5 | 9.38 | 96.12 |

Zorrilla: Don Juan Tenorio | 233 | 32 | 100.5 | 87.1 | 101 | -13.90 | -0.5 | 13.4 | 87.10 |

Series | Entropy at 16 bins |
---|---|

Music | |

Debussy, Études | 3.20296 |

Di Capua, O Sole Mio | 2.15 |

Beethoven, Für Elise | 2.07 |

Chopin, Ballade No. 4 in F | 3.54696 |

Denver, Country Roads | 3.638 |

Bach, Fantasia | 3.65822 |

Poetry | |

Hugo, A sunset | 1.75256 |

Pushkin, Eugene Onegin book I & II | 1.8449 |

Goethe, To Luna | 2.99606 |

Mandelstam, Tristia | 2.94824 |

Blake, On Another’s Sorrow | 2.62664 |

Whitman, A Sight in Camp | 2.67581 |

Mathematical series | |

Random | 3.57727 |

Random walk | 3.54639 |

Linearly increasing numbers | 4 |

Leap (process equation, g = 2*п) | 4 |

Sine wave | 3.78594 |

Devil staircase | 3.17955 |

Lorenz chaos | 3.83145 |

Rossler chaos | 3.94242 |

Process equation chaos A _{t+1} = A_{t} + 4.3 * sin(A_{t}) | 3.76694 |

Bios: Process equation A _{t+1} = A_{t} + 4.65 * sin(A_{t}) | 3.62817 |

Bios: Sum of 4 sine waves | 3.75031 |

Bios: Sum of 3 sine waves | 3.93651 |

Natural and social processes | |

Distance between galaxies | 3.96239 |

Schrödinger’s equation | 3.64343 |

El Niño | 3.66817 |

Air Temperatures | 3.23883 |

Clear Water River | 2.48224 |

Earthquakes | 2.37551 |

MSN Temperature | 3.67427 |

Heartbeat intervals | 3.50399 |

Electroencephalogram | 3.19064 |

Human DNA | 3.67782 |

Human RND | 3.96081 |

France population | 3.51594 |

Daily bond yield DAAA | 3.62622 |

Model time series | Slope + standard error | Model time series | Slope |
---|---|---|---|

Random | 0 | Random walk | 0 |

Process Chaos | 0.06 | Logistic chaos | -0.28 |

Bios | -1.76 | Sum of sine waves | -3.64+ 0.02 |

Pink noise | -1.00 | Brownian noise | -2.11 |

Physical processes | Slope + standard error | Human processes | Slope + standard error |

Quantum | -4.08 + 0.11 | Heartbeat intervals | -1.4943 + 0.03 |

Gravitational waves | -0.27 + 0.04 | Economic | -0.1396 + .03 |

Music | Slope + standard error | Poetry | Slope + standard error |

Piano Sonata | -0.33 + 0.06 | Ginsberg: Howl | 0.4087 + 0.04 |

April in Paris | -0.49 + 0.06 | Wordsworth | -0.1182 + 0.04 |

Fantasia | -0.29 + 0.06 | Hugo | 0.3746 + 0.04 |

Chopin Ballad | -0.151 + 0.06 | Mandelstam: Tristia | 0.065 + 0.12 |

Asymmetry | Dyadic asymmetry and symmetry | Triadic asymmetry and symmetry | |
---|---|---|---|

Integers | 1 | 2 | 3 |

Examples of numerical form | Uni-directional time | Bi-polar and bi-dimensional electro-magnetic charge | Tri-polar nuclear forces |

Numerical archetypes | Divine proportion φ = 1.618… | Perfect circular symmetry: Π = 3.1415… | e = 2.71828 |

Form | Asymmetry < | Opposition, asymmetric and symmetric | Triadicity |

Bourbaki’s structures | Order: lattices | Group opposition and other algebras | Structure: 3D geometry |

Physics | Action: energy flow in time | Communication (two-valued information: opposites) | Tri-dimensional matter |

Physical forces | Gravitation (attraction) | Electromagnetic (attraction and repulsion) | Strong nuclear force creates structure. Weak nuclear force creates asymmetry. |

Light | Linear propagation | Sine wave | Color Spatial radiation |

Schrodinger equation | Energy, total E and potential U | Sine and cos. if E> U; sinh and cosh if E < U | Momentum = mass * velocity |

Einstein equation | E | c^{2} | m |

Chemistry | Asymmetric molecules and biomolecules (Pasteur) | Covalent and | Asymmetric carbons |

Biology | Metabolism | Anabolism and catabolism. Sexuality (Linnaeus). | Organisms. Mother, father and child. |

Evolutionary theory | Life, survival, evolution | Competition and (Darwin) and mutual aid (Kropotkin) | Multicellularity, symbiosis, mutual aid, sociality |

Central Nervous System | Asymmetric dorsal- sensory to ventral motor axis | Quasi-symmetric right-left axis | Vertical axis from simple to complex |

Physiology, Medicine | Homeostasis: equilibrium. | Sympathetic and parasympathetic. Periodicity. | Anatomy. |

Social groups | Age, generations | Sexes. Parent / child. Master / slave. | Upper, middle and lower classes |

Social aims/organization | Health of persons and environment | Sexes. Class cooperation (socialism) and competition (Smith) and struggle (Marx) | Executive, Legislative and Judicial powers |

Economics | Consumption | Distribution, trade | Production |

Psychology | Flux (James). Action (Moreno) | Conflict (Freud). Fight or flight (Cannon). Role reversal (Moreno). | Id, ego, superego (Freud). Creativity: (Moreno). Conflict behaviors and emotions [4]. |

Cognition [3] | Mental operations regarding order | Mental operations regarding classes | Mental operations regarding space |

Education [34] | Pragmatic | Emotional | Cognitive and artistic |

Methodology | Priority of the simple and objective [6, 33]. | Analyze synergy and conflict of opposites | Supremacy of the complex or the subjective [6, 33]. |

Research techniques | Time series | Phase plane of opposites [35], trigonometric analysis | N-dimensional recurrence |

Dynamics | Asymmetric factor of catastrophes | Bifurcation. Bifurcating factor of catastrophes | Form and transformation. Complex patterns: fractal, chaos, bios. |

Process philosophy | Process (not isolated events or stability) | Dialectic: coexisting, interacting, opposites. | Material embodiment. Creative synthesis of opposites and triads. Supremacy of complex |

Logic / Biotic logic | Implication. Evolving concepts. | Mutual exclusion (Aristotle, Boole) and implication (Sabelli) of opposites. | Triadic categories (Hegel, Pierce) and operations. |

Process equation | Recursion: A(t + 1) | Sin(A(t)) | Sequence of patterns: convergence, periodicity, chaos, bios |

Music | Rhythm, tempo | Melody | Harmony |

© 2010 by the authors; licensee MDPI, Basel, Switzerland. This article is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Sabelli, H.; Lawandow, A.; Kopra, A.R.
Asymmetry, Symmetry and Beauty. *Symmetry* **2010**, *2*, 1591-1624.
https://doi.org/10.3390/sym2031591

**AMA Style**

Sabelli H, Lawandow A, Kopra AR.
Asymmetry, Symmetry and Beauty. *Symmetry*. 2010; 2(3):1591-1624.
https://doi.org/10.3390/sym2031591

**Chicago/Turabian Style**

Sabelli, Hector, Atoor Lawandow, and Abbe R. Kopra.
2010. "Asymmetry, Symmetry and Beauty" *Symmetry* 2, no. 3: 1591-1624.
https://doi.org/10.3390/sym2031591