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Introduction Symmetry is pervasive in both natural and man-made environments [1-7]. Humans have an innate ability to perceive and take advantage of symmetry [8] in everyday life, but it is not obvious how to automate this powerful insight. The introduction of computers poses challenging tasks for machine representation and reasoning about symmetry and group theory. I make continuous efforts to develop computational tools for dealing with symmetry in various applications using computers [9-11,21,34,35]. This chapter gives a sampler of an emerging area of research and applications, namely computational symmetry. Computational symmetry refers to the practice of representing, detecting and reasoning about symmetries on computers. The reasons to care about computational symmetry in computer science are many-fold: (i) symmetry exists everywhere; (ii) symmetry is intellectually stimulating; (iii) symmetry implies a structure that can be either help]l or harm/}d in applications; (iv) machine

Content uploaded by Yanxi Liu

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All content in this area was uploaded by Yanxi Liu on Jun 04, 2013

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... Given the evidence of the powerful role of symmetry in the history of the natural sciences, we hypothesize that computational symmetry, defined by the first author as using computers to model, analyze, synthesize and manipulate symmetries in digital forms, imagery or otherwise [152], will likewise play a crucial role in the advancement of our understanding in artificial/machine intelligence. ...

... Symmetry, or deviation from symmetry, has been a fascinating topic across many different fields including science, engineering and the arts. Detection of symmetry, as either a pure intellectual pursuit of programmable perception or as a practical tool for data compression, has also fascinated computer scientists for decades, leading to the effort that we call Computational Symmetry [152]. In this survey, we have provided the first, up-to-date (albeit partial) account of research activities in the emerging field of Computational Symmetry in both computer vision and computer graphics. ...

... In computer, computational symmetry is defined as using computers to model, analyze, synthesize, and manipulate symmetries in digital forms, imagery or otherwise [32]. Here, we use computational symmetry to simplify the storage length and then give computational complexity of mathematical expressions. ...

The complexity of a mathematical expression is a measure that can be used to compare the expression with other mathematical expressions and judge which one is simpler. In the paper, we analyze three effect factors for the complexity of a mathematical expression: representational length, computational time, and intelligibility. Mainly, the paper introduces a binary-lambda-calculus based calculation method for representational complexity and a rule based calculation method for algebraic computation complexity. In the process of calculating the representation complexity of mathematical expressions, we transform the de bruijn notation into the binary lambda calculus of mathematical expressions that is inspired by compressing symmetry strings in Kolmogorov complexity theorem. Furthermore, the application of complexity of mathematical expressions in MACP, a mathematics answer checking protocol, is also addressed. MACP can be used in a computer aided assessment system in order to compute correct answers, verify equivalence of expressions, check user answers whether in a simplification form, and give automatic partial grades.

... Symmetry has been of enduring interest in computer vision because it is an important characteristic that profoundly defines the structure, function, and even aesthetics of objects in the real world [4]. Representing or understanding the real world in vision can be made more concise if we can capture the symmetry in the scene. ...

... The real world is full of approximate symmetries appearing in varied modalities, forms and scales. From insects to mammals, intelligent beings have illustrated effective recognition skills and smart behaviors in response to symmetries in the wild [9,15,38,48], while computer vi- * Contributed equally, order chosen alphabetically sion algorithms under the general realm of computational symmetry [24] are still lagging behind [13,25]. ...

Motivated by various new applications of computational symmetry in computer vision and
in an effort to advance machine perception of symmetry in the wild, we organize the third international symmetry detection challenge at ICCV 2017, after the CVPR 2011/2013 symmetry detection competitions.
Our goal is to gauge the progress in computational symmetry with continuous benchmarking of both new algorithms and datasets, as well as more polished validation methodology.
Different from previous years, this time we expand our training/testing data sets to include 3D data, and establish the most comprehensive and largest annotated datasets for symmetry detection to date;
we also expand the types of symmetries to include densely-distributed and medial-axis-like symmetries; furthermore, we establish a challenge-and-paper dual track mechanism where both algorithms and articles on symmetry-related research are solicited.
In this report, we provide a detailed summary of our evaluation methodology for each type of symmetry detection algorithm validated.
We demonstrate and analyze quantified detection results in terms of precision-recall curves and F-measures for all algorithms evaluated.
We also offer a short survey of the paper-track submissions accepted for our 2017 symmetry challenge.

... However, developing computational methods which generate symmetrical patterns is still a challenge since it has to connect abstract mathematics with the noisy, imperfect, real-world; and few computational tools exist for dealing with real-world symmetries (Liu 2002). Applying evolutionary algorithms to produce symmetrical forms leaves the formulation of fitness functions, which generate and select symmetrical phenotypes, to be addressed. ...

Since the introduction of cellular automata in the late 1940s they have been used to address various types of problems in computer science and other multidisciplinary fields. Their generative capabilities have been used for simulating and modelling various natural, physical and chemical phenomena. Besides these applications, the lattice grid of cellular automata has been providing a by-product interface to generate graphical contents for digital art creation. One important aspect of cellular automata is symmetry, detecting of which is often a difficult task and computationally expensive. In this paper a swarm intelligence algorithm—Stochastic Diffusion Search—is proposed as a tool to identify points of symmetry in the cellular automata-generated patterns.

... Despite the above developing computational methods that generate symmetrical patterns is still a challenge; this is partially due to the difficulty of associating mathematics with the noisy, imperfect, real world, resulting in a small number of computational tools dealing with real-world symmetries [24]. ...

The concepts of order and complexity and their quantitative evaluation have been at the core of computational notion of aesthetics. One of the major challenges is conforming human intuitive perception and what we perceive as aesthetically pleasing with the output of a computational model. Informational theories of aesthetics have taken advantage of entropy in measuring order and complexity of stimuli in relation to their aesthetic value. However entropy fails to discriminate structurally different patterns in a 2D plane. In this work, following an overview on symmetry and its significance in the domain of aesthetics, a nature-inspired, swarm intelligence technique (Dispersive Flies Optimisation or DFO) is introduced and then adapted to detect symmetries and quantify symmetrical complexities in images. The 252 Jacobsen & Höfel’s images used in this paper are created by researchers in the psychology and visual domain as part of an experimental study on human aesthetic perception. Some of the images are symmetrical and some are asymmetrical, all varying in terms of their aesthetics, which are ranked by humans. The results of the presented nature-inspired algorithm is then compared to what humans in the study aesthetically appreciated and ranked. Whilst the authors believe there is still a long way to have a strong correlation between a computational model of complexity and human appreciation, the results of the comparison are promising.

... In geometry symmetrical shapes are produced by applying four operations of translations, rotations, reflections, and glide reflections. However developing computational methods which generate symmetrical patterns is still a challenge since it has to connect abstract mathematics with the noisy, imperfect, real world; and few computational tools exist for dealing with real-world symmetries [29]. Applying evolutionary algorithms to produce symmetrical forms leaves the formulation of fitness functions, which generate and select symmetrical phenotypes, to be addressed. ...

In late 1940s and with the introduction of cellular automata, various types of problems in computer science and other multidisciplinary fields have started utilising this new technique. The generative capabilities of cellular automata have been used for simulating various natural, physical and chemical phenomena. Aside from these applications, the lattice grid of cellular automata has been providing a by-product interface to generate graphical patterns for digital art creation. One notable aspect of cellular automata is symmetry, detecting of which is often a difficult task and computationally expensive. This paper uses a swarm intelligence algorithm—Stochastic Diffusion Search—to extend and generalise previous works and detect partial symmetries in cellular automata generated patterns. The newly proposed technique tailored to address the spatially-independent symmetry problem is also capable of identifying the absolute point of symmetry (where symmetry holds from all perspectives) in a given pattern. Therefore, along with partially symmetric areas, the centre of symmetry is highlighted through the convergence of the agents of the swarm intelligence algorithm. Additionally this paper proposes the use of entropy and information gain measure as a complementary tool in order to offer insight into the structure of the input cellular automata generated images. It is shown that using these technique provides a comprehensive picture about both the structure of the images as well as the presence of any complete or spatially-independent symmetries. These technique are potentially applicable in the domain of aesthetic evaluation where symmetry is one of the measures.

... In geometry symmetrical shapes are produced by applying four operations of translations, rotations, reflections, and glide reflections. However developing computational methods which generate symmetrical patterns is still a challenge since it has to connect abstract mathematics with the noisy, imperfect, real world; and few computational tools exist for dealing with realworld symmetries [21]. Applying evolutionary algorithms to produce symmetrical forms leaves the formulation of fitness functions, which generate and select symmetrical phenotypes, to be addressed. ...

In late 1940's and with the introduction of cellular automata, various types of problems in computer science and other multidisciplinary fields have started utilising this new technique. The generative capabilities of cellular automata have been used for simulating various natural, physical and chemical phenomena. Aside from these applications, the lattice grid of cellular automata has been providing a by-product interface to generate graphical patterns for digital art creation. One notable aspect of cellular automata is symmetry, detecting of which is often a difficult task and computationally expensive. This paper uses a swarm intelligence algorithm — Stochastic Diffusion Search — to extend and generalise previous works and detect partial symmetries in cellular automata generated patterns. The newly proposed technique tailored to address the spatially-independent symmetry problem is also capable of identifying the absolute point of symmetry (where symmetry holds from all perspectives) in a given pattern. Therefore, along with partially symmetric areas, the centre of symmetry is highlighted through the convergence of the agents of the swarm intelligence algorithm. This technique is potentially applicable in the domain of aesthetic evaluation where symmetry is one of the measures.

... In geometry symmetrical shapes are produced by applying four operations of translations, rotations, reflections, and glide reflections. However developing computational methods which generate symmetrical patterns is still a challenge since it has to connect abstract mathematics with the noisy, imperfect, real world; and few computational tools exist for dealing with real-world symmetries [16]. Applying evolutionary algorithms to produce symmetrical forms leaves the formulation of fitness functions, which generate and select symmetrical phenotypes, to be addressed . ...

Since the introduction of cellular automata in the late 1940’s they have been used to address various types of problems in computer science and other multidisciplinary fields. Their generative capabilities have been used for simulating and modelling various natural, physical and chemical phenomena. Besides these applications, the lattice grid of cellular automata has been providing a by-product interface to generate graphical patterns for digital art creation. One important aspect of cellular automata is symmetry, detecting of which is often a difficult task and computationally expensive. In this paper a swarm intelligence algorithm – Stochastic Diffusion Search – is proposed as a tool to identify axes of symmetry in the cellular automata generated patterns.

Symmetrical visual patterns have a salient status in human perception, as evinced by their prevalent occurrence in art, and also in animal perception, where they may be an indicator of phenotypic and genotypic quality. Symmetry perception has been demonstrated in humans, birds, dolphins and apes. Here we show that bees trained to discriminate bilaterally symmetrical from non-symmetrical patterns learn the task and transfer it appropriately to novel stimuli, thus demonstrating a capacity to detect and generalize symmetry or asymmetry. We conclude that bees, and possibly flower-visiting insects in general, can acquire a generalized preference towards symmetrical or, alternatively, asymmetrical patterns depending on experience, and that symmetry detection is preformed or can be learned as perceptual category by insects, because it can be extracted as an independent visual pattern feature. Bees show a predisposition for learning and generalized symmetry because, if trained to it, they choose it more frequently, come closer to and hover longer in front of the novel symmetrical stimuli than the bees trained for asymmetry do for the novel asymmetrical stimuli. Thus, even organisms with comparatively small nervous systems can generalize about symmetry, and favour symmetrical over asymmetrical patterns.

1. Cyclic, Dicyclic and Metacyclic Groups.- 2. Systematic Enumeration of Cosets.- 3. Graphs, Maps and Cayley Diagrams.- 4. Abstract Crystallography.- 5. Hyperbolic Tessellations and Fundamental Groups.- 6. The Symmetric, Alternating, and other Special Groups.- 7. Modular and Linear Fractional Groups.- 8. Regular Maps.- 9. Groups Generated by Reflections.- Tables 1-12.- Appendix for Chapter 2.

A direct relation between the first hyperpolarizability, beta, of noncentrosymmetric molecular structures and the metric centrosymmetricity content, S(i), of such structures is shown for the first time. For a series of systematic, in-plane distortions (stretch, pull, shift, and squish deformations) of the model NLO chromophore benzene, we find a convincing monotonic relationship between calculated values of beta and S(i). These results suggest that the dominant variation in beta for these structures arises from the change in oscillator strength. More striking, these comparisons demonstrate the utility of the S(i) metric in correlating observable behavior with symmetry content.

Tilings and patterns have been made and enjoyed for thousands of years. Their mathematical treatment was begun by n J. Kepleren but was then forgotten until the nineteenth-century development of crystallography. In this unique book, with its abundant illustrations, the authors explain exactly what one means by "tiling" and "pattern", restricting the treatment to two dimensions. There are many surprises; for instance, Figure 1.2.2 shows the 24 "heptiamonds", with the remark that only one of them cannot be repeated by congruent copies to fill and cover the whole plane. Chapter 2, on "Tilings by regular polygons", includes n A. J. W. Duijvestijn'sen "squared square", in which 21 different squares, with sides $2,4,6,7,8,9,11,15,16,17,18,19,24,25,27,29,33,35,37,break 42,50$, are fitted together to fill a square of side 112. Any solution to the slightly simpler problem of "squaring a rectangle" can be extended to a tiling of the whole plane by infinitely many squares of different sizes. Chapter 3, on "Well-behaved tilings", tells us precisely when a tiling can be called "normal". One counterexample is the remarkable Figure 1.0.1 (repeated as a cover design) which is monohedral (all tiles congruent) but is abnormal in that some pairs of tiles share a disconnected set of boundary points. Euler's theorem is used to prove that, if every tile of a normal tiling has $k$ vertices, where the valences are $j_1,j_2,cdots,j_k$, then $sum^k_i=1(1-2/j_i)=2$. Figure 3.8.6 illustrates a nice paradox: it shows a particular pentagon which is entirely surrounded by seven congruent replicas although the arrangement cannot be extended to a monohedral tiling of the whole plane. Chapter 4 describes the transition from metrical to topological tilings. Chapter 5 introduces the subject of patterns, beginning with a fascinating example (Figure 5.0.1) based on a maze. Except in some of the exercises, a discrete pattern means a planar family of mutually disjoint congruent copies of a motif with the property that for each pair of copies, say $M_i$ and $M_j$, there is an isometry of the plane that maps the whole pattern onto itself and $M_i$ onto $M_j$. According to this strict definition, the abnormal Figure 1.0.1 is not a pattern: every two of its infinitely many tiles are related by an isometry that maps one onto the other, but in no case is this isometry a symmetry of the whole pattern! The authors have undertaken the almost incredibly difficult task of classifying patterns so that one can say in what sense any two given patterns are of different types. Table 5.2.1 lists the 3 types of finite patterns, each type-symbol involving an integer $n$ which is the smallest period of a rotatory symmetry; Table 5.2.2 lists the 15 types of frieze patterns; Table 5.2.3 lists the 52 types of discrete periodic patterns. The number of types becomes smaller when the arbitrary motif is replaced by a dot or other symmetrical shape. Further complications arise when the motif is allowed to be infinite, or when copies of the motif overlap. Chapter 6 combines the two topics (tilings and patterns) by making even subtler distinctions: it can happen that several tilings are "really different" even though they have the same topological type and the same pattern type. There is a historical account of the classification of tilings. Attempts by some highly respected crystallographers, such as n A. V. Shubnikoven and n V. A. Koptsiken ref[ Symmetry in science and art, English translation, Plenum, New York, 1974], "led to an almost unbelievable number of errors". A different method of classification is developed in Chapter 7. Chapter 8 describes the complications that arise when tiles are distinguished by being variously colored. Chapter 9 deals with tilings by polygons, not necessarily regular; for instance, there are 24 types of tilings by congruent pentagons. The most exciting developments are reserved for Chapters 10 and 11, on "Aperiodic tilings". Before 1966, nobody could imagine the existence of a set of prototiles which would admit infinitely many tilings of the plane although no such tiling is periodic. Obviously, even such a simple prototile as a domino admits a nonperiodic tiling; but the exciting new idea, embodied in the term "aperiodic", is a set of $n$ prototiles which cannot possibly be arranged in a periodic fashion. n R. Bergeren ref[Mem. Amer. Math. Soc. No. 66 (1966); MR0216954 (36 #49)] discovered the first aperiodic tiling, with $n=20426$. Berger himself soon reduced this fantastic number to 104, n D. E. Knuthen to 92, n H. Läuchlien to 40, n R. M. Robinsonen to 35, n R. Penroseen to 34, n Robinsonen (again) to 32, and later to 24, n R. Ammannen to 16, and later to 6, then Penrose (again) to 5, and ultimately to 2! Many of the amazing ramifications of this theory, including some by n J. H. Conwayen, are published here for the first time. Chapter 11 deals with "Wang tiles": square tiles having colored edges which must match with their neighbors, only translations being allowed. These aperiodic tilings are relevant to questions of logic and computing, because it is possible to find sets of 16 Wang tiles which mimic the behavior of any Turing machine. Finally, Chapter 12 relaxes the restriction that the tiles should be topological disks, or that the tiling should cover the plane only once. The book ends appropriately with a 42-page bibliography and a 6-page index. Reviewed by H. S. M. Coxeter

The Stone−Wales rearrangement is analyzed using a newly developed continuous chirality measure. In enantiomerization reactions of chiral fullerenes we find an approximately linear correlation between π-energy and the chirality content of the molecule. These correlations show that the sensitivity to chirality change increases for larger fullerenes. We show its predictive properties and provide an explanation for it on the basis of another observation; namely, that the chirality value decreases monotonically with fullerene size. Comparison of the enantiomerization to other isomerizations of fullerenes is made.

We studied the way in which the binding of inhibitors of human immunodeficiency virus (HIV) protease causes the protein to deviate from its original C2 symmetric structure. We extended to C2 symmetry our findings that quantitative chirality is a useful, predictive parameter in enzymatic structure−activity correlations (J. Am. Chem. Soc. 1998, 120, 6152−6159). We provide a quantitative assessment of this deviation, the degree of C2-ness, S(C2), by employing the continuous symmetry measures methodology. The data analyzed was for a group of 13 inhibitor/protease complexes, for which the structures and the binding energies are known. S(C2) was determined for the inhibitors before and after binding, for each pair of amino acids of the protein, and for the whole protein complexes. We were able to track the spreading of the C2 distortion throughout the protein to various zones. Maps of iso-symmetry value proved to be a powerful descriptive tool for protein structure−deviation visualization. The main findings are the following: (i) For most inhibitors, the active site imposes its C2 symmetry on the bound inhibitor, rendering it more C2 symmetric than its free form and confirming the picture of enzymes as mechanical devices. (ii) The binding energy of the inhibitors correlates with this imposed C2 symmetry change: the smaller the symmetry change, the better the inhibition. (iii) Analysis of the enzyme's mutant strain V82A (raised against the inhibitors) shows that it has “learned” to cope better with an inhibitor by “following” this symmetry/binding energy correlation. (iv) Symmetry deviations occur in the protein upon binding at remote zones from the active site. Despite variations in the details of these deviations for the different HIV protease/inhibitor complexes, the protein as a whole responds to the various inhibitors with a very similar C2 symmetry change: a global symmetry-well for this protein, has been identified.

A method of detecting natural ``plateaus'' (equals maximal intervals of approximately constant value) in a one-dimensional pattern is described. The method is based on examining neighborhoods of each point having a range of sizes; rejecting neighborhoods of each size for which the standard deviation (of pattern values over the neighborhood) is not a local minimum; and further rejecting any neighborhood N whose subintervals of conparable size do not have mean pattern values approximately equal to that of N.