Unlocking the Secret of the Golden Ratio: Symmetry, Isomorphism, Abstract Geometry, and Harmony
  

Unraveling the Mystery of Beauty

In the pursuit of understanding beauty, two prominent theories come to the forefront. The first posits that humans are innately drawn to objects with proportions based on the Golden Ratio, as this is believed to epitomize balance and harmony. The second theory contends that symmetrical objects and patterns are perceived as aesthetically pleasing due to their intrinsic sense of equilibrium and order. In this post, we will formulate a theory of beauty that unifies these two theories. Our unified theory is elegantly simple yet powerful enough to predict the beauty of the Golden Ratio and the division of segments into halves through pure logical deduction. For millennia, renowned artists and architects have employed the Golden Ratio in crafting exquisite forms. However, it is now that we present a theory of beauty that finally unveils the enigma of the Golden Ratio.

The Beauty of the Golden Ratio

The mathematical essence of the Golden Ratio lies in its unique properties as an irrational number with a simple yet fascinating geometric interpretation. Often denoted by the Greek letter φ (phi), it is approximately equal to 1.6180339887. The Golden Ratio can be defined in various ways, but one common geometric interpretation involves dividing a line segment into two parts.

Imagine a horizontal line segment A, divided into two unequal parts, B and C. There is only one way to divide segment A such that the ratio of the entire segment  A to the larger segment B is equal to the ratio of the larger segment B to the smaller segment C. Mathematically, this relationship is expressed as (B + C) / B = B / C. In this case, the Golden Ratio, φ, is the value of B / C when the equation holds true.

The Golden Ratio has long been linked with beauty across various fields, including art, architecture, and even human facial features. The beauty of the Golden Ratio centers on the notion that certain proportions, specifically those adhering to the Golden Ratio, possess inherent beauty and aesthetic appeal. Numerous studies and works aim to understand and explain this captivating connection. Gary B. Meisner’s book, “The Golden Ratio: The Divine Beauty of Mathematics,” provides a comprehensive overview of the Golden Ratio’s role in the realm of beauty and aesthetics. Furthermore, Meisner’s website, https://www.goldennumber.net/, serves as an invaluable resource for those interested in delving deeper into the subject.

By the end of this blog, we will uncover the true reason behind the Golden Ratio’s significance in aesthetics. As we explore the captivating world of Abstract Geometry, symmetries, and patterns we’ll gain a deeper understanding of the fascinating properties that make the Golden Ratio so essential to the realm of beauty. So, let us embark on a journey that will unveil the mystery of the Golden Ratio and its enduring impact on aesthetics.

Hermann Weyl’s Research on Symmetry and Aesthetics

Hermann Weyl’s research explores the relationship between symmetry and aesthetics from a mathematical perspective, providing an intellectually engaging view on beauty. Weyl’s study of symmetry groups reveals the mathematical structure behind the visual appeal of artistic and natural patterns. He explores reflection, rotation, and shift symmetries, each contributing distinct aesthetics. 

Reflection symmetry, or mirror symmetry, involves an axis along which a shape or pattern is mirrored, creating a symmetrical image. Common in art, architecture, and nature, examples include human faces, classical buildings, and butterfly wings.

 

Rotation symmetry, also referred to as radial symmetry, occurs when a shape or pattern can be rotated around a central point (the center of rotation) by a certain angle without altering its appearance. This type of symmetry is frequently found in nature, such as the radial symmetry in flowers, and in many artistic designs, including mandalas and geometric patterns.

 rotation symmetry

Shift symmetry, or translation symmetry, is characterized by the repetition of a shape or pattern along a specific direction without rotation or reflection. The shape or pattern appears to be “shifted” along the plane in equal intervals, maintaining its appearance throughout the transformation. This form of symmetry is prevalent in tessellations, wallpaper designs, and crystallographic patterns found in nature. 

Weyl’s investigation of symmetry groups, their properties, and classifications elucidates the connection between mathematical structures and aesthetic allure. Central to his analysis is the concept of isomorphism, a mathematical principle that deepens our comprehension of symmetry and its intricate association with aesthetics.

Isomorphism: Revealing the Mathematical Essence of Symmetry

Isomorphism is a mathematical concept that unveils the structural similarities between distinct objects. The term “isomorphism” stems from the Greek words “isos” (equal) and “morphe” (form). Isomorphism refers to a transformation (or mapping) that relates an object to itself while maintaining its geometrical structure intact. In the context of symmetry, isomorphism is a powerful tool to describe and analyze the transformational relationships within symmetrical structures.

To better appreciate the relationship between symmetry and isomorphism, let’s consider the symmetries of an equilateral triangle. An equilateral triangle possesses several symmetries, including reflection and rotation symmetries. Each symmetry operation can be thought of as a transformation that moves the triangle in such a way that it remains unchanged. These transformations can be combined, and the resulting set of all possible transformations forms a structure that captures the essence of the symmetries.

As seen in the above image, there are three axes of reflection symmetry (labeled 1, 2, and 3) and three rotational symmetries (120°, 240°, and 360°). The symmetries of the equilateral triangle can be organized in a structure, known as the dihedral group of order 6, that contains six elements, representing the identity element (no transformation), the three reflections, and the two non-trivial rotations.

The concept of isomorphism enables us to identify and study the relationships between the symmetries of different objects or structures. Isomorphism connects the abstract world of mathematics with the tangible realm of art and nature, offering valuable insights into the aesthetic appeal of symmetry.

Introducing Abstract Geometry as a Framework for study of Symmetry and Patterns

Abstract Geometry Structures

In Abstract Geometry, we generalize the notion of a point ‘a’ to a more abstract concept of an object ‘a’. We also extend the idea of a segment connecting two points ‘a’ and ‘b’ to a relationship between two objects, R(a,b). Consequently, the equilateral triangle discussed in the previous section is represented as a set of objects and a set of relations in the following structure:

< a, b, c ; R(a, b), R(a,c), R(b,c) >

In this representation, we lose the information about the position of points and the length of the segments connecting points. However, if two pairs of objects are connected by the same relation, we assume that the relationship between the pairs is of the same kind. Therefore, if we are told that relation R represents a line segment connecting two points, we would know from the description above that we are dealing with an equilateral triangle. Further, we know that the relation should be reflexive, meaning R(a, b) holds if and only if R(b,a) holds. What if the sides of our triangle were all unequal? Then we would need three symbols for relations, and our structure would look as follows:

< a, b, c ; R(a, b), Q(a,c), G(b,c) >

If we wish to retain some information about the position of the object, we can use predicates or unary relations. For example, we can introduce predicates Left, Right, Center, Up, Down and use them to describe the position of the points in our equilateral triangle as follows:

< a, b, c ; R(a, b), R(a,c), R(b,c), UpCenter(a),  DownLeft(b), DownRight(c) >

If we wish to use more refined descriptions, we can introduce a richer language. The objects, relations, and predicates capture what is important in the picture or geometric shape for the analytical part of our mind which operates with language. Abstract geometry structures represent concise mental models of real objects or geometric forms and prepare visual forms for analytical processing. Think of Abstract Geometry structures as a bridge between the perception of a continuous geometrical form and a description of that form in human language.

The relation between structures and visual objects they represent is a many-to-many relation. One structure can represent several visual objects and vice versa. For example, the structure describing an equilateral triangle could also be used to describe three circles touching each other. In this case, R(a,b) would mean that circles with centers ‘a’ and ‘b’ are touching each other.

In Abstract Geometry, we can be very creative in choosing what we take as our objects and our relations. For example, we can conceive a structure where our objects are segments, and our relations are a numerical ratio between the segments. Our symbols for relations would be numbers, and in the case of equilateral triangles, we would only need the number 1. Here is a structure describing our equilateral triangle:

< (ab), (ac), (bc) ; 1( (ab), (ac)), 1((ab),(bc)), 1((ac), (bc)) >

Historical Origins of Abstract Geometry: A Brief Note

In the early 20th century, mathematicians grappled with understanding the underlying nature of numbers and geometric shapes. Set Theory emerged as a systematic approach to describe the foundations of mathematics and its effectiveness in the real world. Formal predicate logic and semantics were developed as methods to study and formalize mathematical proofs and ensure the validity of set-theoretic constructions.

The structures of Abstract Geometry introduced here are used in Model Theory of formal semantics to describe the fundamental reality to which mathematical proofs are applied. Representing the world in the form of abstract objects, predicates, and relations has served as the foundation for the older symbolic paradigm of Artificial Intelligence that was widespread in the 20th century. For an application of this framework in modeling the dynamics of general systems, you may refer to our paper, From equations to patterns: Logic-based approach to general systems theory.

Isomorphism, Partial Isomorphism, and Pattern

The concept of isomorphism in Abstract Geometry is fundamentally similar to that of geometric isomorphism. Isomorphism refers to a transformation (or mapping) that connects an object to itself while preserving its geometrical structure. However, we must exercise caution when defining the term  structure.  Consider the following structure corresponding to an equilateral triangle:

< a, b, c ; R(a, b), R(a,c), R(b,c), UpCenter(a),  DownLeft(b), DownRight(c) >

The isomorphism corresponding to rotation would preserve the relational structure, but the properties describing position would be excluded from the preserved structure. The formal definition of isomorphism in Model Theory involves mapping both the names of objects and relations, which could address this issue. Nonetheless, Model Theory’s isomorphism is too powerful, so we instead utilize the concept of partial isomorphism.

Partial isomorphism is a mapping of one subset of an object in an Abstract Geometry structure to another subset of its object while preserving selected subsets of the structure’s relations.

Abstract Geometry involves a degree of subjectivity, as we selectively choose a set of relations or properties to incorporate into our model. Consequently, the notion of retaining relations in partial isomorphism aligns with the inherent subjectivity of abstract geometry. For instance, when examining the symmetries of an equilateral triangle, all relevant aspects for our analysis are encompassed in this structure:

< a, b, c ; R(a, b),  R(a,c), R(b,c) >

The ability to select a subset in a system and map it to another subset of the system while preserving relations leads us to the idea of patterns. The notion of pattern and partial isomorphism are connected.  A pattern is a generic structure preserved under partial isomorphism. We can describe patterns by introducing variables X, Y, Z, etc., and using them in place of objects. The patterns describing structures preserved in the symmetries of an equilateral triangle are as follows:

< X, Y ; R(X, Y) >   – line segment pattern

< X, Y, Z ; R(X, Y),  R(X,Z), R(Y,Z) > – triangle pattern 

When we substitute variables with specific objects in a structure, we obtain an instantiation of a pattern within that structure. In our case, both patterns have six possible instantiations within a structure describing the equilateral triangle. Given the close relationship between pattern and partial isomorphism, patterns like those mentioned above become meaningful when they have at least two distinct instantiations within a system. Each pair of distinct instantiations of a pattern defines a single partial isomorphism.

 Patterns can be composed of smaller, interconnected patterns. For example, a triangle pattern comprises three linked line segment patterns. The line segment pattern embodies the similarity between edges, while the triangle pattern captures the geometric symmetries preserved under isomorphism. Consequently, our notions of partial isomorphism and patterns encapsulate both symmetry and pattern ideas.

Patterns within a system reveal its self-similarity, yet they can also be employed to identify resemblances between parts of distinct systems. For instance, the triangle pattern mentioned earlier highlights the shared features of Platonic solids composed of triangles—tetrahedron, octahedron, and icosahedron. The study of patterns in Abstract Geometry is an enthralling pursuit that we will continue to explore in future blog posts. For now, we have all the necessary tools to take the next step towards the primary objective of this post: unraveling the mysteries of the Golden Ratio.

A Brief Theory of Beauty

Venus: Beauty is subjective, Rene, but it is also objective. Beauty is a form of harmony, and harmony is an objective quality. We can recognize beauty in things, people, and nature when they are in harmony.

From: Amo Ergo Sum: A dialogue between Venus and Descartes about truth, beauty, existence, and love

Here is our theory of beauty:

We define harmony as a set of patterns inherent in a system and propose that this concept of harmony is central to the aesthetic perception of the system.

Our Abstract Geometry pattern captures both symmetry and pattern as observed in real-world systems. Therefore, our concept of harmony encompasses the totality of patterns and symmetries inherent in the system. The proposition that harmony is central to a system’s aesthetic perception is a generalization of the theory proposed by Hermann Weyl.

Our framework of pattern and harmony allows us to examine the link between a system’s structure and its aesthetics. While our theory doesn’t provide a specific method for comparing beauty, in some simple cases, it may be possible. The presence of harmony in a system, as opposed to a lack thereof, helps explain the allure of the Golden Ratio proportion.

Predicting Golden Ratio with Pure Deduction

Armed with the definitions of pattern and harmony, we can investigate harmony in a purely deductive manner. What possible patterns can be discovered when dividing a segment A into two segments B and C? For simplicity, we use the same symbol for the segment and its length. Therefore, we can express our constraint for length as A = B + C.

We have a system with three objects <A, B, C>. To find a pattern, we need to establish meaningful relations. One meaningful relation between segments is the ratio of their lengths, while another is the inverse ratio. Let’s define the relation R between objects X and Y as the ratio of their lengths.

R(X, Y) if and only if R = X/Y

The simplest pattern we seek would take the form:

< X, Y ; R(X, Y) >

To be considered a pattern, it must have at least two realizations within the system of objects A, B, and C. For example, let’s consider the following possibility:

<A, B, C; R(A,B), R(B,C) >

To achieve this pattern, we would require the following:

A/B = B/C

The only number that can satisfy this equation is the Golden Ratio φ. Thus, we found one pattern with R = φ and its instantiation:

<X, Y; φ(X,Y)>

<A, B, C; φ(A,B), φ(B,C) >

In the latter system, B > C. By flipping B and C in the relations, we can find another Golden Ratio division of the segment. This time, C > B, but in essence, it is the same Golden Ratio division of segments with exactly the same pattern. Flipping the arguments within the two relations, we can obtain a harmonious system with a pattern where R = 1/φ:

<X, Y; 1/φ(X,Y)>

<A, B, C; 1/φ(B, A), 1/φ(C,B) >

Our pattern is different; however, this again corresponds to the Golden Ratio division of the segment. Are there other harmonious divisions? Certainly! Let’s try to find a number R that would satisfy the following instantiation:

<A, B, C; R(A,B), R(A,C) >

This can be satisfied by R = 2 and division of segments into two equal parts.

<X, Y; 2(X,Y)>

<A, B, C; 2(A,B), 2(A,C) >

Flipping the arguments within both relations, we can obtain a harmonious system with a pattern where R = ½, which again corresponds to the division of the segment A into two equal parts B and C. Interestingly, the division into halves produces a larger pattern with two possible instantiations:

< X, Y, Z ; 2(X, Y), 2(X,Z), 1(Y,Z) >

(X, Y, Z) = (A, B, C) and (X, Y, Z) = (A, C, B)

Thus, the division of segments into half has more harmony than the Golden Ratio division, and it is consistent with the fact that it is more ubiquitous in aesthetics and nature compared to the Golden Ratio.

Are there other possible harmonious divisions of a segment, distinct from the Golden Ratio and the division of segments into equal parts? Well, since this is just a blog post, we are not obligated to provide the answer but leave the question open. We have a recipe for finding them all. There are only 13 possible unordered pairs of ordered pairs, such that all symbols A, B, and C are present. We invite you to try them all to find out if there are other possibilities.

Conclusion

We introduced a framework of Abstract Geometry that allowed us to formalize the concepts of patterns and symmetry. Using this framework, we developed a theory of beauty based on the notion of harmony – encompassing all patterns embedded within a system. Our theory of beauty is a generalization of Weyl’s ideas, which connect beauty to the symmetries and inherent patterns found in a system.

The elegance of our concise theory of beauty lies in its ability to deduce the relationships of the Golden Ratio and division into half through pure deduction, starting from the definition of beauty as harmony. Equipped with our formal definitions of pattern and harmony, we sought to find a harmonious division of segment A into segments B and C. According to our theory of beauty, harmonious implies the presence of at least a simple pattern:

< X, Y ; R(X, Y) >

We discovered meaningful ways to construct harmonious divisions of segments with R equal to φ and 1/φ, which correspond to the Golden Ratio division. Additionally, we found another harmonious division of the segment with R equal to 2 and ½, which corresponds to the division into half.

Our theory of beauty holds the potential for a multitude of applications. For instance, the concepts of pattern and of partial isomorphism could elegantly capture the allure of fractals. While our exploration is merely an initial, modest stride, we hope it serves as a significant leap forward in the realm of aesthetics and mathematics of beauty.

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