Science of Art: Artisan Theory of Meaning, Truth and Beauty

To develop a complete mind: Study the science of art; Study the art of science. Learn how to see. Realize that everything connects to everything else.

― Leonardo da Vinci

 

Introduction

In envisioning the interplay of science and art in contemporary times, how might it materialize? Moreover, what did Leonardo da Vinci understand about the science of art during his time? While it might seem centered on understanding anatomy and nature for accurate art, Leonardo’s approach was much more profound.  Recent investigations by Robert Grant and Alan Green have shed light on the depth of Leonardo’s genius, revealing layers of interpretation within the Vitruvian Man. Their groundbreaking analysis presents compelling evidence that the Vitruvian Man is not just an iconic artwork, but also a precise mathematical representation of the inner chambers of the Great Pyramid of Giza. Such a nuanced weaving of form, meaning, and reference underscores Leonardo’s unparalleled artistry. 

In this post, we explore art’s core: mapping shapes. This idea is central to all art, from kids’ drawings and cave art to masterpieces like the Vitruvian Man. While experts like Leonardo used this method intentionally, many artists tap into a deep intuition. This natural mapping gives art its instant meaning. By analyzing and modeling this, we hope to uncover Leonardo’s insights and lay groundwork for a contemporary Science of Art.

Our discussion will unfold as follows: We’ll start with a thought experiment to explore the most fundamental expressions of art. Using insights gleaned from this experiment, we’ll employ tools from modern cognitive science to establish a mathematical framework for rigorous analysis. Within this framework, the preservation and mapping of structures become key elements. Building on this foundation and initial examples, we’ll outline a theory of Mimetic Truth—a specific version of truth that exists in the context of the mappings created by artwork. We’ll apply the same framework to describe the interpretation of symbols found in artworks, and sketch out a theory of meaning that explains how an artwork’s significance emerges through our imaginative process. Lastly, we’ll discuss the general concept of Beauty, linking it to the mathematical notion of Harmony. To conclude, we’ll demonstrate how this theory, even though developed from basic examples, illuminates the complex methods Leonardo da Vinci used in crafting the Vitruvian Man.

Readers from varied backgrounds stand to gain uniquely from this exposition. Artists may discover new mapping techniques to incorporate into their repertoire. AI engineers delving into art generation can integrate these strategies to enhance their products. Art critics might find this framework a fresh prism through which to interpret the essence of artworks.

Science of Art and the Role of Imagination

Human cognition is multifaceted. Solving a mathematical problem engages our logical reasoning and intuitive faculties. On the other hand, navigating a romantic dinner conversation necessitates a different skill set, like empathy, active listening, and articulate expression. So, which cognitive faculties come into play when we engage with art? Central to this process, arguably, is our imagination. Historically, the intertwined relationship of imagination and art was somewhat overshadowed, particularly by Greek philosophers who held truth and logic in higher esteem. For much of history, until the emergence of cognitive science, studies around imagination lagged behind those centered on logic and mathematics. Yet, is it possible to approach imagination mathematically? Can a mathematical Science of Art evolve from a mathematical perspective on imagination? This article ventures to initiate a dialogue on the mathematical examination of artworks and the interpretative processes rooted in imagination. We seek answers to the following questions:

To scrutinize these questions, we’ll harness a knowledge representation structure encompassing Structures, Partial Morphisms, and Abstract Geometry. Portions of this framework were touched upon in my preceding article on the Golden Ratio. Notably, the concept of partial morphism will shed light on metaphors and analogies, illuminating the way in which the meaning of artwork crystallizes through the workings of our imagination. This analytical framework will empower us to draw parallels between the propositional truth inherent in logic and the mimetic truth of art. In our journey, we’ll also unravel the profound connection between Beauty and Truth, understanding them as essentially intertwined.

The Genesis of Art: Artists at Work

The cornerstone of science often lies in conducting experiments or conceiving thought experiments, followed by rigorous inquiry into the outcomes. Let’s delve into a thought experiment that will underpin our exploration of the Science of Art. Visualize having five wooden sticks and a flat pebble. Their depiction is as follows: 

sixs hapes

Suppose you’re tasked with crafting a few rudimentary artworks using these six objects. The end result may resemble a child’s doodle—uncomplicated, yet bearing the hallmark of art. If I were to rise to this challenge, here’s what I would make:

artforms

Every artwork deserves a title. I’ve christened the first piece “Vitruvian Man,” the next “Fish,” and the final rendition, “Birdhouse.” While rudimentary, that’s precisely the allure. Simplicity often offers the best starting point for scientific inquiry. Now, onto our queries.

We commenced with six elementary geometric forms. To breathe life into our artworks, we played with their arrangement. Consider this: there exist countless ways to organize six shapes on a canvas. This begets our first question:

Question 1: What compelled us to select these particular configurations as our artistic expression?

A plausible response hinges on the idea that these configurations, when perceived collectively, mimic familiar entities from our reality. Art, in essence, mirrors life. This notion aligns with the Mimetic Theory of Art. (For a delightful introduction to the Mimetic Theory of Art, check out this video by Amor Sciendi.) Armed with this insight, we’re poised to lay the cornerstone of our Science of Art, investigating how our artistic elements map to real-world images.

Mathematically, this process of identifying structural parallels between diverse entities is termed ‘morphism’. Originating from the Greek word “morphe” (meaning form), morphism denotes a transformation that bridges one object to another while preserving its inherent geometric integrity. For instance, one can transpose a small triangle onto a larger counterpart, or a square onto another. However, superimposing a square onto a triangle without altering the square’s structure is unfeasible.

Interestingly, the traditional definition of morphism hinges on the notion of “geometrical structure,” a term that can be nebulous. The act of equating a wooden stick with a human hand might not preserve the geometric integrity of either. Yet, we draw parallels because both items have elongated shapes. Similarly, juxtaposing a human head and a pebble resonates because of their rounded nature. This isn’t a geometric mapping in its classical sense. Rather, we draw connections based on semantic categorizations of objects. To traverse this terrain, we require a mathematical scaffold that encapsulates such generalized mappings that succinctly captures both geometric and semantic structures. The next two sections outline our mathematical framework. Concepts introduced therein will empower us to map structures, anchoring our exploration of mimetic truth.

Structures and Partial Morphisms 

Structures and Collages

Structures provide a way to represent systems, scenes, or situations using objects and predicates. These predicates can explain the properties of objects or the relations between them. For example, the situation “George and Mary are married” can be represented as:

< Mary, George; woman(Mary), man(George), married(Mary, George) >

Depending on our focus, we can modify these structures by including or excluding certain properties and relations. A more concise representation might be:

< Mary, George; married(Mary, George) >

Let’s consider another situation where Mary and George are married and they have a daughter named Eliza. While one might visualize a detailed family portrait, the basic representation using our structures is:

<Mary, George, Eliza; married(Mary, George), daughter(Eliza, Mary, George) >

Think of these structures as a bridge connecting our perception of visual content, like photos in a family album, to verbal descriptions of the same.

In a similar vein to how we compile photos in an album, we can compile these structures to track the development or changes of related objects over time. This collection is what we term as a ‘collage,’ and it can simultaneously present different aspects or chronologically map out the evolution of a particular scene, situation, or system of objects.

Partial Morphisms and Patterns

Consider a situation described by the statement: “Edward and Diane are married. They also have a son, Henry.” Here’s a structure collage representing it:

< Diane, Edward; married(Diane, Edward) >

<Diane, Edward, Henry; married(Diane, Edward), son(Henry, Diane, Edward) >

There’s an evident similarity between this structure and the one that describes George, Mary, and Eliza. To articulate this similarity, we introduce the concepts of Morphism and Partial Morphism.

A morphism is a mapping of objects from one structure onto another, preserving all predicates of the structure.

For example, mapping Edward to George and Diane to Mary maintains the predicate ‘married’. Hence, this mapping establishes a morphism between two structures:

<Mary, George;  married(Mary, George) >

      |             |               |                   

<Diane, Edward; married(Diane, Edward) >

If we try to establish a morphism between the subsequent structures in the two collages – describing the children – by mapping Eliza to Henry, we’d falter. The ternary relation son(_, _, _) is distinct from the ternary relation daughter(_, _, _). Therefore, this mapping doesn’t establish a full morphism; it’s only partial. Often, we’ll encounter situations where only a fraction of the structure is preserved, leading us to the concept of a partial morphism.

A partial morphism maps a subset of objects from one structure to a subset in another, preserving only some of the predicates. For instance:

<Mary,  George,  Eliza;   married(Mary, George), daughter(Eliza, Mary, George) >

   |              |                  |        

<Diane, Edward, Henry; married(Diane, Edward), son(Henry, Diane, Edward) >

The presence of either a morphism or partial morphism indicates two structures share a common pattern. We’ll define patterns as structures with variables represented by uppercase letters like X, Y, Z, sometimes with indices. The shared pattern in our morphism and partial morphism examples is:

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

Assigning objects to these variables is termed the instantiation of the pattern. Each pattern should have a minimum of two distinct instantiations.

We initiated our exploration into structures with an aim to discern the structural commonalities among images or situations. By introducing the concept of partial morphism, we effectively capture these similarities. Yet, our current framework is somewhat flexible, allowing a broad spectrum of predicates and relations for representing situations. If, for example, we replaced the ternary relations daughter(_, _, _) and son(_, _, _) with child(_, _, _) in the secondary structures of the collages, we would have full morphisms instead of partial ones for the second structures.

When portraying artworks or images using natural language, descriptions are fluid. Describing an image allows for logical extensions based on the initial description, and it often brings associations to mind. By articulating an artwork through language, by following through with logic, and by considering associations, we construct its meaning. To formulate a theory of artwork’s meaning, our structures need to encompass a similar fluidity.

Transforming Structures and Morphism Degrees

To enhance our formalism, we aim to introduce the capability to transform structures. For example, a structure  with ternary relations “daughter(_, _, _)” and “son(_, _, _)” could be transformed into a structure  with the ternary relation “child(_, _, _).”  Such transformations can be described using the following production rules:

< X, Y, Z; … daughter(Z, X, Y)> ~>  < X, Y, Z; … child(Z, X, Y)>

 

< X, Y, Z; … son(Z, X, Y)>  ~>  < X, Y , Z; … child(Z, X, Y)>

These rules indicate that any structure containing the relations “daughter(, ,)” or “son(, ,)” can be transformed to another structure by appropriately substituting or adding the relations. Consider a structure that captures the essence of an image. Transformation rules guide the natural evolution of its meaning. By employing these rules, the inherent meaning can be broadened. Starting from a select few structures, one can curate a more expansive collage that provides a richer interpretation of the image.

By applying the aforementioned rules to the previously-discussed family structures, we get structures that showcase greater structural congruence:

<Mary,  George,  Eliza;    married(Mary, George),  child(Eliza, Mary, George)>

   |                |               |             |                                              |

<Diane, Edward, Henry; married(Diane, Edward), child(Henry, Diane, Edward)>

This morphism maps three objects while preserving two relations. Prior to the transformation, the mapping involved only two objects and one relation. To capture the extent or richness of a (partial) morphism, we introduce the concept of morphism degree:

Degree = (Number of objects mapped) x (Number of relations preserved by mapping)

Through inference rules, we observed a rise in the possible morphism degree from 2 to 6.

Our ability to assign diverse interpretations to images, transform these descriptions, and subsequently identify patterns in these transformations, mirrors aspects of verbal intelligence. Thus, our logical transformations effectively model verbal intelligence. But to truly grasp art, we need additional transformative patterns that embody pure imagination. Firstly, there’s the mechanism of association, and secondly, metaphorical transfer.

Take association, for instance. If someone mentions the word “violin,” what images spring to mind? Perhaps a violin resting within an open case, or a musician engrossed in playing the instrument on stage. Such imaginative connections aren’t based on deductive reasoning but on associative inference. We can represent these transformations as:

<X ; violin(X)> ~> <X, Y; violin(X), man(Y), plays(Y, X)>

<X ; violin(X)> ~> <X, Y; violin(X), violin_case(Y), in(X, Y)>

Arguably the most fascinating transformation pattern relies on metaphorical transfer. It’s instrumental in describing mimesis and the essence of mimetic truth. However, before delving into that, we need an additional piece of formalism: Abstract Geometry.

Abstract Geometry

Purpose of Abstract Geometry

Previously, we observed that the mapping of the artwork containing six shapes to Leonardo’s painting of the Vitruvian Man is based on the semantics of shapes and their relationships. Elongated shapes correspond to other elongated shapes, round shapes align with round shapes, and so on. Abstract Geometry provides a framework to describe these types of mappings.

We initially introduced structures to represent the semantics of images, defining them in terms of Objects and Predicates. Abstract Geometry narrows its focus to the geometric facets of these images. Specifically, the partial morphisms in Abstract Geometry are morphisms of structures that concentrate on the geometric elements within an object’s structure. For illustration, consider how elements of the mapping from the six-shape artwork to Leonardo’s painting can be portrayed as a morphism of structures:

< stick1, stick3; ElongatedShape(stick1), ElongatedShape(stick3), Connected(stick1, stick3) >

       |         |

< hand,  body;   ElongatedShape(hand),  ElongatedShape(body), Connected(hand, body) >

This morphism upholds the pattern:

< X, Y; ElongatedShape(X), ElongatedShape(Y), Connected(X, Y) >

This section delves into the components of Abstract Geometry, equipping us with the tools to understand the essence of the Science of Art.  For a more comprehensive overview of this topic see the previous article Abstract Geometry.

Objects and Predicates in Six Shapes Drawing

To effectively describe artworks, we need to introduce predicates for the general characterization of shapes, like RoundShape or ElongatedShape. RoundShape does not strictly denote a circle, but something akin to it. Meanwhile, ElongatedShape can describe a variety of objects. A and B are the approximate values of the angles between the shapes measured visually.

The Vitruvian Man

 

Magic of Art: How do we arrive at the meaning of Artwork?

We now have everything at our disposal to draw a correspondence between a primitive drawing of a six-shape artwork and the Vitruvian Man.

The Vitruvian Man Mapping

When viewing these two pictures side by side, we notice many geometric discrepancies between them. Several proportions and angles differ. However, with the aid of Abstract Geometry, we can establish a mapping based on predicates common to both structures. 

For the sake of brevity, we’ll use the following abbreviations for predicates when discussing properties and relations between shapes in our drawings:

ESh: ElongatedShape

RSh: RoundShape

C: Connected

AngA: Angle equal to A

AngB: Angle equal to B

Here’s a pattern that allows the mapping of the six shapes of artwork into a painting of the Vitruvian Man:

< X1, X2, X3, X4, X5, Y; 

   ESh(X1), ESh(X2), ESh(X3), ESh(X4), ESh(X5), RSh(Y), C(X1, X3), C(X2, X3), C(X4, X3), C(X5, X3), C(Y, X3),  AngA(X3, X1), AngA(X3, X2), AngB(X3, X4), AngB(X3, X5) >

The mapping emerges from the resemblance of the types of shapes, patterns of connectivity between them, and angles. The pattern is consistent when we instantiate variables for the six-shape image as follows:


( X1~>stick1, X2~>stick2, X3~>stick3, X4~>stick4, X5~>stick5, Y~>pebble )

It’s also consistent when applied to the structure representing the Vitruvian Man:

( X1 ~>rhand, X2~>lhand, X3~>body, X4~>rleg, X5~>lleg, Y~>head )

It’s essential to emphasize that the Abstract Geometry mapping is solely an artifact of geometric shapes, their connections, and somewhat approximate matching of angles. The wooden sticks, when mapped to the hands and legs of the Vitruvian Man, remain wooden sticks. Yet, something magical transpires. Some sticks transform into legs, others into hands, and one into a body. How is this possible?

Depending on the arrangement of our shapes in the four different pictures, the same shapes adopt different meanings. In the six-shape portrayal of the Vitruvian Man, sticks represent hands, legs, and a body. In contrast, in the Birdhouse depiction, those very same sticks become the walls and roof of the house. This brings us to the next significant question in the Science of Art:

Question 2: How do geometric shapes comprising the artwork acquire non-geometric meaning and get interpreted as hands, legs, walls of a house, etc.?

The exploration of the mechanism of Metaphorical Mapping provides the answer to this question. 

Metaphorical Mapping

What is a metaphor? In Aristotle’s time, metaphors were primarily viewed as poetic expressions. This perspective dominated until the 20th century. The pivotal contribution made by George Lakoff and his colleagues was unveiling the metaphor as the foundational engine driving our thinking and imagination. In modern metaphor research, “metaphor” has come to signify a cross-domain mapping within the conceptual system. For instance, when we employ the metaphor “life is a game,” we map the conceptual domain of games to that of life. This analogy allows us to interpret life using our firsthand experience with games. The game thus becomes a blueprint for navigating life, and we transfer concepts associated with games via metaphorical mapping to life’s conceptual domain.

The book Metaphors We Live By by Lakoff and Johnson, which popularized metaphor research, argues persuasively that metaphors are fundamental to our imagination and cognition. In light of this research, it’s plausible to infer that artistic mimesis is driven by a form of metaphorical mapping. The magic of art might be explained as a dual metaphorical mapping in opposite directions.

Artworks are essentially maps of real objects; they’re representations of reality. However, when interpreting an artwork, the real world becomes a reference for the art. For example, when we fashioned the Vitruvian Man using five sticks and a pebble, we were mapping Leonardo’s masterpiece. But when considering the meaning of our creation, Leonardo’s painting becomes the guide that confers meaning to our artwork.

We organized the sticks and a pebble in a manner that allows for an Abstract Geometry morphism from our configuration to Leonardo’s painting. The object “stick1” is aligned with “rhand,” which is characterized by the predicate Hand(rhand). When deciphering our artwork, “rhand” becomes the reference for “stick1,” endowing it with the predicate Hand via metaphorical transfer. This is an explanation of the magic of Art based on the theory of metaphorical mapping. 

Our structural language and partial morphisms empower us to illustrate the metaphorical transfer process via logical rules. Here’s an initial draft of the rule, shedding light on Question 2:

Assume structure S1 represents artwork and structure S2 symbolizes a real object. If partial morphism F:S1 ->S2 maps the artwork to reality, and morphism F correlates the object a1 from S1 to the real object a2 in S2, and the real object is characterized by predicate P(a2), then we can integrate predicate P(a1) into structure S1, which illustrates the artwork.

Therefore, if our partial morphism aligns the six-shape artwork with the Vitruvian Man painting and links the object “stick1” to “rhand” (characterized by predicate Hand(rhand)), our rule allows for the Hand(_) property transfer via this mapping, integrating the predicate Hand(stick1) into the six-shape artwork structure. We’ve sketched a rule delineating art’s magic with mathematical precision.

However, this rule isn’t without its potential pitfalls. The most glaring is the ontological disparity between literal predicates and those incorporated through metaphorical transfer. This prompts the next pivotal question in our exploration of art:

Question 3: While “stick1” is undeniably just a stick, our metaphorical transfer and mapping rule designates it as a hand, denoted by Hand(stick1). Is it truly a hand, then? Can the predicate Hand(stick1) be deemed true or false?

Answering this query will pave the way for a monumental revelation in the Science of Art.

What is Truth?

The correspondence theory of truth, which has its origins with Plato and Aristotle, posits that the veracity or fallacy of a statement is solely determined by its relation to the world and its accuracy in describing (or corresponding with) that world. The debate over whether art reflects truth has been a contentious issue in art theory.  For a comprehensive review of this subject, consider the article Artistic Truth by Andy Hamilton.

Is Hand(stick1) true or false? It can’t be true, as stick1 in reality is simply a piece of wood and cannot be a human hand. However, in our artistic imagination, this unassuming elongated shape represents a hand. Consequently, Hand(stick1) can’t be outrightly labeled as false.

Mimetic Truth

The proposition Hand(stick1) would be devoid of any significance if found within structures describing a Fish or Birdhouse. Its relevance arises solely from the mapping F, which correlates the arrangement of six shapes to the Vitruvian Man. The implication of Hand(stick1) is entirely attributed to the mapping F. Thus, a fitting evaluation of the statement, and our answer to Question 3, would be:

The object stick1 is perceived as a Hand within the context of mapping F. Since Hand(stick1) is accepted as true exclusively within the confines of mapping F, this context must be explicitly noted when speaking to the statement’s truth value.

We’ll term this contextual truth as mimetic truth. The assertion that Hand(stick1) is true within the scope of mapping F will be symbolized as Hand(stick1 | F).

Having fleshed out the concept of mimetic truth, we can refine our preliminary rule of metaphorical inference and present the rule as:

Assume structure S1 represents artwork and structure S2 symbolizes a real object. If partial morphism F:S1 ->S2 maps the artwork to reality, and morphism F correlates the object a1 from S1 to the real object a2 in S2, and the real object is characterized by predicate P(a2), then we can integrate predicate P(a1 | F) into structure S1, which illustrates the artwork.

Applying this rule to mapping F from six-shape artwork to Leonardo’s Vitruvian Man would integrate the following predicates into a structure describing the six shape artwork:

Hand(stick1|F), Hand(stick2|F), Body(stick3|F), Leg(stick4|F), Leg(stick5|F), Head(pebble|F) 

We have discovered the magic rule that converts simple sticks into hands, legs, and a body, and makes the six-shape artwork an image of the VitruvianMan! This is a mathematical description of the essence of the Magic of Art. We might broaden this rule to include the transfer of binary relations. We could also impose limitations, ensuring mimetic truths don’t directly counter literal truths. It would be insightful to examine the ramifications and potential paradoxes this rule might entail, paving the way for future refinements. But for now, let’s rejoice in our discovery of the rule of artistic creativity!

We’ve crafted a mathematical framework of structures intertwined with Abstract Geometry. This intricate web proved potent enough to charm the mythical Dragon of Art, compelling it to unveil the secrets of Artistic Fire. We’ve uncovered the formula for mimetic truth, shedding light on the intricate dance between art and reality. Metaphorical transfer emerges as the very heart of Art’s magic, a magic potent enough to morph a handful of shapes into the Vitruvian Man, a fish, or a birdhouse. We’ve peeked into the core of Art’s magic and encapsulated its essence in mathematical terms.

Artisan Theory of Meaning

Artistic Symbols and Memes

Our original investigation started with elementary drawings. We explained simple artistic mimesis as double metaphorical mapping and built a mathematical theory for it. Modern Art often includes symbols that add special meaning to an artwork. Meme art popularized by social media belongs to this category. We will explain the interpretation of such art as triple metaphorical mapping.

Here is an artwork by Man Ray called “Le Violon d’Ingres”:

Study of Le Violon d'Ingres

In 1924, Man Ray unveiled a piece that masterfully blended the realms of photography and surrealism. This ingenious juxtaposition served both as a tribute and a gentle critique of the neoclassical artist Jean-Auguste-Dominique Ingres, who was renowned for his fixation on the female silhouette and his stellar depictions of violins. In Man Ray’s creation, the line between the object and the observer blurs, alluding simultaneously to the objectification of women and the intrinsic connection between the visual and musical arts.

Man Ray’s composition seamlessly blends a photograph of a close associate with illustrative violin f-holes. These f-holes, while literal in their depiction, are emblematic in the context of the piece. Given the artwork’s provocative nature, it provides an ideal backdrop for our analysis. In forthcoming discussions, we’ll leverage this contentiousness to shed light on how personal interpretations play a pivotal role in shaping the concept of Harmony. Our immediate endeavor, however, is to unravel the artwork’s embedded symbolism. What influences our understanding of such a piece?

Question 4: What guides our comprehension of artworks imbued with symbolism?

The title of the artwork offers an initial hint towards its interpretation. Cross-referencing this hint with our theoretical framework will further cement the conclusions we derive.

Metaphorical Transfer and Interpretation of Symbols

The initial phases of interpreting a photo echo our previous approach. First, we identify the photograph’s geometry as a woman’s silhouette. Subsequently, we ascribe meaning to various parts of this silhouette, drawing on our knowledge of human anatomy to serve as a guide. Typically, the interpretation of straightforward artwork would conclude following these two mapping phases. Yet, the presence of the f-holes on the woman’s back introduces a layer of complexity. This symbolic inclusion prompts a further leap in our interpretation, spurring our imagination to forge a metaphorical connection between the woman and a violin. 

 Here’s an explanation of the workings of imagination within our formal framework. Consider two structures:

W = <w, fh; woman(w), f-holes(fh), has(w,fh)>

V = <v, fh; violin(v), f-holes(fh), has(v,fh)>

Both structures follow a common pattern:

<X, Y; f-holes(Y), has(X, Y)>

Based on this pattern, we can establish a partial morphism F: W->V. Subsequently, using our rule, we execute a metaphorical transfer of the predicate violin.

<w, fh; woman(w), violin(w | F), …>

This structure conveys the metaphor Woman as a Violin. The artwork’s title affirms our formal inference. The rest of the imaginative process is elucidated by metaphor theory. We employ the metaphor “Woman as a Violin” to harness our experience with the violin as a blueprint for our experience with a woman. Our formalism depicts this explanation as a series of inference rule applications to the structure.

In the section about the transformation of structures, we discussed the alteration of structures through associations. We presented these as transformation patterns:

<X; violin(X)> ~> <X, Y; violin(X), man(Y), plays(Y, X)>

<X; violin(X)> ~> <X, Y; violin(X), violin_case(Y), in(X, Y)>

Utilizing these patterns depicts imaginative associations. They can be applied to structures with predicates like violin(w|F). The subsequent application of these inference rules can produce a structure collage:

<w, fh, x; woman(w), violin(w | F), man(x), plays(x, w)…>

<w, fh, y; woman(w), violin(w | F), violin_case(y), in(w, y)…>

Consider these structures as descriptors for images our imagination might conjure while navigating the metaphor “Woman as a Violin.” These images might then prompt further inferences involving deduction, association, or metaphorical mapping, leading to the continued transformation and expansion of the structure collage representing the meaning we extract from symbol-laden artwork.

To expound on an artwork featuring the symbol of f-holes, at least three metaphorical mapping rounds are necessary. More intricate artwork might comprise multiple symbols, with each potentially generating a distinct metaphorical mapping. Thus, we have a technically precise response to Question 4, succinctly summarized as:

The interpretation of artworks laden with symbols necessitates multiple metaphorical mappings.

Conceptual Space

Conceptual space refers to the collection of structures, patterns, and patterns of inference used during image interpretation. While the associations and inference patterns involved in art interpretation are personal, they often have shared elements. Despite each individual’s unique conceptual space, similarities can emerge within demographic groups and cultures, allowing us to discuss a “Cultural Conceptual Space” associated with specific groups or nationalities.

What’s truly remarkable about the evolving Science of Art is that we’re crafting a language enabling us to articulate conceptual space. When we derive an artwork’s interpretation that can be verbalized, it becomes possible to document the associations and inference rules employed.

Artistic Truth

Art’s meaning often sprouts from mapping a picture to other domains, sometimes multiple ones. One can’t fully grasp the meaning of an artwork’s components without taking in the entire piece. These mappings facilitate the interpretation of geometric shapes and symbols via both geometric and semantic lenses. Such interpretations might align with real-world objects or ideas, but equally, they can evoke imaginary realms. Mimetic truth might mirror reality, but it can just as readily reflect the creations of our imagination. Art isn’t confined to reality; its essence lies in igniting our imaginative flames. This gives rise to a holistic understanding of art’s significance:

The true essence of images we conjure in our minds embodies the artistic truth that artists wish to share.

It’s often posited that artists aim to express specific Artistic Truths. But what drives this expression? This curiosity propels us to our next exploration in the Science of Art.

Question 5: What motivates artists to craft artwork?

Our pursuit of this question will continue in the subsequent sections of this discourse.

Symmetry, Patterns, Harmony, and Beauty

Hermann Weyl delved into the intricate relationship between symmetry and aesthetics, proposing that symmetry is inherently linked with beauty. Let’s consider shapes characterized by rotational symmetry. 

Radial Symmetry

Symmetry can be defined through the lens of automorphism—a transformation that maps an object back to itself while preserving its geometrical structure. Morphism, in this context, can be seen as a broader term, encompassing automorphism. In essence, every automorphism is a self-mapping morphism.

The framework we’re introducing mirrors these ideas but offers a more expansive view. Central to both partial automorphism and partial morphism is the concept of the pattern: the preserved segment or structure. This leads us to our proposed Brief 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.

But what defines the system’s boundaries? From where do these patterns originate? In our previous discussions, our focus remained on patterns intrinsic to the system. However, when discussing a six-shape artwork, external patterns come into play.

Our theory retains its validity when we view the system as permeable, allowing for the import of external patterns. Indeed, this flexibility was always the intent—especially for open systems. Patterns resonate with both automorphism and morphism. And with the introduction of partial morphism, we can reformulate our Beauty Theory using both morphism and automorphism.

We define Harmony as a set of partial automorphisms and partial morphisms associated with a system and propose that this concept of Harmony is central to the aesthetic perception of the system.

This nuanced definition accentuates the Harmony bridging two distinct domains through morphism. In addressing Question 5, our response, although supported by our examples, remains provisional and might falter in outlier situations. Yet, its mathematical precision grants it a foundation for analysis and refinement.

So, why do artists conjure artworks? Ponder the chasm between random six-shape assemblages and deliberate, aesthetic arrangements. The demarcating factor? The presence of a discernible pattern. This introspection propels us to the crux of Question 5: 

Artists are driven by an innate quest for Harmony — a harmony that echoes their personal interpretations and resonates in their unique conceptual spaces. 

The act of creating art is a manifestation of this internal harmony, offering a bridge for like-minded individuals to traverse similar experiential landscapes.

The art piece “Ingress Violin” epitomizes this idea. Its embedded pattern mirrors the artist’s conceptual space. To truly appreciate the artwork requires a resonance of this pattern within our own conceptual realm. However, it might elude comprehension or appreciation by individuals from diverse demographics or cultures. As such, what one perceives as harmonious, another might deem discordant.

Truth and Beauty in Masterpieces

When we contemplate the spheres of Art and Science, an inevitable comparison arises. Science pursues objective truth, while art chases an “artistic truth” that inherently possesses subjectivity. The metrics for scientific success are clear and well-defined. A scientific theory’s veracity — its alignment with empirical facts — stands as its litmus test. Yet, what benchmarks define success in art? Before photography’s advent, the answer was simpler: artists endeavored to create lifelike and realistic representations. However, modern art pivots towards symbolism and impressionism, distancing itself from a strict adherence to reality.

Question 6: What defines artistic success?

One might posit that the answer lies in Harmony. Yet, how does the harmony in renowned masterpieces differ from that in simpler works?

We often hold high-fidelity images in higher regard than rudimentary representations, such as those constructed from basic shapes. To elucidate this difference, we proposed the concept of morphism degree, wherein:

Degree = (Number of objects mapped) * (Number of relations preserved by mapping)

For multiple morphisms, the degrees can be cumulatively considered, leading us to the “degree of harmony.” Greater mappings or higher-fidelity mappings yield a higher degree of harmony.

To address Question 6: the hallmark of artistic success is a pronounced Degree of Harmony. Masterpieces, like those crafted by Leonardo DaVinci, epitomize this heightened Harmony.

Take, for instance, Leonardo DaVinci’s The Vitruvian Man:

The Vitruvian Man

Moreover, recent revelations by Robert Grant and Alan Green have unveiled concealed mappings within the Vitruvian Man. To fathom their insights, the following YouTube video, which explicates how the Vitruvian Man mirrors the Great Pyramid of Giza, is recommended. Click on the image to watch the video. 

Alan Green and Robert Grant Research on The Vitruvian Man

The mapping discourse in this article serves as a conduit to Leonardo’s unique blend of art and science. These intricate mappings, rooted in geometry, measurements, ratios, or symbolism, evoke profound connections.

Based on our symbolic theory, the geometric parallel between the Vitruvian Man and the Great Pyramid of Giza evokes the metaphor Man is a Pyramid. This analogy can also be reversed, leading to Pyramid is a Man, a claim further authenticated by the aforementioned video.

The juxtaposition of the Vitruvian Man with the Great Pyramid showcases a sophisticated mapping. However, with a detailed exploration of both Abstract and Sacred Geometry, as provided in my technical notes on Abstract Geometry, this too fits within our framework. To grasp how the techniques illustrated in the video equate to Abstract Geometry morphisms, one must focus on the pivotal points of the artwork. Breaking down the Vitruvian Man and the Great Pyramid into distinct sections reveals patterns and correlations through geometric notations, including line segment ratios and angles.

Yet, translating the insights of Edward Grant and Alan Green into the language of Abstract Geometry is a complex task, worthy of its own dedicated article. If you’re interested in applying the Science of Art to your own projects, I welcome your feedback and questions. I’m eager to explore potential applications and may be available to collaborate on your initiatives.

In Retrospect: Bridging Art and Mathematics

In this exploration, we’ve constructed the “Artisan Theory of Mimetic Truth, Meaning, and Beauty.” Our journey began with a humble thought experiment and an elemental query: how is art birthed by artists? Grounded in the Mimetic Theory of Art, we championed a structured language that encapsulated the crucial idea of mapping intrinsic to artistic mimesis. This led us to uncover the concept of memetic truth, subsequently building a comprehensive theory of meaning pivoted on metaphorical transfer. Concepts like Pattern, Harmony, and Beauty were unveiled in this discourse.

From this foundation, we’ve formulated a holistic model that delves into the artist’s imagination, illustrating its applicability across art of varying complexity. This model’s robustness is evident in its capacity to address and potentially resolve significant artistic questions, including:

While the answers posited aren’t unequivocal, their precision affords them the capacity to be constructively critiqued and refined. The hallmark of this exploration is its endeavor to merge mathematical precision with artistic interpretation, providing an articulate framework for questions long considered abstract.

One may wonder, why hasn’t this synthesis emerged earlier? Crafting a cohesive theory of art mandates an intricate fusion of personal artistic experience, deep-seated involvement, and mathematical adeptness. The rise of the AI Art movement deserves applause, for it has offered those proficient in mathematics and engineering an avenue to engage with art, fostering a deeper reflection on its essence. Moreover, polymaths like Robert Grant and Alan Green have illuminated the symbiosis between science and art, as epitomized by virtuosos like Leonardo DaVinci. Such revelations have invigorated a resurgence in the convergence of the Art of Science and the Science of Art.

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