The Foundations of Aesthetics

Michael Leyton

Center for Discrete Mathematics & Theoretical Computer Science (DIMACS),

Busch Campus, Rutgers University, New Brunswick, NJ 08904, USA.

mleyton@dimacs.rutgers.edu

Abstract

This paper summarizes the theory of aesthetics that comes from the new foun-dations for geometry developed in my books. The new geometric foundations arebased on two principles: (1) maximizing transfer of structure, (2) maximizing re-coverability of the generative operations. According to the foundations, these arethe two basic principles of aesthetics. This paper shows that the two principlesare fundamental to aesthetic judgement in (1) the arts, where we examine painting,music, and poetry; (2) the sciences, where we examine general relativity and quan-tum mechanics; and (3) computer programming, where we examine object-orientedprogramming. It is shown that all these areas are driven by the same two underlyingprinciples: maximization of transfer and recoverability. Transfer is formalized interms of particular products of groups. It is shown to be the basis of Gestalt. Re-coverability is shown to depend on a new theory of symmetry-breaking, providedin the geometric theory. Together, transfer and recoverability are shown to be thebasis of memory storage; and our rigorous theory of aesthetics says that the rulesof aesthetics are the rules of memory storage. In particular, both the arts and thesciences are driven by the single goal of maximizing memory storage. Finally, theseprinciples are applied to explain core phenomena in object-oriented programming.

1 Introduction

This paper summarizes the theory of aesthetics that comes from the new foundationsfor geometry developed in my books. The new geometric foundations are based on twoprinciples:

(1) maximizing transfer of structure,(2) maximizing recoverability of the generative operations.

According to the foundations, these are the two basic principles of aesthetics. Thispaper will show that the two principles are fundamental to aesthetic judgement in (1)the arts, where we examine painting, music, and poetry; (2) the sciences, where we

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examine general relativity and quantum mechanics; and (3) computer programming,where we examine object-oriented programming. It will be shown that all these areasare driven by the same underlying principles. A full exposition of the theory is givenin my books: (1) Symmetry, Causality, Mind; (2) A Generative Theory of Shape; (3)Shape as History; (4) The Structure of Paintings.

2 Transfer in Art and Science

As stated above, the new foundations for geometry, developed in my books, are based ontwo principles which I claim are the basis of aesthetic judgement in the arts, sciences,and computation. The principles are maximization of transfer, and maximization ofrecoverability. For the next few sections, we will examine the first principle (maximiza-tion of transfer), and then we shall incorporate the second principle (maximization ofrecoverability). The present section will examine some of the evidence that transfer hasa fundamental role in aesthetics.

Let us begin by looking at Holbein’s painting Ann of Cleves, shown in Fig 1. Thisis analyzed in considerable detail in my book The Structure of Paintings. Some of itsbasic composition will be summarized as follows:

The generative history of this painting begins with a circle. This is shown, appro-priately, in the region of the face – both in the top line of the head, and in the neck. Thecircle is then deformed vertically into an ellipse; for example, we see this in the succes-sive downward necklaces. The downward end of the ellipse is a curvature maximum(extreme of bend). It is made successively more extreme in the successive downwardnecklaces.

In the next stage, this downward curvature maximum branches into two copies ofitself, left and right, creating a bay in the dress-band, as shown in Fig 2. The two copiesare at the ends of the two arrows shown.

In the next stage of the generative history, the center of the bay, which is a curvatureminimum (extreme of flatness), itself splits into two copies of itself which move to thesides. The resulting shape is a deepened bay, which is shown as the arm-line in Fig 3.That is, the flattened center of original bay (above the arms), has now become the twoflattened parts of the lower bay, indicated by the two forearms.

Now notice the following crucial point: Holbein transfers any downward action justdescribed onto a corresponding upward action. For example, the downward creation ofthe bay, in Fig 2, now appears as the upward creation of the bay in the veil, as shownin Fig 4. Furthermore, the deepened bay, in Fig 3, now appears as the top-line of thehead. My books have given lengthy analyses of painting, and shown that transfer is thefundamental structuring principle of painting.

I have also shown that transfer is the fundamental principle of music. A movementof a Beethoven symphony has remarkably few basic elements. The entire movementis generated by the transfer of these elements into different pitches, major and minorforms, overlapping positions in counterpoint, etc. For example, Fig 5 shows the famousmotif in Beethoven’s 5th Symphony being transferred to different levels of pitch, in

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Figure 1: Line drawing of Holbein’s Ann of Cleves.

Figure 2: The bay in the dress-band.

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Figure 3: Deepened bay.

Figure 4: The bay in the veil.

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Figure 5: Transfer in a Beethoven Symphony.

eight successive bars from the symphony. Almost the entire score looks like this.Furthermore, transfer is not only the basis of the pitch structure of music, but also

the meter structure. Meter is comprised of an accent hierarchy where the successivelevels are

(1) Primary accent grouping (division into bars).(2) Secondary accent grouping (first subdivision of the bar).(3) Division into beats.(4) Division of beats.(5) Subdivision of beats.

Each level transfers the level below it. For example, Fig 6 illustrates the meter struc-ture of a single bar in 9/8 time. This will be transferred onto the next bar. Furthermore,within this hierarchy, each node transfers the nodes that it dominates, as indicated bythe arrow below it. See Leyton [9], [10] for extensive discussion.

Figure 6: The accent hierarchy of a bar.

I have also demonstrated that poetry is structured by transfer. For example, considerFig 7, which is from a sonnet by Shakespeare. This is propelled forward by an exquisitesequence of transfers: First, the o sound is transferred, as indicated by bracket A. Thenthe l sound is transferred, as indicated by bracket B. Then the i sound is transferred,as indicated by bracket C. Then the th sound is transferred, as indicated by bracket D.Then the i sound is transferred, as indicated by bracket E.

Observe that bracket F marks a powerful phenomenon. It captures the fact thatbracket E is a transfer of bracket C, because both bracket E and C are the transfer of the

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Figure 7: Transfer in a Shakespeare sonnet.

i sound. That is, bracket F transfers bracket C on to bracket E. However, bracket F alsoindicates another transfer. The phrase "lives this" undergoes a mirror transformation tobecome the phrase "this gives". Finally, at the end of the line, the pair of sounds "l- th-",in the phrase, "life to thee" is a transfer of the earlier "l- th-" in the phrase "lives this".

We should also note that the meter structure of poetry conforms to the transfer theoryof musical meter given above.

Having seen that aesthetics in art is based on transfer, we shall now turn to scienceand see that aesthetics in science is also based on transfer.

A comprehensive survey of the use of the term aesthetics, in science, reveals thatthis term is consistently used about the fundamental phenomenon of symmetries of laws.These symmetries have lead to the major discoveries of physics; e.g., the conservationlaws, the existence of particles, the existence of dynamical equations, the unification offorces, the very concept of a force, etc. Let us understand what is meant by symmetriesof laws. It will be seen that this is a powerful example of transfer, and reinforces myclaim that transfer is a crucial part of the meaning of aesthetics.

At the foundations of any branch of physics there is a law which determines howsystems, in that branch of physics, evolve over time. This law is called the dynamicallaw, or dynamical equation, of that branch of physics. For example, in Newtonianmechanics, the dynamical equation is Newton’s second law, F = ma, which determinesthe trajectory of a system in classical mechanics. In quantum mechanics, the dynamicallaw is Schrodinger’s equation which determines how a quantum-mechanical state willevolve over time. In Hamiltonian mechanics, there are Hamilton’s equations whichdetermine how a point will move in phase space.

Let us now consider what is meant by symmetries of the dynamical law.1 Consider

1A dynamical law is always in the form of a differential equation. In the present section, for ease of

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Figure 8: The transfer of an scientific experiment.

Fig 8. The bottom trajectory in the figure shows an experiment being run in a laboratoryin New York, and the upper trajectory shows the same experiment being run in a labo-ratory in Chicago. Let us suppose that we discover a fundamental law that prescribesboth trajectories. Any such law, being a dynamical law prescribes a flow. The twotrajectories shown would be part of the flow prescribed by the law.

Now the most important question one can ask in physics, is this: Is there a trans-formation that takes the flow-line in New York onto the flow-line in Chicago? Let ussuppose that there is, and that the transformation is translation. This translation is shownby the vertical arrows in Fig 8. One says, in this case, that the equation (the flow) hastranslational symmetry; i.e., translation will send flow-lines of the equation onto eachother.

Our illustration used translation as the transformation that sent flow-lines onto flow-lines. But the transformation could have been rotation, in which case the dynamicallaw would have rotational symmetry.

The reason why hunting for symmetries of a dynamical law is so important is thatthe fundamental discoveries come from this. For example, for every symmetry transfor-mation that is discovered, there is a conservation law – e.g., if the discovered symmetrytransformation is temporal translation then one has the conservation of energy; if thediscovered symmetry transformation is spatial translation then one has the conservationof linear momentum; if the discovered symmetry transformation is spatial rotation thenone has the conservation of angular momentum, etc.

It is clear that the phenomenon we have been describing above is one of transfer.That is, a dynamical equation has a symmetry if the flow-lines can be transferred ontoeach other. Thus to state the argument succinctly: The term aesthetics in science isused consistently about symmetries of a law. However, we have seen that symmetriesof a law mean transfer of the law’s flow-lines onto each other.

In this section, we have examined some of the evidence that transfer is basic toaesthetics both in art and science. For much more extensive evidence, the reader shouldconsult my books Leyton, [7], [9], [11], [12].

illustration, we will assume that the equation is first-order. Higher-order equations follow the same basicprinciples, but at higher levels.

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3 Groups

In the next section, we will start to examine the structure of transfer in greater depth.This will require the concept of group, which will be explained in this section. If thereader is familiar with group theory, he or she can go directly to the next section.

Intuitively, one can say the following:

A group is a complete system of transformations.

Examples of groups are:

(1) Rotations. The complete system of rotations around a circle.

(2) Translations. The complete system of translations along a line.

(3) Deformations. The complete system of deformations of an object.

The word "complete" can be explained as follows: Let us suppose we can list thecollection of transformations Ti in a group G, thus:

G = { T0, T1, T2, . . . }.

For example, the transformations Ti might be rotations. The condition that this collec-tion is complete, means satisfying the following three properties:

(1) Closure. For any two transformations in the group, their combination is also inthe group. For example, if the transformation, rotation by 300, is in the group, and thetransformation, rotation by 600, is in the group, then the combination, rotation by 900,is also in the group.

(2) Identity Element. The collection of transformations must contain the "null" trans-formation; i.e., the transformation that has no effect. Thus if the transformations arerotations, then the null transformation is rotation by zero degrees. Generally, one labelsthe null transformation e, and calls it the identity element. In the above list, we canconsider T0 to be the identity element.

(3) Inverses. For any transformation in the group, its inverse transformation is also inthe group. Thus, if the transformation, clockwise rotation by 300, is in the group, thenits inverse, anti-clockwise rotation by 300, is also in the group.

There is a fourth condition on groups, called associativity, which is so simple that itneed not be considered here.

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Figure 9: A deformed cylinder.

4 Generating a Shape by Transfer

Recall that, according to my new foundations for geometry, aesthetics is based on twoprinciples: maximization of transfer, and maximization of recoverability. The firstprinciple was introduced in section 2, and the second principle will be introduced insection 7. We will now begin to examine the first principle in greater depth.

The new foundations for geometry is a generative theory of shape. Such a theorycharacterizes the structure of a shape by a sequence of actions needed to generate it.According to the new foundations, these actions must maximize transfer. That is:

MAXIMIZATION OF TRANSFER. Make one part of the generative history a trans-fer of another part of the generative history, whenever possible.

What we will do now is illustrate the means of generating a shape by transfer. Fig 9shows a deformed cylinder. To generate it entirely by transfer, we proceed as follows:

Stage 1. Create a single point in space.

Stage 2. Transfer the point around space by rotating it, thus producing a circle. This isillustrated in Fig 10.

Stage 3. Transfer the circle through space by translating it, thus producing a straightcylinder. This is illustrated in Fig 11.

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Figure 10: A point is transferred by rotations, producing a circle.

Figure 11: The circle is then transferred by translations, producing a straight cylinder.

Stage 4. Transfer the straight cylinder onto the deformed cylinder by deforming it.

Now observe that the four successive stages created a succession of four structures:

Point −→ Circle −→ Straight cylinder −→ Deformed cylinder.

Most importantly, observe that each of the successive stages created its structure bytransferring the structure created at the previous stage; i.e., there is transfer of transferof transfer. This means that the final object was created by a hierarchy of transfer.Furthermore, the transfer at each stage was carried out by applying a set of actions tothe previous stage, thus:

Stage 2 applied the group Rotations to Stage 1.Stage 3 applied the group Translations to Stage 2.

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Stage 4 applied the group Deformations to Stage 3.

This hierarchy of transfer can be written as follows:

Point T© Rotations T© Translations T© Deformations.

The symbol T© means "transfer". Each group, along this expression, transfers its left-subsequence; i.e., the entire sequence to its left. That is, going successively, left-to-right along the sequence: (1) the group Rotations transfers its left-subsequence Pointto create a circle; then (2) the group Translations transfers its left-subsequence PointT© Rotations (the circle) to create a straight cylinder, and finally, (3) Deformations

transfers its left-subsequence Point T© Rotations T©Translations (the straight cylinder)to create the deformed cylinder.

5 Fiber and Control

Section 4 introduced the transfer operation T©. This operation always relates two groups,thus:

G1 T© G2.

The lower group, that to the left of T©, is transferred by the upper group, that to theright of T©. The lower group will be called the fiber group; and the upper group willbe called the control group. That is, we have:

Fiber Group T© Control Group.

The reason for this terminology can be illustrated with the straight cylinder. Here,the lower group was Rotations, which generated the cross-section, and the upper groupwas Translations, which transferred the cross-section along the cylinder, thus:

Rotations T© Translations.

The thing to observe is that this transfer structure causes the cylinder to decompose intofibers, the cross-sections, as shown in Fig 12. Each fiber, a cross-section, is individuallygenerated by the lower group Rotations. It is for this reason that I call the lower group,the fiber group. Notice, also from Fig 12, that the other group Translations, controls theposition of the fiber along the cylinder. This is why I call the upper group, the controlgroup.

Generally, a transfer structure causes a fibering of some space. As a further il-lustration, consider what happened when we created the deformed cylinder by addingDeformations, above the straight cylinder thus:

Rotations T© Translations T© Deformations.

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Figure 12: Under transfer, a cylinder decomposes into fibers.

Here Deformations acts as a control group, and the group to its left, Rotations T©Translations, acts as its fiber group. In this case, the fibers are now the various deformedversions of the cylinder. For example, the straight cylinder is the initial fiber, and anyof its deformed versions (created by the control group), are also fibers.

My book A Generative Theory of Shape gives a comprehensive mathematical theoryof transfer. The operation T© is formalized in terms of a group-theoretic construct calleda wreath product. The present paper will omit the mathematical technicalities – in orderto make the discussion available to a larger readership. A rigorous definition of wreathproduct is given in the footnote2.

6 Theory of Gestalt

Since the beginning of perceptual psychology, over a hundred years ago, a major un-solved problem has been how the mind forms cohesive wholes, i.e., Gestalts. Using theabove concepts, we can now solve this problem:

2 Consider two group actions: the actions of groups, G(F ) and G(C), on sets, F and C, respectively.The wreath product G(F ) w©G(C) is the semi-direct product {∏

c∈C G(F )c} s©τ G(C), where the productsymbol

∏means the (group) direct product, and the groups G(F )c are isomorphic copies of G(F ) indexed

by the members c of the set C. The map τ : G(C) −→ Aut{∏c∈C G(F )c} is defined such that τ(g)

corresponds to the group action of G(C) on C, now applied to the indexes c in∏

c∈G(C) G(F )c. That is,τ(g) :

∏c∈G(C) G(F )c −→ ∏

c∈G(C) G(F )gc. Finally, we have a group action of G(F ) w©G(C) onF × C defined thus: For φ ∈ G(F ), κ ∈ G(C), and (f, c) in F × C, we have [φ, κ](f, c) = (φcf, κc),where φc ∈ G(F )c.

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THEORY OF GESTALT

The human perceptual system forms cohesive wholes, i.e., Gestalts,by transferring stimuli onto each other. That is, stimuli are boundtogether by transfer. The consequence is that a Gestalt is a n-foldtransfer hierarchy, G1 T©G2 T©. . . T©Gn.

7 Recoverability

Recall that, according to my new foundations for geometry, aesthetics is based on twoprinciples: maximization of transfer, and maximization of recoverability. The previoussections began to examine transfer, and it is now necessary to bring in recoverability.By recovery, we mean the following problem:

Given a data set, recover or infer a sequence of operations that generatethe set.

My book, Symmetry, Causality, Mind (MIT Press), was a 600-page analysis of thisproblem, and one of the main conclusions of this analysis was the following:

ASYMMETRY PRINCIPLE. The only recoverable operations are symmetry-breakingones. That is, a generative history is recoverable only if it is symmetry-breaking on eachof the successively generated states.

It is worth considering a psychological example of this principle at work: In a seriesof experiments [4] [5], I found that, when subjects are presented with a parallelogramoriented in the picture plane as shown in Fig 13a, they see it as a rotated version ofthe parallelogram in Fig 13b, which they then see as a sheared version of the rectanglein Fig 13c, which they then see as a stretched version of the square in Fig 13d. Theremarkable thing is that the only data that the subjects are actually given is the firstfigure, the rotated parallelogram. The experiments found that, on being presented withthis figure, their minds recovered the generative history shown.

The most important thing is that, to carry out this recovery of successively previousstates, their minds were successively removing asymmetries and recovering symmetries.Thus, subjects conjectured that the generative history was symmetry-breaking in theforward-time direction, from square to rotated parallelogram.

8 New Theory of Symmetry-Breaking

The Asymmetry Principle states that recoverability of history is possible only if eachasymmetry in the present goes back to a past symmetry. This means that, forward ittime, the history must have been symmetry-breaking.

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Figure 13: Psychological results found in Leyton [4] [5].

Currently, in mathematics and physics, symmetry-breaking is described by reductionof a group, for the following reason: A symmetrical object is described by a group oftransformations that correspond to its symmetries. When some of the symmetries aredestroyed, then correspondingly some of those transformations are lost, and thereforethe group is reduced.

However, in our system, symmetry-breaking is associated with the expansion of thegroup. For instance, recall from section 4 the case of cylinder. The straight cylinderwas given by the group:

Point T© Rotations T© Translations.

Then the deformed cylinder was given by taking this group as fiber, and extending it bythe group Deformations, via the transfer operation T©, thus:

Point T© Rotations T© Translations T© Deformations.

The added group, Deformations, breaks the symmetry of the straight cylinder. However,the group of the straight cylinder is not lost in the above expression. It is retained asfiber. In fact, it is transferred onto the deformed cylinder, and it is this that allows us tosee the latter cylinder as a deformed version of the straight cylinder.

Thus we have a new view of symmetry-breaking:

NEW VIEW OF SYMMETRY-BREAKING. When breaking the symmetry of anobject which has symmetry group G1, take this group as fiber, and extend it by the groupG2, via the transfer operation T©, thus:

G1 T© G2

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where G2 is the group of the asymmetrizing action.

9 Maximizing Memory Storage

It is now important to understand that the new foundations to geometry directly opposethe foundations that have existed from Euclid to modern physics, including Einstein.In the standard foundations, a geometric object consists of those properties of a figurethat do not change under a set of actions. These properties are called the invariantsof the actions. Geometry began with the study of invariance, in the form of Euclid’sconcern with congruence, which is really a concern with invariance (properties thatdo not change). And modern physics is based on invariance. For example, Einstein’sprinciple of relativity states that physics is the study of those properties that are invariant(unchanged) under transformations between observers.

However, I argue that the problem with invariants, is that they are memoryless. If aproperty is invariant (unchanged) under an action, then one cannot infer from the prop-erty that the action has taken place. In other words: Invariants cannot act as memorystores. Thus I conclude that geometry, from Euclid to Einstein has been concerned withmemorylessness. In fact, since standard geometry, including Einstein’s relativity, triesto maximize the discovery of invariants, it is essentially trying to maximize memory-lessness. These foundations to geometry are inappropriate to the computational age.People buy computers that have greater memory storage, not less. The medical pro-fession fights diseases such as Altzheimer’s because these diseases attack memory, andmemory not only allows intelligence, but is equated with the person’s identity.

As a consequence, I embarked on a 30-year project to build up an entirely new systemfor geometry – a system I recently completed and published as the book, A GenerativeTheory of Shape (Springer-Verlag, 550pages). Rather than basing geometry on themaximization of memorylessness (the aim from Euclid to Einstein), I base geometry onthe maximization of memory storage. The result is a system that is profoundly different,both on a conceptual level and on a detailed mathematical level.

The basic principle of the new foundations is this:

Shape ≡ Memory Storage.

In particular, the claim is that all memory storage takes place via shape.The theory then shows that the maximization of memory storage in shape is achieved

by the two principles: (1) maximization of recoverability, (2) maximization of transfer.These two principles are fundamental to memory because recoverability means thereconstruction of the past from what is available in the present, and transfer meansseeing the present in terms of the past, i.e., as a transfer of the past. The crucialconcept is therefore this: A shape maximizes memory storage, if it is given a generative(historical) description that maximizes transfer and recoverability.

To illustrate, let us go back to the example of the deformed cylinder. Section 4showed how this cylinder can be generated, all the way up from a point, by layers oftransfer: One starts with a point, then one transfers the point by rotations to create a

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circle, then one transfers the circle by translations to create a straight cylinder, and finallyone transfers the straight cylinder by deformations to produce the deformed cylinder.This means that, forward in time, one goes through a sequence of four stages that createa succession of four structures:

FORWARD TIME

Point −→ Circle −→ Straight cylinder −→ Deformed cylinder.

Each stage creates its structure by transferring the structure created in the previous stage.The arrows in the above sequence of four structures represent the forward direction

of time. Now let us consider how one recovers that history. This means that one mustreverse the arrows; i.e., go backward in time.

BACKWARD TIME

Deformed cylinder −→ Straight cylinder −→ Circle −→ Point.

Thus, starting with the deformed cylinder in the present, one must recover the backwardhistory through these stages. We now ask how this recovery of the past is possible. Theanswer comes from our Asymmetry Principle (section 7), which says that, to ensurerecoverability of the past, any asymmetry in the present must go back to a symmetry inthe past.

Now the word asymmetry, in mathematics and physics, really means distinguisha-bility, and the word symmetry really means indistinguisability. Thus the AsymmetryPrinciple really says that, to ensure recoverability, any distinguishability in the presentmust go back to an indistinguishability in the past. In fact, the backward-time sequence,given above, is recovered exactly in this way, as follows:

(1) Deformed cylinder −→ Straight cylinder: The deformed cylinder has distin-guishable (different) curvatures at different points on its surface. By removing thesedistinguishabilities (differences) in curvature, one obtains the straight cylinder, whichhas the same curvature at each point on its surface; i.e., indistinguishable curvatureacross its surface.(2) Straight cylinder −→ Circle: The straight cylinder has a set of cross-sections thatare distinguishable by position along the cylinder. By removing this distinguishabilityin position for the cross-sections, one obtains only one position for a cross-section, thestarting position. That is, one obtains the first circle on the cylinder.(3) Circle −→ Point: The first circle consists of a set of points that are distinguishableby position around the circle. By removing this distinguishability in position for thepoints, one obtains only one position for a point, the starting position. That is, oneobtains the first point on the circle.

We see therefore that each stage, in the backward-time direction, is recovered by con-verting a distinguishability into an indistinguishability. This means that each stage, inthe forward-time direction, creates a distinguishability from an indistinguishability inthe previous stage. Let us check this with the example of the deformed cylinder. Thesequence of actions used to generate the deformed cylinder from a point are:

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Point T© Rotations T© Translations T© Deformations.

Observe that each level creates a distinguishability from an indistinguishability in theprevious level. That is, Rotations produces a cross-section by creating distinguishabilityin position for the single point on the previous level. Then Translations produces astraight cylinder by creating distinguishability in position for the single cross-sectionon the previous level. And finally, Deformations produces a deformed cylinder bycreating distinguishability in curvature on the surface of the straight cylinder of theprevious level.

The fact that each level creates a distinguishability (asymmetry) from an indis-tinguishability (symmetry) in the previous level, means that each level is symmetry-breaking on the previous level. However, we have also seen that each level transfersthe previous level. This is a fundamental point: Each level must act by both symmetry-breaking and transferring its previous level. To fully understand the importance of thispoint, let us state it within the main argument of this section:

MAXIMIZATION OF MEMORY STORAGE

Maximization of memory storage requires (1) maximizing the recov-erability of generative operations, and (2) maximizing the transfer ofgenerative operations.

This means that each stage of the history must fulfill two conditions:(1) It must be symmetry-breaking on the previous stage. (2) It mustact by transferring the previous stage.

That is, each stage must be a symmetry-breaking transfer of the pre-vious stage.

The concept of symmetry-breaking transfer is fundamental to the new foundations forgeometry. Notice that it means that each successive control group must be symmetry-breaking on its fiber.

10 Rigorous Definition of Aesthetics

We are now ready to give our rigorous definition of aesthetics. The new foundationsto geometry proposes that aesthetics is based on two principles: (1) maximization oftransfer and (2) maximization of recoverability. In section 2, we reviewed both artand science with respect to the first principle, maximization of transfer. We put forwardexamples from the arts including painting, music and poetry, and examples from scienceincluding the symmetries of laws. In all cases, we saw that maximization of transferwas basic to aesthetic judgement. Now let us turn to the second principle, maximizationof recoverability.

First consider art. In section 2, it was shown that the composition of Holbein’sAnn of Cleves is based on the following generative sequence: The circle in the neck

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extends downwards to become an ellipse given by a necklace, which has a curvaturemaximum (bottom of the necklace) that did not exist in the circle. Then, this maximumbranches sideways into two copies of itself, to produce a bay. Next, the central curvatureminimum (flatness) of the bay branches sideways into two copies of itself, to producea deepened bay. The history is therefore this:

circle −→ ellipse −→ bay −→ deepened bay

where, at each stage, the number of curvature extremes (maxima or minima) increases.Now it is important to understand that a curvature extreme creates greater distin-

guishability in curvature around the curve, because an extreme involves a fluctuation incurvature. This means that, at each stage in the above history, the introduction of a newcurvature extreme has a symmetry-breaking effect on the previous stage (i.e., createsgreater curvature distinguishability). Therefore, the generative history accords with ourAsymmetry Principle (section 7), which states that a generative history is recoverableonly if it is symmetry-breaking at each of the successively generated states.

The fact that the generative history is recoverable from the structure of the paintingmeans that the painting acts as a memory store for the generative actions. In fact, mybooks have argued this:

Art-works are maximal memory stores.

This is the crucial function of art-works; the reason why they are so valued. Furthermore,I have argued that computer scientists can learn to significantly increase the power ofmemory stores in computers by learning the rules by which artworks are constructed.These rules are given by the new foundations to geometry presented in my books,Leyton, [7], [9], [11], [12].

Now let us turn to science. Current models explaining the physical constitution of theuniverse argue for a succession of symmetry-breakings from the underlying starting state(first to hypercharge, isospin, and color, and then to the electromagnetic guage group).There is considerable puzzlement in physics as to why such backward symmetrizationshould be the case. This is linked to the general bewilderment that Wigner expressedin his famous phrase concerning the "unreasonable power of mathematics" in physics– by which he really meant the unreasonable power of symmetry in physics.

However, according to our theory, backward symmetrization is entirely explicable.It comes from the Asymmetry Principle which states that a generative history is re-coverable only if present asymmetries go back to past symmetries in the generativehistory.

With this in mind, let us return to the issue of aesthetics in science. The termaesthetics in physics is often linked to the use of symmetries to represent past generativestates. Therefore, putting this notion together with the considerations of section 2, thereappear to be two uses for the term aesthetics in physics: (1) the characterization oftransfer, and (2) the characterization of recovered states.

The question therefore is this: To what extent are these two situations of aestheticjudgement separate from each other? Our theory says that they are not separate. Weshowed in sections 8 and 9 that each level of the transfer hierarchy necessarily takes onsimultaneously the role of transfer and recoverability. To use physics as an example:

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The symmetry group acts both as the past state and as the operational structure thattransfers flow-lines of the dynamical law onto each other.

We will illustrate this with both general relativity and quantum mechanics, as fol-lows: First, in general relativity, the gravitational force breaks the symmetry of flatspace-time, making it curved. Corresponding to this it breaks the conservation laws,which act globally in flat space-time and only infinitessimally in curved space-time.This means that, in flat space-time, one can transfer flow-lines onto flow-lines, in thedynamical laws; but in curved space-time this transfer is lost. Thus, as stated in theprevious paragraph, the symmetry group acts both as the past state and as the operationalstructure that transfers flow-lines of the dynamical law onto each other. In other words,the two forms of aesthetic judgement in physics – symmetries of the dynamical laws(transfer), and the description of past states (recoverability) – are made coincident.

Notice the relation of this to memory storage: Curved space-time has an asymmetry(curvature) that stores the effect of the action of the gravitational force.

Exactly the same kind of situation exists in quantum mechanics. As an example,consider the modelling of the hydrogen atom. The atom involves a number of complexfactors, such as the interaction between the electron’s spin and orbital angular momen-tum, and the interaction between the proton and electron spins, etc. The way this ismodelled is as follows: One starts with empty space (called the "free particle" situation).This has the most symmetrical energy function (Hamiltonian potential) possible: simplya flat constant surface, i.e., a surface that is translationally and rotationally symmetric.Then one introduces the simplest form of the hydrogen atom. This is called the Coulombelectrostatic model. This breaks some of the symmetries of the flat energy surface ofempty space, but retains some of its other symmetries. Then, one adds the interactionbetween the electron’s spin and orbital angular momentum. This breaks still more ofthe symmetry. The breaking is called the fine-structure splitting of the Coulomb model.Then one adds the interaction between the proton and electron spins. This breaks stillmore of the symmetry. The breaking is called the hyper-fine splitting.

Now, in these successive symmetry-breakings, one looses the transfer of flow-linesonto flow-lines of the dynamic law (Schrodinger’s equation). Thus, as stated above, thesymmetry group acts both as the past state and as the operational structure that transfersflow-lines of the dynamical law onto each other. Thus once again, the two forms ofaesthetic judgement in physics – symmetries of the dynamical law (transfer), and thedescription of past states (recoverability) – are made coincident.

Notice the relation of this to memory storage. The successively added asymmetries,in building the model of the hydrogen atom, are memory stores for the successivelyadded interactions.

Now let us take stock. I have argued that the term aesthetic is used in science whenthere is maximization of transfer and recoverability. This leads to the rigorous theoryof aesthetics presented in my books:

Aesthetics is the maximization of transfer and recoverability.

Also, we have seen that maximization of transfer and recoverability serves the goal ofmaximization of memory storage. Therefore, this leads to the following related claimin my books:

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The rules of aesthetics are the rules of memory storage.

Let us look at this issue more deeply with respect to science. We know that the mainconcern of science is explaining how things are caused. My book Symmetry, Causality,Mind (MIT Press, 630pages) shows that "explaining how things are caused" is the sameas "converting them into memory stores". That is, extracting the causal history from anobject, is the same as viewing it as a memory store of that history. However, the latterformulation is more powerful, since it is tied to the very concept of computation. Thuswhereas, conventionally, there is a separation between the causal and the computational(calculation) aspects of physics, my new foundations for geometry unifies these twoby showing that the causal aspects are actually a means of setting up computationalcomponents, i.e., memory stores. In fact, I argue:

In science, the concept of causality should be replaced by the conceptof memory storage. In other words, causal constructs in science shouldbe replaced by computational ones.

Under this view, science is the extension of a computational system to encompass theenvironment as extra memory stores. For example, the purpose of general relativityand quantum mechanics is to add curved space-time and the hydrogen atom as extramemory stores.

The argument therefore leads to the following fundamental conclusion: Since sci-ence tries always to maximize the amount of causal explanation in a situation, I thereforeclaim, using my conversion of causal constructs into memory constructs:

Science is the conversion of the environment into maximal memory stores.

For a full elaboration of this theory, the reader should see my books, Leyton, [7], [9],[11], [12].

Now, near the beginning of this section, I said that art-works are maximal memorystores. This means that, both the sciences and the arts are driven by the same goal:producing maximal memory stores. Furthermore, my argument is that this is whataesthetics is.

The issue which then arises is the following: Since my claim is that the sciencesand the arts are driven by the same goal, what is the difference between them? I arguethis: Computation involves two basic operations: (1) reading a memory store, and (2)writing a memory store. The claim then becomes this: Science is the process of readinga memory store; and art is the process of writing a memory store.

To explain this further: According to the above theory, both the scientist and the artistare interested in the maximization of memory information extractable from an object.The scientist focuses on maximizing the memory information obtained by reading.This means that the scientist converts the existing environmental objects into memorystores (e.g., curved space-time, the hydrogen atom). In contrast, the artist focuses onmaximizing memory information by actually creating new objects in the environmentthat will act as memory stores. To state the situation succinctly:

The goal of both science and art is the production of maximal mem-ory stores. Science achieves this by converting existing objects into

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maximal memory stores. Art achieves this by creating new objects asmaximal memory stores.

11 The Aesthetics of Computing

Aesthetics is a major driving force in the organization of computer programs. The reasonis that programs are often such large complex structures that their construction mustaccord with principles of good organization in order that they can be read, understood,and modified by programmers needing to use them.

The new foundations to geometry give considerable insight into the methods oforganizing and using computer programs. My book Generative Theory of Shape givesextensive discussion of this, and the present section will give a brief summary of someaspects of that discussion.

First, it is generally understood that re-usability is a major factor driving program or-ganization. For example, modern computing is largely based on objects (e.g., rectangles,cubes), not only because human beings find that manipulating objects is conceptuallyeasy, but because objects are re-usable items. Not only are programs decomposed intoobjects, but they are decomposed into larger re-usable units, that help the programmersand clients use and adapt them with ease.

In the new foundations to geometry, re-usability is formalized as transfer, i.e., tore-use an item is to transfer it. Since maximization of transfer is one of the two basicprinciples of our geometric theory, we can see that this corresponds to the computer sci-entist’s goal of maximizing re-usability. Notice the relation between this and aestheticsin the arts – for example, the movement of a Beethoven symphony is propelled forwardby the continual re-use of the motival material.

Most crucially, since the new geometry gives an extensive mathematical theoryof transfer – in terms of wreath products – the geometry thereby gives an extensivemathematical theory of re-usability in software (as it does in the arts).

For example, the very notion of an object (class) is modelled by transfer, in thefollowing way: Each geometric class (e.g., a rectangle) consists of an internal symmetrygroup, which we can consider to be specified usually in the invariants clauses of thesoftware text for the class, and an external group consisting of command operations,such as deformations, specified in the feature clauses of the class text. A principle claimof the theory is that the relation between the internal symmetry group and commandstructure, in the software text, is given by the following structure:

Gsym T© G(C)

where Gsym is the internal symmetry group and G(C) is the group of command oper-ations. In other words, because the object is itself a group Gsym, the true action of thecommand operations can be viewed as transferring that group. This transfer followsthe theory of symmetry-breaking in section 8; that is, the command operations act bybreaking the symmetry of the internal group – e.g., deforming the object, moving it

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from its default position (which breaks the symmetry by misalignment), etc. Therefore,the object-structure accords with the theory of recoverability given in section 7. Thatis, the very concept of object is organized by our principles of aesthetics.

Notice also the following: According to section 6, the basis of Gestalt is transfer;i.e., cohesion is formed by transfer. This is being illustrated in the present example. Ourtransfer-based theory of object-orientedness is explaining cohesion in programming.

As another example, the theory provides a deep understanding of inheritance, whichis a basic tool in object-oriented programming. Inheritance refers to the passing ofproperties from a parent to a child. The child incorporates these parent properties, butalso adds its own. This kind of structure covers two types of situation. The first is classinheritance, which is a static software concept, and the second is a type of dynamiclinking created at run-time. The geometric theory gives an algebraic theory of bothtypes of inheritance, but we will have time to deal here with only the latter. This lattertype is fundamental to all computer-aided design, assembly, robotics, animation, etc. Atypical example is a child object inheriting the transform of a parent object, and addingits own.

Now, we saw above that the very structure of an object is organized by transfer. Weshall now see that the inheritance relationship between two objects is also organized bytransfer:

ALGEBRAIC THEORY OF INHERITANCE. Inheritance arises from a transferhierarchy:

Gchild T© Gparent

where Gchild is the command group of the child object, and Gparent is the commandgroup of the parent object.

To illustrate: In many situations, such as robotics and animation, objects can be strungtogether in an n-fold inheritance hierarchy. For example, limbs are put together in aserial-link manipulator; or the sun, earth and moon are put together in an animation ofthe solar system. For this, our geometric theory says the following:

GROUP OF ENTIRE TRANSFORM STRUCTURE. Consider a set of n + 1objects: Object 1 to n, and the World. Suppose that they are linked such that Object iis the child of Object i + 1, and Object n is the child of the World. Then the group ofthe entire transform structure is the transfer hierarchy:

F1GF21 T© F2G

F32 T© . . . T© Fn

GWn

where:

(1) Object i has personal transform group Gi and frame Fi.(2) Personal transform group Gi relates frame Fi+1 of the parent, upperindex, to the personal frame Fi, lower index. (The world frame Fi+1 iswritten as W .)

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Notice that, initially, all frames Fi are coincident, and that the action of the transformsis to move the frames out of alignment; i.e., breaking their symmetries.3 This meansthat the transfer hierarchy is a symmetry-breaking one, on accord with the theory ofrecoverability given in section 7.

12 Conclusion

This paper has summarized the theory of aesthetics that comes from my new foundationsfor geometry. The new geometric foundations are based on two principles: maximiza-tion of transfer and maximization of recoverability. According to the foundations, theseare the two basic principles of aesthetics. This paper has shown that the two principlesare fundamental to aesthetic judgement in (1) the arts, where we examined painting,music, and poetry; (2) the sciences, where we examined general relativity and quantummechanics; and (3) computer programming, where we examined object-oriented pro-gramming. We showed that all these areas are driven by the same underlying principles.

References

[1] Leyton, M., (1974). Mathematical-logical postulates at the foundations of art. TechReport, Mathematics Department, University of Warwick.

[2] Leyton, M. (1984). Perceptual organization as nested control. Biological Cybernet-ics, 51, 141-153.

[3] Leyton, M. (1986a). Principles of information structure common to six levels of thehuman cognitive system. Information Sciences, 38, 1-120. Entire journal issue.

[4] Leyton, M. (1986b).A theory of information structure I: General principles. Journalof Mathematical Psychology, 30, 103-160.

[5] Leyton, M. (1986c). A theory of information structure II: A theory of perceptualorganization Journal of Mathematical Psychology, 30, 257-305.

[6] Leyton, M. (1987a). Nested structures of control: An intuitive view. ComputerVision, Graphics, and Image Processing, 37, 20-53.

[7] Leyton, M. (1992). Symmetry, Causality, Mind. Cambridge, Mass: MIT Press.

[8] Leyton, M. (1999). New foundations for perception. In Lepore, E. (Editor). Invita-tion to Cognitive Science. Blackwell, Oxford. p121 - 171.

[9] Leyton, M. (2001). A Generative Theory of Shape. Berlin: Springer-Verlag.

3In Leyton [9], the symmetries of a Cartesian frame are given by the hyperoctahedral group, which has adecomposition as a wreath product.

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[10] Leyton, M. (2003). Musical works are maximal memory stores. Epos Music (inpress).

[11] Leyton, M. (2004). Shape as History. Basel: Birkhauser.

[12] Leyton, M. (2004). The Structure of Paintings. (Submitted).

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