Math and Aesthetics in Carpets, Faces, Art, and Music           

Asymmetry and Symmetry-Breaking

Article # 1 Symmetry Holds Key to Beauty  W A S H I N G T O N, Sept. 24,1998 — The same trait that makes people love harmonious music may help them choose

a beautiful face, researchers said today. This could be because humans learn symmetrical patterns much faster than asymmetrical ones, said

Michael Ryan, a zoologist at the University of Texas and expert on the subject. "In humans, the perceptual system is biased toward

processing symmetrical signals," he said.


Recent studies indicate the perception of beauty might not be subjective at all but instead arises from a bias hard-wired into the sensory

system. Such studies have shown how symmetrical body features, chiefly  in the face, are viewed as beautiful, while asymmetrical ones are not. Ryan said

another example  is research  that suggests harmonic rather than discordant music strikes humans as more pleasing  because of the way the 

inner ear works.

Senses Stimulated

Writing in the journal Science, Ryan said a number of studies have indicated that physical traits physically stimulate the senses. He says this

means the traditional theories about genetic fitness do not tell the whole  story about why animals and humans choose "attractive" mates. 


The genetic fitness theory holds that pretty traits, such as the elaborate tail on a peacock, signal that the animal is stronger and more fit in

other, invisible ways. But Ryan says his studies show this is not  necessarily true. "Mate choice is important but it doesn’t explain all the

factors we find pleasing or beautiful," he said. 


"In a lot of cases females will prefer more elaborate males but don’t get a reproductive advantage."

 Preprogrammed Response

Water mites, which feed by sensing their prey’s waterborne vibrations, are an example. Their choice of  mates is wholly unrelated to the male’s

reproductive ability, Ryan said. The vibrations the males make mimic the vibrations made by copepods, the prey that mites prefer.

 He said the females are simply already programmed to respond to the vibrations. "Any sensory system is going to be more responsive to

some stimuli than to others," Ryan said.  "If males have a variety of options by which to signal their sexual interest, females will favor those

who use signals they are already keyed into." But he pointed out his "preferred preferences" theory cannot by itself explain natural selection.

Rather, it is a helpful part in a complex puzzle that includes  the more traditional theories.  "The bottom line is the answer isn’t simply going to

 be hypothesis A, hypothesis B or hypothesis C," Ryan said.

Copyright (c)1998  ABCNEWS and Starwave Corporation

SYMMETRY BREAKING article # 2      

(Taken from Mathematics and the Arts URL:

Symmetry breaking exists where symmetry is expected, but that expectation is not met. As we   often see in Oriental

carpets, it is playfulness with symmetry that results in intriguing  patterns.

   In nature, symmetry is imperfect, although mathematicians may  treat it as an ideal.

In art, too, it seems that the approximation of symmetry, rather than its precision,

teases the mind as  it pleases the eye.


 IN CARPETS, BORDER PATTERNS result when any or several  of the basic symmetries  are

repeated in one direction. The constraints of  symmetry are such that there are seven (7)  possible combinations:

 GRIDS and Tessellations

 THE EASIEST WAY TO ANALYZE a pattern is to locate points of rotation, and lines of symmetry. Why? Because the rigid motions

require centers of rotation and axes of repetition or reflection for symmetry to be present.

WHAT IS AN AXIS? An axis is a visible or implied line that is vertical, horizontal or diagonal, along which designs

are repeated or reflected to form  patterns.


 A grid is a visible or implied series of points, or axes that intersect. Grids underlie the structure of all two-dimensional patterns.






Grids are usually based on regular polygons: squares, equilateral triangles, and hexagons. Or they can be based on

rectangles, parallelograms and rhomboids.


THE ARRANGEMENT OF POLYGONS that forms a grid is called tessellation. Other shapes may also



WHAT IS A TESSELLATION? A tessellation is a pattern formed by the 

repetition of a single unit or shape that, when repeated, fills the plane  with

no gaps and no overlaps. Familiar  examples of tessellations are the patterns

formed by paving stones or bricks, and cross-sections of beehives.


Tessellations are not typical of Oriental carpets except as visible grid structures.

Although they often appear in minor borders, only rarely are tessellations used as

field patterns.




                   Conceptions of beauty are chiefly mathematical

 Jaw length is important to conceptions of beauty.


Childlike proportions are highly desirable

 Visit this fascinating website which shows the relationship between PHI, the golden number, and beauty

This is a computer composite of the most desirable features.

                                                                    Synopsis of Article # 4

                                                 Emblems of Mind : The Inner Life of Music and Mathematics
                                                 by Edward Rothstein

From Kepler and the music of the spheres to Einstein and his violin, music and mathematics seem to share a strong relationship. This

pathbreaking book (Emblems of Mind) seeks to unravel the mystery at the heart of that relationship. In this elegant exploration, music

critic Edward Rothstein reveals the profound and intriguing parallels between music and mathematics. Invoking the poetry of Wordsworth,

the theories of Levi-Strauss, the images of Plato and the philosophy of Kant, Emblems of the Mind is "a harmonious virtuoso performance".

Math and a Music Education

In the introduction to his recent book, Emblems of Mind, Edward Rothstein, chief music critic for the New York Times, describes how his

education and interests encompassed both music and mathematics "Before setting out to make my way in the music business," he writes,

"I was in training to become a 'pure' mathematician. Such esoteric subjects as algebraic topology, measure theory, and nonstandard

analysis were my preoccupations. I would stay up nights trying to solve knotty mathematical problems, playing with abstract phrases

and structures. "But at the same time, I would be lured away from these constructions by another activity. With an enthusiasm that could

come only when critical faculties are in happy slumber, I would listen to or play a musical composition again and again, imprinting my ear

and mind and hands with its logic and sense. Music and math together satisfied a sort of abstract 'appetite,' a desire that was partly

intellectual, partly aesthetic, partly emotional, partly, even, physical."


Rothstein goes on to say that such an experience is by no means unique to him. He notes that music and math have been associated

 throughout history. Pythagoras and his followers saw numbers as models of everything in the physical world, and they identified music

with numbers, noting its scales,  tempos, and other regularities. Johannes Kepler envisioned planetary motions as the "music of the

spheres." Galileo Galilei speculated on the numerical reasons why some combinations of tones are more pleasing than others.

Leonhard Euler considered the same problem in a treatise on consonance and whole numbers. Johann Sebastian Bach sometimes

treated the composition of canons and other types of music as exercises akin to solving mathematical puzzles. Frederic Chopin

 described the fugue as "pure logic in music." And 20th century composers have applied sophisticated mathematical theory in their works.


In his 1993 book, The Music of the Spheres, music critic Jamie James examines and ponders the history of the concept of a musical universe --

a cosmos envisioned as a stately, ordered mechanism both mathematical and musical. Music and science were once intimately intertwined and

united by a grand vision, he points out. But music, like much of the most fundamental art and literature of our culture, has now been relegated to the

obscure margins of the curriculum.


"All art, including music, was a much more serious matter before the self-conscious aestheticism of the late nineteenth century took root,"

James argues. "It is a recent notion that music is a divertissement to be enjoyed in comfortable surroundings at the end of the day, far

removed from the hurly-burly of life's business."


I was reminded of these ideas when I read a report in the latest issue of Nature about a study suggesting that a weekly, structured music

program can boost reading and math skills in early elementary school. The study was done by Martin F. Gardiner and Alan Fox of The

Music School in Providence, R.I., Faith Knowles of the Kodaly Center of America, and Donna Jeffrey of the Start with Arts Program.

The study involved 96 first graders between the ages of 5 and 7 in eight public school classrooms. Forty-eight of the students were exposed

to a weekly singing program that emphasized the sequenced development of pitch and rhythm, often through musical games. The second

group of 48 attended music appreciation classes -- musical training for that age typical of the U.S. public school curriculum. Many of the

pupils enrolled in the singing program had performed poorly in kindergarten, compared to those in the second, or control, group. After

7 months of the new program, however, children in the first group had significantly improved their attitude and behavior, caught up in

reading ability, and outstripped the control group in mathematics.


The researchers found comparable results when the study continued into the second grade, with the addition of a few new students to the

structured singing program. Interestingly, students who had received 2 years of extra music showed a higher level of achievement in

mathematics than those not in the program or those in it for only 1 year.


In their report, Gardiner and his colleagues suggest that the pupils responded to the "pleasurable" aspects of the weekly music program,

which motivated them to acquire the necessary skills to progress. Such training forced mental development that was useful in other areas of learning,

particularly mathematics.


"The maths learning advantage in our data could, for example, reflect the development of mental skills such as ordering and other elements

of thinking on which mathematical learning at this age also depends," the researchers conclude. That's not entirely implausible. Earlier

studies by other groups had suggested links between improved math scores and learning to play a musical instrument.


To my mind, however, the study reported in Nature has too many loose ends to provide satisfactory answers to the issues it was trying to

address.  There really isn't enough information to explain how this particular musical program appeared to succeed for this particular group

of pupils. Perhaps the pupils simply benefited from the special attention they received from dedicated teachers who truly loved what they

were doing and conveyed this enthusiasm to their students. Maybe it wasn't the content of the lessons but the spirit that mattered.

Yet the intriguing interplay between mathematics and music, which goes back to antiquity and possibly much earlier in human history, hints

that there may be something deeper and more basic here.


In his conclusion, Rothstein comments that music and mathematics share not only the clarity of their expression but also their beauty and

mystery. "Our attempt to comprehend music and mathematics, to understand their workings and their purposes, is . . . a model for our

coming to know at all -- a model for our education, for the ways we make distinctions and connections.


"We begin with objects that look dissimilar. We compare, find patterns, analogies with what we already know. We step back and create

 abstractions, laws, systems, using transformations, mappings, and metaphors. This is how mathematics grows increasingly abstract

and powerful; it is how music obtains much of its power, with grand structures growing out of small details."


In music and mathematics, it may be just a modest, but mind-opening, step from the kindergarten to the cosmos.

Copyright © 1996 by Ivars Peterson.

  ARTICLE # 5 Symmetry and ASYMMETRY in ART

Excerpt  taken from an article by Natasha Wallace on John Singer Sargent's Lady Agnew


  I  remember reading an article on the nature of beauty. It was trying to

establish if there was some objective yardstick in which we see things as beautiful. Their conclusions were that indeed there is an objective

 measure of beauty, and it seemed to lie within the idea of symmetry. (why symmetry?)

When I step closer to Lady Agnew, it is apparent she is a beautiful woman with a near perfect symmetrical face -- that is, when the face is

at rest -- "almost schematic" is what Ratcliff calls it. But the things that I notice, and what Ratcliff points out are the things that are not

symmetrical and it is these things that give her character and brings the picture to life for me -- Interesting.


Both Charteris and Richard Ormond with Elaine Kilmurry talk in their books about the nervous energy of the women in Sargent's portraits. Lady

Agnew is no exception here. Although she sits with a total comfortable familiarity with her surroundings and takes ownership of the room -- the

"languid pose", her back to the corner of the chair, leg crossed and angled from her left to right, there is an energy (subtle though it is)

which is palatable. Besides the mouth and her cocked eyebrow, I notice also the hand that grips the chair, the ever so slight downward tilt

of Lady Agnew's head (contrasted by the hint of upward tilt to Madame X's -- although it actually doesn’t) -- the tension here is undeniable.

The thing that strikes me over and over about his life is that John Sargent loved women -- women who were strong in character, intelligent

and of course beautiful women. He didn't feel threatened by strong women (as some men can), and above all he truly enjoyed their presence.

Yet John was not, by anyone's measure, a wilting violet. In fact, he was a true man's man (this comes from many sources) -- over six feet tall

and strong in physique and sporting a full beard. His constitution was incredible and he could push himself hard in work and he did. He

was extremely bright, well read and seemed to retain everything he read. He was opinionated, yet self-abasing, and his manner was charming

and humorous. He was a skilled pianist and played often for friends and played while painting with sitters, moving back and forth between

piano and painting. It was from music that he seemed to draw his energy for painting and it was music that occupied many of his sittings.

(Sargent's Musical Talents)

By: Natasha Wallace
Copyright 1998-99

Article # 6  Looking Good: The Psychology and Biology of Beauty

Charles Feng
Human Biology, Stanford University

In ancient Greece, Helen of Troy, the instigator of the Trojan War, was the paragon of beauty, exuding a physical

Model Cindy Crawford, an example of symmetry
Image courtesy of

brilliance that would put Cindy Crawford to shame. Indeed, she was the toast of Athens, celebrated

 not for her kindness or her intellect, but for her physical perfection. But why did the Greek men find

Helen, and other beautiful women, so intoxicating?

In an attempt to answer this question, the philosophers of the day devoted a great deal of time to

this conundrum. Plato wrote of so-called "golden proportions," in which, amongst other things, the

width of an ideal face would be two-thirds its length, while a nose would be no longer than the

distance between the eyes. Plato's golden proportions, however, haven't quite held up to the rigors

 of modern psychological and biological research -- though there is credence in the ancient Greeks'

attempts to determine a fundamental symmetry that humans find attractive.

Symmetry is attractive to the human eye

Today, this symmetry has been scientifically proven to be inherently attractive to the human eye. It has been defined

not with proportions, but rather with similarity between the left and right sides of the face Thus, the Greeks were only

 partially correct.

By applying the stringent conditions of the scientific method, researchers now believe symmetry is the answer the

Greeks were looking for.


Babies spend more time staring at pictures of symmetric individuals than they do at photos of asymmetric ones.

Moreover, when several faces are averaged to create a composite -- thus covering up the asymmetries that any

one individual may have -- a panel of judges deemed the composite more attractive than the individual pictures.

Victor Johnston of New Mexico State University, for example, utilizes a program called FacePrints, which shows

viewers facial images of variable attractiveness. The viewers then rate the pictures on a beauty scale from one

to nine. In what is akin to digital Darwinism, the pictures with the best ratings are merged together, while the l

ess attractive photos are weeded out. Each trial ends when a viewer deems the composite a 10. All the perfect

10s are super-symmetric.


Scientists say that the preference for symmetry is a highly evolved trait seen in many different animals. Female

 swallows, for example, prefer males with longer and more symmetric tails, while female zebra finches mate with

 males with symmetrically colored leg bands.

Female zebra finches prefer males with symmetric colorings.
Image courtesy of

The rationale behind symmetry preference in both humans and animals is

that symmetric individuals have a higher mate-value; scientists believe that

 this symmetry is equated with a strong immune system. Thus, beauty is i

ndicative of more robust genes, improving the likelihood that an individual's

offspring will survive. This evolutionary theory is supported by research showing

that standards of attractiveness are similar across cultures.


According to a University of Louisville study, when shown pictures of different

individuals, Asians, Latinos, and whites from 13 different countries all had the

 same general preferences when rating others as attractive -- that is those that

 are the most symmetric.

Beauty beyond symmetry

However, John Manning of the University of Liverpool in England cautions against over-generalization, especially by

Western scientists. "Darwin thought that there were few universals of physical beauty because there was much variance

 in appearance and preference across human groups," Manning explained in email interview. For example, Chinese men

used to prefer women with small feet. In Shakespearean England, ankles were the rage. In some African tribal cultures,

men like women who insert large discs in their lips. Indeed, "we need more cross-cultural studies to show that what is

true in Westernized societies is also true in traditional groups," Manning said his 1999 article.


Aside from symmetry, males in Western cultures generally prefer females with a small jaw, a small nose, large eyes,

and defined cheekbones - features often described as "baby faced", that resemble an infant's. Females, however, have

 a preference for males who look more mature -- generally heart-shaped, small-chinned faces with full lips and fair skin.

 But during menstruation, females prefer a soft-featured male to a masculine one. Indeed, researchers found that

female perceptions of beauty actually change throughout the month.


When viewing profiles, both males and females prefer a face in which the forehead and jaw are in vertical alignment.

Altogether, the preference for youthful and even infant-like, features, especially by menstruating women, suggest

people with these features have more long-term potential as mates as well as an increased level of reproductive

fitness. Scientists have also found that the body's proportions play an important role in perceptions of beauty

 as well. In general,  men have a preference for women with low waist-to-hip ratios (WHRs), that is, more

adipose is deposited on the hips and buttocks than on the waist. Research shows that women with high WHRs

(whose bodies are more tube-shaped) are more likely to suffer from health maladies, including infertility and diabetes. However, as is

often the case, there are exceptions to the rule.


Psychologists at Newcastle University in England have shown that an indigenous people located in southeast Peru,

who have had little contact with the Western world, actually have a preference for high WHRs. These psychologists

assert that a general preference for low WHRs is a byproduct of Western culture.

Beauty and choosing a mate

Psychological research suggests that people generally choose mates with a similar level of attractiveness. The

evolutionary theory is that by mating with someone who has similar genes, one's own genes are conserved. Moreover,

 a person's demeanor and personality also influences how others perceive his or her beauty.


In one study, 70% of college students deemed an instructor physically attractive when he acted in a friendly manner,

 while only 30% found him attractive when he was cold and distant. Indeed, when surveyed for attributes in selecting

 a mate, both males and females felt kindness and an exciting personality were more important in a mate than good

looks. Thus, to a certain degree, beauty truly is in the eye of the beholder.


Douglas Yu of the University of East Anglia in Norwich, England, agrees. "It's true by definition. Beauty is always

judged by the receiver," he says. At the same time, he says in an email "there is inter-observer concordance, a measure

of objectivity," so that individual perceptions of beauty, factoring in other characteristics such as personality and

intelligence, can often be aggregated to form a consensus opinion. One of the offshoots of Yu's work in ethnobiology

 was a piece in Nature in 1998 that showed that the hourglass-body standard of beauty in women, previously thought

 to be `universally' preferred, was in fact likely swayed by advertising.

The halo effect

In society, attractive people tend to be more intelligent, better adjusted, and more popular. This is described as the

 halo effect - due to the perfection associated with angels. Research shows attractive people also have more

occupational success and more dating experience than their unattractive counterparts. One theory behind this halo

effect is that it is accurate -- attractive people are indeed more successful.


An alternative explanation for attractive people achieving more in life is that we automatically categorize others before

having an opportunity to evaluate their personalities, based on cultural stereotypes which say attractive people must be

intrinsically good, and ugly people must be inherently bad. But Elliot Aronson, a social psychologist at Stanford University,

 believes self-fulfilling prophecies - in which a person's confident self-perception, further perpetuated by healthy feedback

 from others - may play a role in success as well. Aronson suggests, based on the self-fulfilling prophecy that people

who feel they are attractive - though not necessarily rated as such - are just as successful as their counterparts who

are judged to be good-looking.


Whatever the reason, the notion that attractiveness correlates with success still rings true. Yet beauty is not always

advantageous, for beautiful people, particularly attractive women, tend to be perceived as more materialistic, snobbish,

 and vain.


Symmetrical PlantVitruvian Man 1492For better or worse, the bottom line is that research shows beauty matters; it pervades society and affects how we

choose loved ones. Thus, striving to appear attractive may not be such a vain endeavor after all. This isn't to say

plastic surgery is necessarily the answer. Instead, lead a healthy lifestyle that will in turn make you a happier person.