Is beauty in the eye of the beholder or is it a mathematical function? The answer seems to be both. The human face is the ultimate object of beauty, containing symmetry and huge amounts of information about the person. Mathematicians and architects illustrate some of the equations behind beauty, such as the Fibonacci Sequence, logarithmic spirals, fractals, and the Golden Section.
The face is our pathway to perception and communication. We recognize a face we know instantly. Babies' brains recognize and remember faces. A face is the first thing they see.
What causes us to perceive a face as beautiful? We perceive beauty through symmetry and mediocrity. Identifying with the commonalties found in the face we see ourselves. Symmetry is beautiful because it announce physical health. Generally, a symmetrical face is on a symmetrical body, and thus a body with overall good general health. A beauty is a sign post of fitness and sexually health as least partially explained by symmetry.
How have we come to perceive symmetry? If we expose an individual to set of stimuli where the average stimulus is symmetric then an individual will tend to have a symmetric template in their mind. Accordingly, exposure to a set of stimuli containing an average subset builds up preference for this average subset, which becomes a recognizable template. This recognizable template is preferred simply because it is easily recognized. This recognizable template can be mathematically mapped. Thus, the correct mathematics will make you into a model, but it will not make you into a beautiful person.
Galileo said, "[b]eauty is the book of nature written in mathematical characters". Mathematics is the loom upon which God weaved the Universe. The elegance of mathematics underlies the all beauty found in nature. When you see the moon, you probably do not think of Π (Pi). Mathematics can be used to describe much of the world we live in.
The easiest math to see is geometry: the study of points, planes, angles and solids. You can search out structure in nature's diversity. Our mind searches our forms that we clearly recognize. Our pattern recognition circuits in our brains tell us we see circular moon and triangular pines. Even through a microscope we see a world of floating geometry.
How we sense the world depends on its structure. A rose is a rose, is a rose. No, not quite. When you smell its fragrance you do not smell hydrogen, oxygen, and carbon. You smell an arrangement of those atoms into molecules, which create chemicals. With out that structure you would smell water and soot.
Mathematics can explain the smell of a rose, but can it explain breeding rabbits. A pair of rabbits mature, breed and produce a pair of baby rabbits. Next season they produce another pair to join their siblings - 3 pairs altogether. Next season they breed again, and so do the older siblings, now old enough to breed themselves, whilst the younger siblings are not yet old enough to breed - 5 pairs altogether - and so on. The question is, if parent rabbits can breed every season, whilst infant rabbits need a season to mature and become parents, (and assuming no rabbits die!) how many pairs will there be after a given number of seasons? The answer is found in the Fibonaci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind. This sequence can be applied to the cycle of life in plant growth.
Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number:
3 petals: lily, iris
5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
8 petals: delphiniums
13 petals: ragwort, corn marigold, cineraria,
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the asteraceae family
The spiral shapes of a nautilus are called Equiangular or Logarithmic spirals, which are found in the Gold Rectangle. Given a line segment, we can divide it into two segments A and B, in such a way that the length of the entire segment is to the length of the segment A as the length of segment A is to the length of segment B. If we calculate these ratios, we see that we get an approximation of the Golden Ratio, which is expressed as follows:
CB/AC equals AC/AB
The Fibonacci sequence relates closely to the Golden Ratio and to logarithmic spirals. Logarithmic spirals are simply spirals that increase at a logarithmic rate. The Golden Ratio, however, is a special fraction equivalent to about 1: 1.618. A logarithmic spiral can be generated by subdividing a Golden Rectangle into increasingly smaller squares and Golden Rectangles. This subdivision begins by fitting a square within the Golden Rectangle. The remaining space forms a new, smaller Golden Rectangle. By repeating this process, the spiral form soon becomes evident. Furthermore, the subdivided sections can be thought about as Fibonacci numbers.
Regardless of the science, the Golden Ratio retains a mystique, partly because excellent approximations of it turn up in many unexpected places in nature. The spiral inside a nautilus shell is remarkably close to the Golden Section, and the ratio of the lengths of the thorax and abdomen in most bees is nearly the Golden Ratio. Even a cross section of the most common form of human DNA fits nicely into a Golden Decagon. The Golden Ratio and its relatives also appear in many unexpected contexts.
There are interesting symmetries shared by the limerick and ragtime, which can be observed and heard in their family groups of stressed and unstressed syllables, or beats, and which lie at the heart of what gives these forms their characteristic structure or "feel". They possess self-similar qualities which are related to fractal models used by contemporary scientists, and can provide a keen insight into some quite profound inter-relationships between the arts and sciences.
Let's first look at the underlying stress patterns in the metre of the limerick by writing it out using "di" for an unstressed syllable and "dum" for a stressed one like this:
di dum di di dum di di dum
di dum di di dum di di dum
di dum di di dum
di dum di di dum
di dum di di dum di di dum
If we count up the different types of syllable, we discover that out of a total of 34 syllables, 21 are unstressed (the di's) and 13 are stressed (the dum's). There are 8 syllables in the longer lines and 5 in the shorter ones. Out of the total of 5 lines, 3 are long ones containing 3 metrical feet (i.e. 3 stressed dum's), and 2 are shorter ones containing 2 metrical feet. Anyone familiar with elementary mathematics will recognize these numbers as belonging to the well known Fibonacci series.
Despite these numerous appearances in works of art throughout the ages, there is an ongoing debate among psychologists about whether people really do perceive the Golden Shapes, particularly the Golden Rectangle, as more beautiful than other shapes. Yet, most psychologists will accept that we are prime to recognize patterns and symmetry.
The face is our pathway to perception and communication. We recognize a face we know instantly. Babies' brains recognize and remember faces. A face is the first thing they see.
What causes us to perceive a face as beautiful? We perceive beauty through symmetry and mediocrity. Identifying with the commonalties found in the face we see ourselves. Symmetry is beautiful because it announce physical health. Generally, a symmetrical face is on a symmetrical body, and thus a body with overall good general health. A beauty is a sign post of fitness and sexually health as least partially explained by symmetry.
How have we come to perceive symmetry? If we expose an individual to set of stimuli where the average stimulus is symmetric then an individual will tend to have a symmetric template in their mind. Accordingly, exposure to a set of stimuli containing an average subset builds up preference for this average subset, which becomes a recognizable template. This recognizable template is preferred simply because it is easily recognized. This recognizable template can be mathematically mapped. Thus, the correct mathematics will make you into a model, but it will not make you into a beautiful person.
Galileo said, "[b]eauty is the book of nature written in mathematical characters". Mathematics is the loom upon which God weaved the Universe. The elegance of mathematics underlies the all beauty found in nature. When you see the moon, you probably do not think of Π (Pi). Mathematics can be used to describe much of the world we live in.
The easiest math to see is geometry: the study of points, planes, angles and solids. You can search out structure in nature's diversity. Our mind searches our forms that we clearly recognize. Our pattern recognition circuits in our brains tell us we see circular moon and triangular pines. Even through a microscope we see a world of floating geometry.
How we sense the world depends on its structure. A rose is a rose, is a rose. No, not quite. When you smell its fragrance you do not smell hydrogen, oxygen, and carbon. You smell an arrangement of those atoms into molecules, which create chemicals. With out that structure you would smell water and soot.
Mathematics can explain the smell of a rose, but can it explain breeding rabbits. A pair of rabbits mature, breed and produce a pair of baby rabbits. Next season they produce another pair to join their siblings - 3 pairs altogether. Next season they breed again, and so do the older siblings, now old enough to breed themselves, whilst the younger siblings are not yet old enough to breed - 5 pairs altogether - and so on. The question is, if parent rabbits can breed every season, whilst infant rabbits need a season to mature and become parents, (and assuming no rabbits die!) how many pairs will there be after a given number of seasons? The answer is found in the Fibonaci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind. This sequence can be applied to the cycle of life in plant growth.
Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number:
3 petals: lily, iris
5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
8 petals: delphiniums
13 petals: ragwort, corn marigold, cineraria,
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the asteraceae family
The spiral shapes of a nautilus are called Equiangular or Logarithmic spirals, which are found in the Gold Rectangle. Given a line segment, we can divide it into two segments A and B, in such a way that the length of the entire segment is to the length of the segment A as the length of segment A is to the length of segment B. If we calculate these ratios, we see that we get an approximation of the Golden Ratio, which is expressed as follows:
CB/AC equals AC/AB
The Fibonacci sequence relates closely to the Golden Ratio and to logarithmic spirals. Logarithmic spirals are simply spirals that increase at a logarithmic rate. The Golden Ratio, however, is a special fraction equivalent to about 1: 1.618. A logarithmic spiral can be generated by subdividing a Golden Rectangle into increasingly smaller squares and Golden Rectangles. This subdivision begins by fitting a square within the Golden Rectangle. The remaining space forms a new, smaller Golden Rectangle. By repeating this process, the spiral form soon becomes evident. Furthermore, the subdivided sections can be thought about as Fibonacci numbers.
Regardless of the science, the Golden Ratio retains a mystique, partly because excellent approximations of it turn up in many unexpected places in nature. The spiral inside a nautilus shell is remarkably close to the Golden Section, and the ratio of the lengths of the thorax and abdomen in most bees is nearly the Golden Ratio. Even a cross section of the most common form of human DNA fits nicely into a Golden Decagon. The Golden Ratio and its relatives also appear in many unexpected contexts.
There are interesting symmetries shared by the limerick and ragtime, which can be observed and heard in their family groups of stressed and unstressed syllables, or beats, and which lie at the heart of what gives these forms their characteristic structure or "feel". They possess self-similar qualities which are related to fractal models used by contemporary scientists, and can provide a keen insight into some quite profound inter-relationships between the arts and sciences.
Let's first look at the underlying stress patterns in the metre of the limerick by writing it out using "di" for an unstressed syllable and "dum" for a stressed one like this:
di dum di di dum di di dum
di dum di di dum di di dum
di dum di di dum
di dum di di dum
di dum di di dum di di dum
If we count up the different types of syllable, we discover that out of a total of 34 syllables, 21 are unstressed (the di's) and 13 are stressed (the dum's). There are 8 syllables in the longer lines and 5 in the shorter ones. Out of the total of 5 lines, 3 are long ones containing 3 metrical feet (i.e. 3 stressed dum's), and 2 are shorter ones containing 2 metrical feet. Anyone familiar with elementary mathematics will recognize these numbers as belonging to the well known Fibonacci series.
Despite these numerous appearances in works of art throughout the ages, there is an ongoing debate among psychologists about whether people really do perceive the Golden Shapes, particularly the Golden Rectangle, as more beautiful than other shapes. Yet, most psychologists will accept that we are prime to recognize patterns and symmetry.
