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They and their surprising relationship serve as the basis of the rest of notion of order by taking the Greek word for it, cosmos, and giving this name to the universe. Indeed, when Jewish and early Christian thinkers began to view Platonic forms R. David *Nieto of London (–), in his Kuzari Sheni, devoted a. present in Christian neo-Platonic authors who influenced the cognitive context philosophically relevant that this thesis states in relation to its particular hijos tanben los comen, aunque sean sus nietos y hermanos, y a las vezes cosmos created ultimately by God, they could retain hidden in their surface something. We especially want to understand Maimonides in his relations to the science of his time. If the world, the cosmos, were eternal, as described by Plato or. Aristotle Taking exception to an anti-Platonic argument in Aristotle's metaphysics David Nieto However interesting his remarks may be, they do not deal with.
So, the biggest possible golden rectangle, ABCD, can be drawn where the length of the rectangle is same as the base line. Based on this L-shaped form, a golden rectangle can be drawn. Thirdly, a square can be considered to draw a golden rectangle. Considering the midpoint of the base of the square as the centre of a circle arc and the length between the upper corner and the midpoint as the radius of that circle, an arc can be drawn that will intersect the extended base line of the square.
Drawing a rectangle based on the new intersecting point and the square will make a golden rectangle[1,2,15].
So, using the intersecting point the constructed rectangle becomes the golden rectangle, ABCD. The proportionally decreasing squares produce a spiral by using the arcs having radius as the length of the side of the squares[1,2]. The two diagonals intersect at the point O which is called the sink centre of the spiral and all other diagonals of the smaller golden rectangles must lie on these two diagonals. The second biggest square, CGME, have the second arc of the golden spiral while the radius is equal to the side of that square and the point M is its centre.
This process can be continued and this will construct the spiral shape called the Golden Spiral. Based on Figure 4 a and bEquation 1 can be written as, 2 Figure 4. The absolute value of the cosine of any angle of the golden triangle is equal to or ; and. These pentagon and pentagram are possible to characterized using the golden triangle. The decagram and decagon also yield a series of golden triangles by connecting the centre point with any two adjacent edges[1,2,15]. A golden triangle also can be considered as a whirling triangle where a spiral property is identified by subdividing into reciprocal triangles.
A logarithmic spiral, also known as golden spiral, can be produced by joining the arcs having radius as the lengths of the sides of the reciprocal triangles[2,13]. Figure 6 a shows an isosceles triangle, ABC, having two base angles of 72 degrees. There could be only two sink centres of the two possible spirals for a golden triangle. Golden Angle and Golden Ellipse A circle can be divided into two arcs in the proportion of the golden ratio, where the smaller arc marks a central angle of A Golden Section Ellipse is an ellipse drawn inside a Golden Section Rectangle where it has the same proportion of the major and minor axis as 1: Golden spiral formed on the whirling golden triangle 2.
Concept on Dynamic Rectangles Based on the rational and irrational number of the proportion, a rectangle can be considered as either a static rectangle or a dynamic rectangle. Static rectangles do not produce a series of visually pleasing ratios of surfaces while subdividing. On the other hand, dynamic rectangles produce an endless amount of visually pleasing harmonic subdivisions and surface ratios.
Figure 7 shows the various dynamic rectangles and their construction strategies. Representation and construction of various dynamic rectangles 3.
The same concept also can be applied to draw a golden rectangle which could be the geometrical proof of the equation of Phi. If the new rectangle is divided into two equal sections, the newly formed rectangles will be the Golden Rectangles. So Equation 3 can be proofed from Equation 411 13 4. Golden Section and the Beauty of Nature In 12th century, the Leonardo Fibonacci questioned about the population growth of the rabbits under ideal circumstances, such as no predators to eat them or no dearth of food and water that would affect the growth rate.
The answer of the question is the Fibonacci Sequence of Numbers, also known as Fibonacci Numbers that starts from 1 and each new number of the series is simply the sum of the previous two numbers. So the second number of the series is also 1, the sum of the previous 1 and 0 of the series. The sequence of the number looks like the series bellow. Fibonacci numbers are said as one of the Nature's numbering systems because of its existence not only in the population growth of rabbits, but also everywhere in Nature, from the leaf arrangements in plants to the structures in outer space.
The special proportional properties of the golden section have a close relationship with the Fibonacci sequence. Any number of the series divided by the contiguous previous number approximates 1. Some flowers having different number of petals related to Fibonacci numbers, a White Calla Lily having one petal, b Euphorbia having two petals, c Trillium with three petals, d Hibiscus having five petals, e Buttercups with five petals, f Bloodroot with eight petals, g Black Eyed Susan having thirteen petals, h Shasta Daisy having 21 petals and i Daisy with 34 petals Golden section preferences are considered as an important part of human beauty and aesthetics as well as a part of the remarkable proportions of growth patterns in living things such as plants and animals[,13,15,18].
Many flowers have the arrangement in petals that are to the Fibonacci numbers. Some display single or double petals. Three petals are more common like Lilies and Iris. Some have 8, 13, 21, 34, 55 and even 89 petals. All these numbers are consecutive Fibonacci numbers. The petal counts of Field Daisies are usually thirteen, twenty-one or thirty-four. The seed heads are also follow the Fibonacci spiral arrangement. Other flowers having four or six petals also have a deep relation with Fibonacci numbers where they can be grouped into two and three respectively having two members each.
Passion flower also known as Passiflora Incarnata is a perfect example having the existence of the Fibonacci Numbers. Figure 10 a shows the back view of Passion flower where the 3 sepals that protected the bud are at the outermost layer, then 5 outer green petals followed by an inner layer of 5 more paler green petals.
The front view is shown in Figure 10 b where the two sets of 5 green petals are at outermost layer with an array of purple and white stamens, in the centre there are 5 greenish T-shaped stamens and at the uppermost layer has 3 deep brown carpels. Passion Flower Passiflora Incarnata from the back a and front b having the examples of Fibonacci Numbers in nature Romanesco Brocolli is one kind of vegetable that looks and tastes like a cross between brocolli and cauliflower.
It is peaked in shape and has the arrangement with identical but smaller version of the whole thing that makes the spirals related to the Golden Spiral. The well grown Cauliflower has the shape almost as a pentagon which has an intimacy with the Golden Section and Golden Triangles. This enigma can easily be identified as shown in Figure From the cross sectional views of different fruits shown in Figure 12, Banana, Cantaloupe, Cucumber, Kiwano fruit also known as African cucumber, Watermelon have three sections.
Some of the fruits have two subdivisions in each sectional part. Apple seeds are arranged like a pentagram shape that creates five sections. The pentagram structure also can be found in the Star Fruits. Okra also has five sections with the properties of a pentagon. Orange is divided into ten sections which can be grouped into five where each group contains two sub sections. Cross sectional view of some fruits and vegetables, a Banana has three sections, b Cantaloupe also have three sections, c Cucumber having three sections, d Kiwano fruit with three sections, e Watermelon also have three sections, f Apple having five seeds arranged as like a pentagram, g Orange having ten sections can be grouped into five, h Okra have the pentagon shape with ten seeds and i Star fruits have five sections with a pentagram shape There is a direct correlation between the bi-directional spirals of the seed florets and Fibonacci Numbers.
Not only that, the spiral has also a great relationship with the sequence of golden spiral. Figure 13 shows that the sunflower has 34 spirals in clockwise direction and 21 spirals in counter clockwise direction where these two numbers are the elements of the Fibonacci series. Spiral arrangement of the seed florets of a Sunflower Figure Appearance of golden spiral in nature a Orange petals seed florets, b Sunflower seed florets, c Pine Cones, d Broccoli, e Golden Ratio flower and f Petals of a Rose The spiral happens naturally because each new cell is formed after a turn.
Plants grow new cells in spirals format and this pattern is seen on the seeds arrangement of the beautiful sunflower, orange petals seed florets, pine cones, broccoli and even in the petals of rose. Leaves, branches and petals also grow in spirals form too, so that the new leaves don't block the older leaves from the sun ray or the maximum amount of rain or dew gets directed down to the roots.
If a plant has spirals, the rotation tends to be a fraction made with two successive Fibonacci Numbers, for example,or even also common that getting closet to the golden ratio. Loosely distributed leafs structure where the successive leaves in the spiral are numbered and lined from the centre of the spiral Figure Golden Angle and Golden Spiral are also found in some plants and ferns.
Even the eyes of a pineapple follow the golden ratio and golden spiral. Much of the things that are viewed as beautiful by the naked eye establish the factor that possesses the Golden Ratio in one way or another. The term phyllotaxis means "leaf arrangement" in Greek and was coined in by Charles Bonnet, a Swiss naturalist.
In the s, a scientist brothers found that each new leaf on a plant stem is positioned at a certain angle to the previous one and this angle is constant between leaves usually about In the top view of the plant, shown in Figure 15, the angle formed between a line drawn from the stem to the leaf and a corresponding line for the next leaf, is generally a fixed angle which is known as the Divergence Angle or Golden Angle[3,15,18].
Figure 16, a to dshow some succulent plants where this characteristic is clearly visible. Some various Cactuses are showing the existence of the Golden Spiral in their various growth patterns Research has shown that buds placed along a spiral and separated by an angle of InWilhelm Hofmeister suggested that new primordia, cells that will later develop into leaves or petals, always form in the least crowded spot along the growing tip of a plant called a meristem.
This is also known as Hofmeister's rule. Because the plant is continuously growing, each successive primordium forms at one point along the meristem and then moves radially outward at a rate proportional to the stem's growth. Hofmeister's rule tells that the second primordium is placed as far as possible from the first, and the third is placed at a distance farthest from both the first and the second primordia. As the number of primordia increases, the divergence angle eventually converges to a constant value of Various Firn tops and Vine Tendrils show almost the same characteristics of the Fibonacci spiral or Golden spiral.
Not only that, Fibonacci spiral also found on some cactuses and some fruits. Most pineapples have five, eight, thirteen, or twenty-one spirals of increasing steepness on their surface. All of these are Fibonacci numbers. Figure 17 shows various types of Cactuses having almost the same properties of Golden spiral in their growth patterns.
Plants illustrate the Fibonacci sequence in the numbers and arrangements of petals, leaves, sections and seeds. Plants that are formed in spirals, such as pinecones, pineapples and sunflowers, illustrate Fibonacci numbers. Many plants produce new branches in quantities that are based on Fibonacci numbers. German psychologist Adolf Zeising, whose main interests were mathematics and philosophy, found the Golden Section properties in the arrangement of branches along the stems of plants and of veins in leaves.
Figure 18 shows the existence of Golden Spiral and Golden Angle on the leaves of trees. Zeising also concentrated on the skeletons of animals and the branching of their veins and nerves, on the proportions of chemical compounds and the geometry of crystals, even on the use of proportion in artistic endeavours.
He found that the Golden Ratio plays a universal and important role in all of these phenomena. There are a lot of examples of the Golden Section or Divine Proportion found on animals, fishes, birds, insects, and even on some snails. The eye, fins and tail of a dolphin fall at Golden Sections of the length of its body. A penguin body also can be described by the Golden Ratio properties. The Rainbow Trout fish, shown in Figure 19 halso shows the same properties where three golden rectangles together can be fitted on its body where the eye and the tail fin falls in the reciprocal golden rectangles and square.
The individual fins also have the golden section properties. An experiment on Blue Angle fish shows that the entire body of the fish fits perfectly into a golden section rectangle, shown in Figure 19 g. The mouth and gill of the Angle fish are on the reciprocal golden section point of its body height.
Shells like Chambered Nautilus, Conch Shell, Moon Snail Shell, Atlantic Sundial Shell show the spiral growth pattern where the first three have almost like the golden spiral form, shown in Figure 19 a c and d. Tibia Shell spiral growth is not like golden spiral but the sections of the spiral body can be described by the golden mean properties.
Golden spiral also be found on the tail of Sea Horses. The Star fish has the structure like a pentagram which has a close intimacy with golden ratio. The body sections of ants are to the Golden Ratio. The same properties also are found on the beautiful design of butterfly wings and shapes. Weather patterns, Whirlpool have almost the same form like the golden spiral.
Even the Sea Wave sometimes shows almost the same spiral pattern. The three rings of Saturn are designed naturally based on the Golden Ratio. The Galaxy, Milky Way, also has the spiral pattern almost like golden spiral. Relative planetary distances of Solar System also have the golden ratio properties. The orbital distances of planets are generally measured from the Earth.
Mercury is the first planet of the solar system where the Earth is on third. So, to take the measurements from Earth would be like starting the Fibonacci numbers from somewhere in the middle of the sequence. If the measurement is started from the first planet of the solar system, Mercury, a very special and interesting property will be outspread.
The asteroid belt is a part of the solar system and the largest asteroid is Ceres, which is one third of the total mass of all the asteroids. So, Ceres could be the logical orbit. If the average of the mean planet orbital distances of each successive planet is taken in a relation to the one before it, the value will be following the golden ratio number. Relative mean distances where Mercury is considered at the beginning, 1. Total relative mean distance is Some example of the physical universe where Golden Section properties are exists The Fibonacci sequence can be considered more or less the beauty of nature.
The human body also exemplifies the mesmerizing occurrence of Golden Ratio. Total height of the body and the distance from head to the finger tips make the same ratio as Phi. Again the distances from head to naval and naval to hill also express the golden proportion[1,2,4]. The bones of fingers in human hand are related to each other by a ratio of Phi. For instance the proportion between the forearm and upper arm also follows the rule of golden ratio and the same ratio appears between the hand and forearm.
Human faces are also comprised of this ratio within the relationships between the eyes, ears, mouth and nose[17,22]. The human being is the most beautiful and most perfect instauration of Allah. We have indeed created human in the best of modules. It is known that the Arabic text direction is from right to left. The 26 characters can be divided into two sections p and q. The ratio between p and q is 1. Indeed, Quran is a miracle for its eloquence in language, for applicability and relevance of its verses in all spheres of human life, for its inimitable verses and in many other aspects.
Divine Foolishness: August
It does not need to prove as scientific, rather we need to guide by it. The relationship of golden section is observed on the design of the human body shape and structure. It consists of one trunk, one head, one heart etc. Equally important, this phase saw the evolution and partial clarification of axiomatic systems and deductive proofs.
The next major phase, classical mathematics, began more than 1, years later, with the Cartesian fusion of geometry and algebra and the use of limiting processes in the calculus. From these evolved, during the eighteenth and nineteenth centuries, the several aspects of classical analysis. Other contributions of this phase include non-Euclidean geometries, the beginnings of probability theory, vector spaces and matrix theory, and a deeper development of the theory of numbers.
About a hundred years ago the third and most abstract and demanding phase, known as modern mathematics, began to evolve and become separate from the classical period.
This phase has been concerned with the isolation of several recurrent structures of analysis worthy of independent study—these include abstract algebraic systems for example, groups, rings, and fieldstopological spaces, symbolic logicand functional analysis Hilbert and Banach spaces, for example —and various fusions of these systems for example, algebraic geometry and topological groups.
The rate of growth of mathematics has been so great that today most mathematicians are familiar in detail with the major developments of only a few branches of the subject. Our purpose is to give some hint of these topics. The reader interested in a somewhat more detailed treatment will find the best single source to be Mathematics: Other general works are Courant and RobbinsFriedmanand Newman More specific references are given where appropriate.
We do not here discuss probability, mathematical statistics, or computation, even though they are especially important mathematical disciplines for the social sciences, because they are covered in separate articles in the encyclopedia. Ancient mathematics The history of ancient mathematics divides naturally into three periods. In the first period, the pre-Hellenic age, the beginnings of systematic mathematics took place in ancient Egypt and in Mesopotamia.
Contrary to much popular opinion, the mathematical developments in Mesopotamia were deeper and more substantial than those in Egypt.
The Babylonians developed elementary arithmetic and algebra, particularly the computational aspects of algebra, to a surprising degree. An authoritative and readable account of Babylonian mathematics as well as of Greek mathematics is presented by Neugebauer The second period of ancient mathematics was the early Greek, or Hellenic, age. The fundamentally new step taken by the Greeks was to introduce the concept of a mathematical proof.
These developments began around b. The third period is the Hellenistic age, which extended from the third century b. Ptolemy systematized and extended Greek mathematical astronomy and its mathematical methods. The mathematical sophistication of Archimedes and the richness of applied mathematics evidenced by the Almagest were not equaled until the latter part of the seventeenth century.
Classical analysis The intertwined and rapid growth of mathematics and physics during the seventeenth, eighteenth, and nineteenth centuries centered in a major way on what is now called classical analysis: At the basis of all this are two major ideas, function and limit.
Since the most general notion of function can relate any two sets of objects, not just sets of numbers, it is sometimes desirable to emphasize the numerical character of the function. Although a real-valued function has been defined as a set of ordered pairs of numbers, x,ywhere the domain of x is is an unspecified set of numbers, the subsequent discussion of functions is mostly confined to the familiar case in which the domain of x is an interval of numbers.
Even when the discussion applies more generally, it is helpful to keep the interval case in mind. A desire to understand limits was apparent in Greek mathematics, but a correct definition of the concept eluded the Greeks. A fully satisfactory definition, which was not evolved until the nineteenth century by Augustin Louis Cauchyis the following: The calculus The calculus defines two new concepts, the derivative and the integral, in terms of function and limit.
They and their surprising relationship serve as the basis of the rest of mathematical analysis. One of the earliest and most important applications in the social sciences of the concept of a derivative has been to the mathematics of marginal concepts in economics.
Marginal utility, marginal rate of substitution, and other marginal concepts are defined in a similar fashion. Many of the fundamental assumptions of economic theory receive precise formulation in terms of these marginal concepts. The solution, which will not be stated precisely, involves the following steps: For more advanced work, the concept of the length of an interval is generalized to the concept of the Lebesgue measure of a set, and the Riemann integral is generalized to the Lebesgue integral Roughly, the vertical columns used to approximate the area in the Riemann integral are replaced in the Lebesgue integral by horizontal slabs.
Although the interpretation of the integral as an extension of the elementary concept of area is important, even more important is its relation called the fundamental theorem of the calculus to the derivative: This fact plays a crucial role in the solution of many problems of classical applied mathematics that are formulated in terms of derivatives of functions.
Introductions to the calculus and elementary parts of analysis are Apostol — and Bartle Other algebraic equations implicitly define sets of numbers for which they hold.
A functional equation is an equality stated in terms of an unknown function; it implicitly defines those functions as in the algebraic case, there may be more than one that render the equality true.
This is the case of continuous compound interest. This is a simple example of an ordinary differential equation, the solution of which is any function having the property that its derivative is k times the function. A vast literature is concerned with the solutions to this class of equations for different restrictions on P, Q, and R; most of the famous special functions used in physics—Bessel, hypergeometric, Hankel, gamma, and so on—are solutions to such differential equations see Coddington Partial differential equations Many physical problems require differential equations a good deal more complicated than those just mentioned.
For example, suppose that there is a flow of heat along one dimension, x. These are called partial derivatives. According to classical physics, temperature changes due to conduction in a homogeneous one-dimensional medium satisfy the following partial differential equation: Problems involving two or more independent variables usually, time and some or all of the three space coordinates —fluid flow, heat dissipation, elasticity, electromagnetism, and so on—lead to partial differential equations.
Their solution is often very complex and requires the specification of the unknown function along a boundary of the space. This requirement is called a boundary condition.
See Akademiia Nauk S. Some physical problems lead to integral equations. Integral equations arise in empirical contexts for which it is postulated that the value of a function at a point depends on the behavior of the function over a large region of its domain. There is a large body of literature dealing with the solution of various types of integral equations, especially those of interest in physics and probability theory.
Although both differential and integral equations and mixtures of the two, called integrodifferential equations are examples of functional equations, that term is often restricted to equations that involve only the unknown function, not its derivatives or integrals. The choice of K is usually referred to as the selection of the base of the logarithm.
Difference equations are functional equations of special importance in the social sciences. They arise both in the study of discrete stochastic processes in learning theory, for example and as discrete analogues of differential equations. The equation states a relation among values of the unknown function for several successive integers. For example, the second-order, linear difference equation—the analogue of the second-order, linear, ordinary differential equation, described above—is of the form.
An introduction to difference equations is Goldberg Given a functional equation—in the most general sense—the answer to the question of whether a solution exists is not usually obvious. Exhibiting a solution, of course, answers the question affirmatively, but often the existence of a solution can be proved before one is found.
Such a result is known as an existence theorem. If a solution exists, it is also not usually obvious whether it is unique and, if it is not unique, how two different solutions relate to one another. A statement of the nature of the nonuniqueness of the solutions is known, somewhat inappropriately, as a uniqueness theorem.
Some rather general existence and uniqueness theorems are available for differential and integral equations, but in less well understood cases considerable care is needed to discover just how restrictive the equation is. A general work on functional equations is Aczel Three other areas of classical analysis Three other branches of classical analysis will be briefly discussed. Extremum problems For what values of its argument does a function assume its maximum or its minimum value?
This type of problem arises in theoretical and applied physics and in the social sciences. These statements should be intuitively clear for graphs of simple functions.
From these results it is easy to find, for example, which rectangle has the maximum area when the perimeter is held constant: Within the past twenty years new classes of extremum problems have been posed and partially solved; they are mainly of concern in the social sciences, and they go under the names of linear, non-linear, and dynamic programming. An example of a linear programming problem is the following diet problem.
If Xi is the amount of food fi in the diet, the diet will be acceptable only if the following n inequalities are fulfilled: It has been significant in the growth of several two-dimensional, continuous physical theories, including parts of electromagnetism, hydrodynamics, and acoustics, but so far its applications in the social sciences have been mainly restricted to mathematical statistics, as in the concept of the characteristic function of a probability distribution. It is clear from this result that the mere supposition that the derivative exists is a much stronger condition for complex-valued functions than for ordinary numerical functions.
Such functions, which are called analytic, are very strongly constrained—among other things, specifying an analytic function over a small region determines it completely—and this fact has been effectively exploited to solve many two-dimensional problems of theoretical and practical interest. For a right ordering of life, all desire must be ordered and formed according to the hierarchy of goods. The goods of pleasure and economy, so often exalted by the moderns as the principal goods of society, must be desired in subordination to the highest goods, the goods known by the intellect in philosophic contemplation.
This does not mean that men must devote all of their time exclusively to such contemplation; this is not only a practical impossibility, but it would mean neglecting the lower parts of the human being, which, though lower, are capable of being ordered by the higher. The highest activity of man is thus the contemplation of the good, and virtue consists first in this, and secondly in all other activities performed in accordance with this good.
The more the activity of man is regulated by his reason, both in the activity of reason itself and of other faculties in accord with reason, the more perfectly does he tend towards the common good. Oftentimes, moral philosophers find themselves preoccupied with a dichotomy between self-love and altruism, as if this were the fundamental tension in need of resolution in the study of ethics or politics. The classical doctrine of the common good, however, allows the philosopher, and the political man himself, to transcend this dichotomy.
The common good is, by its very nature, the good of all individuals, an eminently personal good, but one which is shared by all without thereby being diminished. Indeed, the joy in possessing such a good is even increased, even made possible, by being thus shared. No individual struggles with the tension between self-love and altruism if he devotes himself to the common good, because thereby he works to ensure the good that by its nature diffuses itself to all equally, to himself and to all others who would receive it in common.
Not all men are willing to receive the common good, however. This indeed is the very basis of sin: Sin also occurs when the common good is desired, not as common, but as if it were exclusively the good of the one desiring it, even to the point of wishing that others not also attain it. The affirmation of the value of each person, as someone individual and unique, prescinds from the fact of commonality between him and other persons, i. DeKoninck argued, on the contrary, that persons receive their true and greatest dignity only from the commonality of the good itself; they are not good apart from the good which communicates itself to many, nor, therefore, apart from the good of the many.
Love, therefore, cannot be ordered aright unless it be also a love of the good as good for the many: The fallen angels did not refuse the perfection of the good which was offered to them; they refused the fact of its being common, and they despised this community.
Thus to love the good in which the blessed participate in order to acquire or possess it does not make man well disposed towards it, for the evil envy this good also; but to love it in itself, in order that it be conserved and spread, and so that nothing be done against it, this is what makes man well disposed to this society of the blessed; and this is what charity consists of, to love God for himself, and the neighbor who is capable of beatitude as oneself.
Quaestiones Disputatae de Caritate, a. A kind of program emerges for the development of communities, which are the most perfect when united by a shared love of common goods. Likewise, the interaction between any two or few individuals, the raising of families, the building of villages, and the regulation of entire cities, cannot be carried out without reference to the common good.
All of the arts and practical scienceseven down to the most mundane and servile, collaborate in common subordination to the political science, for the sake of promoting the common good.