Manual Pythagorean Plato and the Golden Section Sacred Geometry (dissertation)

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The concept of this ratio was first called as the Divine Proportion in the early 's when Da Vinci provided his illustrations for a dissertation which was published by Luca Pacioli in and entitled as De Divina Proportione. This book contains drawings of the five Platonic solids. The Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve balance and beauty. Leonardo Da Vinci used it to define all the fundamental proportions of his painting of The Last Supper, from the dimensions of the table, at which Christ and the disciples sat, to the proportions of the walls and windows in the background.

Johannes Kepler AD who discover the elliptical nature of the orbits of the planets around the Sun also reflects the concept of the Divine Proportion. The term Phi was not used before the 's until the American mathematician Mark Barr who used the Greek letter Phi to designate this proportion in his works. The proportion is said as Golden and Divine because of its unique properties to open the door of deeper understanding of beauty and spirituality of real world and universe. Phi is the first letter of Phidias, who used the golden ratio in his sculptures. Phi is also the 21 st letter of the Greek alphabet, and 21 is one of numbers in the Fibonacci series.

Again 21 holds the eighth position of the series and the number 8 is also a member of the Fibonacci sequence. The characteristics of Phi have some interesting theological implications. The aesthetic primacy of the golden section was established empirically, and it was the very first topic of scientific psychological research as the new discipline emerged in the 19 th century [15].

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Pythagoras, Plato, and the Golden Ratio

Fechner was the first person who fixed his analytical gaze upon this task as early as the 's. Since that time, it has been the focus of a number of research fields such as Structuralism, Gestalt psychology, Behaviourism, Social psychology, Psychiatry and Neuroscience at various points in time [15]. Phi continued to open new doors in the understanding of life, nature and the universe.

It appeared in Roger Penrose's discovery of Penrose Tiles in the 's, which first allowed surfaces to be tiled in five-fold symmetry [37]. It appeared again in in the aluminium-manganese alloy Al 6 Mn , known as quasi-crystals, which was newly discovered form of matter [37]. This study represents the black hole solution in higher dimensions where a sparkling relation is found with the Phi. Anthropomorphic Robot research is one of the interesting fields where the humanoid robot sizing can be established based on Golden Proportion to make the social robot more presentable and acceptable to the general public [16,34,40,41].

Golden Ratio also used in particle swarm optimization [42]. A recent study on ortho-cortex and para-cortex, two types of Cortex in Wool fibre, is conducted to have a better understanding on the relationship between the fibre structure and its properties where the fractal dimension of wool fibre shows the existence of Golden Mean in its structure [43].

A theoretical foundation of applying a new control law, the Golden Section Adaptive Control GSAC law, is established in recent analysis to control a specific aircraft [44]. The next section, section 2, of this writing represents a succinct review on the relation between the Golden Ratio and the Geometry, concept of various dynamic rectangles and their links with the Phi ratio.

Section 3 provides the substantiation of the equation of Phi contingent upon the classical geometrical approach. Phi as the baroque of the natural beauty and natural enigma is presented in section 4 with some representative patterns of the natural phenomena. In section 5, the uses and presence of the golden proportion in various artefacts, arts, portrait, design, architectural works and engineering are briefly introduced in a panoptic manner. Finally, the section 6 abridges the paper with some positive and opposite concept of the relation between the beauty and the Golden Proportion.

Phi and Its Relation with Geometry 2. Concept of Golden Section For a line segment, golden section can be considered as a point where the line is divided into two sections containing a unique property such as the ratio between the bigger segment and the shorter segment should be equal to the ratio between the line and its bigger segment [,26,].

The ratio is approximately 1. Consequently, the idea has been incorporated into many art works, architectural design and mathematical analysis. Figure 1. A line segment AC is divided in the golden section point B 2. ABCD is a golden rectangle shown in Figure 2. This process can be continued for ever which will create smaller to smallest golden rectangles. Based on the different position of the golden section of a line, there can be at most four golden means in a golden rectangle. Figure 2. Firstly, a perpendicular line having the length equal to the bigger segment of the base line divided at the golden section can be considered to draw the next larger golden rectangle.

According to Figure 2 the base line AB is divided into p and q at the golden section point N. The perpendicular line, BC or AD , at any of the end point of the base line is equal to the length of p. 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 [15]. 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. Golden rectangle is also known as whirling square rectangle because of the special property of subdividing into a reciprocal rectangle and a square [1,15]. 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. Figure 3. Based on Figure 4 a and b , Equation 1 can be written as, 2 Figure 4. The absolute value of the cosine of any angle of the golden triangle is equal to 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].

Figure 5. A series of golden triangles construct the Regular Pentagon, Pentagram, Regular Decagon and Decagram 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. Figure 6. Golden spiral formed on the whirling golden triangle 2. 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 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. Figure 7. 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.

Geometry and Masonry: Sacred Geometry

If the new rectangle is divided into two equal sections, the newly formed rectangles will be the Golden Rectangles. Figure 8. So, Equation 3 can be proofed from Equation 4 , 11 12 13 4. Golden Section and the Beauty of Nature In 12 th 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.

Figure 9. 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 [20,23] 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.

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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 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 Passionflower 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.

Figure Passion f lower Passiflora Incarnata from the back a and front b having the examples of Fibonacci Numbers in nature Romanesco Bro c coli is one kind of vegetable that looks and tastes like a cross between broccoli 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 [38] The spiral happens naturally because each new cell is formed after a turn.

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 close to the golden ratio. Plants have distinct characteristics of Golden Ratio where they establish a Fibonacci sequence in the number of leaves.

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 [18].

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 d , show some succulent plants where this characteristic is clearly visible.


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Golden Angle and Golden Spiral are also found in some plants and ferns. In , Wilhelm 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 [3]. 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 Fern 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 [3]. Figure 17 shows various types of Cactuses having almost the same properties of Golden spiral in their growth patterns. Some various Cactuses are showing the existence of the Golden Spiral in their various growth patterns [18,50] The Fibonacci numbers form the best whole number approximations to the Golden Section.

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 [7].

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 [7]. 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 h , also 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 [1].

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 [1]. 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 [1]. 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 as 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. Questo sito raccoglie log per statistiche anonime.

Leggi le privacy information per maggiori dettagli. American Chemical Society, A national historic chemical landmark. The discovery of fullerenes, Bodnar O. Boeyens J. Buis A. Calvin W. Coldea R. El Naschie M. Jakushko S. Jean R. Martineau J. Olsen S. Dissertation], Gainesville, University of Florida, Regardless, it is attributed to Pythagoras and two hundred years later Euclid compiled his "Elements of Mathematics" where this particular 47 Th proposition is found in Book One.

This theorem has been called the root of all geometry and the cornerstone of mathematics. The practical applications alone are worthy of the high esteem that Masonry affords it. And this is the interpretation of the lecture that is most considered when masons speak of it but the meaning of this hieroglyphical emblem does not stop there. The emblem we are usually presented is the 3,4,5 right triangle in this fashion: The vertical line is of 3 units, the horizontal is of 4 units, and the hypotenuse is of 5 units.

Not only is our attention called to this geometrical figure in the Master Mason degree, it is also prominent in the Scottish Rite in the 20th Degree - Master of the Symbolic Lodge and in the 25th Degree - Knight of the Brazen Serpent.


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Geometry treats of the powers and properties of magnitudes in general, where length, breath, and thickness, are considered, from a point to a line, from a line to a superficies surface of a body and from a superficies to a solid. By this science, the architect is enabled to construct his plans, and execute his designs; the general to arrange his soldiers; the engineer to mark out ground for encampments; the geographer to give us the dimensions of the World, and all things within, to delineate the extent of seas, and specify the divisions of empires, kingdoms and provinces; by it, also, the astronomer is enabled to make his observations, and to fix the duration of times and seasons, years and cycles.

In fine, geometry is the foundation of architecture, and the root of mathematics. Many readers may feel like they have been returned to Geometry class. A simple illustration will probably refresh our memories:. The proposition is especially important in architecture. This line is given the value of 3. The builder then marks another point, say point B and draws a line from it at a right angle to line A, and it is given the value of 4.

The distance between line A and B is then measured, and if the distance between A and B is 5, then the room is squared. Before the advent of GPS and Laser measuring tools engineers who tunnel from both sides through a mountain used the 47 th proposition to get the two shafts to meet in the center. The surveyor who wants to know how high a mountain may be, ascertains the answer through the 47th proposition. The astronomer who calculates the distance of the sun, the moon, the planets, and who fixes "the duration of times and seasons, years, and cycles," depends upon the 47th proposition for his results.

The navigator traveling the trackless seas uses the 47th proposition in determining his latitude, his longitude, and his true time. Eclipses are predicted, tides are specified as to height and time of occurrence, land is surveyed, roads run, shafts dug, bridges built, with the 47th proposition to show the way. The Mystical Symbolism of the Pythagorean Triangle.

Earlier in this paper I made reference to the fact that the Egyptians were in possession of the knowledge of the 47 th Proposition. According to Plutarch 46 - C. The vertical line was of 3 units and attributed to Osiris. The horizontal line was of 4 units and attributed to Isis.

And the hypotenuse was, of course, 5 units and attributed to Horus, the son of Osiris and Isis. It is noteworthy that Plutarch studied in the Academy at Athens and was a priest at Apollo's temple at Delphi for 20 years. The units of the triangle's side are significant.


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The three units of the Osiris vertical have been attributed to the three Alchemical principles of Salt, Sulphur and Mercury. All things are manifestations of these three principles according to Alchemical doctrine. The four units of the horizontal line of Isis relate to the so-called four elements: earth, air, water, and fire.

These are of course the four Ancients. The ascending Horus line with its five units represents the five kingdoms: mineral, plant, animal, human, and the Fifth Kingdom. This is the Path of Return. The ascending line finally connects back up with the Osirian line. The Fifth Kingdom symbolizes the Adept as one who has consciously reunited with the Source of all Being. The 47th problem has been referred to as "the foundation of Freemasonry.

Noted cabalist Alan Bennet speculated that the three squares represented the magical squares of Saturn, Jupiter, and Mars. Masonic author Albert Pike, in his Morals and Dogma repeats Plutarch's comments and suggests that the triangle represents matter Isis , spirit Osiris , and the union of the two Horus Sacred Geometry,.

Pythagoras is credited with the discovery of the Golden Rectangle. The Golden Rectangle is built on the "golden ratio" or "golden proportion," which is determined by the irrational number known as Phi. To put it simply, a golden rectangle is a rectangle divided in such a way as to create a square and a smaller rectangle that retains the same proportions as the original rectangle.

To do this, one must create a rectangle based on this ratio. To find the Golden Ratio, one must divide a line so that the ratio of the line to the larger segment is equal to the ratio of the larger segment to the smaller. A is to B as B is to C To get a golden rectangle, you simply turn the larger segment of the line into a square. If you add a square to the long side of the "golden rectangle," you'll get a larger golden rectangle. If you continue to add squares in this way, you'll see the basis for nature's logarithmic spiral patterns. The golden proportion appears in numerous places in nature and in art and architecture.

The face of the Parthenon in Athens is a perfect golden rectangle. The shell of the nautilus is a famous example of a spiral based on the golden mean, as is the spiral of the human DNA molecule. Vitruvian Man.