# Geometry Of Design

Golden section proportion is found in the context of the natural world, and in man-made environment. Throughout history, from the architecture of the Stonehenge from 20th to 16th centuries, B. C, in the architecture, writing, and art of ancient Greeks in the 5th century B. C, to the Renaissance artists and architects, the golden section proportions is evident. Gustav Fechner in the late 19th century who conducted study of the man-made environment context of the golden section proportion, concluded that majority of people preferred the use of golden section rectangle proportion.

He did the experiment by measuring books, boxes, tables, and other man-made things. In 1908, Lalo conducted another study in a more scientific manner, and yet yield very close result to Fechner’s study. Further studies by other scientists yield very close results to Fechner and Lalo thereby supporting the theory that majority of the people really have an addiction to the golden section proportion since early civilizations. This preference can be rooted from the reality that many people have earlier realized the relationship of the golden section proportion to the natural world.

Growth pattern in many living things have golden section proportions as observed by many scientists like Theodore Andreas Cook, who explained that, in new spiral in each spiral growth phase, is very close to the proportion of the golden section square larger than the previous one. A theory known as Theory of Perfect Growth Pattern was formed from the studies of spiral growth patterns of many shells, plants, animals, and even human.

Marcus Vitruvius Pollio, a greek scholar and architect who have written his investigations of human proportions and architecture suggested that the architecture of temples should be based on the proportion of a perfectly proportioned human body. He explained that a well proportioned man has a height equal to the length of his outstretched arms. The height of the body and length of the outstretched arms encloses the man in a square, while the hands and feet touch a circle. Sculptors who sculpted the statues of the Spear Bearer, and Zeus in the fifth century are based on canon of Vitruvius.

Renaissance artists Leonardo da Vinci and Albert Durer both students and scholars of proportioning system and human form have conducted experiments in many proportioning system in their books have identical results to the canon of Vitruvius. The proportioning of the human body by Vitruvius, Da Vinci, and Durer were identical. This ideology was applied to the construction of early architectural design such as the Parthenon in Athens, Gothic Cathedrals, to achieve harmony in all parts.

Regulating lines which can be traced by using the golden section proportion construction helps architects in determining the proportion of the height to the width, and guides the placement of objects such as doors, windows, freeze, and other parts of early temples and cathedrals. The importance of regulating lines was explained by Le Corbusier in his books. The golden section proportion has the power to create harmony. It comes from its unique identity to divide a whole into different parts, with each part’s own identity preserved, and yet blends these different parts into the greater pattern of a single whole.

The golden section is a ration of the Divine Proportion. It is derived from the division of a whole line segment AB, into two parts – respectively AC and CB. To achieve the divine proportion, the ratio of AB to AC, is the same as the ratio of AC to CB. That means AC is longer than CB. This will yield a ratio of approximately 1. 61803 to 1. A Golden Section Rectangle, also known as the Whirling Square, is composed of a square area called a Gnomon and a Reciprocal Golden Section Rectangle joined together.

A golden section rectangle can be further subdivided many times as possible into smaller golden section rectangle having its own square area and reciprocal golden section rectangle. Because of this property it was also called as the whirling square, where its proportionally decreasing square can produce a golden section spiral. A golden section spiral can be constructed on the diagram of the golden section rectangle subdivision by drawing an arc on each square in the subdivision, with radius equal to the length of the side of each square, then connect the arcs together and the resulting figure is a golden spiral section.

Based on the fact that there always remains a square area in subdivisions suggests that the golden section rectangle can be constructed using the square method. First draw a square, from the midpoint A of any side of the square draw a diagonal extending to one of its opposite corner B. Then draw an arc extending beyond the square to C using the radius of the diagonal. The golden section rectangle can also be constructed using the Triangle method. Begin with drawing a right triangle whose sides are in 1:2 proportion.

The line segment AC serves as the base, line segment CD is the hypotenuse. Draw an arc using the length of the line segment AD as the radius that crosses the hypotenuse and label it as point E. From point C, draw an arc along the hypotenuse to point E then intersect it with the baseline line AC using the length of CE, and label the point of intersection at the baseline with B. From point B, draw a vertical line to the hypotenuse. This method constructs a golden section proportion by determining the length of the sides of the rectangle AB and BC, which is a golden section proportion.

The method can also results to a series of circles and squares that are in golden section proportions with each other. There is a notable relationship with the proportioning system of the golden section proportion with the Fibonacci Sequence by Leonardo of Pisa, where a series of number can be produce by adding the two previous numbers to produce the third. The series of numbers beyond the fifteenth divided by the following number approximates 0. 618, and any number divided by the previous number approximates 1. 618. The early numbers in the series below the fifteenth number begin to reach the golden section.

Aside from the golden section rectangle, majority of people also have an addiction to the Golden Section Triangle also known as the Sublime Triangle. It also has similar characteristics with the golden section rectangle. It has an angle of 360 at the vertex, and 720 at the base which is constructed from a pentagon, or from a decagon. To construct a golden section triangle from a pentagon, connect the vertices by diagonal lines. This will not only produce golden section triangles, but also produces a star pentagram with another pentagon at the center which all are in golden section proportions.

This is also known as Pythagoras Lute. The golden section triangle can be further subdivided by connecting the base vertex of the triangle to an opposite side of the pentagon. To construct a golden section rectangle from a decagon, connect any two adjacent vertices to the center of the decagon. This will produce the same triangle having 360 at the vertex, and 720 at the base. A golden section triangle when further subdivided into smaller golden section triangle, also produce a golden section spiral by using the lengths of the sides of each triangle as the radius of the circles.

There are two categories of rectangles: Static Rectangles, having ratios of rational fractions, which do not produce pleasing ratios when subdivided, and Dynamic Rectangles, having ratios of irrational fractions, which produce pleasing amount of harmonic subdivisions. Subdivisions of the Golden dynamic rectangles can be done by drawing diagonals to connect each opposite corner, and then a drawing parallels and perpendicular lines on its sides and diagonals. Another method of constructing a root 2 rectangle is the circle method.

Draw a circle, then, draw a square inscribed in the same circle. Finally, extend the two opposite sides of the square to touch the circle. The result is a root 2 rectangle. The Root 2 Rectangle can be subdivided into proportionally smaller root 2 rectangles. Root two Rectangle can be constructed using the square method. Start by drawing a square, then, draw a diagonal from to connect opposite corner. Use the length of the diagonal to draw an arc that touches the square’s base line. The result is a root 2 rectangle. Subdivisions are done by dividing the whole rectangle in half.

Resulting rectangles can be further subdivided in half. The process can be repeatedly. A root 2 diminishing spiral can be constructed by striking diagonals on the reciprocal root 2 rectangles. The property of the root 2 triangle was applied by many European paper products because of its efficiency in reducing waste papers, and in page sizing. Like golden section rectangles, root 2 rectangles are also dynamic rectangles. The Root 3 Rectangle, like root 2 rectangles, can be further subdivided harmoniously into smaller root 3 rectangles.

The root 3 rectangles can be subdivided into 3 root 3 vertical rectangles, and into 3 root 3 horizontal rectangles. A hexagonal prism which is found in natural crystals, honeycombs and others, can be constructed from root 3 rectangles. To construct a root 3 rectangle, begin with a root 2 rectangle. Draw a diagonal from the two opposite corners. Using the length of the diagonal as radius, draw an arc that touches the baseline of the root 2 rectangle, the result is a root 3 rectangle. To construct a hexagon from root 3 rectangle, rotate the rectangle until each corners meet, repeat the process.

Subdividing is done by dividing the root 3 rectangle into 3. Further subdivisions are done by dividing smaller root 3 rectangles into 3. The Root 4 Rectangle construction begins with a root 3 rectangle, draw a diagonal to connect two opposite corners, use the length of the diagonal as the radius of an arc that touches the baseline of the root 3 rectangle. Subdivisions are done by dividing root 4 rectangles into 4, further subdivisions are done by dividing smaller root 4 rectangles into 4. The Root 5 Rectangle construction begins with a root 4 rectangle.

Draw a diagonal to connect two opposite sides, then, use the length of the diagonal as the radius of an arc that touches the baseline. Root 5 rectangle can also be constructed using the square method. Begin with a square, draw a diagonal from the midpoint of the baseline of the square to opposite corners. Use the length of the diagonals as the radius of an arc extending on the opposite sides of the square touching the baseline of the square. Extend the two opposite sides of the square to enclose the arc. The result is root 5 rectangle.

Subdivision is done by dividing root 5 rectangles into 5 parts. Further subdivision is done by dividing smaller root 5 rectangles into 5 parts. The following chapters were all analysis of classic posters, paintings, and work of arts. The analysis shows how each design was influenced by the golden section proportion and geometrical principles. The most important point of the book is to give each reader a grasp of how geometry was applied to classical works. On the perception of common viewer, a picture may just be some sort of expression of emotions of the target viewer, and of the artists itself.

But the growing evidence of the existence of certain geometrical and mathematical patterns and how does patterns are harmoniously created has brought up some questions. Does this writer generally adhere to common rules- such as the golden section proportion? Or is it only that the patterns were not intentionally created, and that the discovery of these patterns, are just products of the curiosity of those people who are seeking for proofs of the relationship of geometrical principles to the context of both man-made environment and the natural world.

Does adhering to the golden section proportion rule, or on any geometrical principles has something to do with the integrity, and the knowledge of any artist? On my own idea, the application of some form of rules like the golden section proportion rules is to some point intentional, and to some point not. The fact, that ratio and proportion is really can be seen as part of many things, is a good reason for some artists to apply the golden section proportion idea.

For example, most people view an artist work visually, by looking at how it appears real. In that case, any writer who does portray unrealistic proportions in his work would somehow put himself into the door of many critiques. However, there are some abstract ideas which are hidden in some artist works through the use of some form of patterns. Take for instance the encryption code for many organizational data, are hidden in some form of encryption pattern, to block hackers from stealing precious information.

That also works in many arts, some artist like abstract paintings, abstract designs, to capture other type of viewers who sees things objectively. The golden section proportion may or may not be applied to judge the works of many artists, as shown in the above example. The book begins somehow confusing because it focuses early on the theories of the golden section proportions. In the progress of the course, the application of the theories, are presented. Plenty of drawings and illustrations were presented to visually support the ideas being presented within each page.

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