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Historical Background Numbers & Geometry - Primitive Geometry
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Thomas
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Several times it has been mentioned how classical philosophers-, mathematicians and architects were highly occupied by their attempts to solve the problem of immeasurable lengths and indefinable numbers. Besides the many practical situations that required a good understanding of the geometrical relation between squares-, circles and angles (such as architecture), there also existed highly mystical approach towards these problems, where philosophers believed that though the insight into mathematical enigma such as square roots and the number pi, they would be able to unravel the mystery of life itself. As such, typically classical problems such as "squaring the circle", obtained a highly mystical, almost religious status. In order to obtain better insight into the classical approach towards several geometrical problems that by now are considered to be of "high school level" it is necessary to make a hypothetical reconstruction of primitive geometry the way it must have been experienced by people that had no previous knowledge about it. It is relatively easy to make a reconstruction of the "invention" of the basic geometrical forms, assuming that each form can only be invented once the geometrical principals that are needed to draw it are understood. To give an example: the square can only be drawn once the straight-angle has been "invented". This example sound a bit far sought, because many geometrical forms already exists in some kind of "collective subconscious". Even without fully understanding the exact mathematical construction, everybody understands the concept of a square, a triangle and a circle. As such this chapter should be explained as a reconstruction of the first attempts to fully understand these basic geometrical forms. At a practical level, the exact calculation of a centre-point or a straight angle didn't have much use for a primitive tribal community. Long before we started admiring the beauty of architecture, we had already build thousands of primitive shelters and huts that needed no mathematical precision whatsoever. Agricultural terrains were mostly divided by natural borders or marking stones and didn't need exact measurement. Most likely the first thoughts about geometrical problems must have come long after we already mastered basic arithmetic's. It is logical that primitive communities had to do with arithmetic in their daily life, like when they had to count their herd or when trading food and fabrics. Basic geometrical forms were probably understood, but it most likely took many more generations before people started rationalizing about the mathematical methods that were needed to design and work with these forms. It is said that the Egyptians invented geometry (which literally means measurement of the earth), because they were forced to recalculate the division of the fertile terrains close to the Nile after each yearly flood. In fact, considering the Egyptians tremendous architectural achievements, nobody will question their authority when it comes to geometry. The Egyptian land meters most probably made use of the most versatile geometrical instrument of all time, which is a rope! The best available instrument if one has to work with primitive geometric forms is undoubtedly a piece of rope. A simple piece of rope can be used to draw straight lines and divide them in two or more equal parts. The same rope can also be used to draw perfect circles and a triangle, but that's where it stops for the moment. In order to draw a square, a rectangle or a cross, some kind of a 90° right angle is needed. Square It is necessary to start with a circle and the straight line and use them to "invent" a 90° right angle. First of all one needs to draw a circle and draw a straight line through the centre. This can easily be done by means of a rope. Next thing is to draw a cross in the circle by drawing a line in a right angle to the starting line. Here the first problem needs to be resolved, because the right angle still needs to be invented and there are two practical solutions: 1) We know that the distance from the points A & B to the centre point of the circle (M) is equal. We now take a new piece of rope that is slightly longer than A-B and find the middle point of this rope by folding it in two equal parts (length's l). By attaching the extremities of the new rope to points A & B and pulling it tight from the middle of the rope we can now draw a new line (C-D) through the centre of the circle that is a an exact right angle to A-B 2) the second solution (no design available), would be to use the rope to draw two more circles, with A as centre and A-B as radius and with B as a centre and B-A as radius. The points of intersection between the two new circles form a straight line through the centre of the starting circle at an exact right angle to line A-B At this point it is possible to draw a perfect cross and a square within the circle by uniting points A, B, C & D.
What catches the eye is the fact that from the very first attempt to draw a square, one is immediately confronted with the circle. As such it is no surprise that the circle and the square have been experienced as a unique geometric enigma. The rope method might seem a bit unusual at first impression but it definitely is the oldest and most widely used geometrical instrument ever, still frequently used by masons and carpenters on every building site in the world. Triangle Another geometrical body that is closely related to the circle is the equilateral triangle and the hexagram that can be derived from it. A equilateral triangle is a very simple form that can also be drawn by means of the rope method without using a circle. By using circles to construct the equilateral triangle, the hexagram will appear almost automatically. First of all we draw a starting circle with a vertical line through the centre. The intersection points of the vertical line and the circle are the centre points of two new circles of the same diameter. The intersection points of two new circles and the first circle create two equilateral triangles posed opposite each other. This beautiful geometrical form is currently known as the Star of David, symbol of the Jewish people.
The image above shows the intriguing proportions of the hexagram. The relation between the height and the basis of a equilateral triangle can be calculated by means of the proposition of Pythagoras (A2+B2=C2). This way we will find out that the length of the sides relates to the height of the triangle a 2:√3. The height of the hexagram can easily be divided in multitudes of two. In the illustration the hexagram has been divided in four large sections corresponding with the intersection points. Based on a fixed height of 4, the width of the hexagram is √12. Horizontally the hexagram can be easily divided in 6 major sections corresponding with the same intersection points of the two triangles that form the hexagram. Continuing to split-up the hexagram in twelve and sixteen (12+4) subsections, a basic grid becomes visible. On a symbolical level it should be appreciated that the hexagram has such an appealing geometrical relation to the number twelve. The grid that becomes visible is slightly rectangular with a proportion that comes close to 7:8. All in all, the hexagram is an easy to construct but very fascinating figure. It is not surprising that it was widely used in classical times. Its most appealing symbolic quality is without doubt the geometrical symbiosis of the upward and the downward triangle, symbols for concepts like heaven and earth, man and woman, fire and water etc. Gnostic Hebrew tradition claims that king Salomon used the hexagram until his death in 930BC to call on angles and drive out demons. Most sources agree on the fact that the use of the four letter name of God (IHVH) was decisive in these rituals From this time onwards the hexagram was known as the Seal of Salomon or the Shield of David. up |
The highly flexible and
accurate rope method makes it possible to draw an infinite number of
geometrical forms an practically every possible scale. Besides in
architecture and on building sites this method was actually used to make
drawings as well. Medieval Painters used a thigh piece of cord soaked in
ink to draw reference lines on their paintings. They held the rope line
right above the paining and made it "jump" on it, leaving a perfectly
straight ink line on the surface. In classical times, the old Egyptian temple cult was looked upon as the birthplace of geometry and many Greek scholars like Pythagoras were supposed to have obtained their knowledge during a stay in Egypt. Considering their astonishing architectural achievements almost 2000 years before Pythagoras this will not come as a surprise. Apart from the historical origin of the geometric tradition, the incompatibility between the measures of a square and a circle is instantly visible for whoever uses basic geometric forms. As such it is highly likely that the antique fascination with the problem of "squaring the circle" originated contemporarily at different places. Irrational Proportions - The Pentagram The pyramid of Cheops shows a particular interest in the proportion 1:√5. The same √5 also appears in the "golden section" proportions of the five pointed star, used by the Pythagoreans as their "secret" recognition sign. The five pointed star or pentagram is a highly fascinating design that is almost impossible to construct. As easy as its to draw a hexagram, as difficult will be the pentagram. Even drawing a reasonably accurate hexagram by hand proves to be a difficult task. At first sight even the rope method seems to offer few possibilities. The simplest solution is rather improvised and inaccurate. One could draw a circle and experiment with the distances between the five points until a satisfying result is obtained. For a mathematician this is a truly repulsive method because it doesn't follow a systematic approach, but at a practical level it would probably be sufficient to draw a beautiful pentagram that is accurate to the eye. Nevertheless, this method can not have satisfied the classical geometrics, who must have been looking for a more structural approach to draw the pentagram. With the help of an accurate goniometer the task would become much simpler. Divide the circle in five sections of 72° each and the job is done. Another possible approach could be the relation between the radius of the circle and the distance between the points of the pentagram; in this case two times the co-sinus of an angle of 54° or approximately 1,17557.. Not really something for the classics' and perhaps even for us a bit to irrational. In reality their is only one really beautiful geometric construction method for the pentagram, that doesn't even use a circle but the proportions of the golden section. About 300BC, Euclid describes the geometrical proportion of “a line divided in extreme and mean ratio”, better known as the golden section. In the fifth proposition of the thirteenth book of the Elements, he accurately describes how to divide a line in extreme and mean ratio and subsequently uses the proportion to draw a series of complex polygons amongst which the pentagram. Euclid's method is simple but can not be applied in the same simple step by step method as described below.
Illustrations by Leonardo da Vinci in Paccioli's book De Devina
Proporzione Constructing the Pentagram First of all a square ABCD is drawn. This square is divided in two equal rectangles by line EF. The length of the diagonal ED is added to EB in order to obtain a new rectangle HBCI. On side HI we draw a mirroring square that is also divided in two equal rectangles. The pentagram is now closer that is seems. The rectangle HBCI has been constructed according to the proportions of the Golden Section. Presuming that the length of the sides of the starting square (BC) is one, then distance HB equals ½ + ½Φ5 or ≈1,6180339..
The pentagram is now easily constructed as follows: The vertical lines FE & LK are extended downwards. From point B a half circle is drawn with BH as radius. This circle intersects with the extended line FE in the new point O. From O, lines are drawn to G and through A to the new circle intersection point M. The last missing point of the pentagram is point N on the extension of line LK with the same distance as FO. Connect the various intersection points and a beautiful pentagram becomes visible.
The pentagram is a truly fascinating geometric form. Within the small pentagon inside each pentagram (PQRAH), a new pentagram can be drawn that once again relates to the bigger one according to the golden section proportion. |
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