The Golden Section and the Golden Number
The Golden section exhibits unique properties in its proportion related to art, beauty and harmony. Because of these characteristics, painters, sculptors, engineers and architects, have used it extensively in the proportioning of their art-works. We find this proportion in the designs of vases, statues, temples, cathedrals, and many other important structures, like the Parthenon in Athens and the United Nations Building in New York City, N. Y., in the United States.
It is known that the Golden Section proportion is shown in nature by the way in which the leaves are arranged in certain trees and in the form of the shells of certain snails. The proportion can even be found in the proportion of the human figure, and in the shapes of galaxies in the Cosmos. The Golden Number, which represents the Golden section, has a function of 1.618033... value that can be generated mathematically, by the use of algebra, geometry and other means.
One method to produce mathematically the Golden Number is by using the Fibonacci series. This series is named after Leonardo Fibonacci, an Italian mathematician who discovered it. To generate the series, observe the line of whole numbers shown below. The line starts with 0 and the following number is 1. Adding the two previous numbers produces each additional number. To produce the series, add 0 plus 1, which is equal to 1, the next is 1 plus 1, equal to 2, next 1 plus 2, equal to 3, next 2 plus 3, equal to 5, next 3 plus 5, equal to 8, next 5 plus 8, equal to 13, and so on.
0 - 1 - 1 - 2 - 3 - 5 - 8 - 13 - 21 - 34 - 55 - 89 - 144 - 233 - 377
* The function of the Golden Number is equal to 1.618033... and as the function of Pi (p), its decimal numbers never ends. It is generally identified with the Greek letter Phi = (f). For practical use, its value is assumed as 1.618.
One property of the Fibonacci series is that the ratio between the higher and the lower number, of two consecutive numbers, approaches the value of f as the numbers in the series increases.
The higher the numbers in the series, the closer the ratio will be to the function of f. For example, using the numbers in the example of the series, 13 divided by 8 is equal to 1.625... 21 divided by 13 is equal to 1.6153846... 34 divided by 21 is equal to 1.6190476... 55 divided by 34 is equal to 1.6176471... 377 divided by 233 is equal to 1.6180258... as the process continues, the ratio continues approaching the value of f =1.6180339...
The Golden Section can be illustrated geometrically using a square. For example: as shown in figure 132 (a), trace the square
ABCD, any size. Set each side equal to 1. Locate the midpoint E for the side AD (point E). As shown in figure 132 (b), with center at E, trace an arc from point C, to intersect the extension of line AD of the square, at point F.
If a rectangle is drawn with sides AB and AF as its sides, as it is illustrated at figure 132 (c), it will be in proportion to the Golden Section. The side AB will be equal to 1, and the side AC equal to 1.61803.
The angle EDC is 90ļ, since it is a corner of the square. In accordance with the Pythagorean theorem, the hypotenuse (EC) is equivalent to the square root of the sum of the side ED squared = (0.5)≤, plus the side DC squared = (1.0)≤. The result is 1.11803. Therefore, the length of line AF is equivalent to 1.11803 plus 0.5, that is, 1.61803, value that represents the function of the Golden Number.
The length of line DF is equivalent to the distance from point A to point F, less the distance from point A to point D. That is, 1.61803 less 1.0. The result is 0.61803, which is equivalent to the inverse of the Golden Number, that is, 1 /f.
Another method to produce the Golden Number is by means of the proportion between two segments of a line. The line is divided into two unequal segments. It establishes the proportion that the small segment is to the large segment, as the large segment is to the whole line.
In reference to the shown line AB, if point C divides the line in two unequal parts, and segment AC is made equal to one (1), and the segment CB equal to X, the proportion would be:
A _____________C___________________________ B
Small segment (AC) = Large segment (CB)
Large segment (CB) Length of line (AB)
That is, 1 / x = x / (x + 1), this will create the equation
x≤ - x - 1 = 0
Solving the quadratic equation, the values of X will be 1.61803... the Golden Number (f), and the other value will be the inverse value of the Golden Number (0.61803...) = (1 / f). The calculations for the solution to this equation are not shown because my interest is only to show this other method that produces the proportion of the Golden Section.
Among the methods used to create the Golden Number, perhaps the most interesting and my favorite is that generated by the figure of two circles. I consider this method easier to explain and understand.
Figure 134 consists of two circles of equal radius (R = 1), where the circumferences are tangent at point B.
From point K (lowest point of vertical diameter KQ in the left circle), trace a line to the center (L) of the adjacent circle. Extend the line KL to intercept the circumference at point D.
The triangle KOL in the configuration is a right triangle, since KO is perpendicular to OL. Using the Pythagorean theorem, the hypotenuse (KL) can be calculated. The hypotenuse is equivalent to the square root of the sum of the squares of the two sides. Therefore, the hypotenuse is equal to the square root of (OL)≤, plus ( OK)≤. Since OL is equal to 2, and OL is equal to 1, the hypotenuse will be equal to the square root of (2)≤ + (1)≤ =÷5. Consequently, the distance from point K to point D is equal to ÷5 +1, since the radius (LD) = 1.
If the length of line KD is divided in two equal segments, as shown in figure 135, each segment will be equal to f = 1.618033. The geometrical process is equivalent to divide the square root of five plus one, by two, that is, (÷5 +1) / 2 = 1.618033. As it is known, this number corresponds to the Golden Number value (= f).Note: To divide the line KD in two equal segments, from point K, trace an arc with a radius larger than half the length of KD. Then, using an equal radius, from point D, trace another arc. Joint the points of intersection of the circumferences of the two arcs (points (e) and (f). The point where this line intercepts line KD (point G), is the midpoint of line KD.
Note in figure 136 that the distance from point K to point M (the point where line KL cuts the circumference of the circle with center L), is equivalent to the distance K L = (÷5), less the radiusís length (R =1), that is, ÷5 - 1.
If the line KM is divided in two equal segments, the process is equivalent to divide (÷5 - 1) / 2 = 0.61803. This value represents the inverse of f, or (1 / f). The figure also shows other equivalencies in relation to the value of f.
The Golden Number is well known because it has unique properties: its square is equal to itself plus one (f)≤ = (f + 1), while its inverse is equal to itself, less one (1 / f) = (f - 1). An entire range of numeric and geometric relations can be created using the function of the Golden Number.
In reference to an equivalency between f and p , the exact equivalency between those functions has never been found. There are many approximations such a p = 4 / ÷f, p = 1.2 f≤ and p = f / sin 31į (This one I found in my own calculations (see Chapter 11).
Properties of the Right Triangle with Sides
In the Ratio 1:2
When the angles and distances between corridors and chambers inside and under the Great Pyramid are analyzed, it can be noticed their constant relation to the figure of the right triangle which have its sides in the ratio 1:2.
It is known that this triangle was used in many Egyptian constructions. It is the same triangle I used previously to demonstrate how to establish geometrically the Golden Section proportions using the figures of circles. It is formed when the length of the radius of a circle is used as the short side, while the length of the diameter is used as the long side. I will demonstrate that this triangle is an appropriate representative for the function off. In other words; the value of f is present in any right triangle having its sides opposite the hypotenuse, in the ratio 1:2.
The triangle ABC, shown in the upper section of figure 137 illustrates the relation between the radius and the diameter of a circle. The short side (AC) represents its radius, while the long side (AB) represents its diameter.
When I examined the triangle ABC, shown separately in the lower section of figure 137, I observed that the same triangle exposed its relation to the Golden Number and its geometric and mathematical relations. Therefore, the short size could be set as one (1), and the long side equal to (2). The hypotenuse, as it is known, is equivalent to the square root of five (÷5).
Observe that if the hypotenuse (÷5) is added to the length of the short side, and divide the sum by the length of the long side, is equivalent to perform the mathematical process to determine the value of f, that is, (÷5 +1) / 2. Using the triangle ABC, I developed a series of numeric relations concerning the value f, that will help to quickly remember the golden numbers formulas, or to develop them.
1. The hypotenuse (÷5) plus the length of the short side (1) divided by the length of the long side (2), is equal to the Golden Number.
(÷5 +1) / 2 = f
2. The hypotenuse (÷5), less the length of the short side (1), divided by the length of the long side (2), is equal to the inverse of the Golden Number.
(÷5 - 1) / 2 = 1 / f3. The hypotenuse (÷5), plus the length of the long side (2), divided by the length of the short side (1), is equal to the Golden Number cubed. (÷5 +2) / 1 = f≥ 4. The hypotenuse (÷5), less the length of the long side (2), divided by the length of the short side (1), is equal to the inverse of the Golden Number cubed. (÷5 - 2) / 1 = 1 / f≥ 5. The sum of the three sides of the triangle (÷5 +1 + 2), is equivalent to the Golden number cubed, plus one.
(÷5 +1 + 2) = ÷5 + 3 = f≥ + 1
Figure 138 shows a geometric analysis of some Golden Number expressions originated with this angle. Point (t) represents the midpoint of line QO. Trace a circle with center f, and radius fQ (= 0.5). The line segments (Ca) and (bB), in the line CB, is equivalent to 0.618034. This value represents the inverse of the Golden Number (1 / f). Therefore, the line segment (aB) is equal to 1, plus 0.618034, sum that is equal to 1.618034 = f. Note that CB = ÷5, and to the sum of segments (aB and aC). In other form, ÷5 = f + (1 / f).
Figure 139 illustrates also the triangle ABC. With center C and radius equal to 1 (short side of triangle), trace a circle. Trace another circle, with an equal radius, with B as center. The sum of the segments EC and BD is equal to one. The hypotenuse CB is equal to ÷5. Therefore, the length of ED is equal to (÷5 + 2) = 4.236068 = f≥. Since it is known that (1 / f≥) = 0.236068, the expression can be changed to (f≥) - (1 / f≥) = 4.
The segment (jk) of line ED is equal to the square root of five, less two. The expression (÷5 - 2) is equal to 0.236068 and corresponds to the inverse of the Golden Number cubed, that is, 1 / f≥. The length of line ED is equivalent to (4 + 1/ f≥), and also equivalent to f≥.
There are two geometric equations of the Golden Number that are worth to examine. These are the following:
(f) + (1 / f≤) = 2
(f≤) + (1 / f≤) = 3
In reference to the first: (f) + (1 / f≤) = 2, it can be easily demonstrated. As shown in figure 140, trace the square OMNB using OB as one of its sides. From the midpoint of OB (point S), trace an arc with SM as its radius, to intercept line AB (in point h). The length of Bh will be equal to f. The length from point A to point h is 0.381966, which is equivalent to (1 / f≤). The length of AB is equal to 2. Therefore, AB = ( f) + (1 / f≤) = 2.
To demonstrate the second equation f≥ + (1 / f≥) = 3, with center h, project the distance hO vertically and mark point g. The distance hg, will be equal to hO = 0.618034 = 1 / f. The length of hB is equal to f. Using the Pythagorean theorem:
Side hB = f = (÷5 +1) / 2
Side hg = 1 / f = (÷5 - 1) / 2
Since the triangle ghB is a right triangle, applying the same theorem:
(gB)≤ = (hB)≤ + (gh)≤
(gB)≤ = [(÷5 +1 ) / 2 ]≤ + [ 2 / ( ÷5 +1)]≤ = f≤ + 1 / f≤
(gB)≤ = 6 ( 3 + ÷5) / 2 ( 3 + ÷5)
(gB)≤ = 3 ( 3 + ÷5) / ( 3 + ÷5) = 3
(gB)≤ = f≤ + (1 / f≤) = 3
Therefore, it can be seen that f≥ + (1 / f≥) = 3, and that the hypotenuse is equal to ÷3 = 1.732051.
The right triangle with sides in the ratio 1 : 2, besides its direct relation with f, also shows a close relation with the triangle with sides in proportion 3:4:5, as will be demonstrated in next figures, 141 and 142 which illustrates a sectional view of the pyramid' model, which represents the Great Pyramid's geometry. Figure 141 shows the triangle ZIX, which consists of two triangles (iZI and iXI), both with the ratio 1:2 in their sides. If a line is traced from point X, perpendicular to the opposite side (point L), or from point Z, perpendicular to the opposite side (point M), the two triangles created, XLI and ZMI, have their sides in proportion 3:4:5. This shows the relation between the two triangles.
In reference to the model pyramid, figure 142 shows the triangle (ZIX), which is similar to the triangle in figure 141. This triangle is formed between the pyramidís vertical axis, and the descending and ascending passages. The horizontal line (iI), that is, the line that divides in two the triangle ZIX, is equal to 253.73'. Observe that this distance is also equal to the vertical distance between point Z and point X.
The inclined measurement from point X to point S (place where the inclined floor section of the Descending Passage cuts the base line) is also equivalent to 252.73'. Point S is located at 226.94', horizontally, from the pyramidís vertical axis. The vertical distance XO is equal to 113.47'. This means that if from point X, the distance XS (253.73'), is projected to the vertical axis, the projection will coincide with point Z.
Observe in the figure that a similar situation occurs with the triangle (OZp). The distance of 156.81' for the length of the Grand Gallery, is equal to the distance of the inclined line Op, and the distance Wp (corridor to the Queenís Chamber) = OZ (vertical distance from the center of the base line to point Z).
The formulas and geometric relations presented here are of particular interest to those concerned with ancient Egyptian structures and interested in the function of the Golden Number f. To know them, and how they work, allow to a better understanding of the Egyptian structures and their geometrical relations. This is the reason for the repetition of certain measurements inside the Pyramidís structure.
In our times, the Golden Section is almost forgotten and itís teaching in Mathematics, Geometry, Arts and other sciences are very limited. I would recommend that it be included in the appropriate school curriculums. The Golden Section represents a parameter in Arts, harmony, and beauty. Our Creator included it in the formation of the physical universe, and even in the proportions of the human figure (see Chapter 6). It appears that the ancient Egyptians study, find and used some geometrics configurations which exhibits those parameters, to build their pyramids.