© Copyright Lorena Loo
The Circle and the Pentagon:
Wherein the 18 and 19 of Ancient Egyptian Biometrics
Part I
©Lorena Loo
Wherein the 18 and 19 of Ancient Egyptian Biometrics
Part I
©Lorena Loo
The numbers 18 and 19 cropped up frequently in the ancient Egyptians depiction of biometrics, the measure of man or the measure of human. The Egyptians employed a grid of 18:19 in a system of proportion purportedly at the beginning of Dynastic Egypt.
Two examples of this appear in the images below. The image on the left illustrates a grid of 19 squares were employed in scaling the height of the two figures in the diagram. There are 18 squares to the brow and 19 to the top of their heads.
The second image depicts the square grid used in constructing the images. Thoth, the neter of magic, geometry, science, writing and language in the Egyptian pantheon, is illustrated with his arms outstretched to his sides. There are 18 square grid units to the top of Thoth's head whereas there are 19 grid units to the top of the head of the figure to Thoth's left (your right as you look at the diagram). You may notice from the diagram that the ibis-headed Thoth has a flattened skull at the top. The figure on his left, however, has a fuller, rounder skull which adds the extra grid unit to his height.
Two examples of this appear in the images below. The image on the left illustrates a grid of 19 squares were employed in scaling the height of the two figures in the diagram. There are 18 squares to the brow and 19 to the top of their heads.
The second image depicts the square grid used in constructing the images. Thoth, the neter of magic, geometry, science, writing and language in the Egyptian pantheon, is illustrated with his arms outstretched to his sides. There are 18 square grid units to the top of Thoth's head whereas there are 19 grid units to the top of the head of the figure to Thoth's left (your right as you look at the diagram). You may notice from the diagram that the ibis-headed Thoth has a flattened skull at the top. The figure on his left, however, has a fuller, rounder skull which adds the extra grid unit to his height.
The renown mathematician and Egyptologist Rene Schwaller de Lubicz even found the 18:19 square grid in a famous Mayan hieroglyphic tablet illustrated to the left. Such tablets which seem to provide a link between pre-Mayan and Mayan culture to ancient Egypt are rare and to date, no conclusive evidence of such a link have turned up.
So where does this 18:19 ratio arise from? Schwaller de Lubicz's detailed measurements and study of ancient Egyptian art and temples illustrated that the Egyptians almost invariably linked the measurement of phi with that of pi in all their works. Danish engineer Tons Brunes, who himself extensively studied sacred geometry of ancient cultures, concluded that to the Egyptians the circle was absolutely sacred. It is in combining these two lines of thoughts that we find the answer to the why of 18:19 in biometrics.
Below I have constructed a regular pentagon and its circumcircle. A regular pentagon is simply a pentagon with all 5 sides of equal length. Its circumcircle is the circle that just encloses the pentagon. The thick purple line is the "diagonal" of the pentagon. There are 5 such diagonals that can be drawn and when you draw all 5, then you form a pentagram within the pentagon. The thick green line is the height of the pentagon (by pentagon here it is understood I mean regular pentagon). The thin purple line represents the radius of the circumcircle.
Below I have constructed a regular pentagon and its circumcircle. A regular pentagon is simply a pentagon with all 5 sides of equal length. Its circumcircle is the circle that just encloses the pentagon. The thick purple line is the "diagonal" of the pentagon. There are 5 such diagonals that can be drawn and when you draw all 5, then you form a pentagram within the pentagon. The thick green line is the height of the pentagon (by pentagon here it is understood I mean regular pentagon). The thin purple line represents the radius of the circumcircle.
When you form a diagonal of the pentagon, you also create a golden triangle (shaded portion in figure to left). The diagonal forms the base of the triangle and the adjacent sides of the pentagon form the sides. When a golden triangle is mentioned in sacred geometry, it is almost invariably the one where the ratio of the side to its base is phi, the golden ratio. But in the case of the triangle formed by a diagonal of a pentagon with two of its sides, it is the ratio of the base to its side that is the golden ratio (phi). Since all sides of the pentagon are equal in length, then this same ratio holds true for any diagonal of the pentagon to any of the sides of the pentagon.
Using this relationship, you can calculate the ratios of the diagonal of the pentagon to the radius of its circumcircle and the height of the pentagon to the radius of its circumcircle. To 9 decimal place accuracy, these values are 1.902113033 and 1.809016694 respectively. For the circle radius of 10 then, the pentagon diagonal and height respectively are 19.02113033 and 18.09016694 or as whole numbers, 19 and 18. These values of 18 and 19 or 1.8 and 1.9 crop up frequently in biometrics. They can also be discovered embedded into the design of the Great Pyramid (see my article on The Great Pyramid to be posted soon).
In his book How You Can Talk With God, Paramahansa Yogananda said that:
The whole universe—which is God's body—is made of the same five elements that compose man's body. The starlike shape of the human body represents the rays of these five elements. The head, the two hands and the two feet form the five points of the star. So, in this way too, we are made in the image of God.
The five-pointed starlike form of the human body was also known to the alchemist and magician Heinrich Cornelius Agrippa (1486-1535). Considered the most influential writer of renaissance estoerica, Agrippa compiled a three-volume work entitled "De Occulta Philosophia Libri Tres" which combined astrology, magic, Qabbalah, theurgy (divine work), medicine, and the occult properties of rocks, plants, and metals. Probably the most famous of the images from Occulta Philosophia is that of pentagram (or pentalpha) man as illustrated below. Note how Agrippa enclosed his pentagram man within a circle, the circumcircle of the pentagram.
In his book How You Can Talk With God, Paramahansa Yogananda said that:
The whole universe—which is God's body—is made of the same five elements that compose man's body. The starlike shape of the human body represents the rays of these five elements. The head, the two hands and the two feet form the five points of the star. So, in this way too, we are made in the image of God.
The five-pointed starlike form of the human body was also known to the alchemist and magician Heinrich Cornelius Agrippa (1486-1535). Considered the most influential writer of renaissance estoerica, Agrippa compiled a three-volume work entitled "De Occulta Philosophia Libri Tres" which combined astrology, magic, Qabbalah, theurgy (divine work), medicine, and the occult properties of rocks, plants, and metals. Probably the most famous of the images from Occulta Philosophia is that of pentagram (or pentalpha) man as illustrated below. Note how Agrippa enclosed his pentagram man within a circle, the circumcircle of the pentagram.
Agrippa's pentagram or pentalpha man was the Star of the Microcosm, symbolizing Man (understood here that Man denotes human and is not gender specific) within the microcosm which in turn was a representation of the macrocosmic. At each of the five points of the star, Agrippa placed the symbol of a planet. At the center of the circumcircle, located at the genitals, Agrippa positioned the symbol of the Moon.
Forming the pentagram within the pentagon also forms five golden triangles of the type where the ratio of the base to side is phi. But there are also five golden triangles formed where the ratio of the side of the triangle to the base is phi. One such triangle is shaded in the diagram to the left where the side of the pentagon forms its base and its two sides are formed by two successive diagonals of the pentagon. The green line is both the height of the pentagon and the height of the shaded golden triangle. As
mentioned earlier, this is normally the type of triangle referred to when one speaks of a golden triangle. In either case, both types of golden triangles are isosceles (has two sides of the same length). The lines forming the pentagram intersect in such a way that they subdivide any given line into the phi ratio. i.e. The length of the longer line segment to the shorter one is phi, the golden ratio.
Any given line can be subdivided into extreme and mean ratio. That is, subdivided into the ratio of the longer line segment to the shorter one is phi. The length of the whole line to the longer subdivided segment will be phi as well.
An easier way to understand this is depicted in the diagram below. The whole line segment in blue is divided into extreme and mean ratios of a longer line segment (in green) and a shorter one (in orange).
Any given line can be subdivided into extreme and mean ratio. That is, subdivided into the ratio of the longer line segment to the shorter one is phi. The length of the whole line to the longer subdivided segment will be phi as well.
An easier way to understand this is depicted in the diagram below. The whole line segment in blue is divided into extreme and mean ratios of a longer line segment (in green) and a shorter one (in orange).
The following four diagrams illustrate how to phi cut any given line using a 2:1 right angle triangle in sequential order.
The red dot in the last diagram and step determines the phi cut point of our original line.
The diagram to the right shows how a line of length x is divided into extreme and mean ratios of a longer line segment to a shorter one.
At the beginning, I showed an image of Thoth, the ibis-headed neter in the Egyptian pantheon. Here is another image of him from E. Wallis Budge's Gods of the Egyptians. The scene depicts Thoth holding a bowl containing the uas scepter and the ankh and purportedly represents the mystery of the resurrection of order from the sacred ether flux. The ancient Egyptians
seemingly never designed anything just for the sake of decoration. There was always a purpose behind what appeared to be innocent lines in their art and architecture. To illustrate this, in the next diagram I have reversed phi cut the length of the body of Thoth. Normally when you phi cut the body, you do it from the top of the head to the feet so that it is divided into a 1 to phi ratio. But I have reversed the phi cut in diagram so that the cut is phi to 1 from the head to the feet. Note also that in the diagram, I have only shown partial arcs of the circles used in the phi cut. You can see that the reverse phi cut occurs right at the second line at the bottom of Thoth's apron. What on the surface appears to be an innocent decorative line turns out to be far more significant in terms of sacred geometry.
The next diagram illustrates a normal phi cut of the body. The phi cut point occurs at the navel whereas the genitals divides the body into half (i.e. a 1:1 ratio).
Here is our original line divided into extreme and mean proportions:
Phi can be determined algebraically. It is .
