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Polygonal buildings Possess the Ratio of 5:8:8 triangles By Robert Kerson 2/10/2015 This paper is an extension of the work of Leen Ritmeyer whose work on old and ancient Octagonal buildings such as the Dome of the Rock in Jerusalem helps to prove the use of 5:8:8 triangles in designing sacred spaces such as the actual location of the Jerusalem temple. It is advisable to first read my other papers beginning with Discovery of a Sacred Ancient Survey Technique Used In The Middle East and Method By Which Jewish Temple Could Have Been Laid Out which discusses the use 5:8:8 triangles in laying out sacred spaces. This paper is very different in that no 5:8:8 triangles are in this octagonal building but the relative lengths of the sides of this triangle are measuring major features of the building. The ratio of sides 8/5 = 1.6000 are here. This is not the golden ratio of 1.62... but 1.6000. And the ratio pattern is very exact. First the horizontal cross section shown in Figures 1 and 2. There are three outward concentric circles starting at the center O labeled A, B, C. The distance OA= AB = BC. Each of the three circles have a set of two lengths whose ratio is 1.6000. One length is from the center outward, and the other length is from side to side. Figure 1 Is drawn as a tracing off the Ritmeyer horizontal cross sectional drawing of the Dome Of The Rock (see ritmeyer.com. Feb. 2015 entry) This figure is a very simplified rendering of only the important lines shown dashed of the full drawing. The reader needs to go to the original to see the full pattern. My lines are all drawn in color according to the table below. The pattern comprising three circles labeled A, B, C with a center labeled O. These are the three sets of lengths. (drawn staggered to demonstrate these lengths are not connected by any triangle) In an octagon, these three sets are duplicated eight times. In other polygons, these sets are duplicated by the number of sides. For example, in a hexagon, such as the one at Baalbek, there would be six times these three sets.

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Summarizing Fig. 1 in the following table. From left to right are listed the three circles, larger lines of 8 parts, the color code of lines running from center – outward, smaller lines of 5 parts, the color code of lines running from side to side. The pairs are grouped in yellow and orange shading.

From Circle labeled

Ratio number center- outward ratio number Side to side

Circle A 8 Red with black dots

5 Red with black dots

Circle B 8 Red with black dashes

5 Red with black dashes

Circle C 8 Red solid 5 Red solid

Note that all paired lines running from center- outward are the longer of the two lines (8 parts),and all paired lines running from side to side are the shorter of the two lines (5) parts.

Note that major features of this octagon such as placement of piers, columns, openings, wall lengths are all showing layouts in the ratio of 1.6000 or the same ratio as the sides of a triangle having the ratio of 5:8 which is my triangle. Here on the site of the temple, where I have used my triangle to locate same, Dr Ritmeyer had demonstrated my triangle ratio in the Muslim building! There is an interesting pattern occurring: at Circle A, concerning this set, both lines are on Circle A which makes the inner side of piers and columns holding up the dome. At Circle B concerning this set, one line is on Circle B which makes the outside of these piers making the inner octagon and the other line marks out an inside edge of these piers. At Circle C concerning this set, one line is on Circle C which marks out nothing in the building while the other line marks out the length of inner edge of the wall making the outer octagon. There is no connection of Circle C with this wall except being nearby.

The only piers and columns are in Circle A and Circle B. Both circles have side lengths which position the pier and column locations. All side lengths make up the smaller number of the ratio—the length 5, while all center –outward lengths make up the larger number of the ratio—the length 8.

I also have found other examples of these lengths throughout the building. I have labeled four side to side lengths. (There are other examples of these lengths but they are not labeled.) They are drawn as green dots and numbered 1 –1` 2 --- 2` 3 – 3` 4 – 4`

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Example, The distance from green dot 1 to another dark greet dot 1` is the same distance from the side to side line shown as a red line with black dots as part of the pair on circle A. Note that only the last pair passes through the center point (O). The others are on any of the eight lines radiating from the center.

Fig. 1

Next shown is the Vertical cross section seen in Fig. 2. (cross section drawn by de Vogue, 19th

century reprinted in The Quest by Leen Ritmeyer; Carta; Jerusalem; 2006 p.251) Fig. 2a is the

book copy. Fig. 2 shows lines in the same color scheme as in Fig. 1. The length of these lines are

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the same lengths as the lines in Fig. 1 but are drawn in vertical cross section instead of in

horizontal cross section . We have four circles: the lower one is centered at ‘A’, the upper

inner dome is centered at ‘B’, and the outer ovoid dome has two centers labeled ‘C1 ‘ and ‘C2 ‘

which are on design elements of the dome. The same color coding of red with black dots, red

with black dashes, and red solid seen the Fig. 1 is seen in Fig. 2. [This is a 2 dimensional

representation of what is actually a 3 dimensional space. (Circles are actually spheres.) Thus, if

the viewer were to look straight on, they would see the lines as if in front of the actual walls.]

There are two Vesica Pisces: one between centers ‘A’ and ‘B’ (shown), and the other between

‘C1’ and ‘C2’ (not shown).

In Fig. 2 we see as in the horizontal cross section, in the vertical cross section we again have

three pairs of lengths having the ratio of 1.6. The pairs are shown in identical color coding as

the above table. The identical radii of (A) and (B) are shown in red with black dots. This line

pairs with a smaller line whose length is the horizontal distance between C1 and C2 and also is

the vertical distance within the Vesica Pisces which makes one edge of an upper window at the

base of the dome. (The identical radii of C1 and C2 are shown as black dots because these

lengths do not figure in the ratio of 1.6.)

The second pair of lengths having the ratio of 1.6 can be found drawn as a red line with black

dashes. Note how the bottom of one line defines a horizontal band at the base of the dome and

top of the line reaches the top of the dome. The other line runs from the top of the dome to

the top of the (A) circle.

The third pair of lengths having the ratio of 1.6 can be drawn as solid red lines in the vertical

cross section. One line runs from a structural member shown in black at the top of the dome,

and running to a horizontal band at the base of the dome. The other line runs from just below

the open circle at the top of the finial, to a band by the top of the columns within the building.

This finial has a circle believed to represent the sun, but this may not be correct in Islam and

may actually represent the circles seen in Fig. 2 since the measurement of height just below this

open circle is one of the measured pairs of this hidden system of geometry.

Fig. 2

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There are a few interesting facts occurring with these pairs of lines:, a number begin at the

center top, they overlap along the center line, and they have hidden connections between the

two Vesica Pisces (Example a line of in one pair making a Vesica Pisces has the other line of this

same pair involved with the making the other Vesica Pisces.)

I have shown that the ratio of the 5:8:8 triangle is present in this octagonal building. Any

octagonal building laid out in this pattern, would also have this ratio. We may be looking at a

very ancient and lost building technique used in this part of the world.

Another interesting fact which is not on this figure is that he puts this octagon over the western end of the Jewish temple building (the Holy of Holies) near the center of this octagon while I put this octagon over at the eastern end ( Porch, threshold, eastern end of Holy Place, a few northeastern side rooms) of Jewish temple building. He shows in his blog the facade at the eastern end of the Jewish temple building being slightly wider than the width of this octagonal building. His version approximately matches the octagon with the temple's western end, while my version approximately matches the octagon with the Temple's eastern end-- an approximate match of both buildings widths. (The temple's western end was not as wide at its eastern end which had two side extra wings making up the facade. My version is a closer fit to the existing octagonal building.