Circles and Squares
A work in progress. Nov 27.
Some Facts
It is a fact that a square with a perimeter of 900 (sides of 225 each) inscribes within a circle with a circumference of 1000. That is a 9:10 ratio of square to circle. The diagonal of a square with sides of 225 is 318.3, which is the diameter of the 1000 circle (1000/Pi = 318.3). The number of the name Helios (the sun in Greece) was 318. A 2000 circle would contain a square with sides of 450 (perimeter of 1800), and a 400 circle would contain a square with sides of 90 and a perimeter of 360 (as in 360 degrees).
It is also a fact that the circle and square have the same number of degrees, and are twins in this respect. John Michell suggests that all Temples entail 'squaring the circle' somehow; which means different things to different people. One way of doing that is to illustrate a circle with the same area or circumference of a given square. In the case of the areas, as we know that in a circle A = Pi x R squared, dividing the area of the square by Pi will give us R squared. In a 10x10 square the area is 100 and our number is 31.83.
As the diagonal of a 1x1 square is 1.4142, the diagonal of the 10x10 square is 14.142. 8/10 of the diagonal of the square equals the diameter of the circle with the same area as the square. Our radius is 4/10th of the diagonal (14.142) or 5.65. The radius is 56.5% of the side of the sqaure, so the diameter is 113 %. 5.65 x 5.65 = 31.9. One hundred is about 32 times Pi. (.8 x 1.4142 = 1.13. This number times the side of the square gives the diameter of the circle with the same area.)
This means that if we take an 8x8 square (with a perimeter of 32 and an area of 64) and put a circle around it (blue below), that circle will have the same area as a 10x10 square (with a perimeter of 40 and area of 100; red below). As you can see, this will be an increase of 36 or 6 squared, nine per side. Also if we have a circle with a circumference of 10, a square with a perimeter of 9 fits inside that. If we multipy those numbers by 4, we get a circle with a circumerence of 40 and a square with a perimeter of 36.
What we see is nested squares with sides of 8, 9 and 10, and perimeters of 32, 36, and 40. We also see that the circle surrounding the 36 square has a perimeter equal to the 10x10 square (40; purple above), while the circle surrounding the 8 square has the same area as the 10x10 square (100). The circumference of the circle around the 8 square measures 35.54, and the circle inscribed in a square (red above) has a diameter of .7071 times the diagonal of the square.
Geomtery in the Bible
If you were to encode this in a story, how would you do that without using the word circle, which is in the Bible only once. In Ezekiel we see the city described as having sides of 4500 surrounded by suburbs of 250 for a total of 5000 in the larger square, that is a perimeter of 20000. If we make 250 into .25 as our unit square, 5000 into 5 and 4500 into 4.5, then multiply by 4 we get 20:18:1. That is an 18x18 square (9x10) with a one border all round, making 20 for an outside measurement.
If we make 20 a circle, we see that the 18 perimeter square fits that exactly. If we change the ratio to 10:9, there is one part left over that is divided .25 north, .25 east, .25 south and west.
[15] And the five thousand, that are left in the breadth over against the five and twenty thousand, shall be a profane place for the city, for dwelling, and for suburbs: and the city shall be in the midst thereof.
[16] And these shall be the measures thereof; the north side four thousand and five hundred, and the south side four thousand and five hundred, and on the east side four thousand and five hundred, and the west side four thousand and five hundred.
[17] And the suburbs of the city shall be toward the north two hundred and fifty, and toward the south two hundred and fifty, and toward the east two hundred and fifty, and toward the west two hundred and fifty. Ezek.48.
In Exekiel 45 we read of a sanctuary that is 500x500 with suburbs of fifty. Here the perimeter is 20 but the interior square measures 16, 400 per side. This is the square with the same perimeter and area. If we double our numbers we get 1000 and 800 with a border of 1. You may recognize this as the square of Mercury. This is the square that if we put a circle on the four corners, the area will be the same as the square of 10.
2] Of this there shall be for the sanctuary five hundred in length, with five hundred in breadth, square round about; and fifty cubits round about for the suburbs thereof.
Ezek.45.
The oblation is 25,000x25,000, the city is 5,000x5,000 and the sanctuary is 500x500. 5,000 is 1/5th of 25,000, and 500 is 1/10th of 5,000. The altar in front of the Tabernacle was 5x5, and the suburbs of the sanctuary were 50.
72
If you were to interpret that as meaning five hundred plus fifty on every side, you arrive at a 6x6 square around a 5x5 one. The square with sides of six hundred has an area of 36 and a perimeter of 24. Thirty six is the number of cells in the square of the sun, while twenty four is the hours in the day. In Revelation Chapter 21 we see the New Jeruslaem described as a cube having sides measuring 12,000 and a wall which measures 144.
In this case, I suggest that 144 is not a twelve number, but a nine number; 4.5, 9, 18, 36, 72, 144, and I will add 54. You may recognize the numbers as multiples of 72 which is how many years it takes the sky to precess one degree. 72 is also the internal angle of the pentagram. the 36:54:90 triangle relates to the pentagon. In this case (Rev 21) when we divide by two we get a dimension os 6,000 and a wall of 72. The perimeter would be 24 and the area 36.
Notice that 6x12, 3x24 = 72 and 2x36 = 72; 2x72 is 144, so the number definitely ties in with 12. Also 8x9 = 72. Seventy two is divisible by 2, 3, 4, 4.5, 6, 8, and 9.
The Great Pyramid
In terms of a circle with the same perimeter of a given square, we know that the diameter of the circle would be 127.1 percent of the side of the square. The Great Pyramid illustrates this. 1.271 x 755 (pyramid base side) = 960. Dividing by two gives us 480 feet. The height of the pyramid is 127% of half the base.
Dividing by seven gives interesting results. 100/7 = 14.2857. Multpily by 8, you get 114; by 9 you get 128. You will remember that to find the circle with an equal area we multiply by 1.13, and to get the equal perimeter we use 127. 6/7's falls on 31 degrees, and 5/7's falls on 45.5 degrees. We locate those with a vesica. (More on that later.)
Here we use equal overlapping circles to double and halfen circles. This shows that the outer circle is two inner circles wide. The outer circle has twice the circumference of the inner one (C=Pi x D). But since the area function depends on the square of the radius, the inner circle has one quarter (.25) the area of the outer one. That means that the area of the outer ring is .75, so that this image indicates 1:2 and 1:3 in different views. We are reminded of the Temple that was 20x60x30. From the top we see three 20x20 squares; from the side we see two 30x30 squares, and on the ends we see 20x30 (2:3) The Temple is divided into the 1:1 Holy of Holies, and the 1:2 Holy Place. But everything in the Bible is done with squares and rectangles.
You may recall that Psalm 118:22 is where the cornerstone is mentioned; the stone which the builders refused. If you look starting verse 10 you will see it written 'they compassed me about' four times. It sounds like the basic method for laying out a divided square. Below we see that the vesica locates the same lines as diagonal of half the square. The red rectangle below represents the 50x100 courtyard around the Tabernacle, which conceptually generates a 100x00 square.
This is 30 degrees latitude (the location of the pyramid), and half way from the equator to the poles on the axis line. While the sun moves to 23.5 degrees at the solstices, Mercury is the most erratic of the planetary spheres and moves to 31 degrees; that is one degree beyond the red line. Note that the sine function measures a percentage of the axis, and that the sine of 30 degrees is .5, the sine of 31 degrees is .515, and the sine of 23.5 is .4.
The 31 dgeree latitude line is 6/7's as long as the equator. We can use the vesica or the diagonals of half squares in the opposite direction to complete the figure. Notice that the square in the center of the figure above is 4x4 Jupiter square with the same area and perimeter, and that the corresponding square that goes with the big circle is the 8x8 square of Mercury. We would need a border of 1 all the way around that to make the 10x10 square.
Here the area of the square in the center is 1/4th that of the larger sqaure, with half the perimeter. There are four squares on the center surrounded by twleve others. The center square is bounded by 30 degrees, or rather by 31 degrees; the limit of Mercury. Conceptually we have the seven planets in the center area surrounded by the twelve zodiac signs. Some people in India prefer this as the template for their astrology charts. Notice that the circle has been equally divided into 12 parts. (Does that look like a tortoise and four elephants supporting the earth to you?)
Three Gates
We see a 2x2 square with a border of 1 all round, making a 4x4 square; four to the north, four east, four south, and four west. Below we divide each of four squares into triangles with diagonals. Each triangle has the same area as a square above. The four triangles in the center form the sanctuary square again, but this time there are three to the north, three east, three south and west.
In Ezekiel 48 it talks about three gates north, east, south and west, and says that the sides are 4500 each. That is a perimeter of 18000. We know that this relates to a 20000 circle. In Revelation 21 we have the same three gates on each side but our measurement is now 12,000 each side; and there is a wall measuring 144. If we divide by 2, we get sides of 6 and a wall measurement of 72; giving us an area of 36 and a perimeter of 24. 6 is the number of the sun, the magic square of the sun is 6x6 containing the first 36 numbers which total to 666. 24 is the hours in the day. 36 is half of 72 which is the internal angle of a pentagram (360/5 = 72) and is the number of years that it takes the sky to precess one degree.
While we are not finished with the image above, I ask that you compare it to these images. The three by three configuraton is the square of Saturn. Below we see the Pisces constellation near where the March equinox occurs. Note that as the square of Pegasus rises it is horizontal, and when it culminates it is oblique.
We see an interface between 9 and 10. The ten series is 10, 50, 25 , 12.5 and 20, 40, 80, etc. The nine series is 9, 45, 22.5, 18, 36, 54, 72, 144. 18, 36, 54 and 72 are all pentagram numbers. 144 is 12x12. The altar in front of the tabernacle was 5x5, the sanctuary was 500 by 500, the Oblation 25,000 by 25,000. The Tabernacle was 10 wide and the Temple was 20 wide. The altar in front of the Temple was 20x20. The courtyard around the Tabernacle was 50x100x5 tall (curtains).
Feeding 5000
If you aren't reminded of the story about the five loaves and the two fishes, you should be. You may remember that 5000 people were fed, and that they sat in companies of 50, or by fifties and hundreds. 50x100 = 5000. The courtyard was 50x100 (1:2). The Tabernacle was within the courtyard and measured 1:3. While three overlapping circles produce two circles, it takes 5 to produce three (3x1).
Five loaves and Two Fishes
The circles in the center have a diameter of 25, the rectangle is 25x75 with a perimeter of 200. Mark 6 mentions 200 penny worth of bread. (loaves, bread, breadth) The 1:3 section in the middle has a perimeter of 200, the courtyard has a perimter of 300 and an area of 5000. Six diagonal lines divide the square into 12 triangular sections, each one equal in area to a square. Some people use the form below, some use the form above, and some use a circle for an astrology chart.
Astrological Houses: Twelve Baskets of Fragments
Below we add two more circles (loaves) producing a 25x100 rectangle with an area of 2500 and a perimeter of 250. The circles are 30 degrees tall, and they stretch 360 degrees, that is the same 1:6 ratio as in Noah's Ark (300x50 or 360x60), and in the image that the King built in the book of Daniel (6x60). As the twin circles represent the two hemispheres, and the diameter of the earth at the equator is about 8,000 miles, each one of the small circles is about 4,000 wide, thus the story of the seven loaves and the 4000.
Seven Loaves Equal Two Diameters
If we begin with a 1x3 rectangle and add .5 on each side we get 2:4, 3:5, 4:6, 5:7, and 6:8. The last one is the 3:4:5 triangle. 4:6 = 2:3 like the end of the Temple. As we are working with a 25x25 square, we know that 3:5 is 75x125, for a perimeter of 400. Multiplying by 2 we get 150x250 and the dimensions of the Ark of the Covenant are given as 1.5x1.5x2.5. That is, the Temple was 1:3, the courtyard 2:4, and the ark was 3:5. The end of the Temple was 4:6 (20x30).
Here is the image that the John Michell camp (Michell, Schneieder and Fideler) present as illustrating the story of the feeding of the 5000. I am not sure how they accomodate the other two loaves and the 4000? Note the hexagon abitrarily thrown in. Please tell me how the twelve flower at the top relates to the notion of dividing the loaves and fishes? You can see that they were one arc away from Metatron's Cube, but none of them mentions it.
Below we see Michell's illustration of the story of the New Jerusalem in Revelation Chapter 21. Notice how he has bunched his circles into three on the north, three on the east, and three south and west. His circle is equally divided 12 times by two hexagons. He has the figure he needs in the square at the center of his image, but he insists that the text points to 13 circles. Had he 'read' the book instead of projecting his own ideas on it, he would have seen that the altar was square, the Holy of Holies was a square and cubic as the New Jerusalem is described as being, the Oblation was square, the sanctuary was square, and the city was square.
"[16] And the city lieth foursquare, and the length is as large as the breadth: and he measured the city with the reed, twelve thousand furlongs. The length and the breadth and the height of it are equal.
[17] And he measured the wall thereof, an hundred and forty and four cubits, according to the measure of a man, that is, of the angel." Rev 21
In this image a heptagon is arbitrarily placed on the figure since the number 7 is in the Revelation more than once. The story about the New Jerusalem uses on 12, 144, 3 and 4 however. Notice how the circles on the perimeter are each 1/14th of the circle. There are 12 circles with four one half gaps, for a total of fourteen. He translates from squares to circles, then from 12 to 14. And he is suggesting that he is some how revealing clues to the building technique for the Cosmic Temple?
Unfortunately his only examples are from Glastonbury and the Stonehenge. He doesn't attempt to analyze cathedrals or temples, or other figures in the Bible. I offer the following analysis of a Greek temple based on the proportions above. Note especially the three circles that define the temple, and where they cross. This point is 60 degrees on the circle, and 6/7's is at 59 degrees. This means that the building minus the stairs is 6/7's of it with them. The stairs are 1/14th of the circle wide.
See City of Revelation, where Michell writes that the plan of Noah's Ark measures 300x50, "which if the cubit of 1.72 feet is the approporaite unit, is equal to 4440 square feet". He, and others, act as if a real object is being described rather than that a set of numbers is being introduced. The dimensions are 300x50x30 which is 1:6 (50:300) 1:10 (30:300) and 3:5 (30:50). Multiplying by 6/5's we get 360x60x36, where 36 is the sun square, and 60x360 is a band 30 degrees wide at the equator around the globe within which the planets travel.
In the same manner "a desert place" in the story about the fishes is translated to "a lonely place", missing the point of the parable. The story is describing the dimensions of the court of the Tabernacle which was in the desert.
The Michell camp rercommends that these figures are suggestive of astrology charts? Below you will see one of those. Below that you can see the series of nested squares that represents the qaudratum form. The fact that the corners fold over to cover the next square shows that the inner squares are half the size of the larger ones.
Squares in one set reduce as 1, .5. 25, .125. Those in the other the sqaure root of two times 1, .5, .25, etc. To move up in either set you multiply by 2,4, 8; to move down you divide by 2, 4 or 8. To move up to the opposite set you multiply by 1.4142; to move down to the other set you multiply by .7071.
The image above features circles in three sets. 707 and 353 are one half and one qaurter of the square root of two; the next larger one would be 1.4142. The number of the name Hermes is 353. 318 was Helios. As you see a square with a perimeter of 318 fits inside the 353 circle. The 250 circle fits inside the 318 square.
Masonic Lodge Floor by HPH Bromwell
We see the 1000 circle but there is no 500 or 250 circle. 1224 is twice 612, which was the number of the name of Zeus. The famous approximation of the square root of two was 153:265. Zeus is four times 153, and Apollo is four times 265. Three times Hermes is also Apollo. Athena is half of 153 (76.5/77). The 153:265 ratio is based on a circle with a diameter of 306.
We have 306, 612, and 1224; 353, 707 and 1.4142; and 1000, 500 and 250. A square with a perimeter of 450 nests inside the 500 circumference circle, that is 112.5 per side. Instead of a 500 circle and a 450 sqaure Fideler (page 218 Sun of God writes of a 515 circle with a perimeter of 464, pointing out that 515 was the number of the name Parthenos (virgin), as in Parthenon.
While that may be true 515 does not fit either of the three number series. If it IS true you will note that the sine of 30 degrees is .5 and the sine of 31 degrees is .515. The 31 degree latitude line is 6/7's the equator and is marked by placing a 3x5 rectangle in a circle.
I am returning to the notion of the tortoise. The six directions in space correlate to his six appendiges. You will note that many cosmologies put a tortoise at the base of their system. The flat bottom represents the surface of the earth and his domed shell the heavens. Note especially the three 'hexagons' across his back. If you have never examined the inside of a turtles shell, you will be suprized by how much the bottom of it (which is elliptical not round) resembles the image above, especially the four points.
A square inscribes inside a circle at 45 degrees from the equator and the poles.
Below we see an archcitect's rendering of a Masonic Lodge floor in which we see nested circles, squares and rectangles. In the second image, I have added 'astronomical lines' to show how the floor is designed.
The image above shows that while the two outer circles represent a 'side' view of the globe (looking at the equator), the center circle represent the 'top' view, from the poles. The square at the center represents alternately, a location on the equator and a platform at the pole (symbolic mountain top/high place). Typically the conceptual procession is from the outer to the inner, and from the lower to the highest. See the images of the Pantheon and the turtle's shell again. You can imagine a square platform near the top of a pyramid.
It has been pointed out several times that two overlapping circle touch at 60 degrees from the equator. The image below uses a circle with a diamter of 306. As the sine of 60 is .866, we know that 265 is .866 of 306. The sine of 59 degres is .857. The 59 degree latitude line meausres 6/7's of the circumference of the circle (equator), and is just short of 60 degrees.
What this means is that if we make small circles on the red line, they will be 1/7th of the diameter each. Looking at the image of the lodge floor you can see that he did not use this method to obtain his border,as they did in the Parthenon. Instead he uses some version of squaring the circle (either a 8 or a 9 square in a 10 circle).
Above you can see that eight points of the Sri Yantra fall on the 59 degree circle, while four fall on the circumference. Measuring from the equator the red line falls at 31 degrees latitude, which is the limit of Mercury's range on the globe. In the image below the 59 degree circle is in blue and the 45 degree circle is in red. Looking from 'the side' we see that the blue cirlce is falf way up the vertical axis while the red circle is at .7. The small circle that we drew above touches the 45 degree circle. You can almost completely divide a circle into sevenths with just these three circles.
To determine where these points fall we divide 1 by seven then multiply. The cosine function tells you the angle. 6/7's of 1 = .857 for 31 degrees (reciprocal of 59). 5/7's is .714 for 44.5/45.5 degrees. 4/7's is .57 for 55/35 degrees. 3/7's = .428 for 64.5/25.5 degrees. 2/7's = .286 and 77.5/12.5 degrees. 1/7 is .1428 for 81.7 degrees.
You can also divide the number of degrees on the globe by seven. 90/7 = 12.857. Thebes was located at two times that number of latitude, Delphi was located at three times that number, and Greenwich is located at three times that number, 51.4 degrees. These are the side view, of course.
Circles and Rectangles
Below we see the image where both the diagonals of a half square and a vesica are used to locate 30 degrees on a circle. The sine function is a measure of the perentage of the height of the axis. The sine of 30 is .5, as it is half way.
If we draw the diagonals of the red rectangle above, they touch the circle at 26.5 degrees meaning that we can use the 1:2 rectangle incribed in a square or a circle to loacte 26.5 and 30 degrees, as we can use the vesica to locate 30, 60 and 45 degrees. Remember that we can rotate the circle and locate 60 and 63.5 degrees with the rectangle too.
It has already been shown that a 3:5 rectangle inscribed in a circle touches the circle at 31 (or 59 depending on the orientation) degrees; that is within one degree of the location determined by the vesica. A square inscribed in a circle touches at 45 degrees; those are at 5 and 6/7's of the circles diameter. 4/7's falls at 55/35 degrees, and the diagonal of the 5:7 rectangle is 35.5 degrees (tan 35.5 = .713; 5/7 = .714). We can use that rectangle to locate 4/7's on the globe. Three sevenths (.428) is 64.5/25.5 degrees, and we know that a 1:2 rectangle has diagonals that form 26.5 degree angles.
The location between 30 and 31 degrees is a very important place in terms of geometry and geography which can be located by triangle, recatangle or vesica. Another important location is around 36 degrees, half the pentagram angle of 72 degrees. The 5:7 rectangle produces a 35.5 degree diagonal, and the 3:4:5 triangle (3x4 rectangle) produces 36.8 degrees. These are virtually indistinguishable by looking at them.
A diagonal angle of 31 degrees 43 minutes produces the golden ratio, or phi (.618). In terms of whole number ratios 309:500 works well. 500/309 = 1.618 , and 309/500 = .618. 309 was the number of the name of Ares in Greece.
Speculation
I am suggesting that since the 3:5 rectangle was so close to the points indicated by the vesica and hexagon etc, that it was used as a conceptual substitute for half an equilateral triangle. I recommend that the 5:7 rectangle and the 3:4:5 triangle were substituted for the 36 degree angle that is half the interior angle of a pentagram. The diagonal of the 3x4 and 5x7 rectangles are both 'useably' close to 36 degrees.
We see that the Tabernacle and Temple, as well as the central part of the Greek temple was made as a 1:3 are that was divided into one and two. As temples were intended to represent the form and situation of the world as much as possible, we can imagine that this form was used because it presents two focal points that are needed to draw
an ellipse. I have already shown that the two circles are the two hemispheres, and that the central circle is the polar view.
Below we see an illusration of the stretching the cord ceremony that entailed driving two stakes in the ground with gold mallets. Note the ellipse. As the stakes here are about a foot and a half apart, the meaning is symbolic and not operative, meaning, you probably don't start to lay out a building with markers less than two feet across.
The Masonic Lodge Floor differs from the above form in that it places the square on the center rather than on the end. Questions arise as to what form to use based on what logic.
Continue 1188 and the Cutting of the Elm, also a work in progress.
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