When I studied math and geometry in school, it wasn't taught in terms of real world applications, you had to take physics for that; but they could have. When they teach ratios and proportions and trigonometry, they could show you how it relates to putting siding on the wall of a house, for instance. The 'pitch' of a roof refers to the angle of the roof 'slope', and it is expressed as x/12, where x is some number; such as 6 over 12. This means that for every 12 inches of 'run' the roof 'rises' 6 inches; a 1:2 ratio. A roof 24 feet wide is 6 feet tall in the middle. The roof truss for that would be two 6/12 triangles butted together.

As to siding, it comes in 4 foot sheets, so 6/12 is 24/48. Each sheet is 24 inches longer on one side. We know this 1/2 ratio to produce a rectangle with diagonals of 26.5 degrees. 100x225 produces 24 degree angles like the tropics. 3x5 produces 31 degrees, the limit of Mercury's orbit.

As it turns out, the 31 degree latitude line is 6/7 of the equator. What this means is that the sides of a hexagon inscribed inside a circle at 31 degrees are the same as the sides of a heptagon inscribed inside the equator. The inscribed hexagon has sides equal to the radius of the circle. The radius of the 31 degree circle equals the sides of the heptagon in the equator. Here is a large image of the geometric demonstration of that.

  

As the corners of a triangle mark 30 degrees, we know that the 31 degree circle fits just inside of that. The top red triangle on the right shows two radii and one side of the hexagon inside the top red circle. The small red circle indicates the bottom of that triangle, and the 51.4 degree traingle that indicates two points of the heptagon. (Remember that when we use the vesica to reduce we end up with a seven degree radius, and the heptagon uses an eight degree radius, so use the 'other side' of the line. Make it a little long, as we say.)

As 31 degrees is 6/7's of the equator, and the reciprocal of 31 is 59, the 6/7's latitude line is 59 degrees from the pole. We note that a vesica demonstrates the location of 30 and 60 degrees. The 31 degree line is just inside the 30 degree line. The gap between that line and the equator is one seventh. More circle and lines produce the familiar nested rectangles and globes. Overlapping circles indicate one seventh.

Consider what Michael Schneider calls the 12 part celestial canon. As you can see he has divided the circle into 14 parts. He has 12 circles and four half circle gaps, for a total of 14. As was shown above, a circle with a circumference of 14 has a diameter of 4.56. His middle circle is neither 1/4 the diameter, nor is is 1/5 of that, as the diameter and circumference of the circle are not commensurate (they don't match up in whole number ratios).

What he is illustrating is the seven unit circle and the one seventh circle at the center. Above we saw a seven circle on the outside. The vesica locates 59 (60) degrees; this is the 6 circle. The square in the circle locates 44.5 (45) degrees; this is the 5 circle. Overlapping circles from the vesicas locate the one and two circles. Remember that these are correct to within a degree. The four circle is twice the two circle, and the three is half the six.

The sides of a hexagon inside a circle are equal to the radius of that circle. The hexagon inside the six circle has six sides equal to that radius. Seven of these sides makes a heptagon that fits inside the seven circle. A line tangent to the two circle indicates the location of two points of that heptagon. Connecting those two points to the top of the circle produces a 51.4 degree triangle.

When we look at the elevations of cathedral plans or rose windows, we are seeing the real world applications of this geometry and number system. I have tried to show that the canon of the arts that every one talks about, insists that the squares, circles and triangles not be haphazardly sized and positioned. I recommend that two major factors as far as design is concerned, are the system of doubling or halving the circle (or square), and the system of dividing the circle into seven parts (which is easier than it sounds).

What this means is that in 'sacred geometry' we can expect 1) to see circles, triangles and squares that are twice and four times each other, and 2) to see one seventh (sevenths) expressed somehow. We bear in mind that this is real world, and what matters is 'the illusion' and not mathematical perfection.

The two axes of interest in this scheme are 45 degrees and 30 degree diagonals. The 45 degree angles cut the square in half, and 30 degree angles cut the sides of a triangle and a circle in half. One produces a square in the circle (below red) and the other produces a rectangle (above). Remember that the rectangle is not as wide as the circle is; it is .866/1.

We will call one ad quadratum, by the square, and the other ad triangulum, by the triangle. The triangulum is used for elevations and the quadratum for floor plans (except in rare occasions). You don't see triangulum floor plans. The floor is considered as representing the earth, which we divide with squares and rectangles if possible.

The quadratum form is based on this figure. As you can see folding the corners in produces a square half the size of the big square. Each oblique square has half the area of the next right square. If the large square has an area of 100, the next oblique square has an area of 50, the next right square is 25 and so on. the square roots of 100 and 25 are round numbers, but the square root of 50 is 7.071 or half the square root of 2.

In this quadratum figure, the right squares are 1,2,4 and 8 wide, or 1. 1/2, 1/4. 1/8, etc. The oblique squares are multiple of the square root of 2. Below we can see the relationship of the length of the sides. Notice that we are dealing with 45-45-90 triangles (or an equal armed square). The square root of 2 divided by four is 353.5.

Once again, we point out that there are two sets of squares above, one set is rotated in relationship to the others. In each set, each successive square is half the width of the first. One set begins at 1, and one set begins at 1.4142 (sqrt of 2). The one series would be 100, 50, 25, 12.5. The other would be 1.414. .707, .353. The image below shows that you can do the same thing with circles.

We know that the 353 circle is half the 707 circle, and the 612 circle is half the 1224 circle. This is a David Fideler image, and he does not show the circle that goes with the 1000 circle but he describes it as having a circumference of 515 feet? Recommending that that number derives from the word parthenos (virgin). This is clearly not true. The sine of 30 degrees is .5, the sine of 31 degrees is .515; he is one degree off. The circle measures 500 feet, half of 1000.

He further suggests that a square with a perimeter of 464 fits inside the next circle. Dividing 1000 by 2 we get 500. Dividing that by Pi we ge the diameter of that circle which is 159, or half of 318, the diameter of the 1000 circle. The number of the name of Helios (the sun). The diameter of the circle is the diagonal of the square in the circle.

; that is c in a sqd + b sqd = c sqd. C squared equals 25321.5. The square root of half of that is 112.5, the sides of the square in the 500 circle. 464/4 is 116. He fudged his number for some reason?

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The name of the number of the Greek god Hermes is 353. The Roman name for Hermes was Mercury. The planet Mercury has an 88 day orbit of the sun and makes four orbits a year. 88x4=352. Twelve lunar months of 29.5 days each equal 354 days.

The triangulum form depends on the 30-60-90 triangle that is half the equilateral traingle. If we double the equilateral triangle we get a rhombus, which when we cut that in half, we get four 30-60 triangles. These are squares with unequal arms. The axies of a rhombus are in a ratio of 1:1.732 (sqrt 3) to one another.

A long time ago, it was figured out that 265/153 was very close to 1.732. If we multiply this ratio by four we get 1060/612. The number of the name of Apollo was 1060, and the number of the name of Zeus was 612. Also Apollo/Zeus = Zeus/Hermes AND 3 x Hermes = Apollo.

Twice 612 yields 1224, and 1225 is the sum of the cells in the 7x7 square of Venus. 1225 = 5 squared plus 7 squared. 153 divided by 2 yields 76.5. The number of the name of Athena is 76. Apollo was the brother of Athena, who acquired his lyre from Hermes.

Continue to a discussion of Michael Maiers Philosopher's Stone image.

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