Jerry & Bob & MetPhys,

I'd like to catch you up on correspondence between me & Michael about the torus geometry.

Tom

From: TomBuoyed
To: Milamo

Michael,

I'm overwhelmed by all your grid-point insights and I feel like we need to speak on the phone just so I can catch up a little.

You have connected a number of things in my life and work, especially concerning a play that I have been rewriting recently called "Noah's Newfangled Ark."

So there is a synchronicity with Noah's Ark and as well, the 90 deg angle which is so crucial both to Arthur Young and to Rudolf Steiner.

As well the 2*PI*PI is the factor in the volume of a torus with infinitely small hole, also called the umbilicoid of revolution, which you can generate by having a circle hinge around itself using a point of tangency on the circumference.

The torus is the expression of double curvature and expresses Stage 7 in Young's 7 stage theory of process. More on that later.

So the volume of a torus with radius R is

2*PI*PI*R^3

whereas the volume of a sphere of radius R is:

(4/3)*PI*R^3

Anyway my number is (xxx) xxx-xxxx and I'll be home today. If you like, e-mail me your phone # and I'll call you.

Thanks. Tom

From Michael,

Hi, Tom ...

Great !! I'm really glad to hear this !! Synchronicities and all !!

This is exciting ... I left a phone message on your machine.

I was out and about today .. so didn't get a chance to see your email until 7:00 PM or so.

I've_already_Tom .. found a direct connection .. using the formula you gave me for Vol. of a torus .. to the exact, and obviously-intended .. diameter of "The Circle of Churches" in South of France .. in terms of_regular_inches !!!

REG-U-LAR .. as in REG-U-LUS !!

I substituted 57.29577951 for "R" .. radius.

In fact .. I'm thinking that the "R" the shepherds are pointing to .. in the Poussin painting .. "Les Bergers d'Arcadie" .. is standing for "radius" and for "Regulus"... either or both. Or .. "Radian (arc)", maybe.

The Regulus "ASM" Grid POINT Value for Jan.1, 2000 .. is .. 19.7392088 ... 2Pi X Pi.

(Morton, 1999, Internet).

(57.29577951 X 57.29577951 X 57.29577951) X 19.7392088 .. = 3712766.512 Cubic arc-degrees ?? Or .. "toroidal arc-degrees" ??

I'm amazed at how much larger the volume is for a torus .. as compared to volume of a sphere !!

I didn't even_know_there was a formula for volume of a torus !!

What about surface-area on a torus ?

371276.6512 .. regular ("British") inches is the diameter of "The Circle of Churches" in South of France !!

(Morton, 1998, Internet).

That is equal to .. as per my proposal on The Internet .. 18,000 Royal Cubits .. of .. 20.62648063 regular inches each. That's my proposed true, precise, and intended length for .. The Royal Cubit !!

(Morton, Internet, 1998).

It is straight from the Radian (arc) numerical value .. assuming the 360 arc-degrees system .. of .. 57.29577951 (arc-deg) !!

It's the Area-on-a-Hemisphere formula ... 2Pi X (57.29577951 X 57.29577951) = 20626.48063 Square arc-deg.

You might have already seen this on one of the websites with my material .. but I thought I'd put it here for you, anyway.

Direct link .. from Royal Cubit to 360 arc-degrees system numerical value of the Radian Arc .. to the regular inch and regular foot !!

And ... it's the DIAMETER of "The Circle of Churches" ... and the classic Pi ratio approximation is .. 22 / 7 ... the 7 is the diameter !!

-- Michael L.M.

In a message dated 06/02/2001 12:27:56 AM Pacific Daylight Time, TomBuoyed writes:

Michael,

The surface area of the torus is very interesting.

It is equal to 4*PI^2*R^2

Compared to the surface area of a sphere. 4*PI*R^2, you see that the torus has an extra factor of PI in there. Now this has ramifications for your recent discoveries about the 90 degree angle, which is involved in the process of "squaring the circle.:"

Below I copy for you a post I made to the Arthur Young list in Oct 1999 abut "squaring the circle."

Tom

From Tom:

Squaring the circle in the traditional way means having to deal with a construction based on the irrational number PI but also taking the square root of PI. Very messy indeed.

However, "squaring the circle" has a much more elegant meaning if we do it with the circles involved in the torus. Let's start with a circle of radius R. Its circumference is [2*PI*R]

We set up a Cartesian xyz coordinate system, x-axis horizontal (left-right); y-axis vertical (top-bottom), z-axis coming out of the screen toward you (front-back).

We "hinge" the circle on the vertical y-axis, i.e. we make the circle tangent to the vertical axis at the origin. Thus the center of the circle is the point (-R,0,0) Let it swing around that "hinge" in space, thus making the 3-D torus with the infinitely small hole, called the "umbilicoid of revolution" (More about the belly-button reference later).

Now look at the inner circle of the torus which is made by the center of the original circle being rotated around completely. It sits in the xz plane (on your horizontal tabletop) perpendicular to the plane of the original circle in the xy plane (on your vertical screen). It also has a radius of R and its center is at the origin (0,0,0).

These are the two independent curvatures of the torus that Arthur mentions.

What about squaring these circles? In the traditional problem, you start with just one circle of radius R and try to construct a square that has the same area as the original circle. The square then has to have sides equal to:

[(SQ RT PI) * R] to give the same area as the circle [PI*(R^2)]

But let's forget that traditional problem. What I propose here is to take the original circle, snip it at one point and lay it out straight. We then get a straight line whose length is the circumference of the circle: [2*PI*R].

Then take the other circle and do the same thing. We get a straight line of the same length [2*PI*R]. Since the original circle is perpendicular to the other circle, then let's lay out one straight line perpendicular to the other one. Thus, we have the makings of a nice square. Let's take the area of that square.

Each side is [2*PI*R], so the area of this new square is:

[4*(PI^2)*(R^2)]

Amazingly enough, the area of this 2-D square turns out to be identical to the surface area of the 3-D torus!

Voila!

Tom Mellett

From Michael Morton,

Tom ...

That's great stuff !! I really like that !! Thanks !

Yeah ... 129600 .. square of 360 !! That's the Grid LONG of the North Bimini Shark Mound ..

(Munck, 1992, "The Code").

110 (deg) X 22 (min) X 53.55371901 (sec) W.Giza .. = 129600 W.Giza. [ W.Greenwich 79 deg 14 min 52.75371901 sec ].

I love that analysis you did on that torus surface area !!

I noticed the torus volume is 1.5*Pi more than vol. of a sphere.

-- Michael L.M.

Date: Saturday, June 2, 2001 4:35:32 PM

From: TomBuoyed
Subj: Re: squaring the circle
To: Milamo

Michael,

The transition from the sphere to the torus is critical to Arthur Young's whole theory of Process.

I'll give you a short outline.

The Volume of a sphere is (4/3)*PI*R^3

The Volume of the torus is 2*PI^2*R^3

You noted the difference as an extra 1.5*PI.

But Arthur splits up the difference in two parts.

First there is the factor of 2*PI. This represents total uncertainty, total freedom. It is the 360 degrees of full freedom, literally "full circle."

This then leaves a factor of 3/4 which the physicist Eddington called a "scale factor," but which Arthur recognizes as going round the cycle 3/4 which in his theory is the exercise of conscious control. Thus the transition from sphere to torus is the transition from unconsciousness to consciousness and control by that consciousness.

The 2*PI is consciousness, the 3/4 is control by that consciousness.

Then the torus volume is seen as the sphere volume multiplied by 2*PI and then by 3/4.

The torus is an abstract representation of a vortex and the property of any vortex is that form and content are the same. For example, a whirlpool in water is the only form water can take to be its own form. If you pour water into a glass, it takes the form of the glass container. There substance (water) and form (glass) are different. But in a vortex, the substance (water) has a form made out of water itself.

This is true for air, tornadoes, fruit bearing seed within itself, e.g. apple tree, and also the human being with the umbilical cord which is the form of a vortex between mother and fetus. In other words, wherever you find a vortex, you find self-reference and thus unity of substance and form. That's what I believe is the explanation of the saying: "Buddha contemplating his navel." He's contemplating the self-sustained, self-generating nature of the vortex, abstractly symbolized by the torus with infinitely small hole, known in mathematics as the "umbilicoid of revolution."

I'm going to let Jerry & Bob Friend in on our discussion. I'll forward them the stuff about the torus and see what they can come up with.

Tom

PS Visit Arthur Young's site at

http://www.arthuryoung.com/

and Nick Thomas' Projective Geometry site for more about the vortex.

http://www.anth.org.uk/NCT/

From Tom:

Michael,

You're really "cookin' with gas", as they say!

And now you have researched the one crop circle formation that caught my fancy when it first appeared 10 years ago, and still does now -- the Barbury Castle glyph.

Anyway, here are some more things about the torus, tetrahedron and sphere.

I want to separate the torus out from the other two because the torus is topologically speaking, a doubly-connected surface while the other figures, sphere, 4-hedron, cube, other Platonic solids are all simply or singly connected. That means they have a definite inside and outside which are separated by the surface boundary. That is not true of the torus, but that's a minor point now.

The sphere to the tetrahedron or 4-hedron represent two extremes of the volume to surface ratio of simply connected solids. That is to say, the sphere is the 3-D figure in which you have the MINIMUM surface area for any given volume (or the MAXIMUM volume for any given surface area). That means the Volume/Area ratio will be relatively the highest.

Take the Volume of a sphere: V = (4/3)*PI*R^3

The surface area of the sphere: A = 4*PI*R^2

The V/A ratio = (1/3)* R

(If we use unit radius, R=1 then the ratio is simply 1/3 = .33333....)

From Tom:

At the other extreme is the tetrahedron, in which the opposite holds true. There you have the MINIMUM volume for a given surface area (or the MAXIMUM surface area for a given volume).

Since the 4-hedron is a pyramid, the volume of any pyramid is:

(1/6)*[base area]*[altitude]

If we have the length of a side as unit base = 1 foot, then the area of the base will be

sqrt(3)/4 = 0.43301270 sq.ft.

The altitude is sqrt(6)/3 = 0.81649658

Multiplying these all together, we get

V = (1/6)*[sqrt(3)/4]*[sqrt(6)/3] = sqrt(2)/12 = 0.11785113 cu.ft.

Now the surface area of the 4-hedron will be 4 times the area of one face, which we've already calculated as sqrt(3)/4.

So A= sqrt(3) = 1.732050808 sq.ft.

Now the V/A ratio is then sqrt(6)/36 = 0.06041381 ft.

From Tom:

Look how small this ratio is compared to the sphere V/A = 0.333333

As I said before, these numbers represent the extremes of V/A for simply connected solids. All the Platonic solids will have a V/A between the 2 extremes, close to the middle of the range and toward the sphere. For example, take the cube. Its V/A is 1/6 = .16666...

I believe the dodecahedron has a V/A of 0.191

From Tom:

But now look at the V/A for the tetrahedron. I expressed it as sqrt(6)/36

However, I could write it another way as the square root of the inverse of the number you use as the human number 6*6*6 =216.

That is, 4-hedron V/A = sqrt (1/216) = 0.068041381

From Tom:

Now look at the 4-hedron angle, the famous 19.5 degrees, or more exactly 19.47122063 degrees.

(By the way, my calculator give 3 as the last digit, while yours gives 1. I have a simple Sharp pocket calculator which may not have the accuracy of yours. But no matter for this line of research. I'll look in a math handbook for the exact figures.)

This angle is better expressed as the inverse sine (arcsine) of the ratio 1/3, which happens to be the V/A ratio for a sphere. (And Richard Hoagland first noticed the 19.5 deg when he inscribed a tetrahedron inside a sphere. It is the angle formed by a line from the center of the top 4-hedron face meeting any line in the plane of the sphere's equator.)

Now this angle, in a simple right triangle, means that, if the hypotenuse is 3 units, the short leg of the triangle is 1 unit, so that:

sine 19.47 deg = 1/3.

But the long leg of the triangle is sqrt (8) units or 2*sqrt(2) units = 2.828427125

That means:

cosine 19.47 deg = sqrt(8)/3 = 0.942809041

Let me write this number another way:

cosine 19.47 deg = sqrt(8/9) = sqrt(0.888888...) = 0.942809041

There you find your 8/9 fraction expressed as the repeating decimal 0.8888....

also expressed as 144/162 and

multiplied by 10 as the ALDEBRAN grid point = 80/9 = 8.8888...

From Tom:

Now to the torus. The V/A for the torus falls outside the range of the figures above, but let's calculate it.

torus volume = 2*PI*PI*R^3

torus area = 4*PI*PI*R^2

torus V/A = (1/2)*R = 0.50 (with unit radius R=1)

Hence we see that the torus allows for 50% more surface area than the sphere does for a given volume. I don't know what the significance of that is, but perhaps it will become apparent in the future.

Tom Mellett

99.7.2 Comment from MetPhys: 

(Note: 6/14/01 It should be the "Growth Factor" of Young, representing the "doubling" and the 1 / 2 spin of the electron. Intuitively, lets explore what "Growth Factor" means. Young, to me, is saying the concept of growth, not meaning degrees of freedom but life properties become more apparent as we approach the torus attractor and finally the chaotic attractor. This 7 symmetry of the torus provides for 50% more surface area for some reason. That reason will be elusive until we begin to understand the Oroboros "seemingly" swallowing its own tail. It begins to make sense as we rotate the Oroboros 45 degrees or 90 degrees from its common side view and what do we see? We see that the Oroboros mouth is not swallowing its tail but......true, the tail circles back toward the mouth but it passes to the side of the mouth! Now, we begin to realize that the circle of the torus becomes a helix, never returning to its start point ever again. The circle becomes a spiral of life like unto the DNA of our cells. This is what Young would recognize as "growth". The torus evolving. Now you might ask, what does Robert mean when he speaks of "doubling" of this erroneously assumed Oroboros-torus, now a helix? Let me ask you a question. What would you see if you opened a toroid, doubled (stretched) the circle into two helical coils and attached the ends? Would it become an annalema or the figure-8-on-its-side denoting infinity? It would still be a double-toroid. I believe it would still have the properties of a toroid, but its doubling becomes the 720 degree-mobius-strip-rotation of the electron or 1/2 spin defined. Does logic tells us that if one torus has a 50% area reserved over the sphere, then two tori have a 100% area reserved? Nevertheless, this is hinting at the invisible half of life, the Other Side of This Side. The electron has to spin twice (720 degrees) to return to its original beginning point. Now you can see how growth, doubling and 1/2 spin may relate).

Refer to File,100 Electrons and the Diatonic / Pentatonic Scales for a more detailed connection of growth, doubling and 1/2 electron spin, and its relationship to Music.

From File,106 Primes / Gematric Numbers / Fibonacci Numbers and 137 (Fine-structure Constant) I noticed the doubling may be applied in a different way. Below, in Note 6/12/01, I postulate that any irrational "constant" may have to be reversed against itself (doubled), implying that we see only 1/2 of the whole:

Quote: THE RABBIT TREE &THE LOGIC OF LIFE'S BEGINNING by D. G. Leahy.

"The essential middleness of the logical sequence which is the perfect balance point of the infinite Fibonacci sequence is evident in the fact that the logical product of the left side of any generation times the logical value of the dead middle position times the logical product of the right side -- in that order or the reverse, but with the middle in the middle -- always equals the logical value of the dead middle position".

6/12/01 (Note: Using Trinary Logic here, we see the possible reason for "the doubling" that I speak of. This is the 1/2 spin of the electron. This is the "Other Side" and "This Side". What it tells us, is to mirror-reverse-image the Fibonacci series against itself to create the matrix that generates from its own center (0-point). Of course there is no symmetry in irrational constants such as Phi, pi, e, Fibonacci or any constant!!! We may be looking at only 1/2 of it!!!)

Robert Grace. MetPhys@aol.com

© Copyright. Robert Grace. 2001