99.8.1 Torus, Sphere and 4-hedron (Michael Morton)

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Sections List- 99 Electrons and Mythologies
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Tom ... Much thanks !

Thanks for this info ... it is so solid and helpful to me ! I'm thrilled to see the "repeating 8s" there !! .. the Aldebaran Jan.1, 2000 ASM Grid POINT Value. Very exciting ... the "Bull's Eye" in Taurus !! And then up-pops the "bull's eye" crop formation .. 33 days after they take the recent photo of 'The Face' !!! Much thanks to you, Tom, again.

-- Michael L.M.

In a message dated 06/05/2001 7:57:41 AM Pacific Daylight Time, TomBuoyed writes:

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

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

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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.

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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

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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

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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..8

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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

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From Tom Bouyed

Michael,

Another way of expressing the 8/9 is to look at the basic trigonometric identity--- For any angle: sine squared + cosine squared = 1

In the triangle with angle 19.5 deg, then

(sine 19.5)^2 + (cosine 19.5)^2 = 1

1/9 + 8/9 = 1

Tom

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From Tom Bouyed

Michael,

Your use of the radian to degrees equivalent of 360/2*PI = 57. 2957795 as the radius in some of your equations is intriguing. It shows the importance of looking at the polarity of curves or circles versus straight lines.

Rudolf Steiner makes a great deal of this polarity in his anthroposophy. For example, Steiner teaches that our heads are formed in a spherical manner because the universe has great cosmic antipathy to our heads and therefore projects them out as images of itself. On the other hand, our arms and legs are radial, rather straight. They are in deep cosmic sympathy with the universe. (Our chest & torso is the hybrid of the two extremes. To bring this more into the human s=oul, whatever you hate, you will project outside you as an image because you can't stand to be one with what you hate. On the other hand, something you love will draw your arms and fingers to touch or caress. That's the antipathy-sympathy pole.)

Back to grid stuff. If we have a circle with a unit radius equal to 1 foot, then a radian measure is to take that straight line distance of 1 foot and bend it into a curve and place it on the arc of the circle. When you draw lines from each end of the arc to the center of the circle, those lines make an angle of 57.2967795 degrees. In radian measure the angle is 1/2*PI (since the circumference is 2*PI*R and R=1 foot here).

1/2*PI = 0.159154943

Now let's think of it another way. Suppose you make an equilateral triangle, with each side 1 foot long and made out of a malleable metal, like copper. Each angle in the triangle is exactly 60 degrees. But now suppose you artfully start hammering one leg from the inside out and form it into a circular arc. You would then reduce the angle from exactly 60 to 57.295 degrees because you are hammering the once straight leg into an arc. But the point is that you haven't lost any metal. The leg is still 1 foot long, only now it is bent into a perfect arc which squeezes the 60 deg angle down a little more than 2 degrees to get to the radian angle.

Now to the torus:

If you replace your 57.295 deg for the radius by its radian equivalent (1/2*PI), then you get a volume for the torus of (1/4*PI). If you then use this radius in the surface area formula, you get a surface area of unity!

So that if you have a radius of 1/2*PI = 0.159154943 feet, then the volume of the torus will be 1/4*PI = 0.79577471 cubic feet. And the surface area will be exactly 1.00000 square feet.

(Remember the V/A ratio of the torus is (1/2)*R. Since R=1/2*PI here, then the V/A ratio equals 1/4*PI.)

Again, I don't know what this signifies, but it is interesting to look at.

Tom

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From Michael,

Tom ...

See your number there .. of .. 0.79577471 ... ??!!!

That's a decimal harmonic of my Orion Belt-stars Composite Ratio .. as of Jan.1, 2000 .. (MINTAKA X ALNITAK) / ALNILAM .. = (31.00627668 X 43.63323131) / 170.010936 ... = 7.957747155

Too much !!!!!!

-- M.L. Morton

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From Michael Morton

In a message dated 06/05/2001 1:51:43 PM Pacific Daylight Time, TomBuoyed writes:

Michael,

Another way of expressing the 8/9 is to look at the basic trigonometric identity--- For any angle: sine squared + cosine squared = 1

In the triangle with angle 19.5 deg, then

(sine 19.5)^2 + (cosine 19.5)^2 = 1

1/9 + 8/9 =1

Tom

Aha !! Yeah ... I see !! Nice !! We're on a roll .. and how ...

You know .. I don't know if you've read my write-up on ab analogue of the Cydonia (Mars) "Tholus" .. ??

It's just-to-the-south of the runways of Stewart Air Force Base in the lower Hudson Valley of New York, USA.

Dr.Bruce Cornet, PhD., Geologist and Paleontologist .. had .. in 1998 .. a very detailed website .. discussing several "Cydonia Analogues", as he called them .. in the lower Hudson Valley in the vicinity of Pine Bush, Middletown, Warwick, Wallkill, etc.

He had investigated this area in great detail, personally .. taking numerous readings with various instruments .. magnetometers, etc., surveys, etc. he even had official USGS topo map-overlays on his website .. so I decided to email him .. which I did .. and he gave me the catalog numbers/names of the maps to order .. which I did .. and I proceeded to figure-out what turned-out-to-be ... major "AMS" figures for the_exact_centers of his "analogue sites" !! This was in May of 1998.

I emailed him all my results and figures .. and he was initially enthusiastic, but he then seemed to "lose interest" completely. Next thing I knew .. he had completely_pulled_all of the material on his website that had to do with any investigation about "Cydonia analogues" whatsoever !!! And that was in 1999.

Then ... last year .. 2000 .. he emailed me and said that he "had wanted to make friends with certain scientists at JPL and NASA".. and so he had therefore "decided to remove the 'wildly-speculative' material from his website". Hmmmmmm.

Anyway .. I am convinced that those "Cydonia analogues" are very real .. I mean their remains, of course .. much of which is heavily-eroded.

The ASM numbers I found for them are just too precise, too correlative .. for them to be "natural".

The Square of the Sine of 19.47122063 ? 0.1111111111 ... is a decimal harmonic of .. the Grid POINT Value of "The Tholus II" .. just to the south of the runways of Stewart AFB !!

(Morton, 1998, Internet).

"Tholus II" Grid LAT .. 41 (deg) X 29 (min) X 26.64423886 (sec) North .. = 31680 North.

"Tholus II" Grid LONG .. 105 (deg) X 14 (min) X 23.94557823 (sec) W.Giza .. = 35200 W.Giza

"Tholus II" Grid POINT Value .. 35200 / 31680 = 1.111111111 ..

(Morton, 1998, Internet).

Maybe the Jan.1, 2000 Grid POINT Value of ALDEBARAN and the circa 2000 Grid POINT Value of the remains of "Tholus II" are related to what you are showing in your email, Tom ... in some say involving "new physics" or a new understanding of something important .. ?

I also .. to my satisfaction, at least .. have figured-out the "Tholus" ASM figures .. the one at Cydonia.

"Tholus" Grid LAT .. 41 (deg) X 03 (min) X 10.53658537 (sec) North .. = 1296 North.

"Tholus" Grid LONG .. 01 (deg) X 05 (min) E.CYDONIA .. = 5 E.CYDONIA. [ W.NASA 08 deg 27 min 0.8 sec ].

"Tholus" Grid POINT Value .. 1296 / 5 = 259.2

(Morton, 2000, Internet).

There's a decimal harmonic of the surface-area on a Torus !! ... 1296 .. assuming "conventional" 360 arc-degrees system.

And there's a decimal harmonic of Earth's precession cycle !! .. 259.2 .. in terms of Earth years.

Look at this ...

Take the Grid of "Tholus II" .. 31680 .. and the Grid LAT of "Tholus" .. 1296 ...

31680 / 1296 = 24.44444444 ..

Take that result times the Bruce Cathie light-speed harmonic .. (24.44444444 X 162) = 3960 .. average (mean) radius of Earth .. in statute miles.

Now ... take the Grid LONG of "Tholus II" and the Grid LONG of "Tholus" ..

35200 / 5 = 7040 ..

Then divide that by Grid POINT Value of "Tholus II" ..

7040 / 1.111111111 = 6336.

Now .. divide that by the Buckminster Fuller "DNA/RNA Un-Zip Angle" ..

6336 / 7.333333333 = 864.

That's a major gematrian number.

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Multiply "Tholus" times "Tholus II" ...

259.2 X 1.111111111 = 288 .. "double light" in classic gematria.

864 / 288 = 3.

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"35200" is a decimal harmonic of the Grid LONG at the center of the large limestone boulder still lodged in the western side of the remains of the Ark of Ziusudra .. the pre-Sumerian king.

Grid LONG Limestone "intrusive" Boulder .. 13 (deg) X 06 (min) X 4.512820513 (sec) E.Giza .. = 352 E.Giza. [ E.Greenwich 44 deg 14 min 5.312820513 sec ].

Grid LAT Limestone "intrusive" Boulder .. 39 (deg) X 26 (min) X 26.03550296 (sec) North .. = 26400 North.

Grid POINT Value Limestone "intrusive" Boulder .. 26400 / 352 = 75.

***BULLETIN*** ... I just now .. as I write this ... the time on my computer screen says .. "8:33" PM EDT. Just a second .. a_second_before I looked at that time on my computer screen ...

I tried multiplying the "Tholus II" Grid POINT Value .. times this "75" Grid POINT Value for that boulder ...

75 X 1.111111111 = 83.33333333 .. = (50 / 0.6) = (656.56127 / 7.87873524) = (27.58106915 X 3.021396048)

SEE MY MOST-RECENT paper .. regarding that ratio of 83.33333333 .. PLEASE !!!

Talk about an incredible synchronicity !!!!

Wow ... that's all for now ... I need a ... drink. (Of pure grape-juice). And then .. more coffee !!

-- Michael L.M.

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From Michael Morton

In a message dated 06/05/2001 2:14:19 PM Pacific Daylight Time, TomBuoyed writes:

If you replace your 57.295 deg for the radius by its radian equivalent (1/2*PI), then you get a volume for the torus of (1/4*PI). If you then use this radius in the surface area formula, you get a surface area of unity!

So that if you have a radius of 1/2*PI = 0.159154943 feet, then the volume of the torus will be 1/4*PI = 0.79577471 cubic feet. And the surface area will be exactly 1.00000 square feet.

Tom ...

See your number there .. of .. 0.79577471 ... ??!!!

That's a decimal harmonic of my Orion Belt-stars Composite Ratio .. as of Jan.1, 2000 .. (MINTAKA X ALNITAK) / ALNILAM .. = (31.00627668 X 43.63323131) / 170.010936 ... = 7.957747155

Too much !!!!!!

-- M.L. Morton

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From Tom Bouyed

So that if you have a radius of 1/2*PI = 0.159154943 feet, then the volume of the torus will be 1/4*PI = 0.79577471 cubic feet. And the surface area will be exactly 1.00000 square feet.

Tom ...

See your number there .. of .. 0.79577471 ... ??!!

That's a decimal harmonic of my Orion Belt-stars Composite Ratio .. as of Jan.1, 2000 .. (MINTAKA X ALNITAK) / ALNILAM .. = (31.00627668 X 43.63323131) / 170.010936 ... = 7.957747155

Too much !!!!!!

Michael,

From Tom,

I only left out a zero after the decimal point. It does not affect anything about your numbers, but just to make it correct, the value of

1/4*PI is 0.079577471 not 0.79577471

Tom

(Note: Is the inclusion of the 0's the same as Leahy's use of numbers 10,100,1000, etc?)

From Michael,

Tom ...

Also .. I noticed .. you mean (0.5 / Pi) .. not 1/2*Pi ... isn't that right ? And .. (0.25 / Pi) .. not 1/4*Pi .. right ?

-- Michael L.M.

From Tom.

The expressions are equivalent since 0.5 = (1/2) and 0.25 = (1/4). It's the awkwardness of writing fractions in ASCII and knowing which arithmetical operation takes precedence. To be clearer, I should write [1/(2*PI)] and then you are definitely sure that I mean the 2 and the PI to be in the denominator. What you have ambiguity about is whether I meant [(1/2)*PI] by writing 1/2*PI. I did mean [1/(2*PI)]

That's why it's also good to put the equivalent decimal expressions of the fractions.

[1/(2*PI)] = 0.5/PI = 0.159 (inverse of 2*PI)

[(1/2)*PI) = 0.5*PI = 1.5708 (half of PI)

Tom

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From Michael,

Tom ...

Thanks for clarifying that for me. I just needed to be clear on it .. thanks.

-- Michael L.M.

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From Michael Morton

Tom ...

Also .. I noticed .. you mean (0.5 / Pi) .. not 1/2*Pi ... isn't that right ? And .. (0.25 / Pi) .. not 1/4*Pi .. right ?

-- Michael L.M.

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In a message dated 06/05/2001 7:42:00 PM Pacific Daylight Time, TomBuoyed writes:

Subj: slight correction
Date: 06/05/2001 7:42:00 PM Pacific Daylight Time
From: TomBuoyed
To: Milamo, Peace2go, JerryIuliano, MetPhys

So that if you have a radius of 1/2*PI = 0.159154943 feet, then the volume of the torus will be 1/4*PI = 0.79577471 cubic feet. And the surface area will be exactly 1.00000 square feet.

Tom ...

See your number there .. of .. 0.79577471 ... ??!!!

That's a decimal harmonic of my Orion Belt-stars Composite Ratio .. as of Jan.1, 2000 .. (MINTAKA X ALNITAK) / ALNILAM .. = (31.00627668 X 43.63323131) / 170.010936 ... = 7.957747155

Too much !!!!!!

Michael,

Page 8

Sections List- 99 Electrons and Mythologies
Impossible Correspondence Index