Sphere & 4-hedron

Robert,

Here is the latest I sent Michael & Robert Friend. It is a comparison of the volumes and surface areas of the tetrahedron inscribed in a sphere which generates the famous angle of 19.5 degrees.

Tom
Forwarded Message:

Date: Wednesday, July 4, 2001 2:52:48 AM
From: TomBuoyed
Subj: Sphere & 4-hedron

To: Milamo, Peace2go
cc: maryweav@hotmail.com

Michael,

Here is my analysis of the tetrahedron inscribed in the sphere.

I'm going to compare the volume ratios and the the surface area ratios and something will come out about the approximation to e, the base of natural logarithms and thus the e/PI ratio that Hoagland notes.

I take as my unit length the edge of a 4-hedron. Let's call it 1 foot because we'll be delving into area & volume and it's easier to see with a length unit.

If I start with one of the equilateral triangular faces of the 4-hedron, then the altitude of the face (vertex to opposite edge) is equal to SQRT(3)/2 = 0.866025403.

This number is the cosine of 30 deg and notice how close this number is to the e/PI ratio:

e/PI = 0.865255979 SQRT(3)/2 = 0.866025403

This closeness is what I believe is responsible for Munck's little error about e/PI. But let's go on to the volumes.

The volume of the 4-hedron is calculated from the general formula for volume of a pyramid, which is (1/3)* Base Area * Height

The Base area is SQRT(3)/4 = 0.433012701 sq.ft. and the height is SQRT(6)/3 = 0.81649658 ft.

The volume comes out to be SQRT(2)/12 = 0.11785113 cu.ft.

Now the trick is to find the radius of the sphere, which we can do from the lengths of the 4-hedron, since the center of mass of the 4-hedron coincides with the center of the sphere. Here is where you'll see the 19.5 deg derived. In a 4-hedron, the 2-D faces are all equilateral triangles with angles of 60 deg. But in a 3-D 4-hedron, the angle of elevation from the base to the vertex is a bit less than 60. It is 54.73561032 deg

(54* 44' 08.2" which multiplies to a grid value of 19483.2)

The angle from the vertex and the altitude line is 35.26438968 deg

(35* 15' 51.8" giving a grid value of 2719.5

If you subtract these 2 angles you get the angle from the base to the center of the 4-hedron, the center of the sphere and this is the famous 19.5 angle

54.73561032 - 35.26438968 =19.47122063 =19* 28' 16.39"

Grid value 8719.48

Now, the length from the center of the 4-hedron to any vertex is also the radius of the sphere since the 4 points of the 4-hedron are inscribed in the sphere.

So, figuring out that length we get: SQRT(3/8) = SQRT(6)/4 = 0.612372435

Now we can get the volume of the sphere

Vs = (4/3)*PI *R^3 = 0.961912372 cu. ft.

From above Vt = SQRT(2)/12 = 0.111785113 cu.ft.

Taking the volume ratio we get Vs / Vt = 8.162097134

Now to the areas.

The surface area of the sphere is As=4*PI*R^2 = 1.5 * PI = 4.71238898 sq.ft.

The surface area of the 4-hedron, At, is 4 times the area of 1 face or At = SQRT(3) = 1.732050808 sq.ft.

The ratio of areas is As/At = 2.720699046

Notice how close this is to natural log base e= 2.718281828.

This is due to the fact that the ratio of areas is SQRT(3)*PI/2. Remember how close the value is of SQRT(3)/2 = 0.866025403 and e/PI = 0.865255979

I don't know if 2.720699046 is the value that Munck used in his e/PI ratio, but it's close.

You give the actual LAT of the pyramid apex as 29*58' 53.09" Recall also that SQRT(3)/2 is the cosine of 30 deg. Where in the Giza complex do you find the exact LAT of 30*00' 00"N? 6785 feet north of apex or 1.285 miles e/PI is the cosine of 30*05' 17"N Its location would be 7.36 miles north of apex.

One last thing about the ratios. It turns out that the ratio of ratios is exactly 3.

Vs / Vt = 8.162097134
As / At = 2.720699046

Divide those numbers and you get exactly 3. Perhaps it's an indication that we are working with 3-D figures: sphere & tetrahedron.

One reason I think that is: if we go to 2-D, circumscribing a triangle in a circle, then the area ratio divided by the perimeter ratio is exactly 2, indicating we are working with 2-D figures.

I don't know what all this means, but I hope you can figure out something.

Tom

MetPhys suggests:

Since Tom has almost guessed the answer that the number 2, here, seems to be dealing with 2D figures and 3 seems to be dealing with 3D figures, I will only add that, over the years I have sensed that the power exponents ^2 and ^3 mean the same thing as saying, for example, c^2 is light on a 2D grid and c^3 is light on a 3D grid or a spherical surface, for example, a spherical surface with no hole. I would go farther with c^4, as upon a spherical surface with a hole (a torus) but this is pure speculation based upon the "curious doubling" again when seeing the exponent ^8 (2 x ^4) in gravity formulae, implying a "double electron-torus as in mobius-superconducting Cooper "pairs" (^8 or ^4 (torus) + ^4 (torus)).

"Einstein's formula is usually quoted as follows:
E = m * c * c.

But it should actually be written as:
E * E = (m * m) * (c * c * c * c)."
Bonnie Hill

(c*c*c*c) is c to the power of 4 (c^4). See the corrected "doubling"? (Grace, 1999). It will be discovered that by doubling 36 to 72 and doubling 4 to 8 will yeild the correct gematric numbers.

Robert Grace
MetPhys@aol.com

From: MetPhys

Toms Ratios and quote:

"One last thing about the ratios. It turns out that the ratio of ratios is exactly 3.
Divide those numbers and you get exactly 3."

8.162097134 / 2.720699046 = 2.9999999985 (Close enough?)

Vs / Vt ratio = 8.162097134
As / At ratio = 2.720699046

Reference page:
http://hometown.aol.com/MetPhys/99Uelectrons.html (bottom of the page)

I was curious to see if a straight series of unique numbers like these showed up anywhere within the list of prime numbers, so:

Noting Toms ratios and removing all decimals, I noticed 8162, within the Vs / Vt ratio 8.162097134, which appears in prime line number 218 along with several harmonic numbers 9, 18 and one 576. Also, the first three digits of 8162 are phi reversed (816 is 618). No other combination of the full number showed up in any prime line number up to 500.

See Prime line 218
http://hometown.aol.com/MetPhys/106primes.html

Prime line 2..18.. :
1621140..18..8..9...21..9..444470..18..8162..576..173..18..0757..18..7780..9..

No prime line numbers, of 500 were revealed until the As / At ratio 2..720..699046 was reduced to 27, the first two digits. Also noted is the number 720 within this As / At ratio.

What do you all think this means?

Robert Grace
MetPhys@aol.com

© Copyright. Robert Grace. 2001