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"How to use a circle to relate 'e' to Phi " 1st Posted: Feb 22, 2008
This mathematics discovery allows us to relate all mathematical and physical constants logically and causally together. It is needed to navigate between two sides of surface of a Moebius and across its surface. A test: You can square the circle using only a straight edge by knowing how to create the 8 e|Phi points on a circle.

1-17-2009 Found an error thanks to my son which refined the question. Please remember, this was deduced from an observation as I used the process described below. The question of relating e to phi using a circle never occurred to me before, nor to anyone else to my knowledge.



5-09-09 My first method is below the and the Mathematical and scientific IMPLICATIONS are below the "3 question recipe" Found error on 1-17-09. I changed in topic but not here until now.




First QUESTION

1. Given that any circle of radius = 1 unit, construct the left quadrant of a circle and denote left end of horizontal radius by A, center end by B, and the up 3/4 point on vertical radius by C; then
2. draw a perpendicular at C that intersects left circumference at e|phi; then
3. from e|phi drop a line perpendicular to horizontal radius AB at D; so does
4. DC/AD = e?

Second QUESTION

1. Create the hypotenuse of triangle DBe|phi = Be|phi which equals radius of circle; then
2. rotate up line length DB keeping B at same location and
lay it on hypotenuse e|phiB ending at E; then
3. from E drop a line perpendicular to horizontal radius AB at F; so does
4. AF/FB = phi?

Third QUESTION

1. Draw a line from E to A intersecting line e|phiA at G; so does
2. DG/Ge|phi = phi?



Definitions:

"e" is Napier's Constant or Euler's Number. It describes how fast growth or decay occurs per n units of interval like time. e.g. compound interest. e = 2.718..... and it is irrational

"Phi" is the Golden Ratio first shown by Euclid in his definition of the Extreme to Mean ratio; "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.".

Phi = 1.618.... or little phi = .618..... so phi x Phi = .618 times 1.618 = 1. From irrational to rational. While 1.618/2.618 = .618.

It is these boundaries of Phi and phi defined by the basic math operations between the two and the results that end in whole numbers or include .618 with a whole number that are interesting and define the margins of a Moebius. More later.

Using Phi ensures that our logical units are of equal size. No equal sized units = no logic = no geometry = no science.

A "circle" is based upon pi = 3.1415...., also irrational, and obtained by dividing Circumference(Cir) by its diameter(Dia).

"Bowtie Point" of a Moebius strip: There is one and only one point on the outer edge of any Moebius strip that you can draw a perpendicular line down to the middle of the bottom side of a Moebius. It looks like a perfect 'bowtie' centered on one point. Look and you will see.

Ergo, a Moebius has a built in origin 'it' can use to see itself with; i.e. to use as the center of its built-in circle based Cartesian co-ordinate system.

Ergo, tying these two apparently independent base logical constants, 'e' and Phi, to our circle based geometric operation constant, "pi", through the unit 1 defined by Phi, is a very big scientific discovery.



This Prediction Requires Independent Verification Fast.


DIRECTIONS

1. Make a perfect right angled cross.

2. Center a circle on the cross's intersection; thus creating a horizontal and vertical diameter for any circle.

3. The following operation works in either direction for both vertical and horizontal diameters, i.e. left/right or up/down. Showing the operation with one does for all = eight points can be generated.

4. Sub-divide the vertical radius twice, i.e. first in half; then the upper half again = 3/4 radius up.

5. At upper 3/4 point on radius, make a perpindicular bisector extending to the Circumference of circle on both sides. (What shall we call these eight points? Lets try 'e|phi' points for base logical constant points.).

6. DEFINE 'e' by dropping a perpndicular down to horizontal diameter's left radius AB at D from our left hand e|phi point on Circumference. Multiplying DB times 2.718 = radius where B is left end of horizontal diameter.

7. DEFINE Phi by drawing hypotenuse e|phiA of triangle based upon center A. Hypotenuse e|phiA over AD = Phi.

8. Since eight of these points can be generated, a left and right one for each of the four radii; then these e|phi points can be used to combine or locate all other points and there geometric relationships to any other point(s) on the surface of a sphere(circle) or Moebius strip, both outside or inside these circle based geometries. Thus, we have FTL space navigation in our Moebius geometry universe by going between the two sides of its surface.

A fact: This geometric operation has never been demonstrated before on Earth in ancient or modern mathematical literature: Using only a straight edge, do a double sub-division on the radius of a circle, centered on a cross, to find the eight single point(s) on the Circumference that define 'e' with respect to Phi for the four radii.

This is the fundamental geometric, and thus logical discovery that shows how we are tied into the universe. It directly ties our symbols made of reality to the Moebius strip geometry that we spirits and our universe are based upon. Thus, a Moebius' bowtie point, by inspection, gives us an 'objective' common to all, neutral origin to see ourselves from.

Questions Implied:
1. When does the percentage(%) decrease in a vertical radii, e.g. AC equal the % decrease in AB, the base radii?
It almost certainly must be a several more sub-divisions, my guess is the 12th. but that is an even number, and the farthest we can go in a Moebius must always end in an odd number. More thoughts in equation log.

2. Viewpoint is crucial here. Vertical and horizontal are defined with respect to the observer, the manipulator of the circle. Remember there are eight possible e|phi points.

Clearly we must explicitly include this distinction in our choice of which radius is first shortening the most; so "How do we explicitly include where and how the observer fits into this observation?". Answer seems to me would require left/right, up/down, and front/back(see/not see) be explicitly assigned or assumed in starting your operations. e.g. the cross in the circle is built by observer.


Implications
1. Showing how these "e" and "Phi/phi = 1" can be related together with a simple straight edge operation on a circle ties 1, 0, pi, e, and phi together. These are our base numbers(0 and 1), the base geometric operation that generates a circle(pi), and the two logical constants(e and Phi) upon which all scientific descriptions of our universe rest. Ergo, how we can use these logical constants to create and compare scientific observations we can all 'agree we see' = 'objective' at the base level of our universe.

2. We people of Earth have been UNCONSCIOUS of how we work. Since it is done by straight edge = any unit; then we 'Moebius based spirits' are using these constants "unconsciously" to create and live in our life forms.

3. By implication, all the other known constants in our Moebius geometry universe are also implied.

4. Unconscious ensures death sooner or later.

What Was I Initially Looking For and Why?

I was looking for the angles of the triangle that measured the distance between the two sides of the surface of a Moebius at its midline = a circle. I started with a triangle based on the upper left quadrant of a circle. Lets call this triangle ABC with A at center of circle, B at left Circumference(Cir) and C at top vertice. Along the way, I asked the question:

When you shorten the vertical radius AC of this initial ABC triangle by one unit; then the horizontal base AB initially does not shorten as much as AC.

So I asked this question:

At what point on the circle's Cir does the absolute amount of shortening along both arms of this triangle go equal?

The Answer turned out to be one of the circle's e|phi points.

I did this by eyeball on a actual circle. When I drew the line from the e|phi point on the Cir down to the horizontal diameter at point D, I realized that DB times 2.718 might equal AB. After rechecking, I concluded it does. Next, I checked for Phi by using the hypotenuse e|phiA of the triangle ADe|phi and dividing by AD, i.e. e|phiA/AD and it equaled 1.618 = Phi.

I flash hunched about the double sub-division of the vertical radius generating the e|phi point the next day.

My trigonometry and calculus suffer from 40 years of disuse. However, I can visualize complex geometries and rates of change in my head. Tests showing a great ability for doing that, got me into Seattle University college at age 15 in 1957. It certainly was not my grades nor attendance in High School.


This is, of course, a falsifiable assertion. I at first said here, MAKE MY DAY, show this prediction wrong and my whole theory would be wrong, and I could live with that. That's not true. I would not like living in a universe that makes no sense, where there is no accountability, no conservation, no meaning. So MAKE MY DAY and confirm this fast please. The future of life on Earth depends upon your fast confirmation.