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 20201008 revised, 20201012 reread and still Need Independent Confirmation to say this as a Named Natural Law. = As Above, So Below; so it follows that as a Moebius is a circle based geometry, then a smaller circle will reflect its geometry. What is fascinating is Phi/phi = 1, from the irrational to the rational. ToDo Separate article below so combine these 2 topics. Possible named Scientific Natural Law when others verify. In rereading below, it seems improbable I saw this relationship But I was born with the gift of seeing relationships between things geometrically. Scoring in the one in a billion plus on a geometry test is what got me into Seattle University at 15. 12415 Intsead of 3/4 radius used below, use 3.6 of any radius of 3.718 length. You will find taking the square root of (3.6^ + 2.718^) and dividing by 2.718 will equal .618, this is the ephi ratio using the AB length described below as your unit of measure. All tied together with the Pythagorean Theorem. There are implications for FTL navigation and how we decide on putting numbers on our objective units of measure. 102215 Why Does This "possible" Geometric Discovery Make Interstellar FTL Navigation Possible?? Answer comes out of doing the below work. The 1172009 how/to is inaccurate, but captures the gist. This needs a diagram. I am going to restate the 3 questions below later today and redescribe method. What is important is what it means if this relationship is true. It was not a deduction, but an observation as I was working with a imperfect hand drawn circle. In short my unconscious saw the relationship. What it means is we can use not just the radius as our base unit of measuring a circle, but a more useful, precise, and accurate base unit of geometric comparison for circles and there derivative geometries. 1172009 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. Mathematical and scientific IMPLICATIONS below the recipe for construction. 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 ephi; then 3. from ephi drop a line perpendicular to horizontal radius AB at D; so does 4. DC/AD = e? Second QUESTION 1. Create the hypotenuse of triangle DBephi = Bephi which equals radius of circle; then 2. rotate up line length DB keeping B at same location and lay it on hypotenuse ephiB 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 ephiA at G; so does 2. DG/Gephi = phi? After definitions I describe my initial cumbersome method 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 builtin circle based Cartesian coordinate 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. I was looking for something else on a circle using a straight edge and saw the relationship. I am quite sure it implies you can square the circle with a straight edge and that all the other logical and physical constants can be related using it. BUT that means it has to be publicly tested first. My old eyeballs are not a conclusive test. (Some mathematician asserted to prove that one could not square a circle with a straight edge about two years ago. I wonder if he used a straight edge to do so?) INITIAL DIRECTIONS posted several months ago. I did not do the rotation of the line 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. Let A denote center of circle, B = left end of horizontal diameter, and C = top end of vertical diameter. 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. Subdivide 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 the vertical radius (or any of the other three radii), make a perpendicular line extending to the Circumference of circle on both sides. What shall we call these eight points? Lets try 'ephi' points for base logical constant points. 6. DEFINE 'e' by dropping a perpendicular down to the horizontal diameter's left radius AB where A = center of circle at D from our left hand ephi point on circumference. 7. PREDICTION: Multiplying DB times 2.718 = radius where B is left end of horizontal diameter. 8. DEFINE Phi by using left horizontal radius to create another pair of ephi points along left horizontal radius. 9. PREDICTION: The ratio of DB/.25 radius = PHI???(I had forgotton this one. 12909) 10. Since eight of these points can be generated, a left and right one for each of the four radii; then these ephi 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 a Moebius strip, both outside or inside these circle based geometries. 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 subdivision 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 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. Since this these ePhi points are a falsifiable assertion, falsify away. Questions Please. 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 subdivisions, 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 ephi 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 ephi points. I did this by eyeball on a actual circle. When I drew the line from the ephi 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. (MISTAKE made here. In correcting the mistake, I asked the top question.) Next, I checked for Phi by using the hypotenuse ephiA of the triangle ADephi and dividing by AD, i.e. ephiA/AD and it equaled 1.618 = Phi. (WHY WRONG?Hypotenuse is the radius, so AD/radius can not both define e and Phi.) I flash hunched about the double subdivision of the vertical radius generating the ephi point the next day. 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 please confirm this fast . The future of life on Earth depends upon your fast confirmation. Last edited by Dan on Sun Apr 24, 2022 7:20 pm; edited 4 times in total All times are UTC 
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