The Leotta Theorem (revised)

Minor Geometric Proof for the Existence of Epsilon (ε): A Numerical Constant of Ellipsoids

Written by Adam Scott Leotta

BY DEFINITION, AN ELLIPSE HAS A.....................................................................................................6 AB = √(AD + BD )..........................................................................................................................................7 LINE DI, “1,” IS REFERRED TO AS ΕPSILON, “Ε,” .............................................................................7 LINE OP, THE HYPOTENUSE RADIUS “HR,”.....................................................................................10 The salient structural components.........................................................................................................12 Where ε equals 1, the key (k) of all ........................................................................................................12 THE DISTANCE BETWEEN E AND THE FOCUS.........................................................................................................14 IS EQUIVALENT TO THE DISTANCE BETWEEN..........................................................................................................14 G AND THE OTHER FOCUS. ...............................................................................................................................14 THUS, LINE EB IS EQUAL TO LINE DG. FROM.....................................................................................................14 LINE DI IS Ε, THE LEOTTA CONSTANT...............................................................................................................15 Law of Elliptical Establishment (LEE). .................................................................................................15

Contact Information Adam Scott Leotta [email protected] 23116 Cohasset St. West Hills, CA, 91307

The Leotta Constant is referred to as: εpsilon (ε). Just as Archimedes’ Constant reflects an unchanging value whenever a circle’s circumference is divided by its diameter, the Leotta Constant shows how ε, too, is a constant in ellipses, derived from a number of ways. With simple algebraic equations, the Leotta Constant connects the internal values of every ellipse. The remarkable profundity of the Leotta Constant is that its value is always 1, and marks the fundamental numerical value for which we base our understanding of Nature and mathematics. Despite the keen logic of Kurt Gödel, the Leotta Constant is a value that is derived completely within its own system. Derivations of the absolute value of “1” cannot be refuted. The following is a simplistic proof for the constant known as “ε.”

(I simply ask that all readers review this manuscript with patience)

Begin with a line of any length.

Or, an ellipse of any eccentricity (see page 12). Double line IG to DG; double the length of line DG and add it to the segment to create BG.

Construct an ellipse EFGH* with B and D as the foci. DG is the ‘perigee,’ and

BG is the ‘apogee.’ (*Draw line EB to complete the major diameter, and draw

a minor diameter FH as*Picture scaled down for convenience a perpendicular line through the midpoint of BD) At focus D, draw line AD perpendicular to DG. Point A lies on the ellipse. Draw line AB and BF.

By definition, an ellipse has a locus of points, such that the sum of the distances from every point on the locus to two fixed points is always equal. Thus: AB + AD = EB + BG = 2 x BF (Keep in mind points B and D are foci) If line DI equals “1,” line DG equals “2.” By the definition of an ellipse, lines AB plus AD equal eight, “8,” as they are twice line CG, which is twice line DG. Triangle ABD is a right triangle. Thus, line AD equals “3,” line BD equals “4,” and line AB equals “5,” by application of the Pythagorean Theorem. Line BG equals “6,” line EI equals “7,” and line EG equals “8.” If line EG equals “8,”

line BF (which is drawn from a focus to a midpoint) equals “4.” From a line of any length, the Natural integer values of the first eight integers are established from an ellipse; such that, a value of “one” is established, within the system of numbers, without aEFGH predetermined base.if: Thusfar, is an ellipse

line BE plus line BG equals line DE plus line DG equals line AB plus line AD equals 2 times line BF equals line EG. BE + BG = DE + DG = AB + AD = 2 x BF = EG. 2

2

AB = √(AD + BD ) Substituting the above line values: 2+6=6+2= 5 + 3 = 2 x 4 = 8. And, 5 = √(9 + 16). Line DI equals the constant: one, “1;” line GI, “1,” the key, “k,” = x, which can be any natural integer; line BE, “2,” the perigee, “p,” = k + 1; 2 2 line BF, “4,” the vector, “v,” equals k2 +2k +1; alternatively, v = p line BC, “2,” the scale, “s,” equals k + k; line BD, “4,” the wave, “w,” equals 2s; line AD, “3,” the radius, “r,” equals 2k + 1; line AB, “5,” the hypotenuse, “h,” equals w + 1; line BG, “6,” the apogee, “o,” equals w + p; line EI, “7,” the glyph, “g,” equals o + 1; line oP, “1,” the hypotenuse radius, “Hr,” equals k; (see next figure) line EG, “8,” the major diameter, “M,” equals o + p; Line DI, “1,” is referred to as εpsilon, “ε,” and is the elliptic constant. For any given ellipse, ε (the ‘DI’ line segment)

equals 1, which is equivalent to the difference between the hypotenuse and the wave. This is one of the elliptical equations: h–w=1=ε

εpsilon, “ε,” is the Leotta Constant: [ εpsilon = one ] Amazingly, when εpsilon = one; all the above equations remain true for any ellipse and return integer values when the key, “k,” is any Natural integer. Moving on; observe the next figure.

A circle inscribed within a right triangle has a radius equal to the product of the legs that are opposite the hypotenuse divided by the sum of all the sides. oP = (BD x AD) / (AB + AD + BD) Or, more simply, the diameter of a circle inscribed inside a right triangle equals the sum of the legs that are opposite the hypotenuse minus the hypotenuse.

2 x oP = AD + BD – AB Line oP is called hypotenuse radius, “Hr.” It equals: (3 x 4) / (3 + 4 + 5) = 1. 2 times Line oP, hypotenuse diameter, “Hd,” equals: 3 + 4 – 5 = 2. Line oP, the hypotenuse radius “Hr,” equals the key, “k,” which is 1 in this presentation, which is also the perigee minus εpsilon. This is one of the elliptical equations. p – Hr = 1 = ε

Thus, “Hr” is an integer, as are the: perigee, “p,” the scale, “s,” the vector, “v,” the apogee, “o,” the radius, “r,” the wave, “w,” the hypotenuse, “h,” the glyph, “g,” and the major diameter, “M,” whenever the key, “k,” is a natural integer and εpsilon, “ε,” equals One. Draw line GH.

GH is the chord, “c.” For any ellipse, the square of the chord “c” equals two times the square of the vector “v” minus the square of the scale “s.” This is known as the “Leotta Theorem”: 2

2

2

c = 2v – s

Again, all the above equations remain true, regardless of the shape or eccentricity for any ellipse, if line DI, the difference between the hypotenuse, “h,” and the wave, “w,” is the Leotta Constant, “ε,” equals One, “1.”

The Leotta Constant has, for any ellipse, many forms. The following are the ‘elliptical equations’ (two of which we have discussed): ε = h – w; ε = 2p – r; ε = 2h – r2; ε = p – Hr. For any ellipse, when any of the above differences equal one, or are set to one (ε), then, all of the equations that relate the different parts of any ellipse are always the same. They can be referred to as the ‘natural set of equations.’

The point to remember: For any ellipse, where the Leotta Constant is 1 and the key is a natural integer, the aforementioned structural components are always integers. From this statement, we can assert: The Theory of Elliptical Establishment (TEE) The salient structural components of all ellipsoidal shapes are systematic multiples of a fundamental numerical value, ε, established within its own system. The Law of Elliptical Establishment (LEE) Where ε equals 1, the key (k) of all ellipsoidal figures is a natural integer, from which all other salient structures can be defined.

Alternate Derivations (determining ε from outward-in): Conversely, the Leotta Constant can also be acquired from any ellipse. Begin with any ellipse EGFH, of any eccentricity.

Create major and minor axes within the ellipse.

Measure line CE or CG, and draw that length from F to a point B on the line CE. So, line FB = EC.

Mathematically, point B is now the focus of this ellipse, since 2 x BF = EG. The distance between E and the focus is equivalent to the distance between G and the other focus. Thus, line EB is equal to line DG. From point D, draw DA, where A is on the edge of the ellipse. Then proceed to draw line AB.

From the elliptical equations, the hypotenuse minus the wave will always equal 1. Thus: h–w=1 BA – BD = 1 1=ε

Line DI is ε, the Leotta Constant. The key, “k,” which is line IG, equals line DG minus line DI. The rest of the internal structures can be obtained accordingly, now that k is established as a natural integer, thereby supporting and confirming the Law of Elliptical Establishment (LEE).

Addendum: Interesting Notes: If the scale, “s,” equals “ε”; then, the perigee, “p,” equals the Golden Ratio, “Φ.” If the perigee, “p,” equals ε (1), the ellipse is a circle.

We have thus proven that the Leotta Constant, ε, appears in every ellipse, and always equals 1.

Contact Information Adam Scott Leotta [email protected] 23116 Cohasset St. West Hills, CA, 91307