the leotta theorem (revised)

8/14/2019 The Leotta Theorem (Revised)

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Minor Geometric Proof for the Existence of Epsilon ():

A Numerical Constant of Ellipsoids

Written by

Adam Scott Leotta


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BY DEFINITION, AN ELLIPSE HAS A.....................................................................................................6

AB = (AD + BD )..........................................................................................................................................7

LINE DI, 1, IS REFERRED TO AS PSILON, , .............................................................................7

LINE OP, THE HYPOTENUSE RADIUS HR,.....................................................................................10

The salient structural components.........................................................................................................12Where equals 1, the key (k) of all ........................................................................................................12

THEDISTANCEBETWEEN E ANDTHEFOCUS.........................................................................................................14ISEQUIVALENTTOTHEDISTANCEBETWEEN..........................................................................................................14G ANDTHEOTHERFOCUS. ...............................................................................................................................14THUS, LINE EB ISEQUALTOLINE DG. FROM.....................................................................................................14LINE DI IS, THE LEOTTA CONSTANT...............................................................................................................15

Law of Elliptical Establishment (LEE). .................................................................................................15

Contact Information

Adam Scott Leotta

[email protected]

23116 Cohasset St.West Hills, CA, 91307
mailto:[email protected]:[email protected]


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The Leotta Constant is referred to as: psilon ().

Just as Archimedes Constant reflects an unchanging value whenevera circles circumference is divided by its diameter, the LeottaConstant shows how , too, is a constant in ellipses, derived from anumber 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 isalways 1, and marks the fundamental numerical value for which webase our understanding of Nature and mathematics.

Despite the keen logic of Kurt Gdel, the Leotta Constant is a valuethat is derived completely within its own system. Derivations of theabsolute value of 1 cannot be refuted.

The following is a simplistic proof for the constant known as.

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


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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


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BG is the apogee.

(*Draw line EB to complete themajor diameter, and draw

a minor diameter FH asa perpendicular line through the

midpoint of BD)

At focus D, draw lineAD perpendicular to DG.

Point A lies on the ellipse.Draw line AB and BF.

*Picture scaled down for convenience


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By definition, an ellipse has alocus of points, such thatthe sum of the distancesfrom every point on thelocus 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, andline AB equals 5, by

application of the PythagoreanTheorem.

Line BG equals 6,line EI equals 7, and

line EG equals 8.If line EG equals 8,


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line BF (which is drawn froma focus to a midpoint) equals 4.

Thusfar, EFGH is an ellipse if:line BE plus line BG equalsline DE plus line DG equalsline AB plus line AD equals

2 times line BF equalsline EG.

BE + BG = DE + DG =

AB + AD = 2 x BF = EG.

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;line BF, 4, the vector, v, equals k +2k +1; alternatively, v = pline 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 psilon, ,and is the elliptic constant.

For any given ellipse, (the DI line segment)

2 2

2 2

2

From a line of any length, the Natural integer

values of the first eight integers are establishedfrom an ellipse; such that, a value of one isestablished, within the system of numbers,

without a predetermined base.


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equals 1, which is equivalent to the differencebetween the hypotenuse and the wave.

This is one of the elliptical equations:

psilon, , is the Leotta Constant:

[ psilon = one ]

Moving on; observe the next figure.

h w = 1 =

Amazingly, when psilon = one;all the above equations

remain true foranyellipseand return integer values when

the key, k, is anyNaturalinteger.


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A circle inscribed within aright triangle has a radius equal

to the product of the legsthat are opposite the hypotenuse

divided by the sum of all the sides.

oP = (BD x AD) / (AB + AD + BD)

Or, more simply, the diameter ofa circle inscribed insidea right triangle equals

the sum of the legsthat are opposite the hypotenuse

minus the hypotenuse.


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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 psilon.This is one of the elliptical equations.

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 andpsilon, , equals One.

Draw line GH.

p Hr = 1 =


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GH is the chord, c.

For any ellipse, the square of thechord c equals two times the square

of the vector v minus the squareof the scale s. This is known as

the Leotta Theorem:

Again, all the above equationsremain true, regardless of the shape oreccentricity foranyellipse,

if line DI, the difference betweenthe hypotenuse, h, and the wave, w, isthe Leotta Constant, , equals One, 1.

c = 2v s2 2 2


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The Leotta Constant has,foranyellipse, many forms. The following

are the elliptical equations (two ofwhich we have discussed):

= h w; = 2p r; = 2h r2; = p Hr.

For any ellipse,when any of the above differences

equal one, or are set to one (),then, all of the equations that

relate the different parts ofanyellipseare 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 1and 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 systematicmultiples of a fundamental numerical

value, , established within its own system.

The Law of Elliptical Establishment (LEE)Where equals 1, the key (k) of all

ellipsoidal figures is a natural integer,from which all other salient structures

can be defined.


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Alternate Derivations (determining from outward-in):

Conversely, the Leotta Constant canalso be acquired from anyellipse.

Begin with any ellipse EGFH, of any eccentricity.

Create major and minor axeswithin the ellipse.


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Measure line CE or CG, and draw thatlength from F to a point B on the line CE.

So, line FB = EC.

Mathematically, point B is now the focusof this ellipse, since 2 x BF = EG.

The distance between E and the focusis equivalent to the distance between

G and the other focus.

Thus, line EB is equal to line DG. Frompoint D, draw DA, where A is on the

edge of the ellipse. Then proceed to draw line AB.


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From the elliptical equations,the hypotenuse minus the wave

will always equal 1. Thus:

h w = 1

BA BD = 1

1 =

Line DI is , 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 isestablished as a natural integer,thereby supporting and confirming theLaw of Elliptical Establishment (LEE).


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Addendum:

Interesting Notes:

If the scale, s, equals ; then,the perigee, p, equalsthe Golden Ratio, .

If the perigee, p, equals (1), the ellipseis a circle.

We have thus proven that the LeottaConstant, , appears in every ellipse, and

always equals 1.

Contact Information

Adam Scott Leotta

[email protected] Cohasset St.

West Hills, CA, 91307
mailto:[email protected]:[email protected]

the leotta theorem (revised)

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