Epsilon=One
08-16-2005, 07:57 PM
The Mystique of the Ellipse
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http://i.g2d.us/Weinberg_thtfl.gif
This post honors
Steven Weinberg....[1933 -......]
Philosopher, Physicist Laureate
Truth, beauty . . .
is simple, elegant;
and, subconsciously
. . . quickly recognized.
The ellipse is unique among all natural phenomena
because it defines the Elliptical Constant (www.101123.com/EC) (EC),
which is the most important discovery
in the history of mathematics
because the EC, heuristically, describes
the Conceptual Unit (www.101123.com/CU) (CU) that makes possible the emergent,
hyper-relativistic, pulsing, ellipsoidal quantum of space
that evolves to all fermions and bosons.
Pulsoid Theory (PT) is based entirely upon the geometry and simple algebra of the ellipse. PT’s basic premise is that everything that exists is fundamentally composed of pulsing, ellipsoidal quanta that moves, fundamentally in an elliptical manner.
It is the fundamental geometry of (SM) that is predicated by the singularity of its source that creates the CU that makes possible the evolution of:
orthogonal [URL=www.101123.com/D]dimensions (www.101123.com/SM[seminal motion[/URL),
fundamental, intrinsic time (www.101123.com/FIT) (FIT),
the Inverse Square Law (www.101123.com/ISL) (ISL),
and all the subsequent phenomena of that which exists.
The ellipse is defined by:
the Brunardot Theorem (www.101123.com/BT) (BT) , c² = 2v² – s²,
and the Elliptical Constant (www.101123.com/EC) (EC)
that is defined by: the Pulsoid Theorem (www.101123.com/eP2), v = εP² or ε = v/ P² or . . . ε = P²/v.
Such concepts as: the Pythagorean Theorem,
the Golden Ratio (www.101123.com/GR),
the Fibonacci Sequence (www.101123.com) that is revised,
and Brunardot Triangles (www.101123.com/Btr)
are corollaries of the Brunardot Theorem (www.101123.com/BT) (BT).
(If no image appears below, "Click" your browser "Refresh" icon.)
http://b.g2d.us/let500.gif
When, and why,
are all of these structural lines,
at the same time
. . . integers ?
. . .with the identical algebraic relationship
to one another regardless of
the elliptical eccentricity?
(See: Legend below, near the end of this post.)
http://b.g2d.us/ue.gif
The basic Unimetry (www.101123.com/Uni) equation for an ellipse is referred to as
the Brunardot Theorem (www.101123.com/BT), which is:
http://b.g2d.us/bt-eq150.jpg
The Brunardot Theorem (www.101123.com/BT) is a general theorem of which the Pythagorean Theorem (www.CQthus.com/Pyth) and the Elliptical Constant (www.101123.com/EC) are corollaries
Regardless of the elliptical shape, every structural element of an ellipse has the same algebraic relationship to its perigee (or any other part). This relationship is dependent upon the Elliptical Constant (www.101123.com/EC), which is the escapement (www.CQthus.com/Esc) for the metronome of Reality (www.101123.com/metro).
The significance of the ellipse is that it underlies everything that exists. That which exists is either a particle or a wave. All particles, being subject to the illusion of gravitational attraction-at-a-distance (www.101123.com/AAA), move in an elliptical manner. And, all particles evolve from waves at Critical Coalescence (www.101123.com/CCo), which waves are the consequence of the resonating Pulsoidal Ellipses (www.101123.com/PE) of Pulsoids (www.101123.com/P). And, of course, the ellipse describes the internal geometry of a Light wave (www.101123.com/Lightg).
Thus, the Mystique of the Ellipse with its unique, simple constant that determines all of its structural relationship that is referred to as the Elliptical Constant (www.101123.com/EC) (EC), underlies all the complex, physical phenomena of Reality (www.101123.com/R) including mathematics, Consciousness, and . . . anthropoidal, illusionary enigmas.
..................Mathematics does not
..............................explain Nature;
.........................................Nature explains
.................................................. ..mathematics.Concerning mathematics: One must always ask, "Why"? Mathematics as a function of Nature; in the form of pulsing, hyper-relativistic, elliptical motion; reflects all the fundamental concepts of the physical world.
When the Elliptical Constant (www.101123.com/EC) is set, every structural part of every ellipse has the same relationship to its perigee, “p” (Line EB); such as: the apogee, "o," (Line ED) equals: 2p² – p; or, the amplitude, “a,” (Line CF) equals: the square root of (2p³ – p²) or the square root of (r times p²), which is a Natural integer (www.101123.com/NI) if the perigee, "p," equals (any Par1 (www.101123.com/NI) square integer + 1)/2.
Of interest, r = 2p – 1; and h = (2r² + 1) / 2 = 2(v – p) + 1.
Also, if the radius, "r," (Line AD) equals any Par1 (www.101123.com/NI) square integer, the Chord diameter, "Td," (twice Line oT) equals the amplitude, "a," [the square root of (2p³ – p²)] – p + one, which is a Natural integer (www.101123.com/NI).
All Ultron Ellipses (www.101123.com/UE) have Natural integer (www.101123.com/NI) values for the amplitude, apogee, Chord diameter, glyph, lemma, Lemma diameter, hypotenuse, hypotenuse radius, major diameter, major radial, perigee, radius, soliton, vector, vector radius, wave, and diameter chord.
Also see: The Mystique of the Ultron Ellipse (www.101123.com/UEm).
>>..For selected Ultron Ellipse Cycle-Time Arrays "Click" this Line (www.101123.com/UA)...<<
There are four groups, of four tangent circles that are internal to the ellipse, one group for each quadrant of the ellipse, all of which are Natural integer (www.101123.com/NI) value diameters, for a total of sixteen, internal tangent circles.
These sixteen circles, heuristically, represent Resoloids (www.101123.com/R-loid), which are analogous to quarks in the composition of "dark" matter (www.101123.com/DM).
Hypotenuse radius = Line oP = Hr
= perigee, "p," minus 1.
Hypotenuse diameter = 2 x Line oP = Hd
= radius, "r," minus 1.
Vector radius = Line oV = Vr
= (amplitude, "a," minus perigee, "p") / 2.
Radial diameter = 2 x Line oR = Rd
= radius, "r," minus 2.
Chord diameter = Line oT = Cd
= amplitude, "a," minus perigee, "p," plus 1.
Thus,..................................
Hr = p – 1;
Hd = r – 1;
Vr = (a - p) / 2;
Rd = r – 2; and,
Cd = a – p + 1 (or: 2Vr + 1).
Or, more amazing,..................................
...........the inscribed circle diameters are:
Hd = r – 1;
Rd = Hd – 1
Vd = a – p
Cd = Vd + 1
One must ask:........................................
Why ?
Ultimate simplicity (www.101123.com/S) has been found;
at the crux of . . . complexity !!!
The, above, simple equations of Unimetry (www.101123.com/Uni), that represent radii and diameters, are all integer values for Pulsoidal Ellipses (www.101123.com/PE), which, heuristically, represent the most fundamental "strings" that comprise the more complex "strings" that are the four states of Light (www.101123.com/4) and mass; all of which are generated by the family of Conceptual Ellipses (www.101123.com/CE), Pulsoids (www.101123.com/P) and Taisaids (www.101123.com/showthread.php?t=102).
Of course, these four or five equations are meaningless without an adequate universal Proof of One (www.101123.com/PoO) . . . which must be applicable for any system.
"Whatever the final theory may be... Even if (it) turns out to be...strings that can be expressed in a few simple equations...that can describe gravitation...with other forces without mathematical inconsistencies, we will have to ask why there should be such a thing..."
--Steven Weinberg
--Dreams of a Final Theory, 1992.Naturally, the answer to Weinberg's plaint is found in the nature of Infinity (www.101123.com/I).
For any Pulsoidal Ellipse (http://www.101123.com/PE), the three inscribed circle radii, when the Elliptical Constant (www.101123.com/EC) (EC) is set as in Conceptual Ellipses (www.101123.com/CE) (CE), are all Natural integers (www.101123.com/NI) (NI) found with simple addition and subtraction that involves only One, "1" (the difference between the hypotenuse and wave); the amplitude (minor radius), “a”; and the perigee.
The three inscribed circles, that are internal to the ellipse, represent twelve inscribed circles (three for each quadrant of the ellipse).
As time and/or cycle count increase, eight of the twelve circles merge to the center of the ellipse, while the remaining four circles merge, two each, toward each end of the major diameter; in the manner of a Hylotron (www.101123.com/Hy).
See: The Mystique of the Ultron Ellipse (www.101123.com/UEm).
One must ask:........................................
Why ?
And, again,
Why ?
And, again,
Why ?
Over and over,
until . . . Infinity.
And, still, another bit of amazement:The hypotenuse, "h," (a major element of Natural prime numbers (www.101123.com/NPN)) of a Conceptual Ellipse (www.101123.com/CE) (CE) is the perigee, "p," of a Pulsoidal Ellipse (www.101123.com/PE) (PE) with the same Time and Cycle values. (CE h = PE p)
Of course, the relationship of the Conceptual Ellipse (www.101123.com/CE) wave, "w," to the Pulsoidal Ellipse (www.101123.com/PE) perigee, "p," may be more significant, if slightly less elegant.
Ultron Ellipses (www.101123.com/UE) are Pulsoidal Ellipses (www.101123.com/PE) when the Cycle/Time values exceed the threshold of 225 units.
See: Conceptual Ellipses (www.101123.com/CE), Brunardot Ellipses (www.101123.com/BE),
.......Pulsoidal Ellipses (www.101123.com/PE), Ultron Ellipses (www.101123.com/UE) and,
.......Lemma Ellipses (www.101123.com/LE), the latter four of which,
.......heuristically, represent natural ellipsoidal
.......manifestations.
In summary: Not only does the mystique of the ellipse’s dynamic geometry generate structural spheres; but, also, the reflection of the elliptical curve is part of the ellipse’s mystique that applies to the enigmas of light waves.
If energy, such as a light wave, emanates from a focus of an ellipse, in any direction, said energy will be reflected by the elliptical curve directly to the other focus of the ellipse.
Thus, energy that is quaquaversally radiated at a focus (to and from a point in all directions) of an ellipsoid is reflected, in its entirety, back to the ellipsoid’s other focus.
Thus, all energy that is so radiated within an ellipsoid is perpetually conserved within the ellipsoid at the foci which is intrinsic to an Infinity line (www.101123.com/I).
It is this reflective characteristic of a light wave's internal structure (www.101123.com/Lightg) as an ellipsoid, that powers the continually expanding Triquametric motion (www.101123.com/TM) of light waves.
It is the continual hyper-relativistic, pulsating, expansion, with the increase of time, while the Elliptical Constant (www.101123.com/EC) remains relativistically unchanged, that accounts for the Pulsoid's (www.101123.com/P) resonance and increased focal length (wave, “w”) which accounts for light’s Cosmic red-shift of frequency with an increase in time.
That is: the Cosmic red-shift of light’s frequency is a result of an expanding light wave while the locus of the Universe is unchanged . . . or “steady-state” as rationalized by the Equilibrium Theory of Reality (101123.com/ETR).
Legend for a Lemma Ellipse (www.101123.com/LE)
Line AB = FK = hypotenuse = h
Line AD = BJ = radius = r
Line AJ = diagonal = D
Line BC = CD = soliton = s
Line BD = wave = w
Line BE = DG = perigee = p
Line BF = vector = v
Line BG = ED = apogee = o
Line BK = lemma = l
Line CA = CJ = diagonal radial = d
Line CF = CH = amplitude = a
Line CE = CG major radial = m
Line DI = AB – BD = εpsilon = ε = One
Line EG = major diameter = M
Line EI = glyph = g
Line FH = minor diameter = A
Line GH = diameter chord = c
Line IG = key = k
Line oL = Lemma diameter = Ld = Natural integer (www.101123.com/NI)
Line oP = Hypotenuse radius = Hr = Natural integer (www.101123.com/NI)
Line oR = Radial diameter = Rd = Natural integer (www.101123.com/NI)
Line oT = Chord diameter = Cd = Natural integer (www.101123.com/NI)
Line oV = Vector radius = Vr = Natural integer (www.101123.com/NI)
Triangle ABD = Hypotenuse triangle = Brunardot triangle (www.101123.com/BTr5)
Triangle BCJ = Radial triangle = Pythagorean triangle (www.CQthus.com/PTr)
Triangle BFK = Lemma triangle = Pythagorean triangle (www. CQthus.com/PTr)
Triangle CBF = Vector triangle = Pythagorean triangle (www. CQthus.com/PTr)
Triangle CGH = Chord triangle = Pythagorean triangle (www. CQthus.com/PTr)
Inscribed circle ABD = Hypotenuse circle
Inscribed circle BCJ = Radial circle
Inscribed circle BFK = Lemma circle
Inscribed circle CBF = Vector circle
Inscribed circle CGH = Chord circle
p, s, v, o = the first four terms of a Brunardot Series (www.101123.com/BS)
All lines shown in the above image of a Lemma Ellipse (www.101123.com/LE) are integers derived from simple algebraic relationships for an unending sequence of Lemma Ellipses (www.101123.com/LE) that are generated from two unending sequences of Natural Integers (www.101123.com/NI).
©Copyright 2005-2008 by Brunardot. All rights reserved.
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If images don’t display, "click" the Refresh Icon.
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/Weinberg_thtfl.gif
This post honors
Steven Weinberg....[1933 -......]
Philosopher, Physicist Laureate
Truth, beauty . . .
is simple, elegant;
and, subconsciously
. . . quickly recognized.
The ellipse is unique among all natural phenomena
because it defines the Elliptical Constant (www.101123.com/EC) (EC),
which is the most important discovery
in the history of mathematics
because the EC, heuristically, describes
the Conceptual Unit (www.101123.com/CU) (CU) that makes possible the emergent,
hyper-relativistic, pulsing, ellipsoidal quantum of space
that evolves to all fermions and bosons.
Pulsoid Theory (PT) is based entirely upon the geometry and simple algebra of the ellipse. PT’s basic premise is that everything that exists is fundamentally composed of pulsing, ellipsoidal quanta that moves, fundamentally in an elliptical manner.
It is the fundamental geometry of (SM) that is predicated by the singularity of its source that creates the CU that makes possible the evolution of:
orthogonal [URL=www.101123.com/D]dimensions (www.101123.com/SM[seminal motion[/URL),
fundamental, intrinsic time (www.101123.com/FIT) (FIT),
the Inverse Square Law (www.101123.com/ISL) (ISL),
and all the subsequent phenomena of that which exists.
The ellipse is defined by:
the Brunardot Theorem (www.101123.com/BT) (BT) , c² = 2v² – s²,
and the Elliptical Constant (www.101123.com/EC) (EC)
that is defined by: the Pulsoid Theorem (www.101123.com/eP2), v = εP² or ε = v/ P² or . . . ε = P²/v.
Such concepts as: the Pythagorean Theorem,
the Golden Ratio (www.101123.com/GR),
the Fibonacci Sequence (www.101123.com) that is revised,
and Brunardot Triangles (www.101123.com/Btr)
are corollaries of the Brunardot Theorem (www.101123.com/BT) (BT).
(If no image appears below, "Click" your browser "Refresh" icon.)
http://b.g2d.us/let500.gif
When, and why,
are all of these structural lines,
at the same time
. . . integers ?
. . .with the identical algebraic relationship
to one another regardless of
the elliptical eccentricity?
(See: Legend below, near the end of this post.)
http://b.g2d.us/ue.gif
The basic Unimetry (www.101123.com/Uni) equation for an ellipse is referred to as
the Brunardot Theorem (www.101123.com/BT), which is:
http://b.g2d.us/bt-eq150.jpg
The Brunardot Theorem (www.101123.com/BT) is a general theorem of which the Pythagorean Theorem (www.CQthus.com/Pyth) and the Elliptical Constant (www.101123.com/EC) are corollaries
Regardless of the elliptical shape, every structural element of an ellipse has the same algebraic relationship to its perigee (or any other part). This relationship is dependent upon the Elliptical Constant (www.101123.com/EC), which is the escapement (www.CQthus.com/Esc) for the metronome of Reality (www.101123.com/metro).
The significance of the ellipse is that it underlies everything that exists. That which exists is either a particle or a wave. All particles, being subject to the illusion of gravitational attraction-at-a-distance (www.101123.com/AAA), move in an elliptical manner. And, all particles evolve from waves at Critical Coalescence (www.101123.com/CCo), which waves are the consequence of the resonating Pulsoidal Ellipses (www.101123.com/PE) of Pulsoids (www.101123.com/P). And, of course, the ellipse describes the internal geometry of a Light wave (www.101123.com/Lightg).
Thus, the Mystique of the Ellipse with its unique, simple constant that determines all of its structural relationship that is referred to as the Elliptical Constant (www.101123.com/EC) (EC), underlies all the complex, physical phenomena of Reality (www.101123.com/R) including mathematics, Consciousness, and . . . anthropoidal, illusionary enigmas.
..................Mathematics does not
..............................explain Nature;
.........................................Nature explains
.................................................. ..mathematics.Concerning mathematics: One must always ask, "Why"? Mathematics as a function of Nature; in the form of pulsing, hyper-relativistic, elliptical motion; reflects all the fundamental concepts of the physical world.
When the Elliptical Constant (www.101123.com/EC) is set, every structural part of every ellipse has the same relationship to its perigee, “p” (Line EB); such as: the apogee, "o," (Line ED) equals: 2p² – p; or, the amplitude, “a,” (Line CF) equals: the square root of (2p³ – p²) or the square root of (r times p²), which is a Natural integer (www.101123.com/NI) if the perigee, "p," equals (any Par1 (www.101123.com/NI) square integer + 1)/2.
Of interest, r = 2p – 1; and h = (2r² + 1) / 2 = 2(v – p) + 1.
Also, if the radius, "r," (Line AD) equals any Par1 (www.101123.com/NI) square integer, the Chord diameter, "Td," (twice Line oT) equals the amplitude, "a," [the square root of (2p³ – p²)] – p + one, which is a Natural integer (www.101123.com/NI).
All Ultron Ellipses (www.101123.com/UE) have Natural integer (www.101123.com/NI) values for the amplitude, apogee, Chord diameter, glyph, lemma, Lemma diameter, hypotenuse, hypotenuse radius, major diameter, major radial, perigee, radius, soliton, vector, vector radius, wave, and diameter chord.
Also see: The Mystique of the Ultron Ellipse (www.101123.com/UEm).
>>..For selected Ultron Ellipse Cycle-Time Arrays "Click" this Line (www.101123.com/UA)...<<
There are four groups, of four tangent circles that are internal to the ellipse, one group for each quadrant of the ellipse, all of which are Natural integer (www.101123.com/NI) value diameters, for a total of sixteen, internal tangent circles.
These sixteen circles, heuristically, represent Resoloids (www.101123.com/R-loid), which are analogous to quarks in the composition of "dark" matter (www.101123.com/DM).
Hypotenuse radius = Line oP = Hr
= perigee, "p," minus 1.
Hypotenuse diameter = 2 x Line oP = Hd
= radius, "r," minus 1.
Vector radius = Line oV = Vr
= (amplitude, "a," minus perigee, "p") / 2.
Radial diameter = 2 x Line oR = Rd
= radius, "r," minus 2.
Chord diameter = Line oT = Cd
= amplitude, "a," minus perigee, "p," plus 1.
Thus,..................................
Hr = p – 1;
Hd = r – 1;
Vr = (a - p) / 2;
Rd = r – 2; and,
Cd = a – p + 1 (or: 2Vr + 1).
Or, more amazing,..................................
...........the inscribed circle diameters are:
Hd = r – 1;
Rd = Hd – 1
Vd = a – p
Cd = Vd + 1
One must ask:........................................
Why ?
Ultimate simplicity (www.101123.com/S) has been found;
at the crux of . . . complexity !!!
The, above, simple equations of Unimetry (www.101123.com/Uni), that represent radii and diameters, are all integer values for Pulsoidal Ellipses (www.101123.com/PE), which, heuristically, represent the most fundamental "strings" that comprise the more complex "strings" that are the four states of Light (www.101123.com/4) and mass; all of which are generated by the family of Conceptual Ellipses (www.101123.com/CE), Pulsoids (www.101123.com/P) and Taisaids (www.101123.com/showthread.php?t=102).
Of course, these four or five equations are meaningless without an adequate universal Proof of One (www.101123.com/PoO) . . . which must be applicable for any system.
"Whatever the final theory may be... Even if (it) turns out to be...strings that can be expressed in a few simple equations...that can describe gravitation...with other forces without mathematical inconsistencies, we will have to ask why there should be such a thing..."
--Steven Weinberg
--Dreams of a Final Theory, 1992.Naturally, the answer to Weinberg's plaint is found in the nature of Infinity (www.101123.com/I).
For any Pulsoidal Ellipse (http://www.101123.com/PE), the three inscribed circle radii, when the Elliptical Constant (www.101123.com/EC) (EC) is set as in Conceptual Ellipses (www.101123.com/CE) (CE), are all Natural integers (www.101123.com/NI) (NI) found with simple addition and subtraction that involves only One, "1" (the difference between the hypotenuse and wave); the amplitude (minor radius), “a”; and the perigee.
The three inscribed circles, that are internal to the ellipse, represent twelve inscribed circles (three for each quadrant of the ellipse).
As time and/or cycle count increase, eight of the twelve circles merge to the center of the ellipse, while the remaining four circles merge, two each, toward each end of the major diameter; in the manner of a Hylotron (www.101123.com/Hy).
See: The Mystique of the Ultron Ellipse (www.101123.com/UEm).
One must ask:........................................
Why ?
And, again,
Why ?
And, again,
Why ?
Over and over,
until . . . Infinity.
And, still, another bit of amazement:The hypotenuse, "h," (a major element of Natural prime numbers (www.101123.com/NPN)) of a Conceptual Ellipse (www.101123.com/CE) (CE) is the perigee, "p," of a Pulsoidal Ellipse (www.101123.com/PE) (PE) with the same Time and Cycle values. (CE h = PE p)
Of course, the relationship of the Conceptual Ellipse (www.101123.com/CE) wave, "w," to the Pulsoidal Ellipse (www.101123.com/PE) perigee, "p," may be more significant, if slightly less elegant.
Ultron Ellipses (www.101123.com/UE) are Pulsoidal Ellipses (www.101123.com/PE) when the Cycle/Time values exceed the threshold of 225 units.
See: Conceptual Ellipses (www.101123.com/CE), Brunardot Ellipses (www.101123.com/BE),
.......Pulsoidal Ellipses (www.101123.com/PE), Ultron Ellipses (www.101123.com/UE) and,
.......Lemma Ellipses (www.101123.com/LE), the latter four of which,
.......heuristically, represent natural ellipsoidal
.......manifestations.
In summary: Not only does the mystique of the ellipse’s dynamic geometry generate structural spheres; but, also, the reflection of the elliptical curve is part of the ellipse’s mystique that applies to the enigmas of light waves.
If energy, such as a light wave, emanates from a focus of an ellipse, in any direction, said energy will be reflected by the elliptical curve directly to the other focus of the ellipse.
Thus, energy that is quaquaversally radiated at a focus (to and from a point in all directions) of an ellipsoid is reflected, in its entirety, back to the ellipsoid’s other focus.
Thus, all energy that is so radiated within an ellipsoid is perpetually conserved within the ellipsoid at the foci which is intrinsic to an Infinity line (www.101123.com/I).
It is this reflective characteristic of a light wave's internal structure (www.101123.com/Lightg) as an ellipsoid, that powers the continually expanding Triquametric motion (www.101123.com/TM) of light waves.
It is the continual hyper-relativistic, pulsating, expansion, with the increase of time, while the Elliptical Constant (www.101123.com/EC) remains relativistically unchanged, that accounts for the Pulsoid's (www.101123.com/P) resonance and increased focal length (wave, “w”) which accounts for light’s Cosmic red-shift of frequency with an increase in time.
That is: the Cosmic red-shift of light’s frequency is a result of an expanding light wave while the locus of the Universe is unchanged . . . or “steady-state” as rationalized by the Equilibrium Theory of Reality (101123.com/ETR).
Legend for a Lemma Ellipse (www.101123.com/LE)
Line AB = FK = hypotenuse = h
Line AD = BJ = radius = r
Line AJ = diagonal = D
Line BC = CD = soliton = s
Line BD = wave = w
Line BE = DG = perigee = p
Line BF = vector = v
Line BG = ED = apogee = o
Line BK = lemma = l
Line CA = CJ = diagonal radial = d
Line CF = CH = amplitude = a
Line CE = CG major radial = m
Line DI = AB – BD = εpsilon = ε = One
Line EG = major diameter = M
Line EI = glyph = g
Line FH = minor diameter = A
Line GH = diameter chord = c
Line IG = key = k
Line oL = Lemma diameter = Ld = Natural integer (www.101123.com/NI)
Line oP = Hypotenuse radius = Hr = Natural integer (www.101123.com/NI)
Line oR = Radial diameter = Rd = Natural integer (www.101123.com/NI)
Line oT = Chord diameter = Cd = Natural integer (www.101123.com/NI)
Line oV = Vector radius = Vr = Natural integer (www.101123.com/NI)
Triangle ABD = Hypotenuse triangle = Brunardot triangle (www.101123.com/BTr5)
Triangle BCJ = Radial triangle = Pythagorean triangle (www.CQthus.com/PTr)
Triangle BFK = Lemma triangle = Pythagorean triangle (www. CQthus.com/PTr)
Triangle CBF = Vector triangle = Pythagorean triangle (www. CQthus.com/PTr)
Triangle CGH = Chord triangle = Pythagorean triangle (www. CQthus.com/PTr)
Inscribed circle ABD = Hypotenuse circle
Inscribed circle BCJ = Radial circle
Inscribed circle BFK = Lemma circle
Inscribed circle CBF = Vector circle
Inscribed circle CGH = Chord circle
p, s, v, o = the first four terms of a Brunardot Series (www.101123.com/BS)
All lines shown in the above image of a Lemma Ellipse (www.101123.com/LE) are integers derived from simple algebraic relationships for an unending sequence of Lemma Ellipses (www.101123.com/LE) that are generated from two unending sequences of Natural Integers (www.101123.com/NI).
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Terms: PhysicsMathForums.com, Brunardot, and Pulsoid Theory must be cited.
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