BRUNARDOT  HARMONIC  ELLIPSES

with radius "r" and vector "v" as squares of integers; and, diagonal "d," perigee "p," and Brunardot Iteration "i," above the first iteration, as Natural Prime numbers; also, force "F," energy "E," and the Harmonic Ratio "HR" are integers; soliton "s" is an even integer and Light "L" is an integer divisible by 4; and also, included is a simple series and a right triangle; all components, so described, are generated by . . . any, single integer . . . with limitless iterations.
(Except for "0" and +1, which generate the quaquaversal axes of  Reality.)

h=f(A), i=f(A), T=f(h,i); therefore, T = f(A)

 

      Limitless Series,
          BHE CONSOLIDATED TABLE:
                For Integers -8 through +5; and,
                      For Iterations: One, Two, and Six

BRUNARDOT  HARMONIC  ELLIPSES

with radius "r" and vector "v" as squares of integers; and, diagonal "d," perigee "p," and Brunardot Iteration "i," above the first iteration, as Natural Prime numbers; also, force "F," energy "E," and the Harmonic Ratio "HR" are integers; soliton "s" is an even integer and Light "L" is an integer divisible by 4; and also, included is a simple series and a right triangle; all components, so described, are generated by . . . any, single integer . . . with limitless iterations.
(Except for "0" and +1, which generate the quaquaversal axes of  Reality.)

h=f(A), i=f(A), T=f(h,i); therefore, T = f(A)


First Iteration of Integers -8 to +5

A

 =

 -8

-7

-6

-5

-4

-3

-2

-1

+2

+3

+4

+5

Alpha (Any lnteger)

h

 =

144

112

84

60

40

24

12

4

4

12

24

40

Harmonic f(A)

i1

 =

17

15

13

11

9

7

5

3

3

5

7

9

Brunardot Iteration f(A)

inc.

=

-18

-16

-14

-12

-10

-8

-6

-4

2

4

6

8

Increment f(i)

T

 =

1

1

1

1

1

1

1

1

1

1

1

1

Tau: f(h,i); f(A)

p

 =

145

113

85

61

41

25

13

5

5

13

25

41

Fibonacci

s

 =

20880

12656

7140

3660

1640

600

156

20

20

156

600

1640

Sequence

r

 =

289

225

169

121

81

49

25

9

9

25

49

81

Generating

d

 =

41761

25313

14281

7321

3281

1201

313

41

41

313

1201

3281

Right

L

 =

41760

25312

14280

7320

3280

1200

312

40

40

312

1200

3280

Triangle

HR

=

290

226

170

122

82

50

26

10

10

26

50

82

Harmonic Ratio

F

=

2465

1695

1105

671

369

175

65

15

15

65

175

369

Force


Second Iteration of Integers -8 to +5

A

 =

 -8

-7

-6

-5

-4

-3

-2

-1

+2

+3

+4

+5

Alpha (Any lnteger)

h

 =

144

112

84

60

40

24

12

4

4

12

24

40

Harmonic f(A)

i2

 =

161

127

97

71

49

31

17

7

1

7

17

31

Brunardot Iteration f(A)

inc.

=

178

142

110

82

58

38

22

10

-2

2

10

22

Increment f(i)

T

 =

90

72

56

42

30

20

12

6

0

2

6

12

Tau: f(h,i); f(A)

p

 =

12961

8065

4705

2521

1201

481

145

25

1

25

145

481

Fibonacci

s

 =

167974560

65036160

22132320

6352920

1441200

230880

20880

600

0

600

20880

230880

Sequence

r

 =

25921

16129

9409

5041

2401

961

289

49

1

49

289

961

Generating

d

 =

335949121

130072321

44264641

12705841

2882401

461761

41761

1201

1

1201

41761

461761

Right

L

 =

335949120

130072320

44264640

12705840

2882400

461760

41760

1200

0

1200

41760

461760

Triangle

HR

=

2332980

1161360

526960

211764

72060

19240

3480

300

0

100

1740

11544

Harmonic Ratio

F

=

2086721

1024255

456385

178991

58849

14911

2465

175

1

175

2465

14911

Force


Sixth Iteration of Integers -8 to +5

A

 =

 -8

-7

-6

-5

-4

-3

-2

-1

+2

+3

+4

+5

Alpha (Any lnteger)

h

 =

144

112

84

60

40

24

12

4

4

12

24

40

Harmonic f(A)

i6

 =

305

239

181

131

89

55

29

11

5

19

41

71

Brunardot Iteration f(A)

inc.

=

178

142

110

82

58

38

22

10

-2

2

10

22

Increment f(i)

T

 =

323

255

195

143

99

63

35

15

3

15

35

63

Tau: f(h,i); f(A)

p

 =

46513

28561

16381

8581

3961

1513

421

61

13

181

841

2521

Fibonacci

s

 =

2163412656

815702160

268320780

73624980

15685560

2287656

17680

3660

156

32580

706440

6352920

Sequence

r

 =

93025

57121

32761

17161

7921

3025

841

121

25

361

1681

5041

Generating

d

 =

4326825313

1631404321

536641561

147249961

31371121

4575313

363641

7321

313

65161

1412881

12705841

Right

L

 =

4326825312

1631404320

536641560

147249960

31371120

4575312

363640

7320

312

65160

1412880

12705840

Triangle

HR

=

30047398

14566110

6388590

2454166

784278

190638

29470

1830

78

5430

58870

317646

Harmonic Ratio

F

=

14186465

6826079

2964961

1124111

352529

83215

12209

671

65

3439

34481

178991

Force

A

=

Alpha (motion = y-axis)

l

=

Light

a = apogee M = Mean (s2 / v)
c = chord, major o = opposing wave crest
d = diagonal p = perigee
E = Energy, Internal r = radius
f = Force s = soliton
H = Harmonic T = Tau (time, etc. = x-axis)

HR

 = Harmonic Ratio

CU

 = Conceptual Unit
i = Brunardot Iteration v = vector
I = Infinity line (x-axis) w = wave crest
inc. = increment between "i"s x = foci
K = Unimetric Factor


The following is true for:

         Every integer value of
                        Alpha (A) except zero and +1

d

 = A Natural Prime number (Nature's Scale)

r

 = A Perfect square

L

 = An Integer divisible by four

d,r,L
=
A Right Triangle that Generates BHE's


a, c2, d, f, i, L, p, and s and  are integers.
E, r, v, and U are perfect squares.
d, i, and p are Natural Prime numbers.
p, s, v and a are consecutive terms of a series.
p, s, and Tau are functions of Alpha.


An Infinity line "I" and a motion line "m" are at
right angles to one another and are both infinite in length and infinitesimal in width.

Corollary formulas derived from
               the Brunardot Theorem:  c2 = 2v2 - s2

          Primary Corollaries:

U = d - L = 1

 The Natural Unit, U, establishes the relative unit value for all other integers.

U = A Proof of One.

 The Integer One is a Relativistic Function of Alpha (Motion) and Tau.  (Speed, Spin, Space, and Time)


h = 2(A2 - A); or, A2 - A


i1 = 2A - 1


i1 = square root of (2h + 1)

i2 = 2A2 - 4A + 1;

i2 = h - (2A - 1) = h - i1

i3 = 2(A2-A) - 1;

i3 = h - 1

i4 = 2(A2-A) + 1;

i4 = h + 1

i5 = 2A2 - 1;

i5 = h + i1;

 i5   = i1 + h . . .

i6 = 4A2 - 6A + 1;

i6 = 2h - i1;

 i6   = i2 + h . . .

i7 = 4A2 - 4A - 1;

i7 = 2h - 1;

 i7   = i3 + h . . .

i8 = 4A2 - 4A + 1;

i8 = 2h + 1;

 i8   = i4 + h . . .

i8 = (2A - 1)2 = i2 i12 = i8 + h . . .

in   = in-4 + h . . .


Tau (T) = ((i2 - 1) / 2) / 2A(A - 1)

Tau (T) = (((i2 + 1) / 2) - 1) / h

Tau (T) = L / 2hp

Tau (T) = (p - 1) / h

p = (i2 + 1) / 2

s = p2 - p

v = p + s

a = s + v

M = s2 / v

r = i2; or, v - M

d = (i4 + 1) / 2; or 2M + r

L = d - 1

HR= L / h

E = v2 - s2
F = square root of E
c2  = E + v2


And, thus, some Secondary Corollaries:

a = 2p2 - p

a = 2s + p

c2  = 2E + s2

c2  = F2 + v2
d = L + 1
d = L + 2p - r
d  = r2 - L
d = 2v - r
d = i4 - L
E = rv
E = c2 - v2
E = ( c2 - s2) / 2
E = F2
F = square root of c2 - v2
F = square root of v2 - s2
F = square root of rv
F = ip
HR = T(r + 1)
HR = 2pT
L = i4 - d
L = (i4 - 1) / 2

L = 2hpT
p = 2AT (A - 1) + 1 = hT  + 1
p = (i2 + 1) / 2

p = L / 2hT
r = 2v - d
r = 2p - 1
r2 = d + L
s = (i4 - 1) / 4
s = p(p - 1)
s2 = v2 - E
U = 2p - r
U = i4 - 2L

U = i4 - 4hpT
U = p - 2AT (A - 1)
U = HR /T - r

U = p - hT
v = p2
v = E / r
v2 = E + s2
v2 = (c2 + s2) / 2









E-mail :  Brunardot@Brunardot.com







There is one Universe.

It is perpetual, in equilibrium;

and, a manifestation of the
 Unified Concept; thus;

. . . the Fundamental Postulate.


also,

are a single discipline, Philogic,
which proclaims perpetuity

and the nexus of Life; such is


. . . Conceptualism.

 

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