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.