.., ..

O - ..O - ..

6 McCreary Trail, Bolton, ON, Canada L7E8 2C

. . . , , , , , , , .

1.

, . , , 1827 . , . 20- . (1905 .). 1908 ., , , .

, , , . , [1-3]. 1993 . .. .. [4]. [4,5] , . [6], 2005 . Chaos, Solitons & Fractals. . [6] [7], , . , , . [8], [8], [9], [10], [10], [11], [12] [13]. [8, 14, 15], [16], [17-20], [8, 21-23], [24-25], [13].

. [4-25].

2. , &

.

[1-3]. :

x2 = x + 1. (1)

(1) . , .

(1) :

n = n-1 + n-2 = n-1> (2)

n : 0, 1, 2, 3, .

Fn = {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, }, (3)

Fn = Fn-1 + Fn-2 (4)

:

F1 = F2 = 1, (5)

Ln = {1, 3, 4, 7, 11, 18, 29, 47, 76, }, (6)

Ln = Ln-1 + Ln-2 (7)

:

L1 = 1; L2 = 3 (8)

n (. 1).

1.

n 0 1 2 3 4 5 6 7 8 9 10Fn 0 1 1 2 3 5 8 13 21 34 55F-n 0 1 -1 2 -3 5 -8 13 -21 34 -55Ln 2 1 3 4 7 11 18 29 47 76 123L-n 2 -1 3 -4 7 -11 18 -29 47 -76 123

.1, . , n=2k+1 Fn F-n , , F2k+1 = F-2k-1, n=2k , , F2k= -F-2k. Ln , , L2k = L-2k; L2k+1 = -L-2k-1.

, [2, 3] , 19 . . , , Fn Ln

.

:

,

(9)

:

.(10)

k : 0, 1, 2, 3, .

3.

3.1.

(9), (10)

,(11)

.(12)

, (11), (12). , [4]. k (9), (10) x, . , [4]:

.(13)

(14)

.

. (15)

. (16)

Fn Ln (13)-(16) :

sF(k) = F2k; cF(k) = F2k+1; sL(k) = L2k+1; cL(k) = L2k, (17)

k= 0; 1; 2; 3,.... [5] (13)-(16)

. () , (17), (13)-(16). , , , . (17).

3.2. ( )

(13)(16) (11), (12). , , (11), (12), (14) y, (15) . , (13)-(16).

[6] , .

.(18)

.(19)

. (20)

. (21)

:

; .(22)

. 1 2 (18)-(21).

>>> >>>

1.

2.

. 1 2, (18)-(21) (11), (12). , x=0

cFs(x) , cLs(x) cLs(0) = 2. , Fn (n = 0, 2, 4, 6, ) sFs(x) x = 0, 2, 4, 6, , Fn (n = 1, 3, 5, ) cFs(x) x = 1, 3, 5 . , cLs(x) x = 0, 2, 4, 6 , sLs(x) x = 1, 3, 5 .

(11) (12) :

; ;

; .

:

; .

3.3.

(18)-(21) , , . , (11) (12) , , .

(18)-(21) . 2.

2.

Fn+2 = Fn+1+Fn sFs(x+2) = cFs(x+1) + sFs(x) cFs(x+2) = sFs(x+1) + cFs(x)

Fn 2 Fn+1 Fn-1 = (-1)n+1 [sFs(x)]2 cFs(x+1) Fs(x-1) = -1

[cFs(x)]2 sFs(x+1) sFs(x-1) = 1

Ln+2 = Ln+1 + Ln sLs(x+2) = cLs(x+1) + sLs(x) cLs(x+2) = sLs(x+1) + cLs(x)

Ln2 2(-1)n = L2n [sLs(x)]2 + 2 = cLs(2x) [cLs(x)]2 2 = cLs(2x)

Fn+1 + Fn-1 = Ln cFs(x+1) + cFs(x-1) = cLs(x) sFs(x+1) + sFs(x-1) = sLs(x)

Fn + Ln = 2Fn+1 cFs(x) + sLs(x) = 2sFs(x+1) sFs(x)+ i>cLs(x) = 2cFs(x+1)

, [26]

Fn 2 Fn+1 Fn-1 = (-1)n+1, (23)

, , :

[sFs(x)]2 cFs(x+1) Fs(x-1) = -1 (24)

[cFs(x)]2 sFs(x+1) sFs(x-1) = 1, (25)

(23) .

3.4.

. 3.

3.

[ch(x)]2 [sh(x)]2 = 1

[cFs(x)]2 [sFs(x)]2 =

[cLs(x)]2 [sLs(x)]2 = 4

ch(xy) = ch(x)ch(y)

sh(x)sh(y) cFs(xy) = cFs(x)cFs(y)

sFs(x)sFs(y)

2cLs(xy) = cLs(x)cLs(y)

sLs(x)sLs(y)

sh(xy) = sh(x)ch(y) ch(x)sh(y)

sFs(xy) = sFs(x)cFs(y)

cFs(x)sFs(y)

2sLs(xy) = sLs(x)cLs(y)

cLs(x)sLs(y)

ch(2x) = [ch(x)]2 + [sh(x)]2

cFs(2x) = [cFs(x)]2 + +[sFs(x)]2

2cLs(2x) = [cLs(x)]2 +

+[sLs(x)]2

sh(2x) = 2 sh(x)ch(x)

sFs(2x) = sFs(x)cFs(x)

sLs(2x) = sLs(x)cLs(x)

[ch(x)](n) = [cFs(x)](n) = [cLs(x)](n)=

,

[ch(x)]2 [sh(x)]2 = 1 (26)

:

[cFs(x)]2 [sFs(x)]2 = ,(27)

[cLs(x)]2 [sLs(x)]2 = 4. (28)

, (. 3), () , (. 2). , , , (22) , x : 0, 1, 2, 3, . , (24)-(28), , . 2 3, .

3.5.

, [27]. ( , , , ..) , :

.(29)

. , (29), .

, : ? , . , , (29). , , , , : , . , () :

(30)

(30) [27]. , , , . , .. , .

, (30) . , (30) , , , . [27]. , , , , . , , .

, , , .

4.

4.1. -

[2,3] :

,

(31)

n= 0, 1, 2, 3, . (31)

(18) (19), , - (18) (19)

n -n (31), n = 0, 1, 2, 3, . (31) (-1)n, -1 +1 n = 0, 1, 2, 3, . cos(x), -1 +1 x = 0, 1, 2, 3, . , (31) .

1. - :

.(32)

3. -

, Fn, (31) - , (32):

,(33)

n= 0, 1, 2, 3, . - -

(.3), x = 0, 1, 2, 3, , (33). (. 1) - ( . 3 ).

- (32) , (. 4).

4. -

- Fn+2 = F n+1 + Fn: FF(x+2) = FF(x+1) + FF(x)

Fn 2 F n+1 Fn-1 = (-1)n+1 [FF(x)]2 - FF(x+1) FF(x-1) = cos(x)Fn+3 + Fn = 2 Fn+2 FF(x+3)+FF(x) = 2FF(x+2)Fn+3 Fn = 2Fn+1 FF(x+3) - FF(x) = 2FF(x+1)Fn+6 Fn = 4Fn+3 FF(x+6) + FF(x) = 4FF(x+3)F2n+1 = Fn+12 + Fn2 FF(2x+1)=[FF(n+1)]2 + [FF(x)]2

(23) - :

[FF(x)]2 - FF(x+1)FF(x-1) = cos(x)

4.2.

, , 1, , . ,

f(x) = cos(x) + isin(x), . , - (32)

, , (. 4), , .

Figure 4.

2. :

FF(x) = + i .(34)

, (. 4).

[7] (34) :

FF(x+2) = FF(x+1) + FF(x). (35)

, (35) (4).

4.3

(34):

Re[CFF(x)] = ; (36)

Im[CFF(x)] = .(37)

(36) y(x) (37) z(x), :

y(x) = - ,(38)

z(x) = . (38) . y z

, (38) . 3.

(. 5):

+ z2 = .(39)

5. , .

. 5, . . ( ).

>>> >>> 6.

XOY 7.

XOZ

(39) (18) (19):

z2 = [cFs(x) y][sFs(x) + y]. (40)

XOY . 6. (. 1). (34) - (32).

(34) XOY , (. 6).

XOZ . 7.

.

5.

2004 . [28] . , 2003 NASA's Wilkinson Microwave Anisotropy Probe (WMAP) [28] . , .

[28] :

1. (.8). . 8

.

>>>

8.

, , , . [4-7] , , .

, , .

1 [7] .

6.

6.1.

[8], , . =0, 1, 2, 3,... :

Fp(n+1) = Fp(n)+Fp(n-p) n>p+1; (41)

Fp(1) = Fp(2) =... = Fp(p+1) = 1 (42)

, (41), (42), [8] p- . [8], p-

:

(43)

, (41), (42) , . , =0 (41) (42) : 1, 2, 4, 8, 16, 32,..., =1 (3). =1 (43)

+ + + = 2n, (44)

. =1 (43)

,(45)

(3) .

6.2. -

[1-3], Fn/Fn-1 . [8] , - Fp(n)/Fp(n-1) , :

x+1 = x + 1. (46)

[8] - , =1 . (46) [9] , =1 (46) (1).

, =0 = 0 =2, = = =1. , 2 1 , .

(46) , -:

(47)

(46) (+1)- , , +1 x1, x2, x3, , xp, xp+1. , x1 -

p, , x1 = p. (46) , x1, x2, x3, , xp, xp+1:

(48)

n = 0, 1, 2, 3, ; xk (46); k = 1, 2,..., +1. [9] (46):

1+2+3+ 4+... +++1 =1 (49)

123 4...-1 +1 = (-1)p (50)

, [9], =1, 2, 3,... k=1, 2, 3, , p (46) :

1k + 2k + 3k + 4+... +k + +1k = 1. (51)

[9] , , (46), - p. :

(52)

Fp(n-+1) Fp(n--t) - , (41), (42), n +1. , =1 (52) :

n = Fn + Fn-1, (53)

Fn, Fn-1 . n=4 (53) :

x4 = 3x + 2. (54)

(54) , . , (54) , . , (54) : ! .

, (52) (53). , .

6.3.

-, (47).

(47) . , :

. (55)

(47), (55), -:

(56)

, [13], , -:

,(57)

, =0 (57) :

(58)

[29], (58) , ( ) .

, =1 (57) :

(59)

, (59) [29], .

, (57) , [13] . , (=0) (=1).

6.4. -

[10] , . , >0 - Fp(n) (n=0, 1, 2, 3, ) (46) :

Fp (n) = k1(x1)n + k2(x2)n + + kp+1(x+1) n (60)

x1, x2, , xp+1 (46), k1, k2, , kp+1 , :

Fp (0) = k1 + k2 + + kp+1=0Fp (1) = k1x1 + k2x2 +...+ kp+1x+1=1Fp (2) = k1(x1)2 + k2(x2)2 + + kp+1(x+1) 2=1......................................................................Fp () = k1(x1) + k2(x2) + + kp+1(x+1) =1

(61)

, =1 - Fn (60) [2, 3] :

Fn = k1(x1)n + k2(x2)n, (62)

x1, x2 (1), :

x1 = ; x2 = ,(63)

k1, k2 :

F0 = k1 + k2 = 0F1) = k1 + k2(-1/) = 1

(64)

(64) k1 k2:

k1 = ; k2 = .(65)

(63), (64) (62), :

.

(66)

[2, 3], Ln x1, x2 :

(67)

[10] , n- x1, x2, , xp+1 (46) , [10] - Lp(n), =0, 1, 2, 3,..., ,

Lp (n) = (x1)n + (x2)n + + (x+1)n (68)

(46), (48)-(51) , , (68), :

Lp (n) = Lp (n-1) + Lp (n--1) (69)

:

Lp (0) = +1 (70)

Lp (1) = Lp (2) =... = Lp () = 1. (71)

. 5 - =1, 2, 3, 4.

5. -

n 0 1 2 3 4 5 6 7 8 9 10 11 12L1(n) 2 1 3 4 7 11 18 29 47 76 123 199 322L2(n) 3 1 1 4 5 6 10 15 21 31 46 67 98L3(n) 4 1 1 1 5 6 7 8 13 19 26 34 47L4(n) 5 1 1 1 1 6 7 8 9 10 16 23 31

, [10] , , (68)-(71), , , .

, (13)-(16) (18)-(21) (9)-(10) . [10] (60), (68), - - (46), - - , . [11]. , , (13)-(16) (18)-(21), , [11] , , , - .

6.5.

. , Q- [2]. 2 2 :

(72)

, Q- -1, ,

Det Q = -1. (73)

Q- ? , Q- n- . :

(74)

Fn-1, Fn, Fn+1 . (73), , (74) :

Det Qn = (-1)n, (75)

n . , (74) :

Det Qn = Fn-1Fn+1 = (-1)n. (76)

, (76), , 17- ; (76) [26]. , Q- , (76), Q-, (75), !

Q- (72) . [12] , Qp-:

(77)

p : 0, 1, 2, 3, ., Qp- (p+1) (p+1)-.

p p , , , , , . p = 0, 1, 2, 3, 4 Qp- :

Q0 = (1); ; ;

; .

[12] Qp-, n:

(78)

=0, 1, 2, 3, , n = 0, 1, 2, 3, , - , (41) (42).

[12] , (78) :

Det = (-1)pn, (79)

p = 0, 1, 2, 3, ; n = 0, 1, 2, 3, . , [12] ,

: (79) 1, p n ( =0, 1, 2, 3, , n = 0, 1, 2, 3). , (78) +1, -1.

, (77), (78) . (79)

(76). , =2 :

Det = F2(n+1)[F2(n-2) F2(n-2)- F2(n-1)F2(n-3)] +

+F2(n)[F2(n)F2(n-3)- F2(n-1)F2(n-2)] +

+ F2(n-1)[F2(n-1) F2(n-1)- F2(n)F2(n-2)] = 1.

(80)

(78), , (80) 2- F2(n-3), F2(n-2), F2(n-1), F2(n) F2(n+1) n (n = 0, 1, 2, 3, ).

, , (80) (79), , , , =1, 2, 3,....

6.6.

[13] . (74) , (n=2k) (n=2k+1) n:

(81)

(82)

(22), (81), (82) (18), (19):

(83)

(84)

k , k=0, 1, 2, 3, . k (83), (84)

x, , x:

(85)

(86)

, (85), (86) Q- (74) .

. , (85) :

(87)

, Q-, (87)!

(85), (86) , (24), (25).

(85), (86), (24) (25), (85), (86), x:

Det Q2x = 1 (88)

Det Q2x+1 = - 1 (89)

, (75) (76), , (88), (89) , (75) !

, (78), (85), (86), (79), (88), (89). [13, 24, 25].

7.

, [4-25] . :

1. . , (12), (13), . , ( , .), , . , ( ), . , , , , , , , , .. , , ( ). , [4-7], , -, , .

2. . , , . Hyperbolic Universes with a Horned Topology and the CMB Anisotropy [28] , , .

3. , - , -, - - , , - , , , , ,

. , , .

. , - , . , , , . , -, 20- , , . [30-38] -. , (Shechtman), (Mauldin), (William), (El Naschie), , . . [36], , . , - . ( ) [38] : , , , .

1. .. . .: , 1978. 144 .2. Hoggat V.E. Fibonacci and Lucas Numbers. - Palo Alto, California: Houghton-Mifflin, 1969. 92 .3. Vajda S. Fibonacci & Lucas Numbers, and the Golden Section Theory and Applications. Ellis

Horwood limited, 1989. 190 .4. .., .. // .

. 1993, . 7. . 9-14.5. Stakhov A.P. Hyperbolic Fibonacci and Lucas Functions: A New Mathematics for the Living

Nature. Vinnitsa: ITI, 2003. ISBN 966-8432-07-X. 240 .6. Stakhov A., Rozin B. On a new class of hyperbolic function // Chaos, Solitons & Fractals. 2004,

23(2). P. 379-389.7. Stakhov A., Rozin B. The Golden Shofar // Chaos, Solitons & Fractals. - 2005, 26(3). P. 677-684.8. .. . .: , 1977.

288 c.9. Stakhov A., Rozin B. The golden algebraic equations // Chaos, Solitons & Fractals. 2005, 27

(5). P. 1415-1421.10. Stakhov A., Rozin B. Theory of Binet formulas for Fibonacci and Lucas p-numbers // Chaos,

Solitons & Fractals, 2005, 27 (5). P. 1162-1177.11. Stakhov A., Rozin B. The continuous functions for the Fibonacci and Lucas p-numbers.

Chaos, Solitons & Fractals, 2006, 28 (4). P. 1014-1025.12. Stakhov A.P. A generalization of the Fibonacci Q-matrix // . , 1999, No 9.

. 46-49.13. Stakhov A. The Generalized Principle of the Golden Section and its Applications in Mathematics,

Science and Engineering // Chaos, Solitons & Fractals, 2005, 26 (2). P. 263-289.14. .. . .: , 1979. 64 c.15. Stakhov A.P. The Golden Section in the measurement theory // An International Journal

Computers & Mathematics with Applications, 1989, Vol.17, No 4-6. P. 613-638.16. A..

// , 2004, 56, 8. . 1143-1150.17. Stakhov A.P. The Golden Section and Modern Harmony Mathematics // Applications of

Fibonacci Numbers, 1998, Vol. 7. P. 393-399.18. .. // .:

, 77-6567, .11176, 26.04.2004.

19. .. // .: , 77-6567, .12371, 19.08.2005

20. Stakhov A. Fundamentals of a new kind of Mathematics based on the Golden Section // Chaos, Solitons & Fractals, 2005, 27 (5). P. 1124-1146.

21. A.. . .: , 1984. 152 c.22. Stakhov A.P. Brousentsovs ternary principle, Bergmans number system and ternary mirror-

symmetrical arithmetic // The Computer Journal, 2002, Vol. 45, No. 2. P. 222-236.23. .. ,

- // .: , 77-6567, .12355, 15.08.2005

24. Stakhov A., Massingue V., Sluchenkova A.A. Introduction into Fibonacci coding and cryptography. Kharkov: Publisher Osnova, 1999. ISBN 5-7768-0638-0. 236 p.

25. .. , // , , . : - , 2003. C. 311-325.

26. .. // .: , 77-6567, .12542, 01.11.2005.

27. .. . : , 1994. 204 .

28. Aurich R., Lusting S., Steiner F., Then H. Hyperbolic Universes with a Horned Topology and the CMB Anisotropy // Classical Quantum Gravity, 2004, No. 21. P. 4901-4926.

29. .. . : 2000. 352 c.30. . // , 1988, . 156. C. 347-363.31. .. //

, 1978, 7. . 475-500.32. Mauldin R.D. and Willams S.C. Random recursive construction // Trans. Am. Math. Soc., 1986,

295. C. 325-346.33. El Naschie M.S. On dimensions of Cantor set related systems // Chaos, Solitons & Fractals,

1993, 3. P. 675-68534. El Nashie M.S. Quantum mechanics and the possibility of a Cantorian space-time // Chaos,

Solitons & Fractals, 1992, 1. P. 485-48735. El Nashie M.S. Is Quantum Space a Random Cantor Set with a Golden Mean Dimension at the

Core? // Chaos, Solitons & Fractals, 1994, 4(2). P. 177-17936. El Naschie M.S. Elementary number theory in superstrings, loop quantum mechanics, twistors

and E-infinity high energy physics // Chaos, Solitons & Fractals, 2006 27(2). P. 297-330.37. ... , //

, , . : - , 2003. C. 69-79.

38. .. . M.: , 2002. 550 .

.., .. // , ., 77-6567, .12616, 22.11.2005

1. 2. , & 3. 4. 5. 6.