Creative Cogitation

Num­bers, there are many kinds of numbers:

• The Nat­ural Num­bers, 1, 2, 3… are count­ing num­bers that a child can fig­ure out with their fin­gers and toes etc.  Depend­ing on who is doing the def­i­n­i­tions the nat­ural num­bers may or may not include the num­ber 0,. Nat­ural num­bers do NOT include num­bers such as that have dig­its after the dec­i­mal place such as 2.25 , nor do the nat­ural num­bers include neg­a­tive numbers.
• Whole Num­bers are the Nat­ural Num­bers plus Zero plus the Neg­a­tive Coun­ter­parts of the Nat­ural Num­bers,  …-3,-2,-1,0,1,2,3…; Whole Num­bers are also called Inte­gers.  Again Integers/Natural Num­bers do NOT include num­bers with dig­its after the dec­i­mal point.
• Ratio­nal Num­bers are num­bers that can be expressed as a ratio of two Inte­gers (except­ing divi­sion by zero), there­fore are num­bers like 1/1, 2/3, –5/8 etc.  Ratio­nal Num­bers can include num­bers whose dec­i­mal equiv­a­lent ter­mi­nates such as –5/8 = –0.625, or num­bers whose dec­i­mal equiv­a­lent does not ter­mi­nate, but repeats instead such as 2/3=.66666…
• Irra­tional Num­bers are num­bers that can­not be expressed as a ratio of two Inte­gers, for exam­ple, the square root of two.  Irra­tional num­bers do not include num­bers involv­ing the Imag­i­nary Unit.
• Com­plex Num­bers are num­bers num­bers involv­ing the square root of neg­a­tive 1, also called the Imag­i­nary Unit, i.  Num­bers that are mul­ti­ples of the imag­i­nary unit are called Imag­i­nary Num­bers.
• Both Ratio­nal and Irra­tional num­bers can be Alge­braic.  An Alge­braic Num­ber is any num­ber that can be found as a root of poly­no­mial equa­tion with Inte­ger coef­fi­cients, such as the square root of 5, which would have the inte­ger coef­fi­cient 2, or neg­a­tive cube root of 10, which would have the inte­ger coef­fi­cient –3.   Num­bers like the square root of 2 are irra­tional num­bers, but they are alge­braic.  If n is the low­est pos­si­ble degree of such a poly­no­mial, the roots are alge­braic of order n. The square root of two is alge­braic in order 2, nev­er­the­less the square root of two is an Irra­tional Num­ber.
• Tran­scen­den­tal Num­bers are NOT Alge­braic.  The most famous tran­scen­den­tal num­bers are Pi, the ratio of a circle’s cir­cum­fer­ence to its diam­e­ter, and Euler’s Num­ber, which is the base of the nat­ural log­a­rithms among other things.
• Just to com­plete the pic­ture, Real Num­bers are num­bers that are NOT Com­plex Num­bers.  They can included Ratio­nal and Irra­tional Num­bers; they may be Alge­braic or Tran­scen­den­tal.  Real Num­bers can­not include num­bers involv­ing the Imag­i­nary Unit, but Com­plex num­bers can have Real Num­bers as com­po­nents.

Although some Tran­scen­den­tal Num­bers, such as Pi and Euler’s Num­ber are cool, they are not mys­ti­cal, nor are they par­tic­u­larly rare; tech­ni­cally almost all Real and Com­plex Num­bers are Tran­scen­den­tal, since the Alge­braic Num­bers are count­able, but the sets of Real and Com­plex Num­bers are uncount­able. All Tran­scen­den­tal Num­bers are Irra­tional, since all Ratio­nal Num­bers are Alge­braic, but NOT all Irra­tional Num­bers are Tran­scen­den­tal as some Irra­tional Num­bers are Alge­braic (such as the square root of 2).

Now that some of lan­guage of num­bers has been defined let’s con­sider the Golden Ratio, Phi. The Golden Ratio can be found in the pro­por­tions of the human body, the pro­por­tions of many other ani­mals, plants, DNA, the solar sys­tem, art and archi­tec­ture, music, pop­u­la­tion growth, the stock mar­ket.  It can derived math­e­mat­i­cally, geo­met­ri­cally, or via the Fibonacci Series.

The Golden Ratio: the ratio of the sum of the quan­ti­ties to the larger one equals the ratio of the larger one to the smaller one. Con­sider the whole length of some­thing, lets call this length, C, where C is com­posed of two smaller unequal sec­tions, A and B, such that C = A + B. Say B is the larger piece and A is the smaller piece. If C/B = B /A then the divi­sion of C into parts A and B is pro­por­tioned to the Golden Ratio.  This dimen­sion­ing con­sid­ered most pleas­ing to the human eye, prob­a­bly because these dimen­sions are found so fre­quently in nature. If (A+B)/B = B /A, then both equal Phi, the Golden Ratio.  Phi is approx­i­mately 1.6180339887498948482045868343…

Since the Golden Ratio, Phi, is the solu­tion to the equa­tion, x² - x-1 =0, which is Alge­braic in order 2, Phi is NOT a Tran­scen­den­tal Num­ber.

The Fibonacci Series is a series of num­bers with a recur­sive rela­tion­ship; it sim­ply expressed as the sum of the pre­ced­ing two num­bers in the series: start­ing at 0 and 1, then the next num­ber is 1=1+0, then num­ber after that is 2=1+1, then 3=2+1, then 5=3+2, then 8=5+3, etc.  The first 20 terms of the Fibonacci Series are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181…

The Fibonacci Series also fre­quently occurs in nature. We have 2 hands, 5 fin­gers, each fin­ger has 2 knuck­les and 3 segments-all Fibonacci num­bers. The num­ber of flat sur­faces on a banana — usu­ally 3 or 5 — again a Fibonacci Num­ber. Flower seed heads often have a cer­tain num­ber of spi­rals to pack seeds in such a man­ner that the seeds have the same amount of space-the num­ber of spi­rals is usu­ally a Fibonacci Num­ber. Plants also fre­quently arrange their leaves accord­ing to the Golden Sec­tion; If the entire cir­cum­fer­ence is pro­por­tional to 1.618 then the angle of rota­tion is usu­ally 0.618, which is the lesser of the sec­tions in the Golden Ratio 1.618 = 1/0.618. When the lesser sec­tion refers to an angle of rota­tion it is called the Golden Angle. The ten­dency of plants to wind them­selves using the Golden Angle and oth­er­wise arrange them­selves with Fibonacci num­bers is called Phyl­lotaxis; an esti­mated 90% of plants exhibit the ten­dency.    But not all plants fol­low phyl­lotaxis, some­times sweet pep­pers have 4 cham­bers instead of 3, some flow­ers have 4 petals, such as a fuch­sia,  or 6 petals, such as a cro­cus. Still it would seem that the Golden Ratio and Fibonacci num­bers must have a connection.

Indeed, the Golden Ratio, Phi, and the Fibonacci Series are inti­mately con­nected.  As the num­bers in the Fibonacci Series increase, the ratio of suc­ces­sive terms tends toward a cer­tain num­ber and that num­ber is the Golden Ratio; in math­e­mat­i­cal terms the limit of the ratio of suc­ces­sive terms in a Fibonacci Series as the terms go to infin­ity is the Golden Ratio. Lets see this in action: Keep­ing 3 dig­its accu­racy behind the dec­i­mal, the ratios of suc­ces­sive terms in the Fibonacci Series are  Unde­fined = 1/0 (divi­sion by zero is not per­mit­ted), 1.000 = 1/1, 2.000 = 2/1, 1.500 = 3/2, 1.667 = 5/3,  1.600 = 8/5,  1.625 = 13/8, 1.615 =21/13, 1.619 = 34/21, 1.618… 55/34, 1.618 = 89/55, 1.618 = 144/89 etc.  As you can see by the 9th iter­a­tion the ratios of suc­ces­sive terms of the Fibonacci series have con­verged to the Golden Ratio to 3 dig­its of accu­racy behind the decimal.

From an artist stand­point using 1.618 as an approx­i­ma­tion to the Golden Ratio should develop pleas­ing ratio in the scales most of us work in.  There­fore if you wish to sec­tion 36-inch-long can­vas into the golden ratio then 36/1.618 = 22.25 inches will be the larger dimen­sion and 36–22.25 =13.75 inches will be the smaller dimen­sion. Also  22.25/13.75 = 1.618 as it should.

Yours in art – Jake

Artist, AKAJake.com Come Expe­ri­ence the Art!