Creative Cogitation

In the midst of the Binary Birds series I was con­tacted out of the blue by an artist out of  Florida.  She com­mented on my paint­ing Mother Night, which made me think about it some.  Since I am in the midst of my 100 Paint­ings projects I do not turn down any ideas that feed the muse.  So I revis­ited Mother Night this week­end while I was fin­ish­ing hoot­suite.

Both paint­ings had me in the night mode.  I am noc­tur­nal.  I love the night, the stars, the moon, and the gen­eral qui­etude.  I am mar­ried to a morn­ing per­son, so from about 8 PM to when­ever I decide to rest, I get a big block of unin­ter­rupted time; such a thing is pre­cious to all artists-it allows you to get deep into your zone.

Mother Night’s Embrace, by Jake Beck­man. An abstract dragon unfurls its starry wings, embrac­ing the moon. The neg­a­tive space of the night sky forms the watch­ful eye of Mother Night.

I have been work­ing in a square for­mat for the last cou­ple of paint­ings.  The NYC art show is com­ing up and the max­i­mum dimen­sions of the works allowed are 25 x 25 inches, or other vari­ant, so long as they do not occupy more than 50 x 25 inches.  25 x 25 is an odd size, but 24-inch square can­vases are easy enough to come by. I don’t usu­ally work in  a square for­mat; I pre­fer rec­tan­gu­lar can­vases which lend them­selves bet­ter to apply­ing the golden sec­tion. So I embarked on a new ver­sion of Mother Night work­ing in this alien square for­mat.  After cre­at­ing sev­eral sketches using my Wacom tablet in Pho­to­shop, the image of the dragon cre­at­ing neg­a­tive space in the sky sug­ges­tive of an eye started to emerge.  Once I had that idea, every­thing sort of fell into place.

I think it is a such a cool idea I may have to explore it fur­ther. I love the images of the night, the moon and stars.  I also have an affin­ity for drag­ons and other fly­ing crit­ters as well.  And the idea of using neg­a­tive space in this way also intrigues me some.

Mother Night’s Embrace clearly has noth­ing to do with the Binary Birds, but belongs to another series Abstract Con­cepts, so I have gone off the reser­va­tion for this one paint­ing, and I hope you like it.

If you would like to pur­chase Mother Night’s Embrace, please see its page on my web­site; you can pur­chase it using Pay­pal.  Thanks for stop­ping by.

If you are in the New York City area between Sep­tem­ber 16 and Sep­tem­ber 30, 2010 — you can see Mother Night’s Embrace and a work from the Binary Birds series, hoot­suite, at the Broad­way Gallery, 473 Broad­way in SOHO.  The Gallery is open Mon-Fri 10 AM-6 PM. The artists’ recep­tion is from 3–6 PM Sat­ur­day Sep­tem­ber 25, 2010.  Look­ing for­ward to see­ing you there. 

Jake

Artist, AKAJake.com Come Expe­ri­ence the Art!

The art work in this blog is fed­er­ally copy­righted. All repro­duc­tion and pub­lish­ing copy­rights are retained by the artist. Images are not to be copied, re-distributed, imi­tated, derived OR oth­er­wise used in any form with­out the explicit writ­ten per­mis­sion of the artist.

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!