Creative Cogitation

One job that I really stink at and I think I could improve upon is mar­ket­ing me.  This blog is an exten­sion of that effort, at least as far as adding vis­i­ble, search-able, timely con­tent online.

But what have I learned over the years?

No. 1 — Always be ready with the 30-second com­mer­cial.  When­ever you meet new peo­ple you need to be able to con­vey who you are and what you do in 30 sec­onds or less.  Prac­tice it in front of the mir­ror and get it nailed down.

• Mine depends on whether I am mar­ket­ing my day job as a knowl­edge­able small busi­ness tax accoun­tant, OR my acrylic abstract rep­re­sen­ta­tional art & mag­i­cal real­ism to my prospec­tive clients.  I don’t bring up both unless it hap­pens to come up in sub­se­quent conversation.
• If price point is an issue then I also have tai­lored com­men­tary aimed at sell­ing trin­kets, such as greet­ing cards, refrig­er­a­tor mag­nets and greet­ing cards based on my orig­i­nal art designs as well.
• The point is to know your audi­ence and what mes­sage you are try­ing to con­vey to them before you open your mouth and be pre­pared with a short spiel that addresses this concern.
• Toward this end look for cues from you cus­tomers about what they are inter­ested in — fol­low their eyes, ask them what they want, then lis­ten to their answers and get them engaged in conversation.

No. 2 — He who quotes price first loses.  When it comes to sell­ing my art I try to get my con­sumers to tell me what they think my art is worth rather than the other way around.  For trin­kets though I am will­ing to lose as my time also has value.

No. 3 — Always have mar­ket­ing mate­ri­als on hand.  I usu­ally carry around glossy busi­ness cards with an image of my art on the front and my con­tact infor­ma­tion on the back. (I have account­ing busi­ness cards too…) On the back of my art busi­ness cards, I have left a space on the back so I can make notes for my prospects on the card I give them if required.  I have sev­eral ver­sions and I usu­ally let my poten­tial patrons select from among my jew­els.  It is also a great way to find out which of my images is more pop­u­lar at the moment as well.

• In addi­tion to the busi­ness card, I also carry post­cards and full-sized glossy brochures around with me too. Whether you actu­ally need these kinds of mate­ri­als is another ques­tion, but I do think that busi­ness cards are a min­i­mum require­ment.
• When­ever I sell art, a print, or my trin­kets, etc. the buyer always gets another busi­ness card.  If I sell some­thing major I will likely throw in a “gift” of a post­card too.
• Mar­ket­ing mate­ri­als should be up-to-date.  So rather than get­ting 1000 of a sin­gle card, which may be dated in a short period of time, spend a lit­tle bit extra on the upload fees and buy lesser amounts.
• I use Adobe Pho­to­shop and a dig­i­tal cam­era to cre­ate my mar­ket­ing mate­ri­als, I have found that the time and money I spent on get­ting good pho­tographs and pro­duc­ing good mar­ket­ing mate­ri­als is worth the effort.
  • Dig­i­tal cam­eras facil­i­tate the trans­fer of images directly to your com­puter; you won’t have to scan them in. When I pur­chased my high-resolution dig­i­tal SLR cam­era — it paid for itself with the first 20 images I took.  I have used it to cre­ate hun­dreds of images since then.
  • If you are a do-it-yourselfer, then you will want to learn how to take good pic­tures, learn basic image manip­u­la­tion skills such as resiz­ing, set­ting up dif­fer­ent file types (.tif, .jpg, .gif), and learn how to place text on your mar­ket­ing images.

I have no affil­i­a­tion with Adobe, I just hap­pen to have used their prod­ucts for more than a decade.

• Most print­ers, online or oth­er­wise will sup­ply a tem­plate of what is required in order for them to print what you want.    Get these tem­plates and use them to help you with your lay­out.  I use VistaPrint.com, which will allow you to upload images & place text for cus­tomiz­ing your brochure or what­ever.  I have never designed a prod­uct online at VistaPrint.com, but merely have uploaded my com­pleted designs so I don’t know how well the online stuff works.  I have no affil­i­a­tion with VistaPrint.com; I have just used their ser­vices for sev­eral years.  I have found them rea­son­ably priced and sat­is­fac­tory; YMMV.
• You may print your own busi­ness cards etc to good effect, but most home-printed jobs do lack the appeal of pro­fes­sion­ally printed mate­ri­als.  Of course I under­stand that you may be oper­at­ing on a lim­ited bud­get, but if pos­si­ble splurge on the glossy, full-color, card stock materials.

No. 4 — Know when to shut up.  Observe shop­pers and learn the cues about when they are ready to buy and then let it hap­pen.  If some­one comes to you as a refer­ral and already has an object in mind the sale may already be made, but you can kill it by talk­ing the buyer to death.

No. 5 — The cus­tomer is not always right.  20% of your cus­tomers will take up 80% of your time.  If an eighty-percenter is also extremely budget-conscious you really should eval­u­ate whether you want to do busi­ness with them.  Often times these eighty-percenters will be the same buy­ers who will cre­ate headaches after the sale, express­ing con­tin­u­ing dis­sat­is­fac­tion with their pur­chase.  On tip-off is a shop­per who does not know what they really want.  Be on the look­out for high-maintenance clien­tele and then fig­ure out a way to diplo­mat­i­cally tell them to shove off.

To have a web­site or not? I have one and I think they are a good idea for any busi­ness.  Some­times peo­ple will just want to be able to go some­where and find out more about you.  If you do not have an online pres­ence then you may be miss­ing out on would-be con­tacts who may shop else­where if they per­ceive they have more knowl­edge about your com­peti­tors than you.

• There are venues where you can post your cur­rent designs and talk about your­self and have a link­able online address, such as MySpace.com, so there is no require­ment to actu­ally pay for a web­site if you are will­ing to work with what is out there.  I am not affil­i­ated with MySpace.com, but I do have a MySpace.com page as jakebeckman.
• But for a lit­tle bit of money you can cre­ate a 1 to 5 page web­site at many places for a rea­son­able price – many of these ser­vices have prepack­aged designs you can just plug your mate­r­ial into.  I don’t do that, but I do under­stand that many peo­ple do not pos­sess the skill set to
• If you want to have a domain all your own then I rec­om­mend get­ting it sep­a­rately from some­place like GoDaddy.com then point­ing it to your web­site – why? Because some unscrupu­lous “free” web­hosts will gladly set up a domain for you, but if you decide to change web­hosts they may decide not to release your domain to you and any brand­ing you may have devel­oped with your domain goes out the win­dow.  I am not affil­i­ated with Godaddy.com but I have sev­eral domains listed with them.

Those are a few of the things I have learned over the years. Now it’s your turn to speak up. What have you learned? I would love to hear some other ideas about how you pro­mote your­self as an artist.

Yours in Art,

-Jake

Artist, AKAJake.com Come Expe­ri­ence the Art!

PS. I am still look­ing for Spon­sors & Con­tribut­ing Patrons to help me pay the esti­mated $8000 it is going to cost me to attend this event.  Every lit­tle bit helps.

I have been called by other artists trav­el­ing to the Bien­nial & asked about hotels in Flo­rence, ship­ping etc.  Since I have done my research I am post­ing the infor­ma­tion here.  Given the sheer num­ber of hotels in down­town Flo­rence I get the feel­ing it has gone the way of Venice.  Peo­ple don’t live there, they only work there.  Most of the occu­pants of the city on any given day are prob­a­bly tourists these days.  I am also post­ing my research on fly­ing as well. 

Although I could have booked my hotel through the hotel site itself, I opted for my travel agent instead.  Since I am an Amer­i­can Auto­mo­bile Asso­ci­a­tion (AAA) mem­ber so it makes sense for me to use them; I like the one-stop shop­ping thing: hotel, air­fare, trip insur­ance etc.   Your expe­ri­ence with regard to using an travel agent may vary.  But even with a travel agent I did my home­work first.  For exam­ple I really do not like trav­el­ing through most the major US hubs (Atlanta, JFK, DFW, Chicago etc.) if I can avoid it.  Also I could request that my con­nect­ing flights be in the same ter­mi­nal, get decent lay-over times, and book aisle seats. It is a chal­lenge per­form­ing all that via an air­line web­site; often times if you get the out-bound trip the way you want it, you often will find the in-bound side is a com­plete night­mare & book­ing each half sep­a­rately tends to really inflate the cost.   Given the sheer num­ber of hotels in Flo­rence, I did need to nar­row down the field some before book­ing a hotel as well.  For me price, loca­tion and ameni­ties, in that order, were important. 

Air­port Research:  Try Googling “<air­port name or city> air­port reviews”.  The fol­low­ing web­sites may be use­ful too.  

•  SleepingInAirports.net  is a hoot to read. 
• Also check out http://www.airlinequality.com/Airports/apt_forum.htm.

Air­line Research: con­sider http://www.airlinequality.com/Forum/seats.htm

Hotel Research: When I went to research hotels in Flo­rence I found the fol­low­ing sites useful:

• booking.com where you can put in the city, check-in/check-out dates, refine your search by star rat­ing, facil­i­ties, hotel type, hotel theme, and district.  
• nyceflorence.com which has a handy lit­tle map so you can get close to the what­ever land­mark you desire – in my case the Fortezza da Basso  in addi­tion to price, star rat­ing & check-in/check-out dates. 
• I used hotel reviews at Expedia.com, to val­i­date my impres­sions once I got some hotel ideas. 
• I hear that Utell prop­er­ties are often rec­om­mended by over­seas trav­el­ers accord­ing to my travel agent. 

I have quite acci­den­tally and  iron­i­cally set­tled on the Hotel Ari­zona (ironic because I am a res­i­dent of Ari­zona).  It is about 1 mile from the Fortezza.  I am not averse to walk­ing so this dis­tance does not seem extreme to me and besides Flo­rence has the rep­u­ta­tion of being a great walk­ing city.  Whether that holds true or not remains to be seen.  There were at least two other hotels which were a bit closer that I did con­sider: Hotel Basilea and Hotel Sem­pi­one but in the end Hotel Ari­zona won out in my research by a hair.  All of these hotels are 3-star hotels. For some that may be just a bit too high-end for oth­ers way too bud­get ori­ented, which is why your own research is required for validation. 

Ini­tially I had a lovely flight set up with Lufthansa, but some­thing hap­pened in the book­ing, it could not be fixed and the price had just about dou­bled oth­er­wise.  Now I am fly­ing Delta Air­lines & part­ners, which turned out to a lit­tle bit less expen­sive so I used the dif­fer­ence in price to buy trip insurance. 

Trip insur­ance: I decided this trip was too impor­tant not to have it.  It cov­ers many things includ­ing ben­e­fits relat­ing to lost or delayed lug­gage, replace­ment flights, hotels at con­nect­ing air­ports when flights are delayed can­cel, since I included my hotel, it has a replace­ment hotel ben­e­fit as well.  I am trav­el­ing in win­ter which may present some weather related chal­lenges.  Still, I con­sider trip insur­ance more of an incan­ta­tion to be per­formed to ensure a good travel expe­ri­ence.  You may or may not think it is nec­es­sary.  My insur­ance is through Access Amer­ica – I think I received a dis­count because of my AAA membership. 

For the record I do not have any busi­ness rela­tion­ship with any of the air­ports, hotels, air­lines, or web­sites men­tioned herein, nor do I have a busi­ness rela­tion­ship with AAA or Access Amer­ica. I am not specif­i­cally rec­om­mend­ing any air­line, hotel, or web­site as I have lit­tle to no expe­ri­ence with them.  I do enjoy my AAA mem­ber­ship, but as with all such things, you have to use it to get your money’s worth.  I can­not say whether Access Amer­ica trip insur­ance is good or bad as I have never filed a claim with them. 

Good luck in your trav­els and I hope to see you in Italy.

Yours in Art,

-Jake

Artist, AKAJake.com Come Expe­ri­ence the Art!

PS. I am still look­ing for Spon­sors & Con­tribut­ing Patrons to help me pay the esti­mated $8000 it is going to cost me to attend this event.  Every lit­tle bit helps.

In as much as I am decon­struct­ing my can­vases in hopes of get­ting the bag length + height + width under 62 inches so I don’t incur the over-sized bag fee, I have been research­ing pos­si­ble con­tain­ers for the art.  Since my stretcher bars are over-sized the pack­ag­ing is not going to be carry-on because of the length issue-I am avoid­ing the use of card­board, pressed-paper type tubes as I do not think these will be durable enough to sur­vive the bag­gage mon­keys.  Since I have done the research into long hard sided bags I am post­ing it.  Please remem­ber as you read this that I am not affil­i­ated with any of these ven­dors nor am I specif­i­cally endors­ing their prod­ucts. I have never used them. My first thought was sports bags, which quickly devolved to gun bags (golf bags are too long to meet the under 62 inch thing).  The hard sided gun bags I found that will accom­mo­date some­thing 36 inches long and still make the air­line bag­gage size limit:

• Plano Mold­ing Com­pany makes sev­eral gun bags that may fit the bill, Model 1437 is an alu­minum case, Model 10101 is plas­tic and Model 10303 is also plas­tic.  Go on line and enter Plano & the model num­ber to research prices.  I found them from $76 to $150 each. If you have some­thing slightly shorter they do have a really inex­pen­sive bag 153500 which has an inside dimen­sion of 35 inches which retails $31-$42.  I was bummed that it was just a wee bit to short for my needs. There are other plas­tic cases made by Plano which are still rea­son­able which have lengths up to about 53 inches, which would incur the over-sized bag fee but which the air­lines would still likely carry for you as checked bags. I found these bags at Ama­zon, Optic­s­Planet and many “case” deal­ers.  PlanoMolding.com is not easy to nav­i­gate although cer­tain search terms will take you to a spe­cific page that lists their gun cases (and archery cases).
• Pel­i­can Bag (casesbypelican.com/thepelicanstore.com/pelican-case.com etc.) makes one bag that may fit the bill, Model 1700 — the least expen­sive one I found was $155 with­out the foam pack­ing. Pel­i­can also makes a longer bag.  These are very tough water­proof, shock­proof, industrial-grade cases.
• Chicago Case (chicagocase.com) makes one Model 37135 — the least expen­sive one I found was $150.  Googling Chicago Case gun case will lead you to a plethora of legal actions/court cases involv­ing firearms — which makes look­ing for their bag at ven­dors bit tougher. Chicago Case also makes bags in other dimensions.
• Amer­i­c­ase makes a model 4012 which retails from $188. Amer­i­c­ase also makes bags in other dimensions.
• In addi­tion to these ven­dors who had bags that fit my specs there were oth­ers who had larger or smaller bags that didn’t suit my needs, but may suit yours.   skbcases.com, tzcase.com, hfc air­soft rifle cases etc.

Next up, tubu­lar stor­age:  I Googled things like art stor­age tube, art ship­ping tube, tele­scop­ing ship­ping tube, large diam­e­ter plas­tic ship­ping tube, some­times the results of a search sug­gest mod­i­fiers you can use to expand your search.  I found a lot of tubes for blue prints and wound up at some draft­ing sup­ply places, but they would likely work equally well.  Most of these tubes are 3″ in diam­e­ter.  It is some­what harder to find larger diam­e­ters and they do tend to be a lot more expen­sive than the smaller tubes — but I did find one 6″-diameter tele­scop­ing tube that was really cheap.  Please remem­ber I have not tried any of these prod­ucts, I don’t know the ven­dors etc.  This is just infor­ma­tion to give you an idea about where to look.

• Vil­lage Sup­plies (artstuff.net) has a tele­scop­ing tube with a 6 inch inte­rior diam­e­ter tele­scop­ing from 29 to 48 inches that retails for $17.
• I found a Plano Mold­ing expand­able tube designed for fish­ing rods that is at least 5 inches in diam­e­ter, and tele­scopes from 47 to 88 inches.  It retails locally for about $56, and I found it on Ama­zon too.
• Alvin Co. (alvinco.com) makes sev­eral prod­ucts, Alu­minum tubes retail­ing for $35  to $45, clear plas­tic (ice) tubes retail­ing for $12 to $16 of lengths up to 43 inches (these are clear plas­tic tubes so I was think­ing if I wind up with a soft sided bag they would be use­ful as the cus­toms would be able to see what’s inside with­out open­ing the tube), they also make a tele­scop­ing black plas­tic tube that goes from 27 to 50 inches retail­ing for $19.  In fact Alvin “makes” a lot more tubes — I don’t know if they buy them from Chart­pak and oth­ers and rela­bel them or if they have pur­chased the rights from Chart­pak and Axel Tube, but many of the tubes listed below also have “Alvin” equivalents. I even found a 6″ diam­e­ter tele­scop­ing tube 29 to 48 inches for $107.  Many inter­net retail­ers carry their products.
• Art­bin (art.artbin.com) makes make a tele­scop­ing black plas­tic tube that goes from 34 to 60 inches retail­ing for $47. Many inter­net retail­ers carry this product.
• Dick Blick (dickblick.com) makes clear tubes as well 25″ are $10, 37 inches are $11.
• portfolios-and-art-cases.com has a vari­ety of 6 and 8 inch diam­e­ter tubu­lar art lug­gage from 31 to 50 inches rang­ing from $90 to $175.
• Chart­pak has tubes that you can daisy chain together to make arbi­trary lengths. Chartpak’s Rapid­de­sign has a tele­scop­ing tube that goes from 26 to 43 inches with a 3.75 inte­rior diam­e­ter — watch out though many ven­dors are sell­ing the smaller 3″ diam­e­ter one.  I couldn’t find the tubes on their web­site but I did find their tubes at var­i­ous ven­dors. Recall­ing my com­ment about Alvin I won­der if they have picked up this line from Chartpak.
• Axel Tube also can be daisy chained together in foot long sec­tions 4″ diam­e­ter — the basic tube is $19 and the sec­tions are $8 each. Again it would seem that Alvin now has this brand as well.   I could not find the axle tube web­site; if you Google you wind up with either auto­mo­tive parts or the YouTube wannabe.
• Flam­beau (flambeaucases.com) makes a 3 ” diam­e­ter tele­scop­ing ship­ping tube that goes from 61 to 85 inches for those of you with really long stretcher bars. But were I you I might con­sider the fish­ing rod tube by Plano Mold­ing as it has a larger diameter.
• Displays2Go.com has sev­eral 12″ diam­e­ter wheeled graph­ics tubes  (and even some larger diam­e­ters), which will accom­mo­date art from 31″ to 71″ rang­ing in price from $124 to $230.  Again much of this stuff is over-sized accord­ing to the air­line, but it may pro­vide a means of pack­ag­ing your art with reusable air­line trans­port.  These are designed for road war­riors who travel from con­ven­tion to con­ven­tion.   I think they are a bit pricey for my needs, but you may find their price rea­son­able if you plan to use the bag over and over.

So that is what I have found. I hope you find this infor­ma­tion useful.

As for me, armed with the infor­ma­tion I had about gun cases I started ask­ing a few gun enthu­si­asts I knew if they had a gun case meet­ing my spec­i­fi­ca­tions.  It seems one of my friends actu­ally has a Plano case and he will let me bor­row it.

Another Artist talked to 2007 par­tic­i­pant Ana Bikic. She said that most peo­ple did not sell any­thing at that Bien­nale.  She dis­cussed the mat­ter with the Ital­ian Con­sulate in Miami and appar­ently did have bet­ter luck than I did with the Ital­ian Con­sulate in LA; the Miami con­sulate said that since the art was not for sale, it was an exhi­bi­tion etc, there would be no VAT — of course that was 2007 and this is 2009.  When she trav­eled, she flew with her art & she told US and Ital­ian cus­toms that the art was not for sale and had no prob­lems (at least that is what I under­stood of the email I received).

Of course if you are still plan­ning to ship your art, then other rules may apply.

Ana hung her art “tapes­try style” with­out stretch­ers, but does sug­gest using some­thing like Vel­cro to pin the raw edges back.  She said many artists did this in 2007.   She also sug­gested hav­ing mini par­ties at the show booths, as  in 07 the spon­sors did not mind cham­pagne being present at the booths.

Ana sug­gested some­thing to hand out; her pref­er­ence was cards.  An Aus­tralian Artist, Joy Engle­man, sug­gested bring­ing 1000 brochures.  Just because the Bien­nale is not a sell­ing oppor­tu­nity does not mean that it is not a mar­ket­ing oppor­tu­nity.  So get your kit together and pre­pare some­thing nice enough to hand out.

Well that is where I am at with respect to fly­ing with art etc.  I hope you are hav­ing a great day.

–Jake

Artist, AKAJake.com Come Expe­ri­ence the Art!

PS. I am still look­ing for Spon­sors & Con­tribut­ing Patrons to help me pay the esti­mated $8000 it is going to cost me to attend this event.  Every lit­tle bit helps.

This blog is about how to incor­po­rate the Golden Ratio into your com­po­si­tions quickly, with­out the use of a cal­cu­la­tor; after all who has time to fig­ure out how much 36 x 0.618 is when you are out there in the fresh air get­ting into the cre­ative groove.  The num­bered indexes cor­re­spond to plates in the com­pos­ite image below.

1. The gold boxes with dot­ted lines in the image below rep­re­sent cen­tered “golden can­vases” with height and width cor­re­spond­ing to the Golden ratio.  There is one ori­ented ver­ti­cally and one hor­i­zon­tally.  The dot­ted lines rep­re­sent the Golden mean, pos­si­ble places to divide the can­vas in a man­ner that is pleas­ing accord­ing to the Golden ratio (1 to 1.618…). This image has been cre­ated with the assis­tance of a cal­cu­la­tor and is pro­vided to show that the trick that fol­lows does work very well.
2. This trick illus­trated will work on any arbi­trar­ily dimen­sioned rec­tan­gu­lar can­vas.  You draw a line split­ting the can­vas in half and then in half again along the hor­i­zon­tal and ver­ti­cal dimen­sions.  You may want to use a light color and hue of the medium you are work­ing in so you don’t have to spend time eras­ing.  In the illus­tra­tion, the black lines rep­re­sent these divisions.
3. If you place a dot in the direct cen­ter of the four cen­ter boxes cre­ated by divid­ing the can­vas you will arrive at poten­tial places where the focal point should be.  The red dots rep­re­sent these pos­si­ble focal point locations.
4. In the illus­tra­tion below notice how close the dots are to the inter­sec­tions of our ver­ti­cally and hor­i­zon­tally ori­ented golden means.  No cal­cu­la­tor is required to incor­po­rate the Golden Mean into your com­po­si­tions. You really can just eye­ball it, fig­ure out where your focal points should be and pick one.  One of these loca­tions will result in a focal point that is most pleas­ing to the eye if the rest of the com­po­si­tion sup­ports it.
5. A Focal point is really the only thing that can flow out­ward off the com­po­si­tion, like a star.  In gen­eral the rest of the com­po­si­tion should draw you to the focal point.  Real­ize that lines inter­sect­ing where the poten­tial focal points will divide the can­vas accord­ing to the golden ratio; these are the bold blue lines in the illus­tra­tion below.  The light blue shad­ings rep­re­sent two pos­si­ble ways of divid­ing the can­vas, but you could just as eas­ily choose another one of the means.
6. The last image is a sim­ple illus­tra­tion based on the prin­ci­ples just discussed.

Illus­trated Les­son: How to Quickly Apply Golden Ratio to Composition

Yours in art – Jake

Artist, AKAJake.com Come Expe­ri­ence the Art!

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!