Mthemticl Puzzle Sessions Cornell University, Spring 202 Φ: The Golden Rtio The golden rtio is the number Φ = + 5 2 represent this number is pronounced fee.).68033989. (The greek letter Φ used to Where does the number Φ come from? Suppose line is broken into two pieces, one of length nd the other of length b (so the totl length is + b), nd nd b re chosen in very specific wy: nd b re chosen so tht the rtio of + b to nd the rtio of to b re equl. b + b b It turns out tht if nd b stisfy this property so tht +b = then the rtios re equl to b the number Φ! It is clled the golden rtio becuse mong the ncient Greeks it ws thought tht this rtio is the most plesing to the eye. Try This! You cn verify tht if the two rtios re equl then b = Φ yourself with bit of creful lgebr. Let = nd use the qudrtic eqution to find the vlue of b tht mkes the two rtios equl. If you successfully worked out the vlue of b you should find b = Φ. The Golden Rectngle A rectngle is clled golden rectngle if the rtio of the sides of the rectngle is equl to Φ, like the one shown below. If the rtio of the sides is Φ = + 5 2 this is lso considered golden rectngle. (Think of turning the rectngle on its side.) Φ
Mthemticl Puzzle Sessions Cornell University, Spring 202 2 It is possible to split up golden rectngle so tht it contins smller golden rectngle. Tke the golden rectngle shown bove nd drw verticl line splitting it up into by squre nd rectngle. The resulting rectngle hs dimensions Φ by. Since Φ = /Φ, the rtio of the sides of the rectngle is /Φ, so the smller rectngle is lso golden. If the smller rectngle is then split into squre nd rectngle, the smller rectngle hs dimensions /Φ by /Φ. Since /Φ = /Φ 2, this is golden rectngle s well! This pttern continues; if the smllest golden rectngle is broken up into squre nd rectngle, the resulting rectngle will lwys be golden. /Φ /Φ 2 Φ /Φ The golden rtio nd golden rectngles re present in wide rry of rt nd rchitecture. The most fmous exmple of golden rectngle in rchitecture is the Prthenon of Ancient Greece. Also, if spirl is drwn inside of golden rectngle which hs been split up into squres nd smller golden rectngles so tht it crosses the corners of the smller squres nd rectngles inside, the result is the fmous golden spirl, which cn been see in rt nd nture.
Mthemticl Puzzle Sessions Cornell University, Spring 202 3... Φ: The Golden Rtio Reltionship to Fiboncci Numbers The Fiboncci numbers re defined recursively, mening the vlue of the n th Fiboncci number depends on the vlue of previous Fiboncci numbers. The n th Fiboncci number is denoted F n. The vlues of the Fiboncci numbers re: F =, F 2 = nd F n = F n + F n 2, for n = 3, 4, 5,.... For exmple, F 3 = F 3 + F 3 2 = F 2 + F = 2, nd F 4 = F 3 + F 2 = 3. The Elvis numbers from the puzzle with the elf running up the stirs re the sme s the Fiboncci numbers! To mke this cler think bout the very first step Elvis mkes when there re n steps: If he goes up only one stir in his first step, the number of wys to climb up the rest of them is E n, nd if he skips the second stir going stright to the third in his first step, then there re E n 2 wys to climb the remining stirs. Mke the connection! First, find the first 5 Fiboncci numbers. Then, write down the rtio of successive Fiboncci numbers F n+ F n for n =, 2,..., 4. Do you notice nything bout the rtios? Are the rtios very different in vlue? (Think bout it before you continue.) The interesting fct this is getting t is tht s n gets lrger, the rtio of successive Fiboncci numbers get closer nd closer to the vlue Φ! Notice, the rtios re very close to Φ even when only considering the first 5 Fiboncci numbers. The Fiboncci Spirl Since the rtios of successive Fiboncci numbers re close pproximtion to the golden rtio they cn be use to crete series of squres inside of rectngle, similr to breking up golden rectngle into squres nd rectngles, nd the spirl drwn through the corners of the squres is very close to the golden spirl. Mke your own!. On piece of grph pper, trce squre nd the squre bove it. 2. Trce 2 by 2 squre whose right edge is the left edge of the two by squres. 3. Trce 3 by 3 squre whose top edge shres the bottom edge of the 2 by 2 nd first by squre.
Mthemticl Puzzle Sessions Cornell University, Spring 202 4 4. Trce 5 by 5 squre whose left edge shres the right of the 3 by 3 nd by squres. 5. Trce n 8 by 8 squre whose bottom edge shres the top edge of the 5 by 5, second by one nd 2 by 2 squres. Do you see the pttern? Ech block will fit nicely long the edge of the two previous blocks since F n = F n + F n 2. It is necessry to keep rotting round s you dd blocks so the shred edge is the bottom, right, top nd then left edge of the new squre. Once the squres re drwn, strt spirlling out from the first squre you drew. It should look something like the bove picture when you re done. (You cn continue on up to whichever Fiboncci number you like once you get the hng of it!) The Golden Rtio nd the Fiboncci Numbers in Nture The golden rtio nd Fiboncci numbers cn be found in mny plces in nture. For exmple, leves wnt to be rrnged so tht lef is not blocked by the leves bove it, this wy ech lef hs the sme ccess to sunlight. In mny plnts, leves spirl round stem ccording to the golden rtio or Fiboncci numbers. Fiboncci himself ws interested in how quickly rbbits breed, nd under his simplified breeding model the number of rbbits present fter ech breeding seson were wht re now known s the Fiboncci numbers. This model ws not very ccurte since it ssumed the rbbits never die, however the Fiboncci numbers do describe the number of prents, grndprents, gret grndprents, etc of honey bees very well. Honey Bee Fmily Tree There is one queen in honey bee colony. The unfertilized eggs of the queen bee result in mle worker bees nd the fertilized eggs of the queen bee become femle worker bees, whom re usully sterile. A femle becomes reproductive queen only if chosen to be fed royl jelly. In ny cse, this mens tht mles hve on prent (the queen) nd femle bees hve two prents (the queen nd mle). Mke fmily tree which trces bck through the ncestry of mle worker bee nd note the number of ncestors in ech genertion. Do these numbers look fmilir? 2 3 Mle Femle
Mthemticl Puzzle Sessions Cornell University, Spring 202 5 Links to More Informtion These links will lso be posted on the Puzzle Sessions webpge: http://www.mth.cornell.edu/community/community.html Rtios: http://cs.gmu.edu/cne/modules/du/lgebr/frctions/frc5 frm.html Qudrtic Equtions: http://www.enchntedlerning.com/mth/lgebr/qudrtic Ptterns nd the Golden Rtio: http://www.mthsisfun.com/numbers/golden-rtio.html (Specificlly, check out the Clculting It nd Most Irrtionl sections.) http://www.miqel.com/frctls mth ptterns/visul-mth-phi-golden.html Short Video: Donld Duck in Mthemgic Lnd http://www.youtube.com/wtch?v=yvodhfle0mw The Golden Rtio nd Fiboncci Numbers in Nture: http://www.mthsisfun.com/numbers/nture-golden-rtio-fiboncci.html http://www.miqel.com/frctls mth ptterns/visul-mth-phi-golden.html Fiboncci s Rbbits: http://scrmble.hubpges.com/hub/fiboncci-numbers-golden-rtio-in-nture