The Phi Power Series
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- Clinton McDowell
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1 The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at - Chad J What is Phi? Start with a equatio that relates the ratio of two umbers u ad v to their sum. u u+ v = (eq.. v u We will defie the umber Phi as the ratio u = (eq.. v How is Phi Calculated? Substitute eq.. ito eq.. ad we have + = (eq.. which gives = + (eq.. This yields the quadratic equatio = 0 (eq.. Solvig for eq.. we get roots that are both irratioal umbers. ( + 5 = (greater root ( 5 = (lesser root Phi is kow as the greater of the roots. There is othig that makes this root more special the lesser root. As we shall see this math leads us to a series of umbers where ay umbers i the series hold equal sigificace. Phi ad the Fiboacci Series Istead of usig the quadratic equatio to solve for eq.. we could have used a iterative method. I a iterative solutio we start with a iitial guess ad the keep substitutig the result back ito the equatio util it coverges to a umber. Settig the iitial guess to ψ = we substitute the result back ito eq... + = = (iteratio
2 + = = (iteratio + 5 = = (iteratio = = (iteratio I 4 iteratios we have ψ=.6 which is covergig to the value of.68. We could cotiue to iterate to get a more exact value but there is a patter. The deomiators of the fractios are [ 5 ] ad the umerators are [ 5 8 ]. Each umber i the series is the sum of the previous two. This is kow as the Fiboacci series. The first 0 umbers of the Fiboacci series are 0,,,,, 5, 8,,, 4. Each umber is the sum of the previous two. fib fib fib (eq. 4. = + Usig eq. 5 we ca defie ψ as the limit of the ratio of the th cosecutive Fiboacci umbers as approaches ifiity. fib = lim (eq. 4. fib If set =9 i eq. 6 we get ψ=4/ which is accurate to decimal places. Iversely, phi ca be used to calculate ay value i the Fiboacci series. fib = ( + + (eq The Phi Power Series If we multiply both sides of (eq.. by ψ we get: + = + This ca be rearraged to variatios = + (eq. 5. = (eq = (eq. 5. What we have is a expressio of a ifiite power series with base ψ. Let s try eq. 5. for =0, -, -. = + (=0, eq. 5.4
3 + (=- = + (=- Oe iterestig thig ca otice is that the two root of eq.. are actually umbers i the power series. We ca rearrage eq. 5.4 to show the relatioship betwee egative ad positive roots. = = We have a series of phi powers where the sum of the previous i the series yield the ext i the series. The values for =..- are give below. = = = = = Power Series Decompositio Usig the basic equatios we derived above we ca decompose a polyomial of phi powers ito a lower order terms as follows + secod ( + + = + third ( + + = + Oe simple result of this is that we ca produce values i the series based o values that are ot cosecutive. The example below shows how a doublig of a term ca lead to a value i the power series. = + (eq. 6. This ca be geeralized to iclude ay multiplier deoted as a. a = a + (eq Phi Frequecy Modulatio Whe frequecies are multiplied together they are be cosidered to be modulatig each other. Cosider the product of the sie ad cosie fuctios. si( α si( β = (cos( α β + cos( α β
4 We ca modulate frequecies i the phi ratio ad get ew frequecies i the phi series. We apply eq. 5. to cosolidate the frequecy terms. si( tsi( t = ( cos(( t + cos(( + t = ( cos( t + cos( t This ca be geeralized to iclude ay cosecutive frequecies i the phi series. si( si( = ( cos( + cos( (eq. 7. Essetially this meas that we have frequecies i the phi ratio ad we modulate them the results will be the ext higher ad lower frequecies i the power series. 7 Phi Wave Mixig Ay frequecies ca be mixed a o-liear medium. It is ofte useful to approximate the o-liear respose curve as a Taylor polyomial. I the example below we have the first 4 terms of the polyomial for a oliear fuctio f(x cetered aroud the umber a. f ( a f ( a f ( a + f ( a( x a + ( x a + ( x a The secod term has order ad is the liear term. The third term has order ad is the first oliear term. We ca approximate the mixig of phi frequecies with the secod order term while igorig the effects of the other terms. I the example below we calculate the secod order mixig of cosecutive phi frequecies. ( si( + si( si ( + si( si( + si ( [( cos( + cos( cos( + + ( cos( ] [ cos( cos( + cos( cos( ] Essetially we have a trasformatio of the followig frequecies i the series.,,,, [ ] [ ] 8 The Madelbrot Set The most popular example of a fractal is called the Madelbrot set. This fractal is geerated by a simple system that may have applicatios i wave mixig. I eq. 8. c is a complex umber ad z is a umber i a series that either coverges or diverges to ifiity. If the umber coverges it is cosidered to be i the Madelbrot set. z + = z + c (eq. 8.
5 This system resembles the system i sectio 7 that was used i mixig the waves with frequecies i the phi ratio. Breakig out this equatio ito complex umbers we have a alterate form. z + z = ( z + z i + c c i x y x y x+ ( x y x x y + y z z + c + (z z c i (eq. 8. y Although these systems look similar further research will be required to determie if such a coectio exists. 9 Refereces. S Fich, The Golde Mea, Mathsoft. Golde Ratio, Wikipedia
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