Phi in nature
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From: Jim Black on 8 Sep 2005 11:59 Jimmy wrote: > What is so special with Phi? What does it mean? > > > In an overwhelming number of plants, a given branch or leaf will grow out of > the stem approximately 137.5 degrees around the stem relative to the prior > branch. In other words, after a branch grows out of the plant, the plant > grows up some amount and then sends out another branch rotated 137.5 degrees > relative to the direction that the first branch grew out of. > > > All plants use a constant amount of rotation in this way, although not all > plants use 137.5 degrees. However, it is believed that the majority of all > plants make use of either the 137.5 degree rotation or a rotation very close > to it as the core number in their leaf or branch dispersion, sending out > each and every leaf or branch after rotating 137.5 degrees around the stem > relative to the prior branch. > > If we were to multiply the value of 1 over Phi to the second power > (0.3819659.) times the total number of degrees in a circle (360), we obtain > for a product nothing other than 137.50. degrees. As an alternate way to > look at the same idea, if we were to take the value of 1 over Phi > (0.6180339) and multiply it by 360, we obtain approximately 222.5 degrees. > If we then subtract 222.5 from 360 we again find 137.5 degrees - in other > words, the complimentary angle to 1 over Phi is 137.5 degrees, which also > happens to be the value of 1 over Phi to the second power times 360. > > So, if we have followed the described mathematics, it is clear that any > plant that employs a 137.5 degree rotation in the dispersion of its leaves > or branches is using a Phi value intrinsically in its very form. Considering > our discussion in the prior section on Phi in geometry, it is interesting to > note the off-balance five-pointed star that can be seen in the leaf pattern > of plants that employ a 137.5 rotation scheme when they are viewed from > above. Interesting indeed! One of John Baez's "This Week's Finds in Mathematical Physics" posts included an interesting discussion of the golden ratio: http://groups.google.com/group/sci.physics.research/msg/34abc79351261b27 One of the properties of Phi most relevant to your question is its position as the "most irrational number," i.e., the hardest to approximate with fractions. More precisely: John Baez wrote: > We can make this more precise. For any number x there's a constant > c(x) that says how hard it is to approximate x by rational numbers, > given by > > lim inf |x - p/q| = c(x)/q^2 > q -> infinity > > where q ranges over integers, and p is an integer chosen to minimize > |x - p/q|. This constant is as big as possible when x is the golden > ratio! The application here is that if the angle between the leaves is some fraction of 360 degrees, for example 360m/n, then after n leaves, one leaf will be directly over one another. By using the most irrational number times 360 degrees as the angle, the leaves of the plant don't block the sunlight to the leaves below any more than they have to.
From: Uncle Al on 8 Sep 2005 12:37 Jimmy wrote: > > What is so special with Phi? What does it mean? http://www.fuckinggoogleit.com/ > In an overwhelming number of plants, a given branch or leaf will grow out of > the stem approximately 137.5 degrees around the stem relative to the prior > branch. In other words, after a branch grows out of the plant, the plant > grows up some amount and then sends out another branch rotated 137.5 degrees > relative to the direction that the first branch grew out of. > > All plants use a constant amount of rotation in this way, although not all > plants use 137.5 degrees. However, it is believed that the majority of all > plants make use of either the 137.5 degree rotation or a rotation very close > to it as the core number in their leaf or branch dispersion, sending out > each and every leaf or branch after rotating 137.5 degrees around the stem > relative to the prior branch. > > If we were to multiply the value of 1 over Phi to the second power > (0.3819659.) times the total number of degrees in a circle (360), we obtain > for a product nothing other than 137.50. degrees. As an alternate way to > look at the same idea, if we were to take the value of 1 over Phi > (0.6180339) and multiply it by 360, we obtain approximately 222.5 degrees. > If we then subtract 222.5 from 360 we again find 137.5 degrees - in other > words, the complimentary angle to 1 over Phi is 137.5 degrees, which also > happens to be the value of 1 over Phi to the second power times 360. > > So, if we have followed the described mathematics, it is clear that any > plant that employs a 137.5 degree rotation in the dispersion of its leaves > or branches is using a Phi value intrinsically in its very form. Considering > our discussion in the prior section on Phi in geometry, it is interesting to > note the off-balance five-pointed star that can be seen in the leaf pattern > of plants that employ a 137.5 rotation scheme when they are viewed from > above. Makes for a nice sprinkler head, too, or a perforated speaker guard. Five-fold symmetry covering the plane is always of interest since a five-fold symmtric tesselation using identical tiles is impossible. Penrose tilings are fun (unless you are a quilted toilet paper manufacturer). -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf
From: Schoenfeld on 10 Sep 2005 07:01 Schoenfeld wrote: > Jimmy wrote: > > What is so special with Phi? What does it mean? > > It is the only number with the recurrence relation: > > phi^n = phi^(n-2) + phi^(n-1) > > That is why it is such a special number. The rest are details. CLARIFICATION: There are actually 2 numbers which satisfy this recurrence relation - both those numbers related to phi. PROOF: Suppose x was a number such that for all n, x^n = x^(n-1) + x^(n-2) Applying the reverse exponent law on RHS, = x^n/x + x^n/x^2 Dividing both sides by x^n, 1 = 1/x + 1/x^2 Rearranging to form a quadratic, 0 = x^2 - x - 1 Solving for x using the quadratic formula, x = [ 1 +/- SQRT(5) ] / 2 Giving two solutions x=phi and x=(-phi + 1)
From: Edward Green on 10 Sep 2005 20:49 Jim Black wrote: > One of John Baez's "This Week's Finds in Mathematical Physics" posts > included an interesting discussion of the golden ratio: > > http://groups.google.com/group/sci.physics.research/msg/34abc79351261b27 > > One of the properties of Phi most relevant to your question is its > position as the "most irrational number," i.e., the hardest to > approximate with fractions. More precisely: > > John Baez wrote: > > We can make this more precise. For any number x there's a constant > > c(x) that says how hard it is to approximate x by rational numbers, > > given by > > > > lim inf |x - p/q| = c(x)/q^2 > > q -> infinity > > > > where q ranges over integers, and p is an integer chosen to minimize > > |x - p/q|. This constant is as big as possible when x is the golden > > ratio! Ok. It's clear that as q -> infinity the approximation on the lhs should get better and better, so the absolute difference tends to zero. What's not so clear is why it should tend to zero in such a way that inf |x - p/q|q^2 tends to a non-zero limit. Oh... I see: the expression could be more explicitly written: lim inf inf |x - p/q| = c(x)/q^2 q -> oo q p That's a start. Then we have to call in John Baez and tell him to be infernally clever, or else.
From: Autymn D. C. on 11 Sep 2005 03:00
Jimmy wrote: > What is so special with Phi? What does it mean? > > > In an overwhelming number of plants, a given branch or leaf will grow out of > the stem approximately 137.5 degrees around the stem relative to the prior > branch. In other words, after a branch grows out of the plant, the plant > grows up some amount and then sends out another branch rotated 137.5 degrees > relative to the direction that the first branch grew out of. > > > All plants use a constant amount of rotation in this way, although not all > plants use 137.5 degrees. However, it is believed that the majority of all > plants make use of either the 137.5 degree rotation or a rotation very close > to it as the core number in their leaf or branch dispersion, sending out > each and every leaf or branch after rotating 137.5 degrees around the stem > relative to the prior branch. > > If we were to multiply the value of 1 over Phi to the second power > (0.3819659.) times the total number of degrees in a circle (360), we obtain > for a product nothing other than 137.50. degrees. As an alternate way to > look at the same idea, if we were to take the value of 1 over Phi > (0.6180339) and multiply it by 360, we obtain approximately 222.5 degrees. > If we then subtract 222.5 from 360 we again find 137.5 degrees - in other > words, the complimentary angle to 1 over Phi is 137.5 degrees, which also > happens to be the value of 1 over Phi to the second power times 360. > > So, if we have followed the described mathematics, it is clear that any > plant that employs a 137.5 degree rotation in the dispersion of its leaves > or branches is using a Phi value intrinsically in its very form. Considering > our discussion in the prior section on Phi in geometry, it is interesting to > note the off-balance five-pointed star that can be seen in the leaf pattern > of plants that employ a 137.5 rotation scheme when they are viewed from > above. No, E=1/e=((I-1)/I)^I is a better number. With it, you and laptop batteries can live forever by escaping death. Take /that/, Winter and Einstein! -Aut <br><div align=center font="face:Apple Symbols;Symbol"><font size=+3 color=white>☍</font><font size=+2 color=grey>♎</font><font size=+2 color=black>⑉</font><font face=Times New Roman color=black>0·36787944117144··1∎</font><font size=+2 color=grey>☥</font><font size=+3 color=white>☠</font></div><div align=center font="face:Times New Roman;size:12pt">"<i>E</i>=1/<i>e</i>=((<font color=white>∞</font>-1)/<font color=white>∞</font>)^<font color=white>∞</font>, fall from grace in infinite space, to be caught in infinite fraught."</div> |