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When you say the real thing, you must be referring to the Golden Spiral?
No, the so-called "real thing" is a true logarithmic spiral, but then the Fibonacci squares don't tile the plane.The "pure artwork" is *starting* with Fibonacci tiles (which is ok), and then drawing the spiral through the corners. That "spiral" is 100% not natural.
Fibonacci is an approx.
All done with this. Consider yourself led to water but I won't teach calculus for free so publicly.EDIT: I see you edited:Okay, you've caught up to the title of the post... :-)
Ah see why you're begging boggled. It's the damn English. "Golden" versus "Fibonacci" needs a clear distiction.That's why you need to look at the math, not the poetry. Read up on curvature and tell me if the so-called "fibonacci" spiral has a chance of showing up in nature or ordinary calculus.
MathematicsA Fibonacci spiral approximates the golden spiral using quarter-circle arcs inscribed in squares derived from the Fibonacci sequence.A golden spiral with initial radius 1 is the locus of points of polar coordinates {\displaystyle (r,\theta )}(r,\theta ) satisfying{\displaystyle r=\varphi ^{\theta {\frac {2}{\pi }}}\,}{\displaystyle r=\varphi ^{\theta {\frac {2}{\pi }}}\,}The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b:[8]{\displaystyle r=ae^{b\theta }\,}r = ae^{b\theta}\, <<<<<<<<<<<<<<<<< see the e!!!!!or{\displaystyle \theta ={\frac {1}{b}}\ln(r/a),}\theta = \frac{1}{b} \ln(r/a),with e being the base of natural logarithms, a being the initial radius of the spiral, and b such that when θ is a right angle (a quarter turn in either direction):{\displaystyle e^{b\theta _{\mathrm {right} }}\,=\varphi }e^{b\theta_\mathrm{right}}\, = \varphiTherefore, b is given by{\displaystyle b={\ln {\varphi } \over \theta _{\mathrm {right} }}.}b = {\ln{\varphi} \over \theta_\mathrm{right}}.The Lucas spiral approximates the golden spiral when its terms are large but not when they are small. 10 terms, from 2 to 76, are included.The numerical value of b depends on whether the right angle is measured as 90 degrees or as {\displaystyle \textstyle {\frac {\pi }{2}}}\textstyle\frac{\pi}{2} radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of {\displaystyle b}b (that is, b can also be the negative of this value):{\displaystyle |b|={\ln {\varphi } \over 90}\doteq 0.0053468\,}|b|={\ln {\varphi } \over 90}\doteq 0.0053468\, for θ in degrees;{\displaystyle |b|={\ln {\varphi } \over \pi /2}\doteq 0.3063489\,}|b|={\ln {\varphi } \over \pi /2}\doteq 0.3063489\, for θ in radians. OEIS: A212225An alternate formula for a logarithmic and golden spiral is:[9]{\displaystyle r=ac^{\theta }\,}r = ac^{\theta}\,where the constant c is given by:{\displaystyle c=e^{b}\,}c = e^b\,which for the golden spiral gives c values of:{\displaystyle c=\varphi ^{\frac {1}{90}}\doteq 1.0053611}c = \varphi ^ \frac{1}{90} \doteq 1.0053611if θ is measured in degrees, and{\displaystyle c=\varphi ^{\frac {2}{\pi }}\doteq 1.358456.}c = \varphi ^ \frac{2}{\pi} \doteq 1.358456. OEIS: A212224if θ is measured in radians.With respect to logarithmic spirals the golden spiral has the distinguishing property that for four collinear spiral points A, B, C, D belonging to arguments θ, θ + π, θ + 2π, θ + 3π the point C is the projective harmonic conjugate of B with respect to A, D, i.e. the cross ratio (A,D;B,C) has the singular value −1. The golden spiral is the only logarithmic spiral with (A,D;B,C) = (A,D;C,B).[\qoute]
When you say the real thing, you must be referring to the Golden Spiral?https://en.wikipedia.org/wiki/Golden_spiral
