Post by: STxAxTIC on November 06, 2020, 02:13:12 pm
The boxes are fine, as the Fibonacci sequence lends itself to a nice box tiling like that... but that spiral always looked off to me. It's made of quarter arcs that abruptly change radius at every vertical and horizontal line. In other words, that's not math, that's art... So in this post, I reinforce this fact.
In this first chunk of code, we have the boxes-with-artsy-spiral image. You can see I had to do some silly business to make the spiral, because it's not really a mathy object:
Code: QB64: [Select]
- TYPE Vector
- CompassCart.x = 1
- CompassCart.y = -1
- CompassTheta.x = pi
- CompassTheta.y = 3 * pi / 2
- Step1 = 0
- Step2 = 1
- LastPoint.x = 0
- LastPoint.y = 0
- CALL StepCompass
- StepTemp = Step2
- Step2 = Step2 + Step1
- Step1 = StepTemp
- x1 = LastPoint.x
- y1 = LastPoint.y
- LastPoint.x = x2
- LastPoint.y = y2
- SUB StepCompass
- xx = CompassCart.x
- yy = CompassCart.y
- CompassCart.x = -1
- CompassCart.y = 1
- CompassTheta.x = 0
- CompassTheta.y = pi / 2
- CompassCart.x = -1
- CompassCart.y = -1
- CompassTheta.x = pi / 2
- CompassTheta.y = pi
- CompassCart.x = 1
- CompassCart.y = -1
- CompassTheta.x = pi
- CompassTheta.y = 3 * pi / 2
- CompassCart.x = 1
- CompassCart.y = 1
- CompassTheta.x = 3 * pi / 2
- CompassTheta.y = 2 * pi
Now, contrast this to a TRUE logarithmic spiral. The Fibonacci boxes are clearly wrong:
Code: QB64: [Select]
- x0 = 0
- y0 = 0
- xp = x0
- yp = y0
- xd = 1
- yd = 1
- r = phi ^ (theta * 2 / pi)
- x = cartx(r, theta)
- y = carty(r, theta)
- xp = x
- x0 = xp
- y0 = yp
- xp = x
- xd = -1
- xp = x
- x0 = xp
- y0 = yp
- xp = x
- xd = 1
- yp = y
- x0 = xp
- y0 = yp
- yp = y
- yd = -1
- yp = y
- x0 = xp
- y0 = yp
- yp = y
- yd = 1
How to reconcile this? I know what I'm doing to do. Dismiss the artwork for what it is, there is NOTHING fundamental about the Fibonacci spiral.
</rant>
Post by: FellippeHeitor on November 07, 2020, 10:53:35 am
Thanks for sharing the real thing.
Post by: bplus on November 07, 2020, 11:05:11 am
https://en.wikipedia.org/wiki/Golden_spiral
Post by: STxAxTIC on November 07, 2020, 02:53:23 pm
Quote
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.
Post by: bplus on November 07, 2020, 04:01:04 pm
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.
No, the Golden Spiral IS logarithmic.
It is probably what you drew as this is Golden part:
Code: QB64: [Select]
phi - 1= 1/phi
Post by: STxAxTIC on November 07, 2020, 07:24:12 pm
I want to explain this to you properly. Do you know what curvature is? Yes or no.
Post by: bplus on November 07, 2020, 09:48:13 pm
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Post by: STxAxTIC on November 07, 2020, 09:49:08 pm
Anyway, in light of curvature, you can see these are way different beasts:
Post by: bplus on November 07, 2020, 09:49:54 pm
Fibonacci is an approx.
Post by: STxAxTIC on November 07, 2020, 09:51:14 pm
EDIT: I see you edited:
Quote
Fibonacci is an approx.
Okay, you've caught up to the title of the post... :-)
Post by: bplus on November 07, 2020, 10:00:13 pm
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Post by: STxAxTIC on November 07, 2020, 10:02:22 pm
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.
Post by: bplus on November 07, 2020, 10:02:35 pm
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... :-)
I was there in reply #2
Post by: bplus on November 07, 2020, 10:12:22 pm
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.
If you read the wiki you will see the equation from which your can see the e
Quote
Mathematics
A 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}}\, = \varphi
Therefore, 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: A212225
An 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 1.0053611
if θ is measured in degrees, and
{\displaystyle c=\varphi ^{\frac {2}{\pi }}\doteq 1.358456.}c = \varphi ^ \frac{2}{\pi} \doteq 1.358456. OEIS: A212224
if 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]
Post by: bplus on November 07, 2020, 10:26:21 pm
When you say the real thing, you must be referring to the Golden Spiral?
https://en.wikipedia.org/wiki/Golden_spiral
From reply #2 Have you caught up with me yet?
The Golden Spiral which YOU drew with phi = .5 * (1 + SQR(5)) IS a special case logarithmic spiral.
Post by: SpriggsySpriggs on November 07, 2020, 10:27:40 pm
Post by: bplus on November 07, 2020, 11:03:18 pm
I am talking about this:
Quote
Nope, you aren't paying enough attention to what I'm saying. The so-called golden spiral, aka Fibonacci spiral, is NOT a logarithmic curve.
It's not, it's a special case logarithmic spiral, the Fibonacci spiral is discrete (not calculus material) but the e in the equation for the Golden Spiral shows itself to be continuous thus subject to differential curve analysis ie 1st derivative tangents blah, blah, blah. It, the Golden Spiral, IS what he drew with the second bit of code using the PHI variable, I think ;-))
That was the meaning behind his little sketch here:
Your degree of smart-assness only makes things worse. I'll take your answer as a NO.https://www.qb64.org/forum/index.php?topic=3220.msg125004#msg125004
Anyway, in light of curvature, you can see these are way different beasts:
Since STx doesn't want to give calculus lessons here is one from 101:
The first derivative of e^x IS e^x what a beauty!