Dween
Dween
Joined: Jan 24, 2010
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February 27th, 2013 at 6:09:26 AM permalink

In the above animation, each circle's radius is 1/3 the size of the next larger circle. Relatively, they rotate at -3 times the speed of the next largest circle. There is a "pen" in the center of each circle, which draws as it moves.

The pink lines show the first few circles' paths, and the blue path is drawn by the (theoretically) smallest circle. It seems to behaves in a fractal-like manner.

Is there a name for this curve?
-Dween!
Ahigh
Ahigh
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February 27th, 2013 at 6:11:02 AM permalink
I think they are called spirograph.

http://en.wikipedia.org/wiki/Spirograph
Dween
Dween
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February 27th, 2013 at 6:22:43 AM permalink
I am aware of the Spirograph toy, and understand that there is a correlation. However, I am interested in the blue curve, which is created with an infinite number of circles embedded within one another, rotating in complex ways, unable to be replicated with a Spirograph.
-Dween!
Ahigh
Ahigh
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February 27th, 2013 at 6:35:16 AM permalink
I apologize. It was a quip/joke. I'm just trying to be like Buzzard. LOL. Sorry for the distraction and good luck finding your answer. I would be surprised if there is already a name for it though. Interesting stuff nevertheless.
DJTeddyBear
DJTeddyBear
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February 27th, 2013 at 10:06:55 AM permalink
Quote: Dween

I am interested in the blue curve, which is created with an infinite number of circles embedded within one another. . .


Firstly, it's not an infinate number of circles. I expect that if you add another circle, it's path will mimic the blue path unless you magnify the image.

I don't know what to call it but I do not think "fractals" is right. I'm leaning more towards "chaos" but don't think that's right either.


On a side note, count the circles. It took me several attempts to count them all, ans to match all of them to their paths.
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁 Note that the same could be said for Religion. I.E. Religion is nothing more than organized superstition. 🤗
Lexinger
Lexinger
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February 27th, 2013 at 2:56:20 PM permalink
Quote: Ahigh

I apologize. It was a quip/joke.


The original post is a joke? Why ask such a question here?

I think that this is a fractal with inbuilt feedback, as a theoretical reverse means to generate basic nuclear processes. Were foamy space-time to give rise to particles.
Those who can, do; those who can't, teach. But those who confuse the two... they wind up on the internet.
Dween
Dween
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February 27th, 2013 at 3:11:58 PM permalink
Ahigh: I thought you were perhaps joking, but at the same time you weren't far off. No harm, no foul! :)

DJ: In the picture, there are only 4 or 5 circles, but I discovered that it is an infinite summation that produces the curve, sort of like a Koch's snowflake. You only need about 7 iterations to get a decent looking curve, though.

I have recreated it using Flash, without the circles.

The blue circle is iteration 1.
The magenta diamond shape is iteration 2.
The red shape with the "bowling pins" near the compass points is iteration 3.
The black curve is iteration 7.

I have determined a Cartesian summation based on the polar coordinates. It is easier to express it in Cartesian calculations than in polar:
Define r0 := 2/3 the size of the radius of the large circle
Define r1 := 1/3 r0
Define r2 := 1/3 r1
Generalize: Define rn := 1/3 rn-1

iteration 1:
for all Θ = 0 to 2π,
x = cos(Θ)*r0
y = sin(Θ)*r0

iteration 2:
for all Θ = 0 to 2π,
x = cos(Θ)*r0 + cos(-3Θ)*r1
y = sin(Θ)*r0 + sin(-3Θ)*r1

iteration 3:
for all Θ = 0 to 2π,
x = cos(Θ)*r0 + cos(-3Θ)*r1 + cos(9Θ)*r2
y = sin(Θ)*r0 + sin(-3Θ)*r1 + sin(9Θ)*r2

Generalize: iteration n:
for all Θ = 0 to 2π, and for all m = 0 to n
x = Σ cos((-3)mΘ)*rm
y = Σ sin((-3)mΘ)*rm
Not the most elegant pseudo-code, but I hope it gets the point across.
-Dween!
Buzzard
Buzzard
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February 27th, 2013 at 3:20:17 PM permalink
SQUIGGLY ?
Shed not for her the bitter tear Nor give the heart to vain regret Tis but the casket that lies here, The gem that filled it Sparkles yet
thecesspit
thecesspit
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February 27th, 2013 at 3:43:27 PM permalink
Wikipedia suggests it is a form of Roulette (more specifically a sort of Centred Trochoid

Maybe?

Kind of neat.
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
kubikulann
kubikulann
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February 28th, 2013 at 5:40:40 AM permalink
Roulettes or trochoids are usually meant to account for only ONE circle (maybe two) attached to the master circle.
Here the idea is indeed that you go to infinity.

I don't know whether this can be technically considered a fractal, in the sense that fractal is supposed to include the notion of pattern reproduction at lower levels. But on the other hand it is certainly in the same family as the Koch flake, which is a fractal.

More semantic than useful, though. Did you investigate the properties of your curve? That's what is interesting!
Reperiet qui quaesiverit

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