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I have found some types of spirals, and when I analysed those I have found, they do not met the criteria to shape the draw desired.

And a observation point, bacause I think spirograph its a wrong name for it, its because spirograph was a tool popularized launched in 1965, so that is that. I dont thinks this ir a fermat spiral (aka fermat coil) or a double archemeadean with polar cordinates.

The spirograph

https://postimg.cc/CnHtXGjz

The desire here is the formula do replicate on the paper, or the formula, to do it on illustrator or on a graphic calculator.

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    Sorry, maybe I'm dense.... what's the question? Math to create an image like your sample? Isn't that a math question, not a design question?
    – Scott
    Commented May 8 at 23:25
  • the question is to replicate this pattern, i will give a math a shot yeah, but I'm pretty sure that are people here that can answer this.
    – FrakTool
    Commented May 9 at 0:01
  • The way your worded your question just looks like a rant. Maybe edit it and ask it as a question? Check How to Ask if you need help with the format.
    – Luciano
    Commented May 9 at 8:45
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    You don't technically need to use any maths for something as simple as this. You could just rotate an ellipse repeatedly around a circle in Illustrator. see example. Is there some particular reason you need to plot it using a formula? You would likely need a plugin to do this in Illustrator, there's nothing built-in as far as I know.
    – Billy Kerr
    Commented May 9 at 9:38
  • 1
    Perhaps the word you are looking for is hypotrochoid
    – Billy Kerr
    Commented May 9 at 12:00

1 Answer 1

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Tie dye makers call it spirograph because it's a graphic pattern achieved by winding the cloth (before the chemical treatment) to something which resembles a spiral. In the video which shows how your example pattern has been made, the folding and winding method creates separate spirals which are placed to different angles around the center. The video: https://www.youtube.com/watch?v=ix8ialCxYgk The same looking spirals can be created with Ikscape's spiral tool by having maybe only one turn, but high divergence parameter, say =10 or more. Unfortunately only Inkscape developers know exactly the math meaning of the divergence parameter. Here https://lists.inkscape.org/hyperkitty/list/[email protected]/thread/CXYYV7HE6NLZV7XU6L7QEV2FJR7VCZXI/ a developer tells something of the math behind Inkscape's spiral tool. I guess programmers can read it.

The next image is a screenshot of a single turn Inkscape spiral with divergence=12

enter image description here

As long as used in tie dye making context, word spirograph is a perfectly valid name for the pattern made by using foldings and windings of the same type. I guess the artisans who use the method are not very interested in how term spirograph has been used in other circumstances.

As you already know, Spirograph is the commercial name of a drawing tool which can be used to draw curves which with proper pen guidance mechanism adjustments may create also something which resemble your tie dye pattern. Spirograph uses an old mechanical principle, so it's well possible that the principle itself can be called spirograph by some people. Strictly it's a registered business name, not a curve, mechanical principle nor a pattern. See this: https://en.wikipedia.org/wiki/Spirograph and this https://www.researchgate.net/publication/361278698_Flattening_the_Curve_of_Spirographs

At least in Inkscape you can easily write paramtric equations for planar curves. You can create the same as a spirograph device can draw if you can understand the math shown in the linked articles above. And much more, because you are not limited by a mechanism. If you bother to use a math program package you will have even more options.

To only draw a resembling pattern you can do as suggested - tile rotated copies of a curve. The curve can have a featureless simple stroke or it can be textured for richer looking results. Here's an example in Inkscape:

enter image description here

The original is in the left. An array of rotated copies is generated by using the "tile clones" function. Other vector drawing programs can do the same.

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