How do I make spirograh diagrams using software?

Question

There are these "string math artworks" I believe they are called "spirographs" where people usually take a circle or other geometric shape and put in nails on the shape and then use yarn and weave the yarn in a particular way such as to make these artworks.

I am wanting to make similar diagrams using software instead of physically making them with nails and yarn. I found this one link where someone is using Tikz with Latex to generate a circle with nodes. This might work, but I don't know exactly how to use Latex and this Tikz (plug-in) software.

If someone can point me in the right direction that would be great. Thanks.

Here are some links:

https://tex.stackexchange.com/questions/415330/make-graph-where-nodes-and-connecting-edges-are-circles?newreg=03724c430452487a8f756a4c311bbdc3

https://www.youtube.com/watch?v=0K2YbYTyEqY

Answer 1

You can draw a regular polygon in GeoGebra Classic, then connect the vertices with line segments.

You can compute the vertices of a regular polygon in a spreadsheet using $$x=R \cos \frac {2 \pi k}n\\ y=R \sin \frac {2 \pi k}n$$ where $R$ is the distance from the origin to a vertex and $k$ runs from $1$ to $n$. You can then calculate as many points as you like on some of the diagonals. You can then plot the points as an $xy$ plot.

Answer 2

Here's a recreation of the figure from that youtube video. It's written in asympote. If you have one of the popular LaTeX distributions like MacTex or tex-live, then you already have asymptote!

/*  Set up the format of the figure.                                          */
import settings;
settings.outformat = "pdf";

/* Size of the output figure.                                                 */
size(256);

/*  Number of points along the circle.                                        */
int N_Points = 1000;

/*  Number of points to skip for a and b.                                     */
int a_skip = 4;
int b_skip = 3;

/*  ints to correspond to the current points on the circle.                   */
int a, b;

/*  The starting values for a and b. Set them with angles 0 and pi.           */
a = 0;
b = (int)(N_Points / 2);

/*  Loop over a and b and stop once the two values correspond to the same     *
 *  point on the circle. This occurs when a-b = 0 mod N_Points.               */
while ((a-b) % N_Points != 0)
{
    draw(expi(2*pi*a/N_Points) -- expi(2*pi*b/N_Points));
    a += a_skip;
    b += b_skip;
}

/*  Draw the outline of the circle.                                           */
draw(circle((0, 0), 1));

Saving this as spiral.asy, you can then run it with asy spiral.asy which will produce spiral.pdf.

Answer 3

The curves generated by a spirograph are epi- and hypo-cycloids, or more generally epi- and hypo-trochoids. They are obtained by rolling a circle around another, with commensurate radii to obtain closed curve. The drawing pen need not be on the circumference.

https://en.wikipedia.org/wiki/Epicycloid https://en.wikipedia.org/wiki/Epitrochoid

Answer 4

Horses for courses.

If I understand correctly this Wikipedia Spirograph article indicates that you are interested in creating various types of roulette curves. This implies that you want to create a curve that corresponds to a mathematical equation [e.g. $r^2 = k,$ or $x^2 + y^2 = k,$ or $r^2 = 2\cos(2\theta)$].

Under the assumption that you will always be able to supply the mathematical equation that corresponds to the image that you are trying to display, your query can be split into two questions:

• How do you create the image?

• What do you do with the image?

For insertion of an image into a pdf file that can be printed or stored (e.g. electronic math notebook), I recommend starting with LaTeX.

From this foundation, you have two choices. The first choice is to create an image using math software that does not dovetail with LaTeX. Then, convert the image into (for example) a jpeg file (i.e. *.jpg) and then issue the LaTeX commands to insert the graphics file into your pdf file.

Within LaTeX, the other choice is to find a graphics package that dovetails with LaTex - see CTAN. For specialized graphics packages you can search for "graphics" from the opening screen at CTAN. However, there are generally three all purpose Latex-plug-in graphics packages: PSTricks, Asymptote, and PGF-Tikz. All three of these packages are capable of creating mathematical graphs directly into your LaTeX generated pdf file.

All three of these all purpose Latex graphics packages have significant learning curves, and are (perhaps) oriented towards LaTeX users with different skills. What most LaTeX users (me included) do is to spend a brief amount of time exploring each one, including the software's gallery of samples, and documentation. Then, the user will typically choose only one of the three, and focus on mastering that particular package.

This begs the question of how to produce various mathematical images totally outside of Latex. I am largely ignorant in this area. In fact, I stumbled on this query simply trying to educate myself.

I decided to start by searching mathSE for all queries with the [math-software] tag. Once I complete that, I will then start with Wikipedia. So far, my two ground-zero Wikipedia articles are Plotting Software and Math Software.

Once I gather links by exhausting mathSE and my two ground zero links then I'll explore the corresponding packages. One key issue to consider is whether it is acceptable to use the application online through a browswer, or whether you wish to download an application for local use. A peripheral issue is compatibility with your operating System (i.e. Windows, Linux derivative, or Apple-Mac).

Another issue is usage. Do you wish to create animated graphics (e.g. *.png or *.gif files)? Further, are you going to collect the graphical images into a folder on your hard drive? Alternatively, do you simply want a method of quickly creating a temporary math image that you don't plan to store. Examples of temporary usage include helping you to understand a specific math problem and illustrating a mathSE query or answer with an embedded image.