# How are the geometry shapes made?

I am trying for a quite of a time to understand how these shapes are created.

Seems this is not something that rotates around its center. I see 10 circles around, if I am not wrong, but how is this done?

Rotating circles is the way to go as per Julian's answer, however for the interlocking shape version with the five pointed star in the middle, you could use Illustrator's shape builder tool to merge the pieces.

This example was made in Inkscape which has a similar shape builder tool, also possible in Illustrator.

click to see larger

This can be done in Illustrator by drawing a simple circle with a stroke,
then choose `Effect > Distort & Transform > Transform` and adjust the settings.

Afterwards, if you need the actual paths, choose `Object > Expand Appearance`. Then you can use Shape Builder or Live Paint to fill areas between the strokes.

## These shapes appear to be modified hypotrochoids

Hypotrochoids are better known as the curves generated by a Spirograph. Note that although it can be hard to discern at a glance, a hypotrochoid is not a collection of overlapping ellipses. Instead, it's formed from one continuous curve. Some illustration softwares have spirograph tools, such as the spirograph extension for Inkscape.

Given a "ring" with radius R and a "gear" with radius r (to use the terms from the Inkscape documentation), if r/R is an irreducible fraction, then R determines the number of lobes, and r determines the interval between connected lobes. For example, with R = 10 and r = 9, there are 10 lobes, and every 9th lobe is connected.

The first shape appears to be a hand drawn pattern inspired by a hypotrochoid with ring radius 10, gear radius 9, and pen radius 0.6:

while the second shape appears to be a hypotrochoid with ring radius 10, gear radius 9, pen radius 0.7, and a few artistic embellishments:

• Upvote for Spirograph Sep 20, 2023 at 16:48
• Do they still make Sprirograph? Just checked Amazon, OMG, they do!!! Sep 23, 2023 at 16:09

You take a cricle and rotate it by 40 degree, around where you want the center to be. That gives you the base geometric shape. In your example some lines have been left out and some circles are added.

There's a lot of variations on this. You can rotate different shapes. Rotate at different degree (as long a devidable by 360).

You should find a lot of tutorials if you look for Arabic geometric patterns.

• Just to add, there's a fairly new radial repeat feature that'll just let you state how many times you want to repeat the shape and does the math for you. Relevant: helpx.adobe.com/ca/illustrator/using/…
– curious
Sep 18, 2023 at 23:12
• I only used the rotate tool for this so far. (alt click to set center, enter to open dialog, set degrees, copy and then cmd D to repeat). Both your new option @curious and Scott's are really cool! thanks for sharing Sep 19, 2023 at 10:19

I have not much to add. There's several well written answers of the subject. Except the side profile in this image was skipped:

I inserted the red lines which show that the projections do nor fit. The side projection is a manually crafted fake which does not present the same 3D item as the upper part. I bet some experienced Illustrator users here have tried to map some artwork on a revolved surface, but failed to get it right and stayed silent of that subject. To stay silent is no wonder because Illustrator's artwork mapping works so inconsistently that it can be difficult to make any difference between wrong settings, program inaccuracy and geometric impossibilities.

Both projections have several manually crafted details. The curves are split and the parts have got non-uniform widths. The side projection is drawn manually to get a pretty shape, but unfortunately the top edge at least belongs to a shape which doesn't have the rotational symmetry. To get more evidence I tried also to project a collection of N x 36 degrees rotated shapes (10 copies of a little distorted ellipse) on a revolved surface:

The left and right edges of my side profile differ radically from the given original. I wouldn't have anything to write about the projection if the original were something like this:

In that case mapping a shape on a revolved surface also would create something worth showing and it would be here before I had written a single line.

Added due a comment which says that my story is valid for an ortographic projection, but there exists also perspective projections which look different.

The commentator (Mentalist) is right. In a perspective projection, where the camera is near enough the 3D item, the dimples at the ends of the top edge could be invisible (behind the horizon).

But the original side view cannot be at the same time a perspective projection and present a rotationally symmetric piece, because there's 2 straight horizontal lines: The bottom edge and the line through the upper edge hilltops. In a perspective image at least one of them should be curved.

• Your red lines seem to show that the projections fit? Sep 19, 2023 at 16:20
• @justhalf the original side projection has highest points at the left and right edges. Properly drawn side projection has dimples at the right and left edges. Sep 19, 2023 at 16:34
• Just a general reminder to everyone, on the topic of projections “fitting”... this assumes that the projections are orthographic. If rendered from 3D software, camera settings may come into play. If you switch from an ortho cam to a perspective cam, then focal length will influence where the lines end up. (Just like with real life photography.) Sep 21, 2023 at 1:38