After finishing this tutorial try to take it one step further, and create a background for the rings like I did here. You could try out an abstract background by copy & enlarging the rings as full circles a couple of times using Object > Transform > Transform Each… and by applying different transparency modes and values for each circle. Maybe you can also import your creation into Photoshop like I did to add some extra effects and textures.
Draw a circle
Create the first ring, by drawing a simple circle. Select the Ellipse tool and click drag while holding down the Shift key. Give the circle no fill and a black stroke. I've used 10 pt, but depending on the size of your circle choose a width that feels the right proportion.
Go to Object > Path > Outline Stroke to convert the stroke into a fill.
With the ring still selected, go to the Appearance panel and click the Add New Stroke icon at the bottom (left) of the panel. Select white from the color menu and give the stroke a thickness that represents the gap between the rings. In the Stroke panel set the stroke aligning to the Outside. To see what you are doing, you can select another color first, set the appropriate width, and then select white again.
Select the Selection tool (black arrow) and make sure the circle is selected. Now hold down both the Shift + Alt/Option key and start dragging the circle a bit to the right to an estimated distance of the 2nd ring. Release the mouse first, then the keys.
Repeat this transformation, by hitting Cmd/Ctrl + D (Object > Transform > Transform Again).
With the Selection tool still selected, select the 1st ring and hold down the Option/Alt key. Start dragging down and to the left to the estimated position as shown in the image above. Make sure to release the mouse first and then the key.
With the 4th ring selected, duplicate this ring to the estimated position of the 5th ring, holding down both Shift + Alt/Option key. Again, release the mouse first and then the keys.
Repeat this transformation twice for the 6th and 7th ring, by hitting Cmd/Ctrl + D twice.
Horizontal Distribute Center
To make sure the rings are in the right position, select all 7 rings and select Horizontal Distribute Center from the Options bar at the top of your canvas. If you don't see the bar, go to Windows > Options.
Add color to each ring
Select each ring and give it its proper color. You can see the values I've used from the Swatches palette. Though, in my final illustration I decided to use linear gradients for each ring to create a little more depth.
Now we'll turn the white strokes around the rings into fills. Select all circles and go to Object > Path > Outline Stroke.
To create the right overlap for each ring, we need to divide all objects into separate segments, and then recolor certain segments and unite them together again into one object, each object representing one ring. So select all circles and select the Divide option from the Pathfinder panel.
Before we start recoloring certain segments, we first need to Ungroup the entire object of rings. This way we can select each segment separately, recolor it, and unite segments that are part of the same ring together. Go to Object > Ungroup or hit Cmd/Ctrl + Shift + G. Now we can select segments and recolor where needed. We do this easily by using the Eyedropper tool. First deselect the rings by clicking somewhere on an empty space on your canvas. Select the Eyedropper tool from the toolbox and hold down the Cmd/Ctrl key. Your cursor should change into the selection tool (black arrow). Now select a segment that needs to become part of the ring to correct the overlap effect (see image above as an example). Use my example illustration at the top (or further down below) in case you need help which ring needs to be above or below etc. Use (also) the Shift key to select multiple segments (in combo with the Cmd/Ctrl key). Once you have the proper selection, release the keys so you get the Eyedropper tool again and click on a segment that you want to extract the color from. So to correct the yellow ring, click on a yellow segment to color the selected segments where the orange ring overlaps the yellow ring as shown in the example above here. Do the same where the yellow ring overlaps the green one at the bottom.
Now we need to merge all the yellow segments together so we end up with one object for the yellow ring. Select all segments of the yellow ring. Do this by selecting one yellow segment, and then click the Select Similar Objects option from the Option bar (at the very right). Now select the Unite option from the Pathfinder panel, to unite these segments together into one object.
Remove segments to create the proper gaps
In the next step remove the segments around the yellow ring to create the proper gap between the rings. In the image above, I have selected the segments of the orange and green ring that touches the yellow ring. Hit the delete key to remove these segments.
Repeat steps for each ring
This recoloring, uniting of segments, and removing unnecessary segments to create the proper gap, are the steps we need to repeat for each ring. So we use the Eyedropper tool to extract a color, while holding down the Cmd/Ctrl key each time we select segments. We also hold down the Shift key after selecting the first segment, to select multiple segments at once.
Select Similar Objects, then Unite
Then we select the paths of the entire ring by selecting one segment and click the Select Similar Objects option from the Option bar, and we unite them together using the Unite option from the Pathfinder panel. After that we delete the segments of other rings touching the current ring, and we go to the next ring…
When you are all finished your end result should look like the above image.
Remove all unnecessary segments
Next is cleaning up the rest of the unnecessary paths that are there but not visible on a white background. While I fixed the overlap effect for each ring, I made sure that each ring is selectable as one object. This will help in selecting the unnecessary objects. First you select everything, by dragging a selection over all the rings using the Selection tool. Then hold down the Shift key and click the colored rings one by one to deselect them. You could also temporarily group the colored rings before you select everything, so you only have to click on one of the rings to deselect them all. After you're done you can then ungroup them. Or maybe you could select just one white segment, and click the Similar Objects button icon on the very right of the Option bar to select all white segments. Though, because of the path segment division, there are also segments that have no fill and stroke color, so these will not be selected. Anyhow, once you think you have selected all the unnecessary segments, hit the Delete key to remove them.
Still, it's recommended you do an extra check to see if only the paths of ring segments are left, by going to Outline mode. Go to View > Outline, or just hit Cmd/Ctrl + Y. To switch back to Preview mode, just hit Cmd/Ctrl + Y again. You should only see the paths that are part of the rings as shown in the image above.
Copy Rotate at 180°
To take it one step further, we're going to duplicate this set of rings below. First draw one horizontal guide in the center of the rings (the center of the red ring). Then draw two horizontal guides as shown in the image above: one that is at the middle of the bottom rings, and one at the bottom of all rings. Now select all rings, and select the Rotate tool from the toolbox (or hit the letter R). Click somewhere in the middle of the two horizontal guides right on the vertical guide. In the window that appears, enter the 180 as angle value and hit the Copy button.
Add white 3pt outside stroke
In the next steps we're going to repeat the steps from before again to connect the circles together like before, keeping also this gap in mind. Select the circles that are part of row two and three (so the rows that have four rings). Apply a white 3 pt Outside Stroke to these rings.
Convert Stroke to Fill
With the rings still selected, go to Object > Path > Outline Stroke to convert the strokes into fills.
From the Pathfinder palette select the Divide option to divide this all into separate segments.
Repeat the same steps again to create the proper gap and overlap for each ring. Select segments, recolor where needed, then Select Similar Objects, and Unite…
Remove unnecessary segments
When you are finished recoloring and uniting. You can select the colored rings and group them all together. Then draw a rectangular selection over the entire illustration (or hit Cmd/Ctrl + A), hold down the Shift key, click on the rings to deselect them again and hit delete to remove all unnecessary segments. Do a final check by hitting Cmd/Ctrl + Y to switch to Outline mode and hit Cmd/Ctrl + Y again to switch back to Preview mode.
That's it! Hope you enjoyed my tutorial. Don't forget to take it one step further and transform this into something different. Experiment and have fun! Also, don't forget to check out the wallpaper design I created from these rings…