This is a step-by-step explanation of how to create a pattern which repeats infinitely on a plane using a rectangular grid. I use Inkscape in this explanation, but the principles apply to any program (or manual creation, which is also a lot of fun). This works best with line drawings.

If you’re not familiar with what makes a pattern tessellate as opposed to repeat, think in terms of Pac-Man; when you go far enough to the right side, you come out the left side.

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step 1

Start out with a new document. Any dimensions will work. For this example I created a ⅛″ grid on a 2″ square document.

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step 2

Draw some objects. The more they break the boundaries of the canvas the better.

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step 3

Draw a rectangle the same size as you canvas. You might want to make it a different color if your design is complex.

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step 4

1. make sure you have snap to grid on
2. select all
3. group
4. copy & paste dragging rectangle created in previous step to all edges and corners of canvas (for a total of 8 pastes)

You should now have 9 copies of your original drawing in a 3 × 3 grid. If your drawing is opaque, some of the repetition will be obscured. Reduce the opacity of your original drawing to, say, 50% and start over.

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step 5

1. draw another rectangle outlining your canvas
2. cut it
3. select all
4. group
5. paste in place
6. select the group you just created, then the rectangle you just pasted
7. set clip (object > clip > set)

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Now your tessellation is ready to be repeated.
Note: If your original drawing is more than 2× the width and / or height of your canvas, you will need to paste your drawing additional times in step 4. I think these are the interesting tessellations because you have to see the pattern repeated many times in order to see the original design.