How do I design an exact circle tiling in Inkscape?

How do I design an exact circle tiling, like the one in this paper (first page, Figure 2, third column) but with the exact mathematical proportions?

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Note that all tiles are identical.

Answer

The central trick is that the entire figure is composed of eighteen identical 60° arcs, which are rotated by simple angles. All you have to do is create these arcs, rotate them, and put them into place (with snapping):

  1. Create a circle using the Create circles, ellipses, and arcs tool (hold Shift while dragging).

  2. Duplicate it (Ctrl + D). I here make the duplicate red for better distinction.

  3. Make it a 60° arc, either by dragging the handles (while pressing Ctrl) or by manually entering the value in the Create circles, ellipses, and arcs tool:

    Creating a 60° arc

  4. Activate Snapping, Snap to nodes or handles and Snap to cusp nodes.

  5. Repeat the following twelve times:

    1. Duplicate your arc.

    2. Apply Object → Transform → Rotate by 30° as often as needed (or directly enter the correct value).

    3. Position the new arc as needed such that it snaps. (Start with arcs adjacent to your original one to have a basis for snapping.)

    Final result

Attribution
Source : Link , Question Author : Bruno Saramago Monteiro , Answer Author : Wrzlprmft

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