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Wave pattern using Ad Quadratum

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Previously I've shown how to plan out the Wave border pattern using offset circles, but what if there is another way, one much more rooted in geometry?
These photos show a possible formula that could have been used.
Ad quadratum is a pattern seen in the ancient world, there is a Mesopotamium terracotta tablet incised with some geometrical diagrams which has been dated to 18th century BC and has the circle within the square.

wave ad quadratum 1 

1. Ad quadratum, seen in examples of ancient geometry. A circle is drawn inside a square. Another square is then drawn inside this circle.
2. Continue this process .
3. Here you can see the squares and circles drawn out.
4. Block in (here using red) starting at the top right and just going one section of pattern at a time and you will naturally be drawn into the centre. This spiral, if you create another one opposite is known as The Baravelle spiral.

wave ad quadratum 2 

  Here I've added a repeat of the pattern on either side and you can see the border being formed.

wave border pattern Here is a copy of a conventional wave border pattern and you can see the similarities. It would not be too difficult to mark a staff, adjusting the size of the curl to create the above from Ad quadratum.

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