division
In the article in Wiskrant prof. Jan Hogendijk describes how the author of the Persian manuscript examines a remarkable division of the square. Name x the edge of the central square and d the distance to the oblique line that connects two points on the edge of the outer square. Clearly there seems to be a division of the square so that x equals d and the author describes very briefly how to construct it. Hogendijk
translated this description:
- Line AD is the diagonal of a square.
- AB equals BG and AG equals AD
- Find E on the extention of GD.
- Both segments EZ, ZH equal AG
- Connect GH and draw, starting from K the line KL parallel to GH.
- L is the requested. Allah knows the best.
The auteur doesn’t describe the logic behind the construction, but remarkably when creating it suddenly a lot of segments in the construction get the same length. The result of it looks like a unit repeat of a tiling with half stars along the edge of the square. When calculating the construction Hogendijk finds an error that’s limited to 3 minutes (=1/20 of a degree) and admits he doesn’t know how the logic of the construction. One can speak disdainfully on the use of approximate construction. As well you can admire the creativity and minimal error that bear witness of a great geometrical insight. Surely you cannot speak of just decorative work by workers without any mathematical knowledge.