The Base Square and its Orientation
An ancient technique for drawing an octagon is to start from a square. Heinz Götze describes one such method rooted in history, in his book: Castel del Monte, the Geometric Marvel of the Middle Ages.
The square is the starting geometric figure in our development, Folio 104:01. The base octagon is drawn around the square, sharing four of its corners with the square (octagon circumscribing the square), folio 104:02.
Drawing an octagon that circumscribes a square is not an easy and trivial procedure. There is a simpler and geometric method to accomplish this. It is difficult to imagine that the medieval mathematicians, so keen and expert in geometry, may have not been aware of this.
There is an orientation problem for the square that needs to be resolved first, especially in the field, where this geometric process would be duplicated for the actual construction. The issue is the orientation of the base octagon with respect to the cardinal directions, because the entrance of the castle will be on one of the sides of the octagon.
Throughout history, nearly all civilizations have oriented their edifices with respect to directions set by the sun, east-west and north-south, or some other celestial references. The same is true at Castel del Monte. It is documented that the portal of the castle, which falls on one of the eight sides of the plant octagon, faces east.
To obtain such a result, the base square is first oriented 22.5° off the north-south direction, folio 104:03. This orientation of the square in the field results in a base octagon with a side facing east. This ensures that the entrance portal to the castle faces east.
The rest of the presentation on the geometric design focuses on the plant octagon itself, wherein the detail of its orientation is irrelevant. As mentioned above, this is a key determination in the field, when construction is first started.