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This, by the way, is the actual fold template for the origami diamond. You can see that the pseudo-diagonals are now bisections...

@bob012345 Yes, the octagon in the very center is what I was looking for. Excellent. :) That's what I think they were trying to do in...

It looks off but I computed the coordinates and checked the edge lengths which are ##2-√2##.

Finally, I think you meant this.

I thought you meant the squares inscribed in the irregular octagon.

For the irregular octagon, there are two squares inside it but they are not equal. The ratio of areas is 1.125.

@bob012345 ok, but try making the ratio of horizontal and vertical lines with spacing 1,root 2,1 and connecting the points on the sides...

Here is a simple drawing of the regular octagon within the square. If the square has corners at (±1,±1), the outside points are root(2)...

I was looking at how to make a regular octagon in the spirit of the OP but it’s difficult to construct the ratio you mentioned without...

Oh I see. The diagram is perfectly four-fold symmetrical - but it is not eight-fold symmetrical. And the 37/53 is reflected in the two...

From symmetry, it is readily seen that the sides come in 4 identical pairs. The angle between the two sides in each pair is not 135...

The edges are the same lengths but parallel edges are skewed wrt each other, they are parallelograms not rectangles.

Interestingly, the miniseries Dune was filmed in Prague when Eastern Europe had opened up after the Cold War. Here is an article about...