Orient a regular octagon so two sides are horizontal and two are vertical. (The other 4 are at 45 degrees.) Mark nodes on each edge subdividing each edge into the same number of segments.
Straightforward would be to draw lines parallel to the edges through the nodes. However, this results in a great many intersections in the interior.
Instead, first draw only the horizontal and vertical interior lines. This divides the interior into a bunch of rectangles and some isosceles right triangles along the edges. Connect the remaining nodes to the node "across" from them by a series of line segments that are diagonals of these interior rectangles. This will create fewer internal intersections because we are reusing the orthogonal intersections. There will be a few more intersections created at the centers of some of the rectangles.
The density of the internal intersections will be much greater near the edges of the octagon compared to the center. Do some sort of energy minimization to move the intersections inward, dragging the lines with them. The initial horizontal and vertical lines will no longer remain that way. Maybe use curved edges.
Color the regions. I suspect only two colors are needed. Inspired by, how can polyhedra with subdivided octagonal faces be colored? Possibly an interesting board for a game.
The Petaminx twisty puzzle (and variations) demonstrate subdividing a pentagon with lines parallel to each edge.
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