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I’m not sure if this flower is really hyperbolic, but all of the faces are really affine heptagons.
Being affine heptagons means that each face has a unique viewing angle (projection) where it looks like a regular equalateral heptagon (7 sides). The three central affine heptagons form mutually orthogonal planes which join at the center of the 3-fold symmetry. These three faces don’t necessarily have to be orthogonal. They could be more acute, flatter, or even coplanar or asymmetrical. The tiling can obviously be continued ad nauseum. I stopped at 48 faces. Faces of the same color are congruent.
If anyone can tell me precisely what curve is formed by the faces, please leave a detailed comment.
I’d also like to identify the pattern formed by the set of lines along which each of the affine heptagons is projected as a regular heptagon. Where do they converge if at all? Are they perhaps related to a parabolic reflector? Any insight is welcome.