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apparently unaware that it was not consistent with Kepler. For example one of his fifty-nine icosahedra is just a loose gaggle of small shards floating in space and entirely disconnected from each other: hardly an "extension" of a core figure which is not even present. Technically one could say that a "stellation diagram" of a polygon is a set of collinear points with some segments drawn in, but it is pretty trivial and non-encyclopedic. If you want to explore them for yourself then I would suggest a uniform or quasiregular polygon, since it has two (alternating) kinds of edge, but I will not be discussing further anything non-encyclopedic. — Cheers,
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point. In fact it specifically says that the {6,2} is two triangles, which is shown. The problem with the 9-gon is that the caption says the {9,3} is three triangles, yet none of the pictures with the coloured areas are anything but variations on the {9,4}. If, as the caption says, the {9,3} is three triangles, then where are the three triangles in the picture labelled as {9,3}??
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I guess I'm just a bit confused about the designation {9/2} etc. I can see how having 9 points and going to the 4th one around can make the star shape shown, but why is the same shape shown with different numbers? All the shapes are {9/4}, just coloured differently. The examples with 7 sided shapes
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Here are two examples of stellations of an octagon. I say "truncated square" rather than octagon because I wanted half symmetry, 2 types of edges to stellate, but unsure if that's really needed in these cases. I colored edges green and purple that represent the boundary of the stellation and showed
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This is another area where it is easy to be misled by bright ideas and poor resources. Coxeter's treatment of stellation was fundamentally different from his predecessors - Bradwardine, Kepler, Poinsot, Cauchy, Cayley. This has not been helped by
Coxeter's use of their definition before blithely
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it is a process to construct new poligons ( in two dimensions) and to construct new polyhedra in three dimensions. The process consist in extending the facial planes of the polyhedron untill they meet each other...the resulting polyhedron is the new stellated polyhedron. by IP 194.183.69.146 at
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Thanks for the efforts, but the caption still doesn't seem to fit in with the examples to the left. On the left example, there are pictures showing how each n-gon is formed. It is clear how the numbers in braces show the total number of points and the number skipped to get to the next connected
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My primary interest if I support including something is to aid understanding of the 3D stellations, so if its trivial, that makes it a good candidate for teaching something new. But I admit the messy disconnected stellations (59 icosahedra) never much attracted me, only those that showed the
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the sides or faces continuously from the original. Wheeler studied stellation diagrams more closely and observed the way that the face planes divided space into cells, whose edges are segments of the lines of intersection. Coxeter then took this cell-based approach to its logical conclusion,
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It seems like stellation of polygons ought to work like stellations of polyhedra. In polyhedra there's one face plane (stellation diagram) for every type of face, and each shows all the intersection boundaries, and each domain can be ON or OFF. In polygons, or especially regular polygons, a
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Yes, The star polygons named {9/p} are the colored ones with the full stellation, the black one. Maybe it would be more clear if the black edges were recolored to be less prominent as lighter gray?
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stellation diagram is a line with intersection points dividing domains. So a general polygonal stellation ought to have some sort of binary ON/OFF for each segment between points.
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The confusion arises because the graphic is misleading. The Large black-lined figure repeated four times is not the enneagon itself but the
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presenting his own work as if it were consistent. Bradwardine studied only cyclic polygons, Kepler expanded the idea to polyhedra by
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going around at different rates make perfect sense. Can any one explain??
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I have clarified the caption a bit. Any better? — Cheers,
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