Abstract:
Poncelet in his introduction to famous porism theorem, studied momentarily the variable polygons having all but one of their vertices on sides of a given polygon and all the sides passing through the vertices of another polygon. He shows that the free vertex describes a conic. Poncelet cites Brianchon, who cites Maclaurin and Braikenridge for the case of triangles (n = 3). Poncelet started to work on these questions in 1813 when he was a prisoner of war in Russia.
In Poncelet porisms infinitude of polygons inscribed into one ellipse and and circumscribed around another ellipse, given that one such polygon exists, is proved. This result inspired discovery of many other porisms commonly known as Poncelet type theorems. In Steiner porism infinitude of chains of tangent circles all of which are internally tangent to one circle and externally tangent to another circle, again provided that one such chain exists, is studied. In Emch's porism Steiner's porism is generalized to chain of intersecting circles, where these intersections are on another circle. In Zig-zag type porisms infinitude of equilateral polygons with vertices taken alternately on two circles (or lines), is studied. Money-Coutts theorem is about chain of tangent circles alternately inscribed into angles of a triangle. Generalizations and connections of these results with each other, were studied by Bauer, Protasov. Similar generalizations for the space with rings of tangent spheres were discussed in Ccoxeter. Surprisingly, the original configuration of Maclaurin about polygons inscribed into one polygon and circumscribed around another polygon did not attract much attention after these powerful generalizations. In the current talk we will study in detail the case of convex polygons inscribed into one convex polygon and circumscribed around another one. The results are surprising as we will show that the maximal number of such polygons is not dependent on the number of sides of the polygons. At the end of the talk the case of generalized polygons, considered as collections of vertices or lines is also considered. Possibility of construction of such regular polygons with Euclidean instruments (straightedge and compass) will also be disscussed. This construction algorithm was very useful for drawing diagrams in the current study. For completeness we also provided an independent proof for generalized Maclaurin-Braikenridge's conic generation method. The results are then extended for polyhedra and the following result is proved:
The number of graphs, that can be inscribed into one graph of a given convex polyhedron and circumscribed around around another graph is at most 4.
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