cp's OEIS Frontend

A322106 Numerator of the least possible squared diameter of an enclosing circle of a strictly convex lattice n-gon.

%I A322106 #29 Oct 19 2022 06:42:48
%S A322106 2,2,50,8,10,10,1250,29,40,40,2738,72,82,82,176900,17810,1709690,178,
%T A322106 11300,260,290,290,568690,416,2418050,488,3479450,629,2674061,730
%N A322106 Numerator of the least possible squared diameter of an enclosing circle of a strictly convex lattice n-gon.
%C A322106 If the smallest possible enclosing circle is essentially determined by 3 vertices of the polygon, the squared diameter may be rational and thus A322107(n) > 1.
%C A322106 The first difference of the sequences A321693(n) / A322029(n) from a(n) / A322107(n) occurs for n = 12.
%C A322106 The ratio (A321693(n)/A322029(n)) / (a(n)/A322107(n)) will grow for larger n due to the tendency of the minimum area polygons to approach elliptical shapes with increasing aspect ratio, whereas the polygons leading to small enclosing circles will approach circular shape.
%C A322106 For n>=19, polygons with different areas may fit into the enclosing circle of minimal diameter. See examples in pdf at Pfoertner link.
%D A322106 See A063984.
%H A322106 Hugo Pfoertner, <a href="/A322106/a322106.pdf">Illustration of convex n-gons fitting into smallest circle</a>, (2018).
%H A322106 Hugo Pfoertner, <a href="/A322106/a322106_1.pdf">Illustration of convex n-gons fitting into smallest circle, n = 27..32</a>, (2018).
%e A322106 By n-gon a convex lattice n-gon is meant, area is understood omitting the factor 1/2. The following picture shows a comparison between the minimum area polygon and the polygon fitting in the smallest possible enclosing circle for n=12:
%e A322106 .
%e A322106     0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
%e A322106   6                          H ##### Gxh +++++ g
%e A322106   |                     #        +      #    *   +
%e A322106   |                 #       +              #        +
%e A322106   |             #       +                 *   #        +
%e A322106   5         I       i                          F        f
%e A322106   |       #       +                    *        #       +
%e A322106   |     #       +                                #      +
%e A322106   |   #       +                     *             #     +
%e A322106   4 J       j                                      #    e
%e A322106   | #     @+                     *                  #  +
%e A322106   | #     +      @                                   #+
%e A322106   | #    +              @     *                      +#
%e A322106   3 K   +                     @                     +   E
%e A322106   |  # +                   *         @             +    #
%e A322106   |   #                                    @      +     #
%e A322106   |  + #                *                        +@     #
%e A322106   2 k   #                                      d        D
%e A322106   | +    #           *                       +        #
%e A322106   | +     #                                +        #
%e A322106   | +      #       *                    +         #
%e A322106   1 l       L                         c        C
%e A322106   |   +       # *                +        #
%e A322106   |     +       #           +        #
%e A322106   |       +  *    #     +        #
%e A322106   0         a ++++ Axb ##### B
%e A322106     0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
%e A322106 .
%e A322106 The 12-gon ABCDEFGHIJKLA with area 52 fits into a circle of squared diameter 40, e.g. determined by the distance D - J, indicated by @@@. No convex 12-gon with a smaller enclosing circle exists. Therefore a(n) = 40 and A322107(12) = 1.
%e A322106 For comparison, the 12-gon abcdefghijkla with minimal area A070911(12) = 48 requires a larger enclosing circle with squared diameter A321693(12)/A322029(12) = 52/1, e.g. determined by the distance a - g, indicated by ***.
%Y A322106 Cf. A063984, A070911, A321693, A322029, A322107 (corresponding denominators).
%K A322106 nonn,frac,more
%O A322106 3,1
%A A322106 _Hugo Pfoertner_, Nov 26 2018
%E A322106 a(27)-a(32) from _Hugo Pfoertner_, Dec 19 2018