consider the vertices of a regular n-gon with side length 1, centered on the origin and an additional "inner" point at Cartesian coordinates (inner, 0). the distance from the inner point to the nearest polygon vertex depends on the rotation of the polygon.
the smallest possible distance is when a polygon vertex is on the X axis, at coordinates (1/(2*sin(Pi/n)), 0). the distance is abs(inner - 1/(2*sin(Pi/n))). (if n=1, then let the vertex be at (0,0)).
the largest possible distance is when there are two vertices symmetrically above and below the X axis, at coordinates (1/(2*tan(Pi/n)), +-1/2). the distance is sqrt((inner - 1/(2*tan(Pi/n)))^2 + 1/4). (if n=2, let the vertices be at (0, +-1/2)).
let the inner point itself be a vertex of a regular polygon with side length 1, centered on the origin and having one vertex (the inner point) lying on the positive X axis. let this inner polygon have m sides, so inner = 1/(2*sin(Pi/m)). in order for inner to actually be inside, m < n.
incidentally, all the coordinates of all the vertices of these regular polygons can be expressed in radicals.
we seek pairs of nested regular polygons such that the gap between them is about 1, or more precisely, that the smallest possible distance as defined above is strictly less than 1 and largest possible distance is strictly greater than 1. this happens for a 2-gon nested inside a 9-gon, and for all (m, m+6) for m >= 7. there are no solutions for m = {1, 3, 4, 5, 6}.
for m=1 and n=6, the distance is always exactly 1, not a solution by our definition with "strictly", but maybe it should be.
for m=6 and n=12, the largest possible distance is exactly 1. this is geometrically surprising: attaching equilateral triangle ears to the vertices of a regular hexagon (and squares to the edges) yields a regular dodecagon (12-gon). trigonometrically, 1/(2*tan(Pi/12))-1 = sqrt(3)/4.
it is a little surprising that solutions (m, m+6) seem to exist for all m >= 7. the offset 6 is probably round(2*Pi): in the limit when polygons are nearly circular, adding 1 to the radius adds 2*Pi to the circumference. future post on the limiting case of two lines of dots.
next, consider fudging to create solutions for m = {3, 4, 5, 6}:
the outer n-gon must still have side length 1, but let the inner m-gon have side length s, different from 1. among all possible relative orientations (rotations), half the time the distance to the nearest vertex should be less than s; half the time greater. this probability distribution is from the point of view of a given vertex of the inner polygon. surprisingly, there are closed form solutions for s, via Mathematica. we start with the following auxiliary functions:
inner[n_,s_]= s/(2*Sin[Pi/n])
l[n_]= inner[n,1]
x[n_]= l[n]*Cos[Pi/(2*n)]
y[n_]= l[n]*Sin[Pi/(2*n)]
py[n_,inx_]= (x[n]-inx)^2+y[n]^2
let us first consider a regular 9-gon of side length 1 with a concentric equilateral triangle of side length s.
s /. First[Solve[py[9,inner[3,s]] == s^2 && s>0, s]] //Simplify //InputForm
(Sqrt[3]*(-2*Cos[Pi/18] + Sqrt[2*(5 + Cos[Pi/9])])*Csc[Pi/18]*Sec[Pi/18])/16
N[%]
0.9349946
that is, a regular nonagon with side length 1 with a concentric equilateral triangle with side length (Sqrt[3] * (-2*Cos[Pi/18] + Sqrt[2*(5 + Cos[Pi/9])]) * Csc[Pi/18]*Sec[Pi/18])/16 ~= 0.9349946 inside it satisfies our requirements on s.
10-gon with inside it a regular triangle of the following side length:
(Sqrt[3]*(Sqrt[(3 + Sqrt[5])*(20 + Sqrt[2*(5 + Sqrt[5])])] - 2*(1 + Sqrt[5])*Cos[Pi/20]))/8
1.03312
10-gon with inside it a square of the following side length:
(Sqrt[(3 + Sqrt[5])*(12 + Sqrt[2*(5 + Sqrt[5])])] - 2*(1 + Sqrt[5])*Cos[Pi/20])/(2*Sqrt[2])
0.9561354
11-gon with square inside it:
((-2*Cos[Pi/22] + Sqrt[2*(3 + Cos[Pi/11])])*Csc[Pi/11])/(2*Sqrt[2])
1.04714
11-gon with pentagon inside it:
-((((-5 + Sqrt[5])*Cos[Pi/22] + Sqrt[85 - 35*Sqrt[5] - 5*(-3 + Sqrt[5])*Cos[Pi/11]])*Csc[Pi/11])/(Sqrt[10 - 2*Sqrt[5]]*(-3 + Sqrt[5])))
0.967348
12-gon with pentagon inside it:
(-Sqrt[(5*(136 + 15*Sqrt[2] + 68*Sqrt[3] - 56*Sqrt[5] + 9*Sqrt[6] - 5*Sqrt[10] - 28*Sqrt[15] - 3*Sqrt[30]))/2] - (1 + Sqrt[3])*(-5 + Sqrt[5])*Cos[Pi/24])/(Sqrt[5 - Sqrt[5]]*(-3 + Sqrt[5]))
1.05153
12-gon with hexagon inside it:
((2 + Sqrt[3])*Sec[Pi/24])/(Sqrt[2] + Sqrt[6])
0.974261
13-gon with hexagon inside it:
(Csc[Pi/13]*Sec[Pi/26])/4
1.05232
no regular polygon that satisfies our constraints fits in a heptagon or octagon. future work: relax constraints further.
motivation is a collection of moving dots filling the plane which remain roughly equally spaced, dots staying below some maximum velocity, and whose configuration never repeats. this can be achieved by having rings of dots rotate independently at speeds none of which are rational multiples of another. future post: points on lines.
previously vaguely similar: concentric regular polygons that touch, not maintaining any gap.
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