Small polygons with large area

Abstract

A polygon is \textit{small} if it has unit diameter. The maximal area of a small polygon with a fixed number of sides $n$ is not known when $n$ is even and $n\geq14$. We determine an improved lower bound for the maximal area of a small $n$-gon for this case. The improvement affects the $1/n^3$ term of an asymptotic expansion; prior advances affected less significant terms. This bound cannot be improved by more than $O(1/n^3)$. For $n=6$, $8$, $10$, and $12$, the polygon we construct has maximal area.

Full text

Small polygons with large area
Christian Bingane∗Michael J. Mossinghoff†
June 4, 2022
Abstract
A polygon is small if it has unit diameter. The maximal area of a small polygon with a fixed
number of sides nis not known when nis even and n≥14. We determine an improved lower
bound for the maximal area of a small n-gon for this case. The improvement affects the 1/n3term
of an asymptotic expansion; prior advances affected less significant terms. This bound cannot be
improved by more than O(1/n3). For n= 6,8,10, and 12, the polygon we construct has maximal
area.
Keywords Polygons, isodiametric problem, maximal area
1 Introduction
A polygon is said to be small if it has diameter 1. Reinhardt [1] first studied two extremal problems
for small polygons a century ago: determining the maximal area for a small polygon with nsides, and
determining the maximal perimeter for a small convex polygon with nsides. These are sometimes
referred to as isodiametric problems for polygons in the literature. See [2,3] for a survey of work on
these problems and related ones. We focus on the area problem in this paper.
Reinhardt proved that the regular small polygon alone has maximal area when nis odd, and that
this polygon is never optimal when nis even and n≥6. It is straightforward to show that there are
infinitely many different small quadrilaterals with maximal area, including the square, and the optimal
hexagon was first determined by Graham in 1975 [4]. The optimal octagon was established by Audet
et al. in 2002 [5], and the cases n= 10 and n= 12 were resolved by Henrion and Messine in 2013 [6].
The problem remains open for larger n.
In 2006, Foster and Szabo [7] proved that the area A(Pn)of a small polygon Pnhaving an even
number of sides nsatisfies A(Pn)< An, where
An=n
2sin π
n−n−1
2tan π
2n−2=π
4−5π3
48n2−π3
24n3+O1
n4.(1)
Since the area of the small regular polygon Rnwith neven is given by n
8sin 2π
n, it follows easily that
An−A(Rn) = π3
16n2+O1
n3.
∗Département de mathématiques et de génie industriel, Polytechnique Montréal, Montreal, Quebec, Canada, H3C 3A7.
Email: christian.bingane@polymtl.ca
†Center for Communications Research, Princeton, NJ, USA. Email: m.mossinghoff@idaccr.org
1

In 2005, the second author [8] described a construction for a polygon Mnwith an even number of
sides nfor which
An−A(Mn) = a3π3
n3+O1
n4,
with
a3=5303 −456√114
5808 = 0.0747679609 . . . .
The first author [9] recently improved this, constructing a polygon Bnfor each even n≥6satisfying
A(Bn)−A(Mn) = a5π3
n5+O1
n6,
with a5= 0.25097 . . . when n≡2 mod 4 and a5= 0.35411 . . . when n≡0 mod 4. In this paper,
we generalize the latter construction to produce small polygons Qnthat exhibit an improvement in the
1/n3term for the area problem. We establish the following result.
Theorem 1. Let n≥6be an even integer, let Bndenote the small n-gon from [9], and let Andenote
the upper bound on the area of a small n-gon given by (1). There exists a small n-gon Qnsatisfying
An−A(Qn) = δπ3
n3+O1
n4<8π3
109n3+O1
n4,
with δ= 0.0733883168 . . ., so
A(Qn)−A(Bn)>π3
725n3+O1
n4.
Moreover, Qnis the optimal small polygon for n≤12.
This article is organized in the following way. Section 2establishes some notation and describes
some prior constructions. Section 3describes the new construction and proves Theorem 1. Section 4
reports on the computation of optimal small n-gons for a number of even nassuming of an axis of
symmetry, and compares the results of our construction with these polygons.
2 Prior constructions
The skeleton of a small polygon Pconsists of the vertices of P, together with all of the line segments
that connect two vertices of Pat unit distance from one another. Let n≥6denote an even integer.
From [4] it is known that the skeleton of an optimal n-gon Pforms a connected graph, and a linear
thrackle: each pair of line segments in the skeleton intersect one another, possibly at an endpoint.
Foster and Szabo [7] proved that the skeleton of an optimal small polygon, considered as a graph,
consists of an (n−1)-cycle, with a single additional pendant edge connected to the remaining vertex.
It is conjectured that this additional pendant edge forms an axis of symmetry in the skeleton of the
optimal polygon; this is in fact the case for n≤12. We thus consider polygons having a skeleton of
the form shown in Figure 1: a star with n−1points on vertices v0, ..., vn−2, an additional vertex
vn−1with distance 1from v0, and the line connecting v0and vn−1forming an axis of symmetry for
the polygon.
We place v0at the origin and v1at the point (0,1) in R2. Let θ0denote the angle ∠vn−1v0v1, and
for 0< k < n/2let θk=∠vk−1vkvk+1. Due to the symmetry of the construction, and the fact that
the star forms a closed path, we have
n
2−1
X
j=0
θj=π
2.(2)
2

v0(0,0)
v9(x5, y5)
v7(x7, y7)
v5(x5, y5)
v3(x3, y3)
v1(x1, y1)
v11(0,1)
v10(x10 , y10)
v8(x8, y8)
v6(x6, y6)
v4(x4, y4)
v2(x2, y2)
θ0
θ1
θ2
θ3
θ4
θ5
Figure 1: Polygon and skeleton for n= 12
If we let (xk, yk)denote the coordinates of vk, then
xk=
k−1
X
j=0
(−1)jsin j
X
i=0
θi!, yk=
k−1
X
j=0
(−1)jcos j
X
i=0
θi!.(3)
In addition, the vertices vn/2−1and vn/2are connected by a horizontal line in the skeleton, so
xn/2−1=−xn/2=(−1)n
2
2.(4)
We can compute the area A=A(θ0, . . . , θn/2−1)of such a polygon by determining the area of
n/2−1triangles Ak:A1is the area of the triangle ∆v0vn−1v1, and Akis the area of ∆v0vk−1vk+1 for
2≤k < n/2. Then
A= 2
n
2−1
X
k=1
Ak.(5)
It follows that
2A1=x0= sin θ0,
2Ak=xk+1yk−1−yk+1 xk−1
= sin θk+ 2(−1)k xksin θk
2+
k−1
X
j=0
θj!+ykcos θk
2+
k−1
X
j=0
θj!!sin θk
2
=
k−2
X
i=0
(−1)i sin i+1
X
j=0
θk−j!−sin i+1
X
j=1
θk−j!!
(6)
for 2≤k < n/2. We thus obtain an expression for the area in terms of the n/2angles θk.
3

In [8,9], this formulation was simplified to employ just three variables, α,β, and γ, by taking
θ0=α,
θ1=β+γ,
θ2=β−γ,
θk=βfor k≥3.
(7)
This configuration was selected to mimic some of the pattern observed in [8] for the small n-gons
with large area for n≤20, which were constructed using heuristic optimization methods over the
parameters θ0,...,θn/2−1. We extend these calculations in Section 4for n≤120 and note that this
pattern continues: see Table 3. There, in each polygon constructed, the angles θishow a pattern of
damped oscillation, with the odd-indexed values for the constructed n-gon appearing to converge from
above to a limiting value in π
n,π
n−1, and the even-indexed angles (after θ0) converging to the same
value from below. Using α,β, and γin this way then allowed approximating the largest variations
one appears to expect in the sequence of angles θi, while keeping the analysis tractable by using only
three variables.
As in [9], we note certain constraints on α,β, and γinherited by the geometry, namely
α+n
2−1β=π
2,(8)
sin(α+β+γ) = sin α+sin(α+ 3β/2)
2 cos(β/2) .(9)
The former clearly follows from (2), and the latter is a consequence of this, combined with (3) and (4).
After using these to eliminate βand γ, the expression for the area in [9] thus relied only on α, and an
asymptotic analysis was performed on this expression after setting α=aπ/n +bπ/n2+cπ/n3. A
similar analysis was performed in [8]. Both of those works found that
An−A(Pn) = (5303 −456√114)π3
5808n3+(192107 −17934√114)π3
21296n4+O1
n5,(10)
with Pn=Mnor Bnrespectively, with the analysis in [9] producing an improvement in the 1/n5term.
3 Proof of Theorem 1
We obtain improved small polygons by generalizing the construction of Section 2, keeping some
additional variables to allow for more variation in the sequence of angles θi, beyond what is captured
by (7). Let rdenote a positive integer, and suppose nis even and n≥2r+ 4. We describe a
construction for a small n-gon which involves r+ 2 variables.
Assume first that ris even. Our variables are α,β,β1,...,βr/2, and γ1,...,γr/2, and we set
θ0=α,
θ2i−1=βi+γi,1≤i≤r/2,
θ2i=βi−γi,1≤i≤r/2,
θk=β, r < k < n/2.
For convenience, let
ϕr=α+ 2
r/2
X
i=1
βi.
4

We derive an expression for the area of the small n-gon in terms of α,β, and the βiand γi. For
k > r, the coordinates (xk, yk)in (3) become
xk=xr+
k−1
X
j=r
(−1)jsin(ϕr+ (j−r)β)
=xr+sin ϕr−β
2−(−1)ksin ϕr+ (2k−2r−1)β
2
2 cos(β/2) ,
yk=yr+
k−1
X
j=r
(−1)jcos(ϕr+ (j−r)β)
=yr+cos ϕr−β
2−(−1)kcos ϕr+ (2k−2r−1)β
2
2 cos(β/2)
(11)
when ris even. The constraint (8) on the sum of the angles is now
ϕr+n
2−r−1β=π
2,(12)
and by combining this with (4) and (11), we deduce a generalization of (9):
xr+sin ϕr−β
2
2 cos(β/2) = 0.(13)
Using (6) and (11), the area 2Akis then
2Ak= sin β+ 2(−1)kxksin ϕr+ (2k−2r−1)β
2
+ykcos ϕr+ (2k−2r−1)β
2sin β
2
= sin β−tan(β/2) + 2(−1)kxrsin ϕr+ (2k−2r−1)β
2
+yrcos ϕr+ (2k−2r−1)β
2+cos((k−r)β)
2 cos(β/2) sin β
2
for r < k < n/2. Combining this with (12) and (13), it follows that
n
2−1
X
k=r+1
2Ak=n
2−r−1sin β−tan β
2−xrsin ϕr+yrcos ϕr+1
2tan β
2.
We thus obtain an expression for the area of our small n-gon having r+ 2 variables, whose number
of terms depends only on r:
A=
r
X
k=1
2Ak+n
2−r−1sin β−tan β
2−xrsin ϕr+yrcos ϕr+1
2tan β
2.(14)
We then eliminate βusing (12) and γr/2with (13).
If ris odd, we employ the same strategy as with r+ 1, except we set βr+1
2=β. The scheme (7)
therefore corresponds to the case r= 1.
For each even n≥6and r≥1, let Qn,r denote the small polygon obtained by maximizing our area
function (14) over the parameters α,β1, ...,βbr/2c,γ1,...,γdr/2e−1, for α∈π
2n−2,π
n,βi∈π
n,2π
n
5

for each i, and γi∈0,π
nfor each i. An asymptotic analysis reveals that the area is maximized for
large nwhen
α(n) = aπ
n+O1
n2,
βi(n) = biπ
n+O1
n2,1≤i≤ br/2c,
γi(n) = ciπ
n+O1
n2,1≤i≤ dr/2e − 1,
where a,b1,...,bbr/2c,c1,...,cdr/2e−1maximize a particular cubic polynomial in rvariables. When
r= 1 this polynomial is
−π3
192(88a3+ 84a2−222a+ 107),
while at r= 2 it is
−π3
192(88a3+ 12a2(8b1−1) −6a(16b2
1+ 21) + 128b3
1−48b2
1−216b1+ 243),
and r= 3 yields
−π3
192(88a3+ 12a2(16b1−12c1+ 7) −6a(32b2
1+ 64b1c1−80c2
1+ 56c1+ 37)
+ 128b3
1+ 192b2
1c1+ 384b1c2
1−384c3
1+ 336c2
1−240b1+ 204c1+ 267).
After removing the factor −π3, we use the NMinimize function in Mathematica to determine the
optimal value for these polynomials by numerical methods, requiring 0≤a≤1,0≤bi≤2for each
i, and 0≤ci≤1/3for each i. This produces an asymptotic estimate for A(Qn,r)having the form
A(Qn,r) = π
4−5π3
48n2−qrπ3
n3+O1
n4.(15)
For completeness we let Qn,0denote the polygon created by selecting αoptimally in the n-gon
with θ0=αand θi=βfor 1≤i < n/2, subject to (8), so when α=π/(2n−2) and β=π/(n−1).
This is the polygon created by simply adding a vertex at unit distance antipodal to one vertex of the
regular small (n−1)-gon, as in Figure 1for n= 12. The polygon Qn,0then has
q0=7
48 = 0.1458333333 . . . .
For r= 1, as reported in [8,9], the optimal choice for ais
a=2√114 −7
22 = 0.6524616592 . . . ,
which produces
q1=5545 −456√114
5808 = 0.1164346275 . . . .
At r= 2, we obtain
q2= 0.1156971503 . . . .
In fact, q2is a root of the polynomial
x4−70705
15876x3+269167127
41150592 x2−3381027871
987614208 x+737985313
2341011456.
6

The case r= 3 yields a further improvement,
q3= 0.1150899130 . . . ,
which is a root of a polynomial with rational coefficients and degree 8:
x8−338067189760423194
232662255261540774x7+1980606171874180754147
22335576505107914304 x6
−158140620301705167575191
536053836122589943296 x5+59647522303796634759434731
102922336535537269112832 x4
−836103610314364495378933003
1235068038426447229353984 x3+52675103710698128327456883067
118566531688938934017982464 x2
−14538141342029184829034957803
105392472612390163571539968 x+442235633612728385344035304147
40470709483157822811471347712 .
We improve this further with subsequent values of r: our results through r= 16 are summarized in
Table 1.
The polygons Qn,r are small by construction, and for even n≥6we set
Qn=(Qn,n/2−2n≤34,
Qn,16 n≥36.
The first statement of Theorem 1then follows by combining (15) at r= 16 with (1) and (10). For
the final statement, we calculate the areas of Q6,1,Q8,2,Q10,3, and Q12,4by optimizing over αand the
relevant βiand γi. The values we obtain are consistent with the optimal areas for n= 6 from [4],
n= 8 from [5], and n= 10 and 12 from [6], and the polygons Qnare optimal in these cases. Values
for the area and angles of these polygons are recorded in Table 2.
Additional small improvements can certainly be obtained using larger values for r. Of course, the
procedure becomes more computationally onerous as rincreases, due to the increasing complexity
of the optimization procedure as the number of variables grows. Such improvements are likely to be
minuscule, however, given the rapid convergence exhibited in the qrvalues we compute.
4 Constructions for small n
We construct some small polygons with large area for particular values of nand display their values
in two tables. First, for even integers nwith 6≤n≤120, we calculate the small n-gon with maximal
area P∗
n, assuming the presence of an axis of symmetry. We employ the skeleton of Foster and Szabo
as in Figure 1, and assume that the polygon is symmetric about the line connecting v0and vn−1.
For each such n, using (5) and (6) we construct P∗
nby maximizing the area Aover n/2variables
θ0, θ1, . . . , θn
2−1, subject to (2) and (4). More precisely,
A(P∗
n) = max
θ0,θ1,...,θ n
2
−1
sin θ0+
n
2−1
X
k=2
2Ak(θ1, θ2, . . . , θk)
s.t.
n
2−1
X
k=0
θk=π
2,
n
2−2
X
i=0
(−1)isin i
X
j=0
θj!=(−1)n
2
2,
0≤θ0≤π/6,
0≤θk≤π/3,1≤k≤n/2−1.
(16)
7

Table 1: Optimal values of qrin (15) for Qn,r, together with values for the free parameters that produce
this coefficient
r qra b1, . . . , bbr/2cc1, . . . , cdr/2e−1
0 0.1458333333333333
1 0.1164346275953378 0.6524616592755737
2 0.1156971503834968 0.6554858160170336 1.022718374818576
3 0.1150899130453658 0.6585214692355722 1.027063969740726 0.06794672543480737
4 0.1150687309140004 0.6586249743177490 1.027209190884176 0.06761473542307868
1.003828109549754
5 0.1150557470337394 0.6586900229138448 1.027300358228207 0.06740615925761905
1.004461819824590 0.01028318684355288
6 0.1150552764425733 0.6586923731079715 1.027303650624826 0.06739862447575472
1.004484647927062 0.01023212957791077
1.000570284354183
7 0.1150549998390641 0.6586937593365635 1.027305592742427 0.06739417976638237
1.004498111789651 0.01020201310839604
1.000662623065463 0.001509019841357918
8 0.1150549897593822 0.6586938098307351 1.027305663478150 0.06739401787457336
1.004498602162219 0.01020091616496605
1.000665984979524 0.001501502526879076
1.000083456862159
9 0.1150549838708233 0.6586938392297916 1.027305704817829 0.06739392319318871
1.004498888799399 0.01020027499791677
1.000667949966222 0.001497108650978320
1.000096926042045 0.0002203516777390261
10 0.1150549836560699 0.6586938403067916 1.027305706321311 0.06739391975105442
1.004498899243624 0.01020025160734295
1.000668021641018 0.001496948418422726
1.000097417195005 0.0002192534425102402
1.000012181578676
11 0.1150549835307224 0.6586938408223755 1.027305707189345 0.06739391767843365
1.004498905348772 0.01020023796370542
1.000668063456477 0.001496854886068534
1.000097703894909 0.0002186123676843603
1.000014146646888 0.00003215289654154845
12 0.1150549835261505 0.6586938408430183 1.027305707214692 0.06739391761102890
1.004498905576070 0.01020023745865263
1.000668064981190 0.001496851472990505
1.000097714354882 0.0002185889876955666
1.000014218316758 0.00003199263572187769
1.000001777358190
13 0.1150549835234823 0.6586938407444091 1.027305707207800 0.06739391752000037
1.004498905698189 0.01020023715826531
1.000668065875004 0.001496849480792517
1.000097720453880 0.0002185753423198028
1.000014260156537 0.00003189910564581757
1.000002064060629 0.000004691124054714891
14 0.1150549835233850 0.6586938407702582 1.027305707207799 0.06739391752829402
1.004498905739078 0.01020023716420131
1.000668065815119 0.001496849406827393
1.000097720770217 0.0002185748364450190
1.000014261621179 0.00003189570671658796
1.000002074524450 0.000004667741696766253
1.000000259301370
15 0.1150549835233282 0.6586938406842894 1.027305707182444 0.06739391752641309
1.004498905675305 0.01020023709721798
1.000668065987254 0.001496849396193236
1.000097720721770 0.0002185745506217027
1.000014262600232 0.00003189368791732473
1.000002080718492 0.000004654108249822218
1.000000301103361 0.0000006844384307984062
16 0.1150549835233261 0.6586938406334803 1.027305707190814 0.06739391746458313
1.004498905736593 0.01020023714027207
1.000668065901726 0.001496849368106921
1.000097720834017 0.0002185745455879002
1.000014262573901 0.00003189363350580009
1.000002080858254 0.000004653599949365328
1.000000302668948 0.0000006810134779486807
1.000000037787016
8

Table 2: Constructing optimal polygons for small n
n A(Qn,n/2−2)α β1,β2γ1
6 0.6749814429301047 0.3509301888703616
8 0.7268684827516268 0.2652408674910718 0.4379295350493946
10 0.7491373458778303 0.2126101953284637 0.3433714044229845 0.02476000789351616
12 0.7607298734487962 0.1770854623284314 0.2827755557037131 0.01982894085863103
0.2763754214389234
Problem (16) was solved on the NEOS Server 6.0 using AMPL with the nonlinear programming
solver Ipopt 3.13.4 [10]. The AMPL code is available in OPTIGON [11], a free package for extremal
convex small polygons available on GitHub. For selected even n≤120, Table 3shows the values θ∗
i
that we calculated for constructing P∗
n, and each value A(P∗
n)is displayed in Table 4. The areas shown
here for n≤20 agree with those from [8], and the values of A(P∗
n)for larger nin Table 4match
or slightly exceed the best value found in the literature [12–14]. Ipopt required less than 1second to
compute each value in Table 4.
Second, for selected even n≤120 we determine the area of Qn,r for 0≤r≤4by optimizing (14)
over the rparameters α,β1, ...,βbr/2c,γ1, ..., γdr/2e−1. These areas are also displayed in Table 4,
along with that of the regular n-gon Rnand the upper bound An. Julia and MATLAB functions that
give the coordinates of the vertices of all polygons presented in this work are provided in OPTIGON.
In all tables in this paper, each numerical value is rounded at the last displayed digit.
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9

Table 3: Angles θ∗
0, θ∗
1, . . . , θ∗
n
2−1of P∗
n.
n i θ∗
8iθ∗
8i+1 θ∗
8i+2 θ∗
8i+3 θ∗
8i+4 θ∗
8i+5 θ∗
8i+6 θ∗
8i+7
6 0 0.350930 0.653342 0.566524
8 0 0.265241 0.470631 0.405228 0.429696
10 0 0.212610 0.368131 0.318611 0.339137 0.332306
12 0 0.177085 0.302604 0.262947 0.279461 0.273290 0.275409
14 0 0.151583 0.257026 0.233904 0.237628 0.232444 0.234442 0.233769
16 0 0.132428 0.223448 0.194967 0.206716 0.202285 0.204013 0.203359 0.203580
18 0 0.117533 0.197661 0.172654 0.182938 0.179070 0.180577 0.179999 0.180218
1 0.180145
20 0 0.105629 0.177228 0.154925 0.164076 0.160640 0.161977 0.161464 0.161661
1 0.161586 0.161611
22 0 0.0959016 0.160633 0.140496 0.148745 0.145652 0.146854 0.146393 0.146571
1 0.146503 0.146528 0.146520
24 0 0.0878067 0.146886 0.128526 0.136037 0.133224 0.134316 0.133898 0.134059
1 0.133997 0.134021 0.134012 0.134015
30 0 0.0700443 0.116891 0.102361 0.108291 0.106075 0.106933 0.106605 0.106731
1 0.106683 0.106701 0.106694 0.106697 0.106696 0.106696 0.106696
40 0 0.0523626 0.0872236 0.0764267 0.0808253 0.0791841 0.0798182 0.0795763 0.0796689
1 0.0796334 0.0796470 0.0796418 0.0796437 0.0796429 0.0796432 0.0796431 0.0796431
2 0.0796431 0.0796431 0.0796431 0.0796431
50 0 0.0418008 0.0695718 0.0609765 0.0644752 0.0631706 0.0636742 0.0634822 0.0635556
1 0.0635275 0.0635382 0.0635340 0.0635355 0.0635349 0.0635351 0.0635350 0.0635350
2 0.0635350 0.0635350 0.0635349 0.0635349 0.0635349 0.0635349 0.0635349 0.0635349
3 0.0635349
60 0 0.0347820 0.0578637 0.0507223 0.0536280 0.0525448 0.0529626 0.0528033 0.0528642
1 0.0528408 0.0528496 0.0528462 0.0528474 0.0528469 0.0528470 0.0528469 0.0528469
2 0.0528468 0.0528468 0.0528468 0.0528467 0.0528467 0.0528467 0.0528467 0.0528467
3 0.0528467 0.0528466 0.0528466 0.0528466 0.0528466 0.0528466
70 0 0.0297804 0.0495296 0.0434206 0.0459055 0.0449793 0.0453364 0.0452002 0.0452521
1 0.0452321 0.0452396 0.0452366 0.0452376 0.0452371 0.0452372 0.0452371 0.0452370
2 0.0452370 0.0452369 0.0452369 0.0452369 0.0452368 0.0452368 0.0452368 0.0452367
3 0.0452367 0.0452367 0.0452367 0.0452367 0.0452366 0.0452366 0.0452366 0.0452366
4 0.0452366 0.0452366 0.0452366
80 0 0.0260361 0.0432947 0.0379568 0.0401276 0.0393185 0.0396302 0.0395112 0.0395565
1 0.0395390 0.0395454 0.0395428 0.0395437 0.0395432 0.0395432 0.0395431 0.0395430
2 0.0395430 0.0395429 0.0395429 0.0395428 0.0395428 0.0395427 0.0395427 0.0395426
3 0.0395426 0.0395426 0.0395425 0.0395425 0.0395425 0.0395424 0.0395424 0.0395424
4 0.0395424 0.0395424 0.0395424 0.0395424 0.0395424 0.0395423 0.0395423 0.0395423
90 0 0.0231282 0.0384545 0.0337147 0.0356421 0.0349236 0.0352001 0.0350945 0.0351345
1 0.0351190 0.0351247 0.0351223 0.0351230 0.0351226 0.0351225 0.0351224 0.0351223
2 0.0351222 0.0351221 0.0351221 0.0351220 0.0351219 0.0351219 0.0351218 0.0351218
3 0.0351217 0.0351217 0.0351216 0.0351216 0.0351215 0.0351215 0.0351215 0.0351214
4 0.0351214 0.0351214 0.0351214 0.0351213 0.0351213 0.0351213 0.0351213 0.0351213
5 0.0351213 0.0351213 0.0351213 0.0351213 0.0351213
100 0 0.0208046 0.0345883 0.0303258 0.0320589 0.0314127 0.0316612 0.0315662 0.0316021
1 0.0315881 0.0315931 0.0315909 0.0315915 0.0315911 0.0315910 0.0315909 0.0315907
2 0.0315906 0.0315905 0.0315904 0.0315903 0.0315903 0.0315902 0.0315901 0.0315900
3 0.0315900 0.0315899 0.0315898 0.0315898 0.0315897 0.0315897 0.0315897 0.0315896
4 0.0315896 0.0315895 0.0315895 0.0315895 0.0315894 0.0315894 0.0315894 0.0315894
5 0.0315893 0.0315893 0.0315893 0.0315893 0.0315893 0.0315893 0.0315893 0.0315893
6 0.0315893 0.0315893
110 0 0.0189055 0.0314290 0.0275563 0.0291308 0.0285436 0.0287692 0.0286828 0.0287153
1 0.0287025 0.0287070 0.0287050 0.0287054 0.0287050 0.0287049 0.0287048 0.0287046
2 0.0287045 0.0287043 0.0287042 0.0287041 0.0287040 0.0287039 0.0287038 0.0287037
3 0.0287037 0.0287036 0.0287035 0.0287034 0.0287034 0.0287033 0.0287033 0.0287032
4 0.0287031 0.0287031 0.0287031 0.0287030 0.0287030 0.0287029 0.0287029 0.0287029
5 0.0287028 0.0287028 0.0287028 0.0287028 0.0287027 0.0287027 0.0287027 0.0287027
6 0.0287027 0.0287027 0.0287026 0.0287026 0.0287026 0.0287026 0.0287026
120 0 0.0173243 0.0287991 0.0252508 0.0266933 0.0261551 0.0263616 0.0262824 0.0263121
1 0.0263003 0.0263043 0.0263024 0.0263028 0.0263023 0.0263022 0.0263020 0.0263018
2 0.0263017 0.0263015 0.0263014 0.0263012 0.0263011 0.0263010 0.0263009 0.0263008
3 0.0263007 0.0263006 0.0263005 0.0263004 0.0263003 0.0263003 0.0263002 0.0263001
4 0.0263001 0.0263000 0.0262999 0.0262999 0.0262998 0.0262998 0.0262997 0.0262997
5 0.0262996 0.0262996 0.0262996 0.0262995 0.0262995 0.0262995 0.0262994 0.0262994
6 0.0262994 0.0262994 0.0262994 0.0262993 0.0262993 0.0262993 0.0262993 0.0262993
7 0.0262993 0.0262993 0.0262993 0.0262993
10

Table 4: Comparing areas of small polygons
n A(Rn)A(Qn,0)A(Qn,1)A(Qn,2)A(Qn,3)A(Qn,4)A(P∗
n)An
6 0.6495190528 0.6722882584 0.6749814429 – – – 0.6749814429 0.6877007594
8 0.7071067812 0.7253199909 0.7268542719 0.7268684828 – – 0.7268684828 0.7318815691
10 0.7347315654 0.7482573378 0.7491189262 0.7491297887 0.7491373459 – 0.7491373459 0.7516135587
12 0.7500000000 0.7601970055 0.7607153082 0.7607228359 0.7607297471 0.7607298734 0.7607298734 0.7621336536
14 0.7592965435 0.7671877750 0.7675203660 0.7675256353 0.7675308404 0.7675309615 0.7675310111 0.7684036467
16 0.7653668647 0.7716285345 0.7718535572 0.7718573456 0.7718611688 0.7718612660 0.7718613220 0.7724408116
18 0.7695453225 0.7746235089 0.7747824059 0.7747852057 0.7747880405 0.7747881160 0.7747881651 0.7751926059
20 0.7725424859 0.7767382147 0.7768543958 0.7768565173 0.7768586570 0.7768587158 0.7768587560 0.7771522071
22 0.7747645313 0.7782865351 0.7783739622 0.7783756055 0.7783772514 0.7783772976 0.7783773302 0.7785970008
24 0.7764571353 0.7794540033 0.7795213955 0.7795226929 0.7795239821 0.7795240189 0.7795240452 0.7796927566
30 0.7796688406 0.7816380102 0.7816725130 0.7816732130 0.7816738921 0.7816739122 0.7816739269 0.7817597927
40 0.7821723252 0.7833076096 0.7833221318 0.7833224422 0.7833227341 0.7833227431 0.7833227495 0.7833587784
50 0.7833327098 0.7840695435 0.7840769608 0.7840771244 0.7840772750 0.7840772797 0.7840772830 0.7840956746
60 0.7839634745 0.7844798073 0.7844840910 0.7844841875 0.7844842749 0.7844842777 0.7844842796 0.7844949027
70 0.7843439529 0.7847256986 0.7847283918 0.7847284534 0.7847285085 0.7847285103 0.7847285115 0.7847351925
80 0.7845909573 0.7848845934 0.7848863952 0.7848864368 0.7848864738 0.7848864750 0.7848864758 0.7848909473
90 0.7847603296 0.7849931681 0.7849944322 0.7849944617 0.7849944876 0.7849944885 0.7849944890 0.7849976272
100 0.7848814941 0.7850706272 0.7850715479 0.7850715695 0.7850715884 0.7850715890 0.7850715895 0.7850738759
110 0.7849711494 0.7851278167 0.7851285079 0.7851285242 0.7851285384 0.7851285389 0.7851285392 0.7851302562
120 0.7850393436 0.7851712379 0.7851717699 0.7851717826 0.7851717935 0.7851717939 0.7851717941 0.7851731162
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11