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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+O1

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+O1

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+O1

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+O1

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+O1

n4<8π3

109n3+O1

n4,

with δ= 0.0733883168 . . ., so

A(Qn)−A(Bn)>π3

725n3+O1

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

2−1β=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+O1

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−1β=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)kxksin ϕr+ (2k−2r−1)β

2

+ykcos ϕr+ (2k−2r−1)β

2sin β

2

= sin β−tan(β/2) + 2(−1)kxrsin 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−1sin β−tan β

2−xrsin ϕr+yrcos ϕr+1

2tan 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−1sin β−tan β

2−xrsin ϕr+yrcos ϕr+1

2tan 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

nfor each i. An asymptotic analysis reveals that the area is maximized for

large nwhen

α(n) = aπ

n+O1

n2,

βi(n) = biπ

n+O1

n2,1≤i≤ br/2c,

γi(n) = ciπ

n+O1

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+O1

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.

References

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Mathematiker-Vereinigung, vol. 31, pp. 251–270, 1922.

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Institute Communications, pp. 1–16, Fields Institute for Research in Mathematical Sciences,

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[4] R. L. Graham, “The largest small hexagon,” Journal of Combinatorial Theory, Series A, vol. 18,

no. 2, pp. 165–170, 1975.

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natorial Theory, Series A, vol. 98, no. 1, pp. 46–59, 2002.

[6] D. Henrion and F. Messine, “Finding largest small polygons with GloptiPoly,” Journal of Global

Optimization, vol. 56, no. 3, pp. 1017–1028, 2013.

[7] J. Foster and T. Szabo, “Diameter graphs of polygons and the proof of a conjecture of Graham,”

Journal of Combinatorial Theory, Series A, vol. 114, no. 8, pp. 1515–1525, 2007.

[8] M. J. Mossinghoﬀ, “Isodiametric problems for polygons,” Discrete & Computational Geometry,

vol. 36, no. 2, pp. 363–379, 2006.

[9] C. Bingane, “Tight bounds on the maximal area of small polygons: Improved Mossinghoﬀ

polygons,” Discrete & Computational Geometry, 2022.

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