Regular Nonagon
Author
Eric W. Weisstein
Title
Regular Nonagon
Description
The regular nonagon is the regular polygon with nine sides and Schläfli symbol {9}. The regular nonagon cannot be constructed using the classical Greek rules of geometric construction, but Conway and Guy (1996) give a Neusis construction based on angle trisection. Madachy (1979) illustrates how to construct a nonagon by folding and knotting a strip of paper. Although the regular nonagon is not a constructible polygon, Dixon (1991) gives constructions for several angles which are close...
Category
Educational Materials
Keywords
URL
http://www.notebookarchive.org/2019-07-0z5jsko/
DOI
https://notebookarchive.org/2019-07-0z5jsko
Date Added
2019-07-02
Date Last Modified
2019-07-02
File Size
189.69 kilobytes
Supplements
Rights
Redistribution rights reserved
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Regular Nonagon
Regular Nonagon
Author
Author
Eric W. Weisstein
July 3, 2018
July 3, 2018
©2018 Wolfram Research, Inc. except for portions noted otherwise
Definitions
Definitions
ineq=1/4axCsc[Pi/9]-1/4Sqrt[3]ayCsc[Pi/9]+1/2ayCos[Pi/18]Csc[Pi/9]+1/8a^2Cos[Pi/18]Csc[Pi/9]^2+1/2axCsc[Pi/9]Sin[Pi/18]+1/8Sqrt[3]a^2Csc[Pi/9]^2Sin[Pi/18]≥0&&-((ay)/2)+1/2axCot[Pi/9]-1/8a^2Csc[Pi/9]-1/4axCsc[Pi/9]+1/4Sqrt[3]ayCsc[Pi/9]+1/8Sqrt[3]a^2Cot[Pi/9]Csc[Pi/9]≥0&&ay+1/2a^2Cot[Pi/9]≥0&&-((ay)/2)-1/2axCot[Pi/9]-1/8a^2Csc[Pi/9]+1/4axCsc[Pi/9]+1/4Sqrt[3]ayCsc[Pi/9]+1/8Sqrt[3]a^2Cot[Pi/9]Csc[Pi/9]≥0&&-(1/4)axCsc[Pi/9]-1/4Sqrt[3]ayCsc[Pi/9]+1/2ayCos[Pi/18]Csc[Pi/9]+1/8a^2Cos[Pi/18]Csc[Pi/9]^2-1/2axCsc[Pi/9]Sin[Pi/18]+1/8Sqrt[3]a^2Csc[Pi/9]^2Sin[Pi/18]≥0&&-(1/2)ayCos[Pi/18]Csc[Pi/9]-1/2axCos[(2Pi)/9]Csc[Pi/9]+1/4a^2Cos[Pi/18]Cos[(2Pi)/9]Csc[Pi/9]^2+1/2axCsc[Pi/9]Sin[Pi/18]+1/2ayCsc[Pi/9]Sin[(2Pi)/9]-1/4a^2Csc[Pi/9]^2Sin[Pi/18]Sin[(2Pi)/9]≥0&&-(1/2)axCsc[Pi/9]+1/2axCos[(2Pi)/9]Csc[Pi/9]-1/2ayCsc[Pi/9]Sin[(2Pi)/9]+1/4a^2Csc[Pi/9]^2Sin[(2Pi)/9]≥0&&1/2axCsc[Pi/9]-1/2axCos[(2Pi)/9]Csc[Pi/9]-1/2ayCsc[Pi/9]Sin[(2Pi)/9]+1/4a^2Csc[Pi/9]^2Sin[(2Pi)/9]≥0&&-(1/2)ayCos[Pi/18]Csc[Pi/9]+1/2axCos[(2Pi)/9]Csc[Pi/9]+1/4a^2Cos[Pi/18]Cos[(2Pi)/9]Csc[Pi/9]^2-1/2axCsc[Pi/9]Sin[Pi/18]+1/2ayCsc[Pi/9]Sin[(2Pi)/9]-1/4a^2Csc[Pi/9]^2Sin[Pi/18]Sin[(2Pi)/9]≥0;
implreg=ImplicitRegion[ineq,{x,y}];
verts=a{{-(1/2)Cos[Pi/18]Csc[Pi/9],1/2Csc[Pi/9]Sin[Pi/18]},{-(1/4)Sqrt[3]Csc[Pi/9],-(1/4)Csc[Pi/9]},{-(1/2),-(1/2)Cot[Pi/9]},{1/2,-(1/2)Cot[Pi/9]},{1/4Sqrt[3]Csc[Pi/9],-(1/4)Csc[Pi/9]},{1/2Cos[Pi/18]Csc[Pi/9],1/2Csc[Pi/9]Sin[Pi/18]},{1/2Csc[Pi/9]Sin[(2Pi)/9],1/2Cos[(2Pi)/9]Csc[Pi/9]},{0,1/2Csc[Pi/9]},{-(1/2)Csc[Pi/9]Sin[(2Pi)/9],1/2Cos[(2Pi)/9]Csc[Pi/9]}};
reg=Polygon[verts];
assum=a>0;
Figure
Figure
In[]:=
Show[LaminaData["FilledRegularNonagon","Diagram"],ImageSize300]
Out[]=
Plots
Plots
Equations
Equations
Properties
Properties
Rehashing named triangle objects...
Area
Area
Area
Area
Assuming[assum,FullSimplify[Area[reg/.a1]]]//Timing
4.113589,Cos-2+6+Csc-2++2Cot6+-4Sin
1
16
π
18
2
Cot
π
9
π
9
3
Secπ
18
π
9
3
Cscπ
9
π
18
Assuming[assum,FullSimplify[Area[implreg/.a1]]]//Timing
$Aborted
NIntegrate
NIntegrate
Block[{a=1},NIntegrate[Boole[ineq[x,y]],{x,-1.5,1.5},{y,-1.5,1.5}]]
NIntegrate::slwcon:Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::ncvb:NIntegrate failed to converge to prescribed accuracy after 18 recursive bisections in x near {x,y} = {-1.40373494146224707390668839934733114205300807952880859375000000000,1.26619}. NIntegrate obtained 6.18716 and 0.0172599 for the integral and error estimates.
6.18716
Integrate
Integrate
(area=Block[{a=1},Integrate[Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞}]])//Timing
...ran overnight—crashed/ran out of memory?...
Area[RegularPolygon[9,1/2Csc[π/9]]]//FullSimplify
9
4
π
9
N[%]
6.18182
Assuming[a>0,Area[ineq[x,y],{x,y}]]//FullSimplify//Timing
RegionMeasure
RegionMeasure
Assuming[assum,FullSimplify[RegionMeasure[reg]]]
RegionMeasure::nmet:Unable to compute the measure of region PolygonaCosCsc,-aCscSin,aCosCsc,aCscSin,0,aCsc,-aCosCsc,aCscSin,-aCosCsc,-aCscSin,-,-aCot,,-aCot.
1
2
π
14
π
7
1
2
π
7
π
14
1
2
3π
14
π
7
1
2
π
7
3π
14
1
2
π
7
1
2
3π
14
π
7
1
2
π
7
3π
14
1
2
π
14
π
7
1
2
π
7
π
14
a
2
1
2
π
7
a
2
1
2
π
7
RegionMeasurePolygonaCosCsc,-aCscSin,aCosCsc,aCscSin,0,aCsc,-aCosCsc,aCscSin,-aCosCsc,-aCscSin,-,-aCot,,-aCot
1
2
π
14
π
7
1
2
π
7
π
14
1
2
3π
14
π
7
1
2
π
7
3π
14
1
2
π
7
1
2
3π
14
π
7
1
2
π
7
3π
14
1
2
π
14
π
7
1
2
π
7
π
14
a
2
1
2
π
7
a
2
1
2
π
7
Assuming[assum,FullSimplify[RegionMeasure[implreg]]]
AreaInertiaTensor
AreaInertiaTensor
Centroid
Centroid
FromCycles::shdw:Symbol FromCycles appears in multiple contexts {Combinatorica`,System`}; definitions in context Combinatorica` may shadow or be shadowed by other definitions.
PermutationQ::shdw:Symbol PermutationQ appears in multiple contexts {Combinatorica`,System`}; definitions in context Combinatorica` may shadow or be shadowed by other definitions.
Permute::shdw:Symbol Permute appears in multiple contexts {Combinatorica`,System`}; definitions in context Combinatorica` may shadow or be shadowed by other definitions.
RandomPermutation::shdw:Symbol RandomPermutation appears in multiple contexts {Combinatorica`,System`}; definitions in context Combinatorica` may shadow or be shadowed by other definitions.
ToCycles::shdw:Symbol ToCycles appears in multiple contexts {Combinatorica`,System`}; definitions in context Combinatorica` may shadow or be shadowed by other definitions.
NIntegrate
NIntegrate
Block[{a=1},NIntegrate[{x,y}Boole[ineq[x,y]],{x,-1.5,1.5},{y,-1.5,1.5}]]
NIntegrate::slwcon:Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::eincr:The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained -7.49092× and 0.00162556 for the integral and error estimates.
-8
10
NIntegrate::slwcon:Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::eincr:The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained -0.0000611432 and 0.001465 for the integral and error estimates.
{-7.49092×,-0.0000611432}
-8
10
Integrate
Integrate
Block[{a=1},NIntegrate[{x,y}Boole[ineq[x,y]],{x,-1.5,1.5},{y,-1.5,1.5},Method"QuasiMonteCarlo"]]
NIntegrate::maxp:The integral failed to converge after 50000 integrand evaluations. NIntegrate obtained -0.0000544329 and 0.0251981 for the integral and error estimates.
NIntegrate::maxp:The integral failed to converge after 50000 integrand evaluations. NIntegrate obtained 0.00112107 and 0.0252054 for the integral and error estimates.
{-0.0000544329,0.00112107}
Assuming[a>0,Centroid[ineq[x,y],{x,y}]]//FullSimplify//Timing
RegionCentroid
RegionCentroid
Assuming[assum,FullSimplify[RegionCentroid[reg/.a1]]]//Timing
{0.237146,{0,0}}
Assuming[assum,FullSimplify[RegionCentroid[implreg/.a1]]]//Timing
$Aborted
Convex
Convex
TimeConstrained[Region`ConvexRegionQ[reg],600]//Timing
TimeConstrained[Region`ConvexRegionQ[implreg],600]//Timing
Diagonals
Diagonals
EdgeLengths
EdgeLengths
s=FullSimplify[Norm/@(Subtract@@@Partition[verts,2,1,1]),assum]
{a,a,a,a,a,a,a,a,a}
GeneralizedDiameter
GeneralizedDiameter
Perimeter
Perimeter
Total[s]
9a
RadiiOfGyration
RadiiOfGyration
Vertices
Vertices
Graphics[MapIndexed[{Point[#1],Text[#2[[1]],#1,BackgroundWhite]}&,p=RegularPolygon[7,1/2Csc[π/7],-π/14][[1]]]]
FullSimplify[Through[{MinValue,MaxValue}[{x,ineq[x,y]},{x,y}]],a>0]//Timing
$Aborted
FullSimplify[Through[{MinValue,MaxValue}[{y,ineq[x,y]},{x,y}]],a>0]//Timing
Cite this as: Eric W. Weisstein, "Regular Nonagon" from the Notebook Archive (2018), https://notebookarchive.org/2019-07-0z5jsko
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