Regular Heptagon
Author
Eric W. Weisstein
Title
Regular Heptagon
Description
The regular heptagon is the seven-sided regular polygon illustrated above, which has Schläfli symbol {7}. According to Bankoff and Garfunkel (1973), "since the earliest days of recorded mathematics, the regular heptagon has been virtually relegated to limbo." Nevertheless, Thébault (1913) discovered many beautiful properties of the heptagon, some of which are discussed by Bankoff and Garfunkel (1973). Although the regular heptagon is not a constructible polygon using...
Category
Educational Materials
Keywords
URL
http://www.notebookarchive.org/2019-07-0z5jrg1/
DOI
https://notebookarchive.org/2019-07-0z5jrg1
Date Added
2019-07-02
Date Last Modified
2019-07-02
File Size
286.81 kilobytes
Supplements
Rights
Redistribution rights reserved
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Regular Heptagon
Regular Heptagon
Author
Author
Eric W. Weisstein
July 3, 2018
July 3, 2018
©2018 Wolfram Research, Inc. except for portions noted otherwise
Definitions
Definitions
ineq=-(1/2)ayCos[Pi/14]Csc[Pi/7]+1/2ayCos[(3Pi)/14]Csc[Pi/7]-1/2axCsc[Pi/7]Sin[Pi/14]+1/4a^2Cos[(3Pi)/14]Csc[Pi/7]^2Sin[Pi/14]-1/2axCsc[Pi/7]Sin[(3Pi)/14]+1/4a^2Cos[Pi/14]Csc[Pi/7]^2Sin[(3Pi)/14]≥0&&-(1/2)axCsc[Pi/7]-1/2ayCos[(3Pi)/14]Csc[Pi/7]+1/4a^2Cos[(3Pi)/14]Csc[Pi/7]^2+1/2axCsc[Pi/7]Sin[(3Pi)/14]≥0&&1/2axCsc[Pi/7]-1/2ayCos[(3Pi)/14]Csc[Pi/7]+1/4a^2Cos[(3Pi)/14]Csc[Pi/7]^2-1/2axCsc[Pi/7]Sin[(3Pi)/14]≥0&&-(1/2)ayCos[Pi/14]Csc[Pi/7]+1/2ayCos[(3Pi)/14]Csc[Pi/7]+1/2axCsc[Pi/7]Sin[Pi/14]+1/4a^2Cos[(3Pi)/14]Csc[Pi/7]^2Sin[Pi/14]+1/2axCsc[Pi/7]Sin[(3Pi)/14]+1/4a^2Cos[Pi/14]Csc[Pi/7]^2Sin[(3Pi)/14]≥0&&-((ay)/2)+1/2axCot[Pi/7]+1/2ayCos[Pi/14]Csc[Pi/7]+1/4a^2Cos[Pi/14]Cot[Pi/7]Csc[Pi/7]-1/4a^2Csc[Pi/7]Sin[Pi/14]-1/2axCsc[Pi/7]Sin[Pi/14]≥0&&ay+1/2a^2Cot[Pi/7]≥0&&-((ay)/2)-1/2axCot[Pi/7]+1/2ayCos[Pi/14]Csc[Pi/7]+1/4a^2Cos[Pi/14]Cot[Pi/7]Csc[Pi/7]-1/4a^2Csc[Pi/7]Sin[Pi/14]+1/2axCsc[Pi/7]Sin[Pi/14]≥0;
implreg=ImplicitRegion[ineq,{x,y}];
verts=a{{1/2Cos[Pi/14]Csc[Pi/7],-(1/2)Csc[Pi/7]Sin[Pi/14]},{1/2Cos[(3Pi)/14]Csc[Pi/7],1/2Csc[Pi/7]Sin[(3Pi)/14]},{0,1/2Csc[Pi/7]},{-(1/2)Cos[(3Pi)/14]Csc[Pi/7],1/2Csc[Pi/7]Sin[(3Pi)/14]},{-(1/2)Cos[Pi/14]Csc[Pi/7],-(1/2)Csc[Pi/7]Sin[Pi/14]},{-(1/2),-(1/2)Cot[Pi/7]},{1/2,-(1/2)Cot[Pi/7]}};
reg=Polygon[verts];
assum=a>0;
Figure
Figure
In[]:=
Show[LaminaData["FilledRegularHeptagon","Diagram"],ImageSize300]
Out[]=
Plots
Plots
Diagram
Diagram
LaminaData["FilledRegularHeptagon","Diagram"]
DiscretizeRegion
DiscretizeRegion
Block[{a=1},DiscretizeRegion[#]]&/@{reg,implreg}
MinValue::ztest:Unable to decide whether numeric quantities Max0,Abs[1],Abs-+-,Max[1],Max[1] are equal to zero. Assuming they are.
2Cos[1]Csc[1]-Cos[1]Csc[1]Cos[1]Cot[1]Csc[1]+Cos[1]Power[2]SecSin
1
2
1
2
1
4
1
4
3π
14
π
7
Times[3]+Times[3]
1
2(Sin[1]+1)
-Cos[1]Csc[1]+11+1+Cos[1]Csc[1]Sin[1]
2(Sin[1]+Sin[1])
,
Polygon
Polygon
Schematic Diagram
Schematic Diagram
Equations
Equations
Properties
Properties
Area
Area
Area
Area
Assuming[assum,FullSimplify[Area[reg/.a1]]]//Timing
2.325647,-12-13Csc++34+CscTan
1
32
π
14
3
Csc
π
14
2
Cot
π
14
π
14
π
14
Assuming[assum,FullSimplify[Area[implreg/.a1]]]//Timing
$Aborted
NIntegrate
NIntegrate
Integrate
Integrate
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
NIntegrate
NIntegrate
Integrate
Integrate
Assuming[a>0,Centroid[ineq[x,y],{x,y}]]//FullSimplify//Timing
RegionCentroid
RegionCentroid
Assuming[assum,FullSimplify[RegionCentroid[reg/.a1]]]//Timing
{0.188971,{0,0}}
Assuming[assum,FullSimplify[RegionCentroid[implreg/.a1]]]//Timing
$Aborted
Circumradius
Circumradius
With[{n=7},a/2Csc[Pi/n]]//FullSimplify
1
2
π
7
TrigToExp[%]//FullSimplify
9/14
(-1)
-1+
2/7
(-1)
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}
GeneralizedDiameter
GeneralizedDiameter
Inradius
Inradius
With[{n=7},a/2Cot[Pi/n]]//FullSimplify
1
2
π
7
TrigToExp[%]//FullSimplify
1+a
2/7
(-1)
2-1+
2/7
(-1)
Perimeter
Perimeter
Total[s]
7a
Vertices
Vertices
Lengths
Lengths
Cite this as: Eric W. Weisstein, "Regular Heptagon" from the Notebook Archive (2018), https://notebookarchive.org/2019-07-0z5jrg1
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