Regular Decagon

Author

Eric W. Weisstein
July 3, 2018

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polygons/RegularDecagon.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RegularDecagon.html.

©2018 Wolfram Research, Inc. except for portions noted otherwise

Definitions

ineq=2 Sqrt[5+2 Sqrt[5]] a+y≥Sqrt[2 (5+Sqrt[5])] x+Sqrt[5] y&&(1+Sqrt[5]) (Sqrt[5-2 Sqrt[5]] x+y)≤2 Sqrt[5+2 Sqrt[5]] a&&Sqrt[5+2 Sqrt[5]] a≥2 y&&2 Sqrt[5+2 Sqrt[5]] a+(1+Sqrt[5]) (Sqrt[5-2 Sqrt[5]] x-y)≥0&&2 Sqrt[5+2 Sqrt[5]] a+Sqrt[2 (5+Sqrt[5])] x+y≥Sqrt[5] y&&2 Sqrt[5+2 Sqrt[5]] a+Sqrt[2 (5+Sqrt[5])] x+Sqrt[5] y≥y&&2 Sqrt[5+2 Sqrt[5]] a+Sqrt[10-2 Sqrt[5]] x+y+Sqrt[5] y≥0&&Sqrt[5+2 Sqrt[5]] a+2 y≥0&&2 Sqrt[5+2 Sqrt[5]] a≥(1+Sqrt[5]) (Sqrt[5-2 Sqrt[5]] x-y)&&2 Sqrt[5+2 Sqrt[5]] a+Sqrt[5] y≥Sqrt[2 (5+Sqrt[5])] x+y;

implreg=ImplicitRegion[ineq,{x,y}];

assum=a>0;

verts=a{{1/2 (1+Sqrt[5]),0},{1/4 (3+Sqrt[5]),Sqrt[5/8+Sqrt[5]/8]},{1/2,1/2 Sqrt[5+2 Sqrt[5]]},{-(1/2),1/2 Sqrt[5+2 Sqrt[5]]},{1/4 (-3-Sqrt[5]),Sqrt[5/8+Sqrt[5]/8]},{1/2 (-1-Sqrt[5]),0},{1/4 (-3-Sqrt[5]),-(1/2) Sqrt[1/2 (5+Sqrt[5])]},{-(1/2),-(1/2) Sqrt[5+2 Sqrt[5]]},{1/2,-(1/2) Sqrt[5+2 Sqrt[5]]},{1/4 (3+Sqrt[5]),-(1/2) Sqrt[1/2 (5+Sqrt[5])]}};

reg=Polygon[verts];

Figure

Show[LaminaData["FilledRegularDecagon","Diagram"],ImageSize->300]

Plots

Diagram

LaminaData["FilledRegularDecagon","Diagram"]

DiscretizeRegion

Block[{a=1},DiscretizeRegion[#]]&/@{reg,implreg}

Polygon

Graphics[RegularPolygon[10]//N]

Properties

TableForm[FullSimplify[RegularPolygonInformation[10]],TableDepth->2]

Equations

Inequality

RegularPolygon[10,1/2Csc[π/10]]//FullSimplify

FullSimplify[Inequalities[RegularPolygon[10,a/2Csc[π/10]],{x,y}],a>0]

ineq[x_,y_]:=2 (Sqrt[5+2 (Sqrt[5])]) a+y≥(Sqrt[2 (5+(Sqrt[5]))]) x+(Sqrt[5]) y&&(1+(Sqrt[5])) ((Sqrt[5-2 (Sqrt[5])]) x+y)≤2 (Sqrt[5+2 (Sqrt[5])]) a&&(Sqrt[5+2 (Sqrt[5])]) a≥2 y&&2 (Sqrt[5+2 (Sqrt[5])]) a+(1+(Sqrt[5])) ((Sqrt[5-2 (Sqrt[5])]) x-y)≥0&&2 (Sqrt[5+2 (Sqrt[5])]) a+(Sqrt[2 (5+(Sqrt[5]))]) x+y≥(Sqrt[5]) y&&2 (Sqrt[5+2 (Sqrt[5])]) a+(Sqrt[2 (5+(Sqrt[5]))]) x+(Sqrt[5]) y≥y&&2 (Sqrt[5+2 (Sqrt[5])]) a+(Sqrt[10-2 (Sqrt[5])]) x+y+(Sqrt[5]) y≥0&&(Sqrt[5+2 (Sqrt[5])]) a+2 y≥0&&2 (Sqrt[5+2 (Sqrt[5])]) a≥(1+(Sqrt[5])) ((Sqrt[5-2 (Sqrt[5])]) x-y)&&2 (Sqrt[5+2 (Sqrt[5])]) a+(Sqrt[5]) y≥(Sqrt[2 (5+(Sqrt[5]))]) x+y

Block[{a=1},RegionPlot[ineq[x,y],{x,-2a,2a},{y,-2a,2a}]]

Properties

Area

Area

Assuming[assum,FullSimplify[Area[reg/.a->1]]]

Assuming[assum,FullSimplify[Area[implreg/.a->1]]]

Integrate

Assuming[a>0,Area[ineq[x,y],{x,y}]]//FullSimplify//Timing

RegionMeasure

Assuming[assum,FullSimplify[RegionMeasure[reg/.a->1]]]

Assuming[assum,FullSimplify[RegionMeasure[implreg/.a->1]]]

AreaInertiaTensor

Assuming[a>0,AreaInertiaTensor[ineq[x,y],{x,y}]]//FullSimplify//Timing

Centroid

Integrate

Assuming[a>0,Centroid[ineq[x,y],{x,y}]]//FullSimplify//Timing

RegionCentroid

Assuming[assum,FullSimplify[RegionCentroid[reg/.a->1]]]

Assuming[assum,FullSimplify[RegionCentroid[implreg/.a->1]]]

Convex

TimeConstrained[Region`ConvexRegionQ[reg],600]//Timing

TimeConstrained[Region`ConvexRegionQ[implreg],600]//Timing

GeneralizedDiameter

GD[n_]:=If[OddQ[n],(1/2) Csc[(π/(2 n))],Csc[(π/n)]]a

GD[10]

ineqs=FullSimplify[And@@ineq@@@{{x1,y1},{x2,y2}},a>0]/.a->1

Maximize[{(x1-x2)^2+(y1-y2)^2,ineqs},{x1,x2,y1,y2},Reals]//Timing

RadiiOfGyration

Assuming[a>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]//Timing

Region

Block[{a=1},Through[{MinValue,MaxValue}[{x,ineq[x,y]},{x,y}]]]

Block[{a=1},Through[{MinValue,MaxValue}[{y,ineq[x,y]},{x,y}]]]

Quit

Lamina

Diagonals

Length[v={{1/2 (1+Sqrt[5]),0},{1/4 (3+Sqrt[5]),Sqrt[5/8+Sqrt[5]/8]},{1/2,1/2 Sqrt[5+2 Sqrt[5]]},{-(1/2),1/2 Sqrt[5+2 Sqrt[5]]},{1/4 (-3-Sqrt[5]),Sqrt[5/8+Sqrt[5]/8]},{1/2 (-1-Sqrt[5]),0},{1/4 (-3-Sqrt[5]),-(1/2) Sqrt[1/2 (5+Sqrt[5])]},{-(1/2),-(1/2) Sqrt[5+2 Sqrt[5]]},{1/2,-(1/2) Sqrt[5+2 Sqrt[5]]},{1/4 (3+Sqrt[5]),-(1/2) Sqrt[1/2 (5+Sqrt[5])]}}]

ds=With[{n=Length[v]},Select[Subsets[Range[n],{2}],Mod[Subtract@@#,n]≠n-1≠1&]]

Length[%]

FullSimplify[Binomial[n,2]-n]

n(n-3)/2/.n->7

Graphics[{
{Yellow,Opacity[.2],EdgeForm[Black],Polygon[v]},
Line[ds]/.Thread[Range[Length[v]]->v]
}]

SortBy[FullSimplify[Norm[Subtract@@v[[#]]]&/@ds],N]