Smoothed Octagon

Author

Eric W. Weisstein
April 15, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComputationalGeometry/SmoothedOctagon.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Packing Density

Octagon

N[4/(3+Sqrt[2]),17]

N[(8- Sqrt[32] - Log[2])/(Sqrt[8]-1),17]

FullSimplify[Sqrt[6Sqrt[2]-8](Sqrt[2]+1)/2]

Simplify[(Sqrt[2]-1)^2]

FullSimplify[(y-#2==(#4-#2)/(#3-#1)(x-#1))&@@Flatten[Table[{3+Cos[#],Sin[#]}&[2Pi i/8],{i,2,3}]]
]

FullSimplify[Solve[%,y]]

FullSimplify[Solve[{
Sqrt[(3-2 (Sqrt[2]))(x-3)^2-1/Sqrt[2]]==y,
y==-(1+(Sqrt[2])) (-1+x)
}]]

N[%]

Solve[FullSimplify[(y-#2==(#4-#2)/(#3-#1)(x-#1))&@@Flatten[Table[{3+Cos[#],Sin[#]}&[2Pi i/8],{i,2,3}]]],y]

FullSimplify[Solve[(-1+(Sqrt[2])) x==1]]

Show[{
ParametricPlot[{
{x,-Sqrt[(3-2 (Sqrt[2]))x^2-1/Sqrt[2]]},
{x,Sqrt[(3-2 (Sqrt[2]))x^2-1/Sqrt[2]]},
{x,4-3 (Sqrt[2])-x+(Sqrt[2]) x},
},{x,2+10^-7,5},
PlotStyle->{{},{},RGBColor[0,1,0]},
AspectRatio->Automatic,
DisplayFunction->Identity],
Graphics[{RGBColor[1,0,0],Line[Table[{3+Cos[#],Sin[#]}&[2Pi i/8],{i,0,8}]]}]
},
DisplayFunction->$DisplayFunction]