Pair of Pants

Author

Eric W. Weisstein
August 12, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Topology/PairofPants.nb.

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

©2026 Wolfram Research, Inc. except for portions noted otherwise

Figrues

Planar Model

A pair of pants is a disk with the interiors of two disjoint disks removed.

pants2d=Module[{holeColor,neutral,surfaceColor},surfaceColor=RGBColor[0.15,0.45,0.8];holeColor=LightDarkSwitched[White,Black];neutral=LightDarkSwitched[Black,White];Graphics[{Directive[surfaceColor,Opacity[0.72]],Disk[],Directive[holeColor,Opacity[1]],Disk[{-0.38,0},0.28],Disk[{0.38,0},0.28],Directive[neutral,AbsoluteThickness[2]],Circle[],Circle[{-0.38,0},0.28],Circle[{0.38,0},0.28]},
ImageSize->250
]
]

Three-dimensional surface

This three-dimensional pair of pants has one upper and two lower boundary components.

pants3d=Module[{a=(18/25),surfaceColor=RGBColor[0.15,0.45,0.8]},
ContourPlot3D[
((((x-a))^2)+(y^2)) ((((x+a))^2)+(y^2))==(a^4)+((19 z)/100),
{x,-(29/20),(29/20)},{y,-(23/20),(23/20)},{z,-1,1},
Mesh->None,ContourStyle->Directive[surfaceColor,Opacity[.7]],Method->{"ShrinkWrap"->True},PlotPoints->60,MaxRecursion->1,Boxed->False,Axes->False,Lighting->"Neutral",ViewPoint->{2.1,-2.7,1.5},ImageSize->Small,ViewVertical->{0,0,1}
]
]

Both

GraphicsRow[{pants2d,pants3d},ItemAspectRatio->-Automatic]