Axis of Symmetry

Author

Eric W. Weisstein
August 14, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/SolidGeometry/AxisofSymmetry.nb.

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

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

Figure

Dashed blue lines show the reflection axes of three plane figures and representative rotational axes of three solids.  The highlighted cylinder and cone axes are their principal axes and have continuous rotational symmetry, while every line through the center of the sphere is a rotational axis.

Module[{axisColor,coneGraphic,cylinderGraphic,neutral,planeGraphic,rectangle,solidColor,sphereGraphic,square,triangle},neutral=LightDarkSwitched[Black,White];axisColor=Blue;solidColor=StandardGray;triangle={{0,1.05},{-0.91,-0.525},{0.91,-0.525}};square={{-0.9,-0.9},{0.9,-0.9},{0.9,0.9},{-0.9,0.9}};rectangle={{-1.2,-0.7},{1.2,-0.7},{1.2,0.7},{-1.2,0.7}};planeGraphic[shape_,angles_,label_]:=Graphics[{Directive[axisColor,AbsoluteThickness[2]],Table[Line[{-1.3 {Cos[angle],Sin[angle]},1.3 {Cos[angle],Sin[angle]}}],{angle,angles}],Directive[neutral,AbsoluteThickness[2]],Line[Append[shape,First[shape]]],Text[Style[label,15,neutral,FontFamily->"Times",Background->LightDarkSwitched[White,Black]],{0,-1.45}]}];cylinderGraphic=Labeled[Graphics3D[{Directive[solidColor,Opacity[0.25]],Cylinder[{{0,0,-0.8},{0,0,0.8}},0.65],Directive[axisColor,Opacity[1],AbsoluteThickness[2]],Line[{{0,0,-1.2},{0,0,1.2}}]},Boxed->False,ImageSize->300],Style["cylinder\nprincipal axis",15,neutral,FontFamily->"Times"],Bottom];coneGraphic=Labeled[Graphics3D[{Directive[solidColor,Opacity[0.25]],Cone[{{0,0,-0.8},{0,0,0.9}},0.7],Directive[axisColor,Opacity[1],AbsoluteThickness[2]],Line[{{0,0,-1.2},{0,0,1.25}}]},Boxed->False,ImageSize->300],Style["cone\nprincipal axis",15,neutral,FontFamily->"Times"],Bottom];sphereGraphic=Labeled[Graphics3D[{Directive[solidColor,Opacity[0.25]],Sphere[{0,0,0},0.7],Directive[axisColor,Opacity[1],AbsoluteThickness[2]],Line[{{-1,0,0},{1,0,0}}],Line[{{0,-1,0},{0,1,0}}],Line[{{0,0,-1},{0,0,1}}]},Boxed->False,ImageSize->300],Style["sphere\ninfinitely many axes",15,neutral,FontFamily->"Times"],Bottom];GraphicsGrid[{{planeGraphic[triangle,{(π/2),(π/2)+((2 π)/3),(π/2)+((4 π)/3)},"equilateral triangle\n3 reflection axes"],planeGraphic[square,{0,(π/4),(π/2),((3 π)/4)},"square\n4 reflection axes"],planeGraphic[rectangle,{0,(π/2)},"nonsquare rectangle\n2 reflection axes"]},{cylinderGraphic,coneGraphic,sphereGraphic}},
Alignment->Bottom,Dividers->All]]