NUMBAT OER - Open Educational Resources

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5. Symmetry in three-dimensional objects

Just as the symmetry of 2-D objects is defined with respect to one or more lines (axes), the symmetry of 3-D objects can be defined with respect to one or more planes. Imagine the object being sliced into two equal pieces through a plane of symmetry. If the cut face is then placed against a flat mirror, the half of the object and its reflection in the mirror look identical to the original:

Symmetry across three dimensions
Figure 2. Symmetry in three dimensions. a: illustration of the principle of symmetry – an ellipsoid has been cut into two halves and the cut face of one half placed against a flat mirror to produce a reflection that recreates the original 3-D object; b: the three planes of symmetry for a triaxial ellipsoid; c and d: two of the nine planes of symmetry for a cuboid; e: any plane of symmetry for a cone passes through the vertex (point) and bisects the base; f: any plane that passes through the centre of a sphere is a plane of symmetry; g: a cylinder has a single plane of symmetry cutting it into two halves across the long axis, and any plane that bisects it lengthways.