2008 NCTM Presentation
on
Three-Dimensional Geometry
Susan Sexton
University of Georgia
SYMMETRY OF A CUBE
The cube has 48 symmetries.
This can be verified by counting
the possible combinations of how vertices are chosen: 8 * 3 * 2 = 48.
24 of them are rotations and are
found by the following:
á An axis exists from the center of one face to the center of
the opposite face. This axis can be rotated four times. Therefore, not counting
the identity, these degrees of rotation are 90¡, 180¡, and 270¡. There are 3 of these axes. This comes
to a total of 9 rotations.
á An axis exists from the midpoint of one edge to the midpoint
of an opposite edge. This axis can be rotated twice. Again, not counting the
identity, this degree of rotation is 180¡. There are 6 of these axes. This
comes to a total of 6 rotations.
á An axis exists from a vertex to the opposite vertex along their
shared long diagonal of the cube. This axis can be rotated three times. Again,
not counting the identity, these degrees of rotation are 120¡ and 240¡. There
are 4 of these axes. This comes to a total of 8 rotations.
á Finally we have the identity which was not counted in any of
the above rotations.
So 9 + 6 + 8 + 1 = 24 rotations.
Now I turn to the symmetries. A
plane of symmetry or a 3D mirror should bisect a segment perpendicularly that
joins two vertices. There are lots of pairs of vertices of the cube. Let me go
through them as systematically as I can.
á There are edges that join adjacent vertices. There are 12
such edges. A plane that perpendicularly bisects one such edge will also bisect
three other edges, giving a total of 4 edges bisected. So there are 3 planes of
symmetry for edges joined by adjacent vertices.
á A little harder to explain are the segments formed from
non-adjacent vertices. But these non-adjacent vertices are opposite on the same
face of the cube and there are 12 of these (2 per face). A plane that perpendicularly bisects
this type of segment (really a diagonal of a face) will bisect the face of the
cube and will do so to the opposite face (and its respective diagonal). So
there are 6 of these planes of symmetry.
á A final type of segment is joined from two opposite vertices
that lie on the same long diagonal of the cube. But there is no such plane of
symmetry that perpendicularly bisects this segment.
So 3 + 6 = 9 planes of symmetry for
a cube.
Finally I will look at the turn reflections.
I know that there are 15 turn reflections since 48 – 24 – 9 = 15.
I will use the strategy of focusing
on the axes of rotations discussed above.
á The axis which exists from the center of one face to the
center of the opposite face can be rotated four times. Therefore, not counting
the identity, these degrees of rotation are 90¡, 180¡, and 270¡. There are 3 of these axes. However since
a 180¡ turn reflection is actually the antipodal symmetry, there are really 6
(3 each for the 90¡ and 270¡ rotations) turn reflections.
á The axis which exists from the midpoint of one edge to the
midpoint of an opposite edge can be rotated twice. Again, not counting the
identity, this degree of rotation is 180¡. There are 6 of these axes. But since
each of these 180¡ rotations are really the antipodal symmetry, there are no
turn reflections for this axis.
á The axis which exists from a vertex to the opposite vertex
along their shared long diagonal of the cube can be rotated three times. Again,
not counting the identity, these degrees of rotation are 60¡ and 300¡. There
are 4 of these axes. Therefore there are of 8 these turn reflections.
á Finally we have the antipodal symmetry which was not counted
in any of the above turn reflections.
So 6 + 8 + 1 = 15 turn reflections.
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