Polymorf - Knowhere - Cubic symmetry

(110) mirror planes, (110) rotational planes, and [110] axes of 2-fold rotational symmetry

The cube also has six (110) planes of mirror symmetry passing through its six sets of

opposite edges. The planes slice diagonally through each face, dividing the cube into

identical halves. These mirror planes also correspond to the cube's six (110) planes of


Fig. 3a - One (110) plane	Fig. 3b - Two (110) planes	Fig. 3c - Six (110) planes
	Required parts
4 small squares	16 small squares	24 small squares
4 right triangles	16 right triangles	12 right triangles
1 rectangle, 14 pinges	8 rectangles, 58 pinges	6 rectangles, 66 pinges
.
Fig. 3 Alternative models for demonstrating (110) cubic planes and [110] axes
.

2-fold rotational symmetry. The cube's six [110] axes of 2-fold rotational symmetry that

correspond to these six rotational planes are perpendicular to the center points of each

respective plane on a line passing through the midpoints of its opposite edges (use axles

colored the same as their respective planes to highlight these [110] axes). Fig. 3a

gives view of the cube looking down a [110] rotational axis to show the 2-fold symmetry.

Fig. 3b shows only two of the (110) planes intersecting due to the complexity of modeling

all six planes simultaneously. Instead, in Fig. 3c each plane is projected outside of the

centermost cube. Viewing the model face on projects the (110) diagonal planes back

into the center of the cube in kaleidoscopic fashion.

Exercise: What properties do the (100) and (110) cubic planes and the

[100] and [110] cubic axes have in common? How do they differ?

Page 2 Geometry rules! - Cubic symmetry