Chapter 1, Problem 12E

To determine

The letters of the alphabet written in uppercase block style have symmetry group with four elements.

The letters H, I, O, and X have horizontal,vertical, diagonal, and cross diagonal symmetry.

For the letters H, I, O and X, there is horizontal,vertical, diagonal, and cross diagonal symmetry.

Horizontal symmetry: If a horizontal line passes through the pattern from right to left, divides the pattern in to identical halves, then such a pattern is said to have horizontal symmetry. Horizontal symmetry in the letters H, I, O, and X are shown in the figure 1.

  

FIGURE: 1 Horizontal symmetry in the letters H, I, O, and X.

Vertical symmetry: If a vertical line passes through the pattern from top to bottom, divides the pattern in to replicating halves, then such a pattern is said to have vertical symmetry. Vertical symmetry in the letters H, I, O, and X are shown in the figure 2.

  

FIGURE: 2Vertical symmetry in the letters H, I, O, and X.

Diagonal symmetry: If diagonal line divides the pattern in to replicating halves, then such a pattern is said to have diagonal symmetry. Diagonal symmetry in the letters H, I, O, and X are shown in the figure 3.

  

FIGURE: 3 Diagonal symmetry in the letters H, I, O, and X.

Cross diagonal symmetry: If two diagonal lines intersecting at the center of the pattern, divides the pattern in to identical portions, then such a pattern is said to have cross diagonal symmetry.Cross diagonal symmetry in the letters H, I, O, and X are shown in the figure 4.

  

FIGURE: 4 Cross diagonal symmetry in the letters H, I, O, and X.