1. The diagram will require paper 20 X 20 cm. or 6 X 6 inches.

Construct the equilateral triangle XYZ, each side 9 cm. (or 4 in.) long. Divide each side into equal parts, each 3 cm. (or 1 in.) long, the points of division being A, B, C, D, E, and F.

Draw AB, CD, and EF, thus completing the interior part of the diagram ABCDEF, which has six dotted sides all equal.

Upon each of the six sides AB, BC, CD, etc., construct an isosceles triangle with the angles at A, B, C, etc., each 75°, thus forming the six-pointed star GHIJKL.

CHAPTER XI

POLYGONS AND SYMMETRY

1. You have been told that pyramids take their names from the shape of their bases. Now the bases, like all faces, take their names as follows:

First, from the number of edges or corners, the number of edges being the same as the number of corners.

Secondly, from equality in the length of the edges.
Thirdly, from equality in the size of the angles.
Fourthly, from equality in both edges and angles.

Fifthly, from peculiarity in the arrangement of the edges or angles.

The general name for a face is polygon (pol'-y-gon), which means "having many corners; but this name is usually applied only to faces which have that is, more than four edges.

more than four corners,

Equilateral Polygons

If the edges of a face are all equal to each other, it is called

an equilateral polygon.

If the angles of a face are all equal to each other, it is called an equiangular polygon.

Equiangular Polygons

If a face is both equilateral and equiangular, it is called a regular polygon.

Regular Polygons

2. A polygon is symmetrical with respect to a straight line when this line divides it into two parts such that, if the figure

ло

Symmetrical Polygons

be revolved on the line, the two parts will exchange places, each exactly covering the space formerly occupied by the other. The straight line is called the axis of symmetry.

You can test this by an experiment. First construct a symmetrical polygon as follows: Draw a square ABCD with edges 4 cm. (or 2 in.) long.

Draw EF connecting the middle points of two opposite edges AB and DC. Divide each edge into four equal parts.

Draw PL and MN connecting the points of division nearest D and C. Draw EP and EN.

A symmetrical polygon LMNEP will thus be formed, of which EF is the axis. Cut out this polygon, using a ruler and knife so as to preserve the edges of the gap left in the paper. Then turn the polygon over, and replace it in the paper in the reversed position. You will see that the ends of the axis EF are in their former position; but N and P, and M and L, have exchanged places; thus all points in the polygon, except those in the axis, have exchanged places with other points.

Draw the following figures, all of which are symmetrical with respect to a line, and then draw their axes:

1. An isosceles triangle.

2. A straight line.

3. An angle with equal sides.

4. An equilateral triangle (three axes).

5. A square (four axes).

6. A straight line met at its middle point by two equal straight lines so as to form three angles each 60°.

7. A rectangle (two axes).

8. A parallelogram with equal angles.

9. A rhombus (two axes).

10. A trapezoid with two equal sides.

Two figures when considered together may be symmetrical with respect to a line.