The Teaching of Geometry

by David Eugene Smith

We therefore have the following regular polygons:

From the equilateral triangle, regular polygons of 2ⁿ · 3 sides;
From the square, regular polygons of 2ⁿ sides;
From the regular pentagon, regular polygons of 2ⁿ · 5 sides;
From the regular pentedecagon, regular polygons of 2ⁿ · 15 sides.

This gives us, for successive values of n, the following regular polygons of less than 100 sides:

From 2ⁿ · 3, 3, 6, 12, 24, 48, 96;
From 2ⁿ, 2, 4, 8, 16, 32, 64;
From 2ⁿ · 5, 5, 10, 20, 40, 80;
From 2ⁿ · 15, 15, 30, 60.

Gauss (1777-1855), a celebrated German mathematician, proved (in 1796) that it is possible also to inscribe a regular polygon of 17 sides, and hence polygons of 2ⁿ · 17 sides, or 17, 34, 68, ..., sides, and also 3 · 17 = 51 and 5 · 17 = 85 sides, by the use of the compasses and straightedge, but the proof is not adapted to elementary geometry. In connection with the study of the regular polygons some interest attaches to the reference to various forms of decorative design. The mosaic floor, parquetry, Gothic windows, and patterns of various kinds often involve the regular figures. If the teacher uses such material, care should be taken to exemplify good art. For example, the equilateral triangle and its relation to the regular hexagon is shown in the picture of an ancient Roman mosaic floor on [page 274].[84] In the next illustration some characteristic Moorish mosaic work appears, in which it will be seen that the basal figure is the square, although at first sight this would not seem to be the case.[85] This is followed by a beautiful Byzantine mosaic, the original of which was in five colors of marble. Here it will be seen that the equilateral triangle and the regular hexagon are the basal figures, and a few of the properties of these polygons might be derived from the study of such a design. In the Arabic pattern on [page 276] the dodecagon appears as the basis, and the remarkable powers of the Arab designer are shown in the use of symmetry without employing regular figures.

Problem. Given the side and the radius of a regular inscribed polygon, to find the side of the regular inscribed polygon of double the number of sides.