Geometry: An Interactive Journey to Mastery

Lesson 28: Tilings, Platonic Solids, and Theorems

Tilings, Platonic Solids, and Theorems

Lesson 28

Topics

x Platonic solids.

x The Pythagorean theorem revisited, Napoleon’s theorem, midpoint areas, and variations.

Summary

The visual beauty of regular tilings provides intellectual beauty, too: We can prove mathematical theorems via

tiling patterns. In this lesson, we pick up on the tiling theme of Lesson 4 and establish mathematical results.

Example 1

Why is it not possible to make a convex three-dimensional polyhedron with faces that are congruent regular

octagons?

Solution

,IVXFKDVROLGH[LVWHGWKUHHRUPRUHRFWDJRQDOIDFHVZRXOGPHHWDWHDFKYHUWH[RIWKH¿JXUH)RUWKH¿JXUHWR

be convex, the sum of the interior angles of the octagons meeting at each vertex must be less than 360°.

However, each interior angle of a regular octagon is 180° 360° 8 135°.

7KUHHRUPRUHRIWKHVHDQJOHVVXPWRJUHDWHUWKDQƒ6XFKD¿JXUHFDQQRWEHFRQVWUXFWHG

Example 2

A square is inscribed in a circle, and a rectangle, with one side

twice as long as the other, is inscribed in the same circle.

ͼ௘6HHFigure 28.1௘ͽ

Establish that the area of the rectangle is^45 the area of the square.

Figure 28.1