Download Geometry For Middle School Teachers - Homework 1 Questions | MA 241 and more Assignments Mathematics in PDF only on Docsity! Shapes and Designs Extensions 1 1. Four squares can fit together perfectly in the plane surrounding a common vertex (since each interior angle of a square measures 90 degrees). Let’s call this a (4,4,4,4) cluster. Similarly, two squares and three equilateral triangles can fit together perfectly surrounding a common vertex. There are essentially two different ways to do this: (4,4,3,3,3) (where the squares are adjacent) and (4,3,4,3,3) (where the squares are not adjacent), and we will regard these as two different clusters. Note that we could have called this last cluster (3, 3, 4, 3, 4) as well—it still refers to the same cluster. However, (4, 4, 3, 3, 3) and (4, 3, 4, 3, 3) are not the same. (a) You have just seen three planar clusters. Determine all possible planar clusters that can be formed by fitting together combinations of regular polygons in the plane surrounding a common vertex. Be systematic in some fashion, so that you can be certain you have found all of them, and explain clearly how you know this. (b) Some of the clusters can be extended to tile the plane so that at every vertex, exactly the same cluster appears—the same sequence of polygons, in either clock- wise or counterclockwise order. For example, if you extend the (4, 4, 4, 4) cluster, you get the familiar tiling of the plane with squares, with four squares meeting at each vertex. Of the clusters you have found, determine which ones can be extended. Make a precise drawing of each one you have found. You may wish to use Geometer’s Sketchpad to make the drawings. (c) Choose one of the clusters that cannot be extended to tile the plane and prove that it cannot. 2. If the measures of the interior angles of a cluster of regular polygons surrounding a common vertex sum to less than 360 degrees, then the cluster will not be planar. Let’s call such clusters space clusters. For example, the (4, 4, 4) cluster, consisting of three squares meeting at a common vertex, is a space cluster. If you try to extend this cluster so that the same cluster surrounds each vertex, you will construct a cube. (a) Identify all possible space clusters consisting of a single type of regular polygon, and prove that you have found them all. For each one, extend the cluster to create a three-dimensional polyhedron. Build the polyhedron using Polydron. Look up the name of each polyhedron. (b) Show that the space cluster (3, 4, 3, 4) can be extended to make a polyhedron. Build the polyhedron using Polydron. Draw a careful picture of it. Look up the name of this polyhedron. 1
