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Now that we have tried some preliminary examples, what can we conclude? Can we predict, without actually trying, which regular polygons will tessellate by themselves? Of the regular polygons that we tried, which ones tessellated? What is distinct about these regular polygons that allows them to tessellate? We discovered that the equilateral triangle, square, and regular hexagon each have the common ability to fill the space around a vertex perfectly, whereas other regular polygons cannot. Therefore, it is likely true that regular polygons that can perfectly fill the space around a vertex will be able to tessellate. Notice that since each interior angle of a equilateral triangle
has 60°, six of these complete a full circle rotation of 360°.
Similar, each interior angle of a square has 90°, so four of these
complete 360°. Finally, each interior angle of a regular hexagon
has 120°, so three of these will complete 360°.
Following is a table summarizing these results. An example from
the table is as follows: Suppose the number of sides of a regular
polygon is 6. Then, the number of degrees in the interior angle
is 180(6-2)/6 which equals 120. (See Regular Polygons if you are confused.) The number of polygons need to fill 360
degrees is 360 divided by 120 which is 3. Thus, the numbers 6,
120, and 3 appear in the table below.
Only 3, 4, and 6 perfectly fill 360° In the last column, you see that some of the numbers have decimals, which represent "parts" of polygons needed to fill 360 degrees; since "parts" of polygons are not useful in tessellations, those polygons other than the 3-, 4-, and 6-sided triangle, square, and hexagon cannot tessellate. But do any regular polygons with more than 11 sides tessellate? As you can see, the number of polygons needed to fill 360 degrees decreases as the number of sides increases. So, as the number of sides increases past 11, the number of polygons needed will fall below 2.4. The next integer below 2.4 is 2. But having only two polygons around a vertex means both polygons have 180 degree angles (straight lines). Since there are no polygons having a straight angle (180°), we can see that 3, 4, and 6 are the only numbers of sides that work. In other words, equilateral triangles, squares, and regular hexagons are the only regular polygons that will tessellate by themselves.
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