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Semiregular Tessellations (2/4)
 

1. Introduction
2. Filling 360 degrees
3. Determining which
Arrangements Tessellate
4. The Semiregular
Tessellations
5. Hands-On Activities
Filling 360 degrees (continued)
Let us use a systematic approach to find all possible arrangements of regular polygons that will perfectly fill the space around a point.

First we will determine all possible polygon arrangements involving three polygons. Then we will determine all arrangements involving four polygons. Then five, six, and so on, until we realize we can go no further.

When we determine all possible polygon arrangements for a certain number of polygons, we will first assume that the minimum number of sides of one of the polygons is three (equilateral triangle). After we are done searching for these, we will then assume the minimum number of sides is four (square). After searching, we proceed to the minimum number being five (regular pentagon). We continue this until the minimum number makes it impossible to have a polygon arrangment that fills 360 degrees.

 

    A Systematic Approach to Find Valid Arrangments of Regular Polygons

    Arrangements with three polygons

    smallest number of sides: 3

    Three Polygons, Minimum: 3

    The three-polygon arrangements with 3 as the minimum

    Arrangements: (3,7,42) (3,8,24) (3,9,18) (3,10,15) (3,12,12)

     

    smallest number of sides: 4

    Three Polygons, Minimum: 4

    The three-polygon arrangements with 4 as the minimum

    Arrangements: (4,5,20) (4,6,12) (4,8,8)

     

    smallest number of sides: 5

    Three Polygons, Minimum: 5

    The only three-polygon arrangment with 5 as the minimum

    Arrangements: (5,5,10)

     

    smallest number of sides: 6

    Three Polygons, Minimum: 6

    The only three-polygon arrangment with 6 as the minimum

    Arrangements: (6,6,6)

    Arrangements with four polygons

    smallest number of sides: 3

    Four Polygons, Minimum: 3

    The combination of 3, 3, 4, 12 results into two distinct arrangements
    Four Polygons, Minimum: 3

    The combination of 3, 3, 6, and 6 produces two distinct arrangements
    Four Polygons, Minimum: 3

    The combination of 3, 4, 4, and 6 produces two distinct arrangements

    Arrangements: (3,3,4,12), (3,4,3,12) --- (3,3,6,6), (3,6,3,6) --- (3,4,4,6), (3,4,6,4)

     

    smallest number of sides: 4

    Four Polygons, Minimum: 4

    The only four-polygon arrangment with 4 as the minimum

    Arrangements: (4,4,4,4)

    Arrangements with five polygons

    smallest number of sides: 3

    Five Polygons, Minimum: 3

    The combination of 3, 3, 3, 3, and 6 produces one arrangement
    Five Polygons, Minimum: 3

    The combination of 3, 3, 3, 4, and 4 produces two distinct arrangements

    Arrangements: (3,3,3,3,6) --- (3,3,3,4,4), (3,3,4,3,4)

    Arrangements with six polygons

    smallest number of sides: 3

    Six Polygons, Minimum: 3

    The only six-polygon arrangement with 3 as the minimum

    Arrangements: (3,3,3,3,3,3)

 

Finally, we are done. In summary, the images on this page represent the 21 arrangements of regular polygons that will fill the 360 degrees around a point. There are actually only 17 different combinations of regular polygons--4 of these combinations each produce two different arrangements.

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Real examples of semiregular tessellations:


TemplatesTo browse full-page templates of the semiregular tessellations that are ready to be printed, proceed to the templates page:

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