TESSELLATIONS: An Application Of Simple Regular Polygons

Mary Racky                     Kenwood
                               5015 S. Blackstone
                               Chicago, Il. 60615
                               312-535-1409

Objective:

    The students will develop basic skills making and identifying homogeneous 
tessellations, both regular and semiregular. 

Materials needed:

    One overhead projector,
    One transparency of tessellation patterns with vertices marked and polygon
name listed below,
    One set of overhead transparency pens,
    Two - four small plastic bingo chips, 
    One set of plastic regular polygon shapes made from a tessellation pattern 
consisting of 10 equilateral triangles, 6 squares, 4 octagons, 4 duodecagons,  
    One set of construction paper regular polygon shapes for each student in the 
class made from the same tessellation pattern as the plastic overhead polygons,     
    One set of 6 to 8 construction paper circles of diameter 1 inch in a color 
to contrast with the floor of the room being used. 

Strategy:

    The first phase consists of various groups with a large surface for a 
working space taking about 5 minutes to "investigate" the contents of an 
envelope containing regular polygon shapes to see what they are and what they 
can do. 
    The second phase consists of students working with their own polygon pieces 
to develop a pattern they can illustrate is repetitive using only one polygon 
shape.  Volunteers should display their results using the plastic display pieces 
for the overhead.  Then conclusion number one is presented by introducing 
regular homogeneous tessellations from their discoveries.                    
    The third phase consists of combinations of regular polygons being used to 
develop various 2 polygon semiregular tessellations.  These can be illustrated 
with the use of the plastic polygon pieces for the overhead machine.  This 
should be a somewhat limited display with emphasis on replication for 
tessellations.  Then conclusion number two is presented by identifying semi-
regular homogeneous tessellations from their discoveries. 
    The fourth phase consists of a summary of discoveries made to this point 
concerning requirements for tessellations gleaned from the previous experiments.        
    The fifth phase consists of combinations of regular polygons created by 
students using 3 polygons in each pattern.  It is "hoped" that a student will 
attempt to use an octagon surrounded by an alternating pattern of squares and
3 triangles.  If not presented and no equivalent is presented, the octagon 
should be suggested for continued experimentation until such a pattern is found 
which contains "holes" or gaps between the consecutive polygons.  This will lead 
to a discussion of the last condition necessary for a tessellation concerning 
the sum of the angles at the vertex of the tessellation. 
    The sixth phase consists of a brief discussion of the patterns of 4 squares 
on the floor outlined with paper tape and having one of the contrasting circles 
at the common vertex of the 4 squares.  This should conclude with the summary
of the use of the circle pattern of rotation (360 degrees) at the common vertex 
to determine true/false tessellation.
    The seventh phase consists of a return to the above mentioned octagon 
pattern, 8-3-3-4, displayed on a chalk board.  Continue with this display by 
inserting the degrees of the angles at the common vertex, 135-60-60-90, to prove 
this pattern is not a valid tessellation.  A reinforcement should be done using 
previous tessellation patterns displayed on the overhead.  An overhead pen or 
one of the small bingo chips can be used to mark the common vertex while 
students compute the total degrees found by rotating in a circle about the 
common vertex.
    The eighth and last phase consists of a brief introduction to the more 
artistic type of tessellations from the regular polygons with ideas concerning 
the "nesting" of patterns necessary to develop a tessellation.  It is possible 
to consider this an optional phase of unit one since it might also be considered 
phase one of unit two on tessellations. 

References:

    Rather than list all the materials used in bibliography form, I would 
suggest that the person interested in creating such a project obtain the 
catalogue from Creative Publications.  Investigate the many materials available 
including the overhead projector polygon pieces and wooden polygon pieces which 
could be used as an alternative to the paper pattern pieces I indicated I used.