TESSELLATIONS: An Application Of Simple Regular Polygons

Mary Racky Kenwood
5015 S. Blackstone
Chicago, Il. 60615


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.


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.


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.
Return to Mathematics Index