Return to Mathematics IndexTESSELLATIONS: An Application Of Simple Regular Polygons

Mary Racky Kenwood

5015 S. Blackstone

Chicago, Il. 60615

312-535-1409Objective:

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.