In the "Finding Patterns"
section of the Beyond Numbers exhibit, visitors use various regular polygons
to "tile the plane" and create periodic motifs. Visitors also add tiles
to an existing quasi-periodic Penrose pattern resulting in self-similar,
but never periodic, patterns that could be extended to infinity.
NCTM
Standards and/or AAAS Benchmarks addressed:
Mathematics as Reasoning: Students understand and apply reasoning processes,
with special attention to spatial reasoning and reasoning with proportions
and graphs.
Mathematics as Communication: Students develop common understanding of mathematical
ideas, including the role of definitions.
Patterns and Functions: Students describe, extend, analyze, and create
a wide variety of patterns.
The Mathematical World, Shapes: Students learn that many objects can be described
in terms of simple plane figures and solids. Shapes can be compared in
terms of concepts such as parallel and perpendicular, congruence and similarity,
and symmetry. Symmetry can be found by reflection, turns or slides.
VOCABULARY
quasi-periodic -- in reference to a tiling,
self-similar but non- periodic patterns that can extend to infinity
polygon -- usually a plane closed figure
having three or more sides
periodic -- appearing at regular intervals;
in reference to tiling, having patterns appearing at regular intervals
tiling -- a collection of similar shapes
laid on a plane that can extend to infinity without gaps or overlaps
Penrose tiling -- a kind of quasi-periodic
tiling devised by mathematician Roger Penrose
"town theorem" -- "In any Penrose tiling,
for every region R of diameter d, there is an exact replica of R within
a distance 2d in some direction." John Horton Conway
translation -- a motion
that involves sliding without rotation or reflection; a composite of two
reflections over parallel lines
rhomb (rhombs) -- a rhombus, an equilateral
parallelagram
INTRODUCTION
In the following activities, students compare and contrast quasi- periodic
tiling with periodic tilings. In a more systematic approach, students usually
explore periodic tilings before learning about quasi-periodic tiling. In
this lesson students learn what periodic tiling is by seeing the absence
of periodicity in quasi-periodic tilings.
OBJECTIVES
Students will compare and contrast quasi-periodic tiling with periodic
tiling.
MATERIALS
compasses
Worksheets A, B, and C
acetate copies of 3 tilings
The lesson plan has been written with the assumption that each student
will receive a worksheet and an acetate copy of the 3 tilings: the Rhombburst,
the Penrose tiling , and the periodic tiling. If you are unable to use
a copier to make acetate copies, use the two copies included in this manual
on an overhead projector.
Our section of the Rhombburst, has been reproduced with the permission
of Alan Schoen. A colorful 25" by 38" Rhombburst Poster, by Alan Schoen,
can be obtained for around $12 (which includes mailing) from:
Kaleidos
316 West Oak St.
Carbondale, IL 62901
(618)-529-4677
PROCEDURE
Activity 1 - Lines of symmetry Use your discretion to decide if
this lesson will take more than one class period. Distribute the Rhombburst
acetate sheet to the class. Let students share their informal observations
about the pattern. Then distribute the Rhombburst worksheet. Have students
follow worksheet directions and write descriptions of what generalizations
they can make. Repeat this procedure with the Penrose tiling acetate and
worksheet and then the periodic tiling acetate and worksheet. The Rhombburst
and the Penrose tilings are not periodic. When a region is repeated or
found in another part of the tiling, the surrounding patterns are not necessarily
the same. Since there is a "quasi-regular" repetition (as illustrated by
the "town theorem"), these tilings are called quasi-periodic.
ASSESSMENT
Have students complete a table comparing and contrasting periodic tilings
and quasi-periodic tilings.
Use this answer key to evaluate the table.
How all three tilings are alike
Quasi-periodic tiling is not like periodic tiling because ...
Periodic tiling is unlike quasi- periodic tiling because ...
Are on paper
Are made of rhombs
Have reflection symmetry
Have rotational symmetry
Can extend infinitely
No gaps between tiles
Infinite repetitions of any region
Regions have the same proportion of different tiles
It does not repeat "regularly".
Has no fundamental region
It can't have five-fold symmetry.
Has fundamental region that translates to tile the plane
WORKSHEET A - Rhombburst
This tiling was created by Alan Schoen. It is composed of four rhombic
shapes and is named Rhombburst.
Overlap the acetate Rhombburst on the one on your paper. Try rotating
the acetate from a selected point to find the rotational symmetry of this
tiling. On another piece of paper, describe what you find.
Flip the acetate over and see where the "reflection" fits over the
Rhombburst. Describe what you find.
What other patterns can you see in the Rhombburst?
WORKSHEET B - Penrose Tiling
This tiling, called the Penrose Tiling, was created by Roger Penrose.
It is composed of two rhombic shapes.
Overlay the acetate Penrose tiles on this paper. Explore the rotation
and reflection symmetries. What do you find?
Draw a circle with a 2 centimeter (or 3/4 inch) radius on any part of
this pattern. In that segment, outline and shade a group of rhombs of any
shape. Use a compass to estimate the diameter of your group.
Around your shaded group, draw an arc with a radius of two diameters'
length. Within that arc you should be able to find an exact replica of
your outlined segment. Here is an example.
Note the surroundings of the replica. Are they also identical?
WORKSHEET C - Periodic Tiling
This is a periodic tiling made of two rhombic shapes. In a periodic
tiling, any region repeats an infinite number of times under a fixed translation.
A translation is made when all points on the region move the same distance
in the same direction.
Move the acetate tiling over the paper tiling, keeping the edges of
both the acetate and the paper parallel. Notice that any group of rhombs
translates many times.
Choose a small group of rhombs at the bottom of the page. How many times
does the group translate on the page? Be sure to try all directions.
Note that when the group of a periodic tiling translates, the surrounding
patterns are always the same.
