|
Part
1:
- Quickly sketch
a pure tessellation using a regular polygon on your paper. Shade in
one of the tessellated polygons, and focus your attention on one of
the polygon’s vertices. How many polygons share that vertex?
Comment on the sum of the measures of all of the angles which share
that vertex. How is this sum connected to the ability to tessellate
your polygon? Compare your findings with your neighbor.
- What is the measure
of each interior angle of a regular pentagon? Can three or more non-overlapping
pentagons share a common vertex? Complete a chart, similar to the one
below, to organize your thoughts.
|
Name
of Polygon
|
.
|
.
|
.
|
.
|
.
|
.
|
.
|
.
|
|
#
of sides of a regular polygon
|
3
|
4
|
5
|
6
|
7
|
8
|
10
|
12
|
|
#
of degrees in each interior angle of the regular polygon
|
.
|
.
|
.
|
.
|
.
|
.
|
.
|
.
|
|
if
tessellated, # of polygons that share a common vertex
|
.
|
.
|
.
|
.
|
.
|
.
|
.
|
.
|
- How is the chart above related to the ability to tessellate a regular
polygon? What are the necessary and sufficient conditions to create
a pure regular tessellation?
- How many pure tessellations can be created using regular
polygons? Justify your answer to your partner.
- What kind of transformations can you use to create a tessellation?
Describe each in detail.
- Using commands under the Transform menu, The Geometer’s Sketchpad
enables you to Translate, Rotate, and Reflect figures.
Practice using each of these commands as you create as many regular
pure tesellations as possible.
- Using your script, construct an equilateral triangle. Tile your sketch
window using equilateral triangles. Compare your tiling process with
your neighbor’s.
When using the following commands, remember that angle measurement units
and distance units can be set by selecting the Preferences
command under the Display menu.
To translate a polygon by a rectangular vector:
- Select the polygon interior, sides and vertices of the figure
you wish to translate.
- Choose Translate from the Transform menu.
- At the bottom left-hand side of the window, mark by Rectangular
Vector.
- Enter in values for the horizontal and vertical translation,
and click OK.
|
To translate a polygon by a polar vector:
- Select the polygon interior, sides and vertices of the figure
you wish to translate.
- Choose Translate from the Transform menu.
- At the bottom left-hand side of the window, mark by Polar
Vector.
- Enter in values for the angle and distance, and click OK.
|
To translate a polygon by a marked vector:
- Mark a vector of the desired translation. To do this, select
two points and under the Transform menu, select Mark
Vector "A® B". Your
order in which you select your points will determine the direction
of the vector.
- Select the polygon interior, sides and vertices of the figure
you wish to translate.
- Choose Translate from the Transform menu.
- In the window verify the desired translation, and click OK.
|
To rotate a polygon by a fixed angle:
- Select the point you want as a center pivot point.
- Under the Transform menu, select Mark Center "X".
- Select the polygon interior, sides and vertices of the figure
you wish to rotate.
- Choose Rotate from the Transform menu.
- [On a Mac computer...At the bottom left-hand side of the window,
mark by Fixed Angle.] [On a PC computer...uncheck the by
Marked Angle box.]
- Enter in values for the rotation angle, and click OK.
|
To rotate a polygon by a marked angle:
- Select the point you want as a center pivot point.
- Under the Transform menu, select Mark Center "X".
- Mark an angle for the desired rotation. To do this, select three
points (in the conventional order point, vertex, point), and under
the Transform menu, select Mark Angle "X-Y-Z".
Your order in which you select your points will determine the
direction of the angle.
- Select the polygon interior, sides and vertices of the figure
you wish to rotate.
- Choose Rotate from the Transform menu.
- In the window verify the desired rotation, and click OK.
|
To reflect a polygon:
- Select any straight object in your sketch window as the desired
mirror.
- Under the Transform menu, select Mark Mirror "q".
- Select the polygon interior, sides and vertices of the figure
you wish to reflect.
- Choose Reflect from the Transform menu.
|
- Using your script, construct a square. Tile your sketch window using
squares. Compare your tiling process with your neighbor's.
- Using your script, construct a regular hexagon. Tile your sketch window
using regular hexagons. Compare your tiling process with your neighbor's.
- Regular tessellations that involve more than one shape are called
semi-pure regular tessellations. How might an equilateral triangle,
a square, and a regular hexagon together tessellate the plane? Demonstrate
using the Sketchpad. (Hint: Start by placing squares on each side of
a regular hexagon.) There are a finite number of semi-pure regular tessellations
(actually less than ten!). Working together as a class, try creating
as many semi-pure regular tessellations as possible. Share your tessellations
with the class.
Part 2:
- Up to this point, you have made tessellations with regular polygons.
You have demonstrated that there are only three pure tessellations of
regular polygons and eight semi-pure regular tessellations. Do you know
of a non-regular figure that tessellates? Will a scalene triangle tessellate?
You might want to attack this problem concretely first (i.e., draw and
cut out some congruent scalene triangles and try to construct a tessellation
physically). Then using Geometer's Sketchpad, try to construct a tessellation
in a new sketch window.
- Observe the angles around each of the triangle's vertices. How many
times did each angle of the triangle fit around each vertex? What is
the sum of the measures of the three angles of a triangle? Compare you
results with the results of your neighbor. State a conjecture about
your findings.
- Record any difficulties you had while trying to tessellate the plane
with a scalene triangle using the Sketchpad. How could you remedy these?
- Do a similar investigation as above, and create a tessellation with
a non-regular convex quadrilateral. Does any convex quadrilateral tessellate
the plane? State a conjecture about your findings.
- Investigate whether a concave quadrilateral will tile the plane. Create
your own concave quadrilateral and try to create a tessellation with
it. Document your attempt by describing your process on paper.
- We have explored some of the beauty of tessellations. M. C. Escher,
an artist and mathematician, has become famous for his wonderful tessellations!
Escher spent many years learning how to use translations, rotations,
and reflections to create his masterpieces. Study the Escher tessellations
below.
- In the left-hand tessellation, focus your attention on one of the
tail’s vertices. How many tadpoles share that vertex? Comment
on the sum of the measures of all of the angles which share that vertex.
How is this sum connected to the ability to tessellate the tadpole?
- Explain Escher's use of symmetry in each of the above tessellations.
What different transformations were used to create these tessellations?
You can view additional works of Escher on the web (see the Additional
Resources below).
- Use "A Tessellation Tutorial" http://forum.swarthmore.edu/sum95/suzanne/tess.intro.html
from The Math Forum to create your own Escher-like tessellations
using the Sketchpad. Write a paragraph about the construction of your
tessellation.
Extension:
- The procedure you used in "A Tessellation Tutorial" can
be generalized to create other Escher-like tessellations. Create an
Escher-like tessellation by starting off with a figure that is not
a parallelogram. Write a paragraph about the construction of your tessellation.
|
