Background and Goal
Although this information was originally written to
assist educators of Grade 6 children, it can easily be adapted
for students in any grade level. Certainly, there is a
tight-curricular fit with the Grade 6 Mathematics area but I
would encourage educators from junior and senior high, as well as
teachers wishing to integrate tessellations into an Art unit, to
become involved.
For those wishing to explore how this tessellation activity integrates into the current curriculum, consider the following goal:
Goal: To create Escher tessellations (which involves using translations - slides) as part of the Grade 6 Mathematics unit on Shape and Space (Transformations)
I would recommend that classroom teachers begin this activity by having students use scissors to construct tessellations on paper as suggested in the Grade 6 mathematics curriculum guide on pages D116 - 119. Once students have grasped the more concrete format, they are better prepared to participate in the more abstract computer-generated tessellations described below.
These instructions are written so that educators and students can use either the Paintbrush software found on all Windows 3.x microcomputers or the Paint alternative found on all computers using the more recent Windows 95 operating system. If you wish to use this article with your students, simply cross out the references to the software application (Paintbrush or Paint) for which your students have no access.
Creating Tessellations Using "Paintbrush" or
"Paint"
In this "Let's Get Connected" activity, students must
share their creations in a standard 512 pixel wide by 342 pixel
high frame. In order to ensure that the final image conforms to
this standard as well as to ensure that the participant has
work-space room for copying the various tessellation building
block components, one should initially configure the image
properties as outlined below:
Paintbrush participants should start the application and then select the "Options/Image Attributes" menu items. Paint users should begin by starting the application and then select the "Image/Attributes" menu items. Select the "in" or "Inches" as "Units" by clicking on the appropriate radio-button. With the attribute of the picture now set to inches, change the "Width" to 8.5 and the "Height" to 11 so that the empty image now conforms to a standard 8.5" x 11" format. Similarly make certain that the radio button beside "Colors" is selected and click on the "OK" button. This layout will provide both space for the resulting 512 x 342 pixel tessellation near the top of the page and leave adequate space for constructing and copying tessellation building blocks in the lower half of the page. Next move your mouse to the top left-hand corner of the resulting 8.5" x 11" blank sheet before proceeding to the steps outlined below.
Step One
Select your favorite color from the menu. In Paintbrush,
use the "Filled Box" rectangle tool to draw a square.
If the shift key is held down as the "rectangle" is
drawn, it will make a square. In Paint, the "Rectangle
Tool" will only draw the outline of the square in the color
chosen, so one must also choose the "Fill With Color"
paint can tool to make the square a solid (for example, blue)
color as shown. It is recommended that you save your work
frequently as you proceed through this activity. Save your
creations as as 256 color bitmap (BMP) files after each step
(e.g. step1.bmp, step2.bmp, step3.bmp ... ). Such a procedure
will allow one to recover from a mishap without the need to start
from the very beginning. Now save this colored square as
"step1.bmp" (without quotes).
Step Two
Using the "Scissors" (Paintbrush) or the
"Free-Form Select" (Paint) tool, start on the top edge,
wiggle around as you are going down and exit the lower edge of
the square. Continue dragging the mouse around the left side of
the square and link up with the starting point. Once the loop is
complete and the mouse button is released, the "cut"
line will become dotted indicating the segment is
"active". In Paintbrush, the dots will outline the path
drawn whereas in Paint the dots will form a rectangular outline.
Depress the mouse button to drag the active left side away from
the initial square as shown. The active element will be
transferred to the right side of the square as illustrated in the
next step.
Step Three
Reposition the "active" portion on the right side
making certain that the two vertical edges align exactly and that
the top and bottom edges are straight. Once the two edges tightly
butt up against one another (without any overlap) and the
alignment is exact, release the mouse button to re-position the
cut area. Save your creation as "step3.bmp" (without
quotes).
Step Four
Repeat the process but this time go from top to bottom
or bottom to top. To aid in the repositioning of the bottom
segment, one may wish to start the cut at a bottom corner. One
can then use this as a reference point when placing the bottom
segment on the top of the shape as illustrated below.
Step Five
Look at the figure. What does it look like to you? To
me, the shape on the left looks like a tropical fish. Remember to
save your creation as "step5.bmp" (without quotes). I
am going to add "internal" contrasting colors to give
this fish some added detail.
Step Six
Use the pencil, line or paint brush tool (with a
contrasting color selected) to decorate the figure. In this case
I have chosen yellow to add detail to the fish body. If you are
viewing this activity on a printed page, the images will appear
as differing shades of gray whereas the true colors will appear
if you are viewing this on-line. Do not add anything to the outer
edges of the silhouette. Save your image as "step6.bmp"
(without quotes).
Step Seven
Use the rectangular scissors tool to draw a rectangle
around the image. When a dotted frame surrounds the image
indicating it is now "active", click on the
"Edit/Copy" option. Next select "Edit/Paste"
to position a second copy of the image on the screen. Drag this
"active" image and position it to the right of the
original as shown. Save the image pair as "step7.bmp"
(without quotes).
Step Eight
At this point, one must exchange colors between the two
fish. Ultimately the blue fish with yellow highlights (on the
left) will be have a partner which will eventually become a
yellow fish with blue highlights (on the right). Do not use the
"Paint Roller" tool (in Paintbrush) or the "Fill
With Color" paint can tool (in Paint) to color the body
yellow on the right fish. If you do, the entire fish shape will
take on a yellow color and you will not be able to distinguish
any details. You must use an intermediary color (different from
either of the two color choices of the original fish on the left)
for the body of the right fish. Use the appropriate tool to paint
the body of the fish on the right with a different color (for
example, light green) which will still provide a contrast between
the original body (blue) and details (yellow). Once this is
accomplished as shown above, one can now paint the details blue
on the right fish. You may find the "View/Zoom" menu
options (in Paintbrush) or the "Magnifier" tool (in
Paint) useful to assist you. Once the detailed portions are
magnified, it is much easier to zoom in and "pour
paint" into the magnified areas. Remember to utilize
"Edit/Undo" (or Ctrl-Z) immediately after any painting
mishap to allow one to correct any mistakes. Once all the details
are painted blue on the fish on the right, it is a simple process
to fill the body of the fish yellow as illustrated below.
Step Nine
Now there should be one darker figure (a blue fish with
yellow details on the left) and one lighter figure (a yellow fish
with blue details on the right) with opposite colors as
indicated. You may find it beneficial to use the "Scissors
Tool" (with "free-form" star shape in the top left
corner of the tool bar) to select the fish on the right and drag
it further to the right. If the two fish are sufficiently
separated, each can be selected by using the "Scissors Pick
Tool" (shown as the top right tool with the rectangle) by
quickly dragging a rectangle around each one as opposed to having
to drag a mouse carefully between the "nose" of one and
the "tail" of the other fish. Once the fish are
separated, save the contrasting image pair as
"step9.bmp" (without quotes). These two contrasting
fish are going to be the basic building blocks for the
tessellation that you will build by copying and pasting each fish
in turn as outlined in the next step.
Step Ten
Using "Scissors Pick Tool" (which can be
identified by the rectangle in the top right corner), select the
blue fish and drag it near the bottom left corner of the
full-sized 8.5" x 11" screen. You may need to click on
the vertical and horizontal scroll bars to move between the top
and bottom portions or left and right sides of this larger page
format. Likewise select the yellow fish and drag it near the
bottom right of the full sized screen. These two "building
blocks originals" will will be positioned near the bottom of
the page and copied in turn to form the first tessellation row
across the top of the screen. Begin by using the "Scissors
Pick Tool" to select the blue fish near the bottom left
corner. Once it is selected (or "captured" within the
dotted rectangle), click on the "Edit/Copy" menu items.
This action (which can be accomplished with the
"Ctrl+C" shortcut command combination) places a copy of
the blue fish in the computer's memory buffer. It is important
that you copy the blue fish to the computer buffer rather than
simply move it, as you want to keep the original available so
that you can make more copies from the two fish in the bottom
corners. Once the blue fish has been copied, move to the top of
the screen by clicking on the vertical scroll bar and select the
"Edit/Paste" menu command (or the "Ctrl+V"
command combination) to paste the copy from the memory buffer on
to the screen. By default, the image will be placed in the top
left hand corner. While this image has a dotted rectangle
surrounding it, one can move it as it is "active" by
dragging it to a new location. Drag it a short distance away from
the top and left hand edge. When you click off the image, the
blue fish will be anchored in place.
Now one must repeat the process by taking a copy of the yellow fish and placing it "tail to nose" with the blue fish near the top of the screen to start the first tessellation row. Use the "Scissors Pick Tool" to select the yellow fish near the bottom right corner. Once it is selected (or "captured" within the dotted rectangle), click on the "Edit/Copy" menu items to copy the yellow fish's image to the computer buffer. Next select the "Edit/Paste" menu command to paste the yellow fish into the top left hand corner of the screen. While the image is "active" (shown by the dotted rectangle surrounding the yellow fish), drag the yellow fish and carefully fit it into position so that the tail and nose interlock. Hint: If the two fish do not interlock exactly, review Steps 1 - 8 and try again. Continue with this procedure until you have created a complete row of alternating fish which span the 8.5" screen width as illustrated below and save the row as "step10.bmp" (without quotes).
Step Eleven
Once a row has been completed, the creator may be able
to copy an entire row (or alternating portions) and position them
in a staggered format to created a series of rows down the page
as illustrated below:
Although this tessellation is starting to develop, it is very important that copies of the blue and yellow "building block" fish from the bottom of the page are strategically positioned around the outside edges of the frame so that there is no white space showing through. Save the resulting tessellation as "step11.bmp" (without quotes).
Step Twelve
Once the entire width has been spanned, the top half of
the page has been completely filled, and all the portions around
the top edges of the frame have been filled, it is necessary to
resize the image to conform to the default tessellation size
dictated in this "Let's Get Connected" activity. To do
this one needs to modify the attributes of the image and change
its dimensions from inches to pixels (pels). Paintbrush users
should select the "Options/Image Attributes" menu items
while Paint users should select the "Image/Attributes"
menu items. Click on the radio-button beside "pels" to
indicate that you want the dimensions changed to pixels. Next
change the "Width" to 512 and the "Height" to
342 pixels and click on the "OK" button. The image will
now automatically crop the right hand and bottom edge so that
your tessellation design fits the default standard picture size.
(Educators who are familiar with graphic conversion programs like
"Lview" may wish to open the larger 8.5" x
11" file named "step 11.bmp" with such an
application and then use the rectangular cropping tool to
diagonally drag and define a 512 x 342 image.) Regardless of the
process used, save this 512 x 342 tessellation image as
"step12.bmp" (without quotes).
Step Thirteen
Time permitting, creators may wish to add the following options
to their tessellation as shown below:
To add text to the creation, Paintbrush users can select the "abc" tool while Paint users can double-click on the "T"ext tool from the tool box. One may now be able to select the Text Style including the font, size and color. Click on the tessellation and begin typing in the information one wishes to include. If the information is too large or is a poor choice of color, it is important to back-space to remove inappropriate text styles. Once one clicks elsewhere, the text will be "fixed" on the tessellation and the only way to recover is to go to a previously saved version (such as "step12.bmp") and re-enter the text. Once the descriptive information has been entered correctly, save the resulting "autographed" tessellation as "step13.bmp" (without quotes).
Step Fourteen
In preparation for sending your tessellation as an e-mail
attachment to the facilitator of this "Let's Get
Connected" activity, one must give it an unique filename.
Save your previously created 512 x 342 pixel defined tessellation
"step13.bmp" one last time using the standard
"8.3" rule in which the filename does not exceed eight
characters and the filename suffix (in this case BMP) contains
three characters. Use your school name (up to 5 characters) and a
three-digit number (which your teacher will assign in consecutive
order) to create a unique file name to save this picture. For
example, some possible filenames might be:
Save your tessellation, as a bit-mapped file, and return to the "Teaching 'n' Techniques" page, by clicking on the link below to continue with the sharing process.
Return to "Teaching 'n'
Techniques" to continue