Tessellations

M. C. Escher

            1. This initial hexagon shape can generate the entire tessellation through a series of translations. Visualize this for yourself. Imagine this hexagon shape being moved to the left and to the right. The pattern should coincide exactly. Then imagine this hexagon shape being moved in a diagonal direction (just as a regular tessellation of hexagons would). The pattern should also coincide exactly. Once you have convinced yourself that this hexagon shape can produce the rest of the tessellation by simple translations, move to step 2.

            2. We notice that three black creatures inside of this hexagon seem to be symmetric about the center of the hexagon. Upon further inspection, we realize that the hexagon can be divided into three sections, each of which is a rotated copy of the other. So, let us outline one of these three sections.

            3. We immediately notice that the diamond section can be divided in half. Each half is a mirror image of the other. So, let us outline one of the mirror halves, which turns out to be an equilateral triangle.

            4. To better understand how the pattern inside of this equilateral triangle was created, let us draw outlines over the important boundary lines of the creatures. After doing so, we recognize the rotational symmetry inside of this equilateral triangle. What this means is that the entire tessellation shown at the top was generated from a simple line that started at a vertex of an equilateral triangle and ended at the center of the triangle.

The following animation illustrates how to recreate the above tessellation. The process involves the following steps: rotate the line design twice around the center of the equilateral triangle (120 degrees each time), reflect entire design across the side of the equilateral triangle, rotate entire design around the center of the hexagon, and translate entire design to tessellate the plane:

Create your own computer tessellation using “Paint” Pull up from Start > Programs > Accessories > Paint

               Step One If the shift key is held down as the "rectangle" is drawn, it will make a square. 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).

Paint tools: “Star” free form tool Paint fill

                                       Step Two Using the “Star - Free-Form Select" 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". 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                       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. 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 "Fill With Color" paint can tool 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

Choose the transparent overlay

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 "Magnifier" tool useful to assist you. Once the detailed portions are magnified, it is much easier to zoom in and "pour paint" into the magnified areas.

Tip: 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 “Star free-form“ tool 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 “Star” tool 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 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.

Once the blue fish has been copied, 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.

Step Ten Using “star” tool, select the blue fish and drag it near the bottom left corner of the full-sized 640 x 480 pixel screen. Likewise select the yellow fish and drag it near the bottom right of the full sized screen. These two "building blocks will be copied in turn to form the first tessellation row across the top of the screen.

Begin by using the “Star" 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

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 “Star” 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.

Tip: 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. Continue building up alternating rows until you have created a tessellation like the one below. Save your creation in a file called "tessell8.bmp" (without quotes).

More internet sites with tessellations: http://library.thinkquest.org/16661/creating.other/index.html http://library.thinkquest.org/11750/index.shtml

http://www.geckostone.com/pavers.html http://mathforum.com/geometry/rugs/symmetry/fp.html

Created for West High School 2000-2001 By Elizabeth Waddington Technology Integration