Tessellation by Reflection and Rotation
Using a 120 Triangle

Polygons that can be used: 

Any 120 degree isosceles triangle

Symmetry: 

Translational symmetry
3 pt. rotational symmetry
Reflectional symmetry
Glide-reflectional symmetry

Technique:

1) Draw the beginning polygon, a 120 degree isosceles triangle (see Figure 1).

2) Use the pencil to draw a line design from the bottom vertex to the middle vertex. To avoid an overlap, stay close to the triangle line near the bottom (see Figure 1).


Figure 1  

3) “Lasso” the line design to select it. Copy it and paste a copy. Go into TRANSFORMATIONS and rotate the copy a NEGATIVE 120 degrees. Then move the copy so it lines up with the top side (Figure 2).


Figure 2

4) “Lasso” the entire piece, copy it, and paste a copy. Go into TRANSFORMATIONS and FLIP HORIZONTALLY. Move the copy so it fits back-to-back with the original. Your piece is now done (see Figure 3).


 Figure 3

5) Erase the lines of your original polygon (in this case, the triangle) and any other extra lines (especially the line down the middle) so only your design can be seen. You now have a tessellating piece. Add any details with pencil at this time (Fig. 4).


Figure 4: Christmas tree Figure

6) “Lasso” your entire piece to select it. Copy it and paste a copy. Rotate the copy a POSITIVE 120 degrees. Move (translate) the copy so it fits into the top right side of the original like a puzzle piece. Paste another copy and rotate it a POSITIVE 240 degrees. This piece should fit against the lower right side of the original. You should now have a “triplet” of your piece (see Figure 5). (If you are coloring, you might wish to color the three pieces in three different ways at this time.) “Lasso” this entire triplet and copy it. Paste a copy and move (translate) it to fit like a puzzle (no more rotating!). Continue to paste and move the triplet until the entire page is covered. (To get pieces to fit off the sides you may need to pick up the piece near its sides.) See completed tessellation.


5: “Triplet”