Math 487 Lab 8

Tessellation Lab

Tessellations need not be periodic. (For example, strips of rectangles randomly displaced at each level.) We will concentrate on tessellations that turn out to be periodic. This means that there is a basic tile M and two vectors v and w so that translates of M by (Tv)j(Tw)k tile the plane. More about this later.

Irregular shapes

Quadrilaterals

• Cut out the quadrilaterals on the sheet that you are given. Start with the two convex congruent quadrilaterals. Working with other students and their quadrilaterals, arrange these quadrilaterals as a tessellation, if possible.
• Take note of how the quadrilaterals are arranged.
• Now begin again and do the same for the non-convex quadrilaterals.
• Sketchpad. Draw a quadrilateral ABCD with Sketchpad (just any four points, nothing special). Then make a construction of a number of quadrilaterals congruent to ABCD so that you have a tessellation that hangs together when you drag A, B, C, and/or D.
• Hint: What kind of operation takes a polygon and forms a congruent polygon elsewhere on the plane?
• SAVE this figure. You will use it later.

Questions.

• What are the angles as they appear at a vertex?
• Why do they fit together to fill up a total of 360 degrees?

Triangles

Now, let's try to tessellate using triangles.

• P-Method. Start a new sketch and draw a triangle ABC. Use a half-turn transformation to "glue" a triangle congruent to ABC to a side of ABC. What is the shape of this figure? Use the ideas from above to tesselate the plane with this shape.
• K-method. Start a new sketch and draw a triangle ABC. Use a line reflection transformation to "glue" a triangle congruent to ABC to a side of ABC.

  Questions.

  • What is the shape of this figure?
  • Use the ideas from above to tessellate the plane with this shape.
  • Save both figures.

Making Connections.

• (1) What kind of polygon do you get with the P-method? Why is it called the P-method? What kind of polygon do you get with the K-method? Why is it called the K-method?
• (2) Angle Sums. What are the angles IN ORDER as they appear at a vertex in the P-method? In the K-method? Is it clear why the angles sum to 360 in each case?
• (3) Look at the triangle tessellations in the big tessellation handout. Which one are P-method and which are K-method?

Symmetries of the tilings by quadrilaterals and triangles

Return to your first quadrilateral figure.

Translations.

Create the polygon interior for ABCD. Observe that some of the quadrilaterals in the tessellation are translations of ABCD. Mark some vectors and fill in the interiors of these polygons BY TRANSLATING ABCD INTERIOR. Do not just fill in the polygons.

Midpoint quads.

Now connect the midpoints of ABCD to form a quadrilateral. What kind of quadrilateral is it? Do the same for the neighboring quads. Do you see the midpoint figures fitting in to a larger pattern? How does this pattern relate to the translations you used to color the tiling.

Half-turns and translations.

The quadrilateral symmetry was built from half-turns. Examine the tessellation and observe that each of these half-turns is a symmetry of the tessellation. What is the symmetry that results from composing two of these half-turns.

Connections.

Connect the facts about composition of half-turns, translations, and parallelograms in one coherent story.

Exercise.

Carry out the same sort of exercise for triangles, coloring triangle the same if one is a translate of the other. How do the turns, the reflections, and the translations fit together?

Building tessellations from regular tessellations

Kaleidoscopes

Explore the files in the Mathday97 folder called 9A, 9B, 9C

Cutting corners

Build a regular tessellation cut off some corners. For example, if 3 polygons meet at a point, cut a triangular hole at the corner.

Bumping out sides (with connections to Escher)

Use rotations to create Escher bumps.