Tuesday 6 December class

Topics:

• Return to regular tilings of the plane:

"regular" in this case meant "with caffeine"

From last time Regular polygons and their exterior and interior angles.
 and Tiling the plane with regular polygons: we show that there are only three regular tilings of the plane: by equilateral triangles, by squares, and by regular hexagons.
But we can exploit these to make beautiful patterns. (Something you may want to investigate for yourself.)
Here's another way these regular tilings are used. Surprising!

• Semiregular tilings (tessellations) of the plane;
here is another source (from Math Forum)
A semiregular tiling is a tiling by some combination of different regular polygons (not just one type). It must fulfill two rules:
1) The arrangement of the types of polygons must be the same at every vertex - we say that each vertex is of the same type
      - and -
2) The sum of the angles of the polygons that meet at that vertex must be 360 degrees (obviously).

It turns out that there are only 8 semiregular tilings. They are named by listing the numbers of sides of the polygons going around each vertex, starting with a polygon with the least number of sides: I usually (in this class) use letter names instead of numbers, but it's not terribly important which way you do it. So instead of writing 3,3,3,4,4 I would write TTTSS (for triangle, triangle, triangle, square, square) for example. Choose whichever makes sense to you.


• Frieze patterns: (that link gives the webpage I handed out in class) - these are related to tilings, but they are tiling in only one direction, so to speak. A basic pattern, or "block" as I call it, is repeated by translation to the left and right. (In principle, the block is repeated infinitely many times: in reality, it is repeated until the end of the wall or whatever.)

We will use the naming convention in the above-linked MAA article: be aware that other sources will use different names for these patterns. It turns out that there are only 7 different frieze patterns based on the symmetries of the individual blocks. (Remember that all frieze patterns have translational symmetry.)

Assuming that the translational symmetry is in the horizontal direction (as it usually is), the symmetries that the block may have are:
       vertical reflection (reflection over a vertical line through the middle of the block)
       horizontal reflection (reflection over a horizontal line through the middle of the block)
       rotation by 180° around a point at the center of the block
       glide reflection ("gliding" in the horizontal direction by half a "block" and then horizontal reflection - this requires looking at the whole frieze pattern, not just the individual block, and it is probably the hardest to see at times.)
The block may, of course, have none of these. Please also note that if the frieze pattern has both horizontal and vertical reflection, then it will automatically have rotation, but the converse is not true. (For example, the Greek Key pattern has rotation but has neither horizontal nor vertical reflection.)

We looked at some of the examples here (EscherMath) to classify them according to our naming convention. Please note that this second site is using a different naming convention: please stay with the names and the F-numbers given in the handout. If you read that whole page, you will see another good explanation of the symmetries in the 7 frieze patterns: can you identify which of our names should be given to each pattern?

Homework:
• As I mentioned, your Project 2 will be to collect real-life (NOT Googled) examples of regular and semiregular tilings, and frieze patterns. Try to find an example of each of the 3 regular tilings, at least 2 of the semiregular tilings, and at least 4 of the frieze patterns. (Some semiregular tilings and frieze patterns are rather rare!) You can and should start doing this now.
Document them by taking nice (i.e. focused and framed) photos or else making a nice sketch, and make a note of the place where you found each one. In addition, identify which tiling or frieze pattern it represents.
Extra credit will be given if you find more than the above specified minimum number of different patterns.
Sources may be patterns you find in the world (on walls or walkways or fences, fabrics, etc.) or in craft patterns (knitting, crochet, embroidery, etc.) - the point is NOT to use a web search to search for "frieze patterns" for example. We want to see how these appear in our everyday (offline) lives.
This Project will be due on the day of the Final Exam. I will provide more detailed instructions on how to submit it in a separate post.

• Look at these three patterns from the MathForum article on tilings (tessellations): they look as if they should be able to be continued into semiregular tessellations. What goes wrong if you try to do that? Describe in your own words (but be specific!) It may help if you make 2 copies of each image into a program that can manipulate images (I did this in TextEdit on a Mac) and then try to move one of the images around to fit it together with the other one. Big hint: remember that each vertex in the tiling must have the same type! Is it possible to make this happen?

      

• In the EscherMath discussion of the frieze patterns, go to the heading "The Seven Frieze Patterns" (or here is a pdf of them) and identify each of those frieze patterns according to our naming convention (given here).

• In the Frieze Exercises from EscherMath, do problems #1, 2, 3, 4, and 9a: use our naming convention, again. I'll be posting answers to these later, or you could do them on Piazza for extra credit!