TessellationsSome ideas for Tessellations in the Classroom.An Imaginary Session between Peter and a student.Peter: We'll leave Pattern Number 12 till later. All my questions relate to the first eleven patterns. Make a list of the shapes in this activity. If you don't know the name - just note the number of sides.Student:
I also saw a diamond but I guess that was just a square in disguise! Peter: What's special about each of these shapes? Student: Their sides are equal in length and their angles are the same size. Peter: Quite right! They're called regular shapes. Peter: Now imagine that you're walking around one of the squares. At each corner you will need to turn through how many degrees? Student: 90 degrees Peter: After you've walked right around the square you will have turned through how many degrees? Student: 360 degrees. Peter: Quite right! Now ... what about the hexagon? How many degrees at each corner? Student: 60 degrees Peter: Yes ... that was quick ... I guess you realised that all you had to do was divide 360 by 6! Student: Sure did! How about I add a column to our table? Peter: Great idea! Go ahead! Student:
Peter: Well done! You'll find that this is a very helpful idea when we look at the Turtle Machine Activity. OK. We've noted that all the patterns are made with regular shapes. You'll also notice that some use just one shape and some use more than one. How many of each?Student: Three use just one shape, six use two shapes and two use three shapes. Peter: What makes these patterns special is the fact that at each corner the number of shapes meeting there is the same no matter where you look. Check this out for yourself. Student: Wow! That's not always obvious but it is in fact true. Peter: What is even more amazing is that, using only regular shapes, these 11 patterns are the only ones possible that obey this rule! Peter: Here's one to think about. Why can't we make a regular tessellation using pentagons (5 sides)? Student: Hmmm ... Let's see ... walking around 5 sides means turning 360 / 5 = 72 degrees at each corner. That means the inside angle must be 180 - 72 = 108 degrees. So if we have 3 pentagons meeting at a corner, that will be a total of 3 x 108 = 324 degrees - this leaves a gap. But 4 x 108 = 432 degrees will overlap. So pentagons are a no go! Peter: Brilliant! Finally, Tessellation Number 12 shows you what is possible when you move away from regular shapes. If you have access to the Internet, do a search for tessellation images - you'll find some stunning pictures! |