Geometry

1) Dissect the following three rep-tiles from the following handout 2) create two examples of rep-tiles on your own (after a while feel free to do some sleuthing on the internet if you find yourself getting too stuck on this!). 3) Find at least two interesting examples of tessellations in the real world - on floors or walls (or ceilings!) - for instance bathroom tiles sometimes are in a 4.8.8 tessellation. Please find tessellations that are more elaborate than simply rectangles or squares. Either snap a quick picture of the tessellations or draw enough of them so that it's clear that their patterns can be in fact be continued forever (i.e. can indeed tile the entire, infinite, two dimensional plane). Your tessellation examples should include more than just one shape (i.e. not just a regular tessellation). 3.1 Do some search and find something that is particularly interesting according about tessellations 4) Do some search on the internet, and simply find and print out and example of best (favorite Escher tessellation pattern) you may select your choice from the web bellow (http://www.mcescher.com/) Or just google Escher to find lots of examples on the internet!) or in GeoGebra (5) And on to semiregular tessellations tessellation we figured out that we might be able to make patterns with 3 triangles and 2 squares as well.

5.1 Now explore semiregular tessellations on your own. 5.2 So for this problem try to find four more on your own (you me try using the illuminations web side for fun( https://illuminations.nctm.org/Activity.aspx) and see how many involve just triangles and squares, and then try replacing 120 degrees worth with a hexagon... or two?! Can you make one with an even higher sided polygon, beyond the octagon? 5.3 And yet one additional semiregular tessellation i.e. find five instead of just four more than the 4.8.8 that we know for fact. 6) We talked about a way to use the power of similar triangles (and parallel line construction) to create lengths of the form A/B where A and B were any integers (or, in fact, any constructible length segments) and also we know that this could be reversed in a sense to create the product A x B, i.e. if you're given two segments, one length A, and one length B, show how you can create a segment of length A x B. Note this is trivial if A and B are integers, - you can just multiply the integers together in your head and just add together that many copies of a unit one length segment. So using this "calculating device"(!) to calculate something more complicated, Phi squared, i.e. start with a one inch segment (or 1 cm if you prefer), and then create a segment of length Phi (using just the usual straightedge and compass all the way through this problem). Next use the approach we multiplying two lengths together to create a segment that's Phi^2 in length.

6.1 And finally show that it comes out as expected (remember that Phi^2 was just Phi plus something!) by comparing the Phi^2 length that you created to the length of Phi - pretty neat!