Geometry

1) Try to solve at least three of the devilish division puzzles this involve cutting up the eight figures on the handout into two equal congruent parts (congruent to each other, but perhaps mirror images). 2) With our work on tessellations with quadrilaterals, pentagons, etc. come up with an example of a tessellating nonagon (i.e. a 9 sided polygon that can be used to tile the plane - note that it clearly won't be a regular nonagon!). (i.e a "row house" design for tessellating with pentagons ) 2.1) Can you use a similar approach for this problem (perhaps more bends in the roofline?!)? 2.2) Next design a tessellating decagon - good luck! - do you find this easier or more challenging than working with the nonagon? 3) As mentioned in class there are several types of convex hexagons that tile the plane. The regular hexagon tessellates, of course. Another type is the one we saw in class where the opposite sides of the hexagon are parallel and equal in length. For this problem decide whether the following type of hexagon also tessellates - this is one where there are two opposite sides that are equal in length and three consecutive interior angles that sum to 360 degrees (note that the three consecutive angles are assumed to be the ones between the segment marked x and the segment marked y). You can see one example of such a hexagon in the diagram below - in this particular example the angle at C is shown as a right angle, but in general it doesn't have to be, so please don't assume that's the case.

Will all hexagons with these two constraints always tessellate? Please give a detailed explanation as to why this works out, or find an example of such a hexagon that doesn't tessellate, explaining why not.

4) Speaking of tessellating polygons, please do some search on internet about pentagonal Tiling Discovering. Can you come up with a new one? Just kidding!! Can you play with existing one to come up with new pattern? Please do not spend to much time on this. 5) Now, start working with unit square (grid point paper attached “aka geoboards). It's easy enough to see that it's possible to create squares on the grid paper with areas 1, 2, 4, and 5 (where the vertices of the square are required to lie on points on the grid paper, and it's assumed that the grid

paper is in a regular square array with dots that are exactly one unit length away from each other) - you'll see that some squares (such as for areas 2 and 5) will need to be "tilted" - not squared off to the grid paper's vertical/horizontal axes. On the other hand you can also check that there's no square with area 3, for example. Now figure out which areas can be represented for all the integers up to 25. ? Try to figure out an approach to answering this question that doesn't just rely on trial and error guessing on a geoboards what is the specific feature of an integer N so that a square with area N can be created, and why? (i.e. what property must the integer N possess? If I asked you to figure out whether you could create a square of area 630, what could you do with the number 630 itself to figure this out?) 6) Pushing forward a bit more on your own - now that you've considered squares on the unit square grid, think about triangles. What types of areas can be represented by triangles on the same unit square grid paper (i.e. triangles with vertices located at points on the grid paper, assuming again that the grid dots are a unit length away from each other)? Is it possible to create an equilateral triangle on the unit square grid paper? This might be difficult to prove, so try to provide a reasonable argument either for or against. 7) Noting to turn in Do some reading and learn about Penrose and the toilet paper scandal! there's a Penrose Tile generator program (for Windows - sorry, Mac users!) that you might be interested in trying out.

8) Please do some research about the NOVA episode on Fractals or you can also go ahead and Google "Fractals - Hunting the Hidden Dimension" please write an statement about what you think is the most interesting use/application of fractals (or anything else you found of interest on it)