![]() ![]() Not only do they not have angles, but you can clearly see that it is impossible to put a series of circles next to each other without a gap. I agree with John Golden, in that you could extend the idea to have student think about the "so what". I am an 11th Grade math teacher and I have done a larger project with my students in which they have to design their own tessellation using Geometer's Sketchpad. This entails an understanding in transformations, interior angles of a polygon and I differentiated by creating different roles: some students had to design a mutated figure that would tessellate with an equilateral triangle, square, regular hexagon, irregular triangle, and irregular quadrilateral. I am stuck in how to make this project more authentic to the students though. I feel something is missing in my project that requires them to take it further than just designing their own.Īlthough it is true that tessellations can be found both in the natural world as well as in more synthetic (man-made) products/ art/architecture. I am thinking about how I could create certain parameters in which the students will have to fill a finite plane of some shape and they will have to make some sort of prediction. I'm still thinking about how to move forward on this though. As for the honey bees an interesting thing to look into is why do honey bees use regular hexagons rather than other regular polygon that tessellates- it has to do with optimizing the amount of honey a regular hexagon stores. Here's a nice article that may give some ideas that students could look into to understand the purpose of tessellations in our natural world. This page displays some common regular and semi-regulartessellations, but with a twist. The polygons are hinged at oneor more vertices so that the appearance of the tessellation maybe changed dynamically. ![]() Thiscould be a fun extension of the study of tessellations in middleor secondary mathematics.Īs a first example, consider a tessellation of squares as shown:Įach blue square is hinged at its vertices so that the tessellationchanges as any blue square is rotated and my look like one ofthe following:Īll implementations are done in Geometer'sSketchpad and can be explored by clicking on the figure. The blue squares have only rotated, but the yellowsquares have become rhombi. Watching the transformation is fascinatingand I suggest you take the time to watch it in GSP. Clicking onany of the figures will take you there (if you have GSP on yourcomputer). I had better luck with hexagons, as the next figure shows: Here's my aborted attemp to make a hinged tessellationwith triangles:Īs you can see, it's no longer a tessellation (andsome of the triangles were incorrectly placed), but you can stillopen the GSP file and see it rotate. Here are hexagons again, but with triangles and squares:Īgain, only the green hexagons are "stable". ![]() This looks like a traditional quilt pattern, but it "morphs"into other patterns while maintaining a beautiful symmetry. This traditional tile pattern is composed of octagons andsquares, but the squares transform into beautiful stars and pinwheels. The octagons rotate, adding triangles to the squaresand then collapsing into a single octagon. The final hinged tessellation hereis composed of dodecagons, hexagons, and squares. Comparethe changing patterns of the squares to the tessellation above.Ĭompare the above tesselation to this one, which appearsthe same, but has "stable" squares which rotate withthe dodecagons. My final attempt was not a tessellation, but a hinged dissectionpuzzle of an equilateral triangle into a square by Dudeney. ![]()
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