To set this up, in class we talked about a new way to look at "point," "line," and "plane," after all, they are undefined terms. What if we defined these things such that a plane was a grid, much like city streets on a NS EW grid. Points would be lattice points, or intersections of the "streets" on the grid, and we could travel only on the grid its self, much like a taxicab would drive. A circle is still the set of all points equidistant to a single point.
In
the grid to the right, the center point of a "circle" is shown with a
circle of radius 2 (diameter = 4). Remembering that you can only travel
on the lines and points are only places where the lines intersect, find
the circumference. Then take the ratio of the circumference to the
diameter. This ratio is known as pi. (On a Euclidean plane, like we
have been using, the ratio is about 3.14159....). Verify that this
calculation of the new "pi" is indeed 4. Make your own grid; identify a circle with a different radius. Is the ratio still 4? The above is our "tangent" in class last week.
So what did he do that was so cool? Trevor asked, What if we have the same rules about lattice points, traveling only on the lines, but we set up the plane to be equilateral triangles. What would pi be? (Actually, he said, "This got me thinking....") To the right is a snip of his new grid. One unit is the length of the side of an equilateral triangle; points are only vertices of the triangle. You can see the center and a circle shown. The radius of this circle is 2. Find the circumference. Find the value that is the ratio of the circumference to the diameter ("pi"). It's not 3.14159..... and it's not 4.
Then he thought some more. What if the grid was tessellating
hexagons. In the process of thinking, he figured out that the only
tessellating regular polygons are squares, triangles, and hexagons. To
the right is his image of the tessellating hexagons. Of course, he used
a radius of 2 and found a value for the circumference, diameter and the
ratio of the circumference to the diameter. Go ahead, find that ratio. (Answer is in his blog :) )So ok. In this next blog, I know Trevor is still thinking about this (DO keep thinking, Trevor!).
For your blog, you can expand on this idea. OR better yet, find something like, "Why are there only three regular polygons that tessellate?" or "What is the relationship between regular tessellations and regular polyhedra?" or "What if we put a circle on a soccer ball?" (Trevor's question, lay off that one, I think he may be writing about that one.....) BTW, a soccer ball is also a "truncated icosahedron." OR even better, think of a tangent we took in class (or maybe your brain just took in class without us being with you on that mental adventure....) As yourself, "What if..." or be like Trevor and just start thinking....
I started poking around in the idea of tessellations, because of the idea you posted about there only being 3 regular polygons that tessellate. I found a link that I'll post below that confirmed this, but it goes on to say that there are 8 semi-regular tessellations, or tessellations of two or more regular polygons. There's one in which you can surround a hexagon with triangles (it's sort of difficult to describe, but follow the link if you're interested.) Anyway, that's sort of a tangent, but I think I've figured out why there are only three regular polygons that tessellate with themselves. At each vertex, there are 360 degrees. (4 squares, 3 hexagons, 6 triangles). So the interior angles need to be a number that divides evenly into 360. Looking for example at hexagons, each interior angle of a regular hexagon is 120 degrees. This allows 3 of them to fit around an arbitrary point with no space.. But if we look at a nonagon (9 sides), each interior angle is 140 and 360/140 is not an integer, meaning nonagons do not tessellate.
ReplyDeletehttp://www.mathsisfun.com/geometry/tessellation.html
I started looking at pictures of tessellations for inspiration, and I realized that there are a lot of cool tessellations found in nature. Most use many different shapes that tessellate, but one of the best examples of regular polygons tessellating is a bee hive. Bees use tessellating hexagons to maximize the efficiency of storage in a hive. Other tessellations include the pattern on a giraffe, the scales on reptiles and fish (although this example isn't perfect since scales technically overlap each other, but the scales do look tessellated), the shells of turtles, dried and cracked mud, spider webs, butterfly and moth wings, pineapples, bark, leaves, and pinecones. So, as usual, we can find something in nature that uses math for everyday life.
ReplyDeleteThinking about tessellations, I remembered back to a calendar my mom had previously had in which each month had a different picture done by Escher. Escher uses tessellations to create interesting images. Throughout his work, he uses tessellations of nature, animals, and architecture. Most commonly, his pictures display birds in flight. However, none of these pictures use simple polygons, which is why I believe they can tessellate. Although his drawings aren't pure tessellations, they utilize tessellations and create quite interesting images.
ReplyDeleteHere are some links to photos which intrigued me:
http://uploads3.wikipaintings.org/images/m-c-escher/day-and-night.jpg
http://uploads2.wikipaintings.org/images/m-c-escher/magic-mirror.jpg
http://www.artlex.com/ArtLex/t/images/tessel_reduc.liz.lg.jpg
When looking for the answer to the question, "Why can only three shapes tessellate?", I found something very interesting. The measures of the interior angles of the three shapes that can tessellate, the equilateral triangle, square, and regular hexagon all have interior angle measures of 60, 90, and 120 respectively. All of these numbers can divide evenly into 360, the number of degrees present in a typical square/rectangular plane. Since there can be no overlap in a tessellation and the entire plane must be filled, only figures whose interior angle measures are divisors of 360 can tessellate. The only figures with these measurements are a square, equilateral triangle, and regular hexagon, making them the only shapes that can tessellate.
ReplyDeleteFor a more in depth explanation you can look at this site:
http://mathandmultimedia.com/2011/06/04/regular-tessellations/
To be honest, I wasn't quite sure what a tessellation was when I started writing this blog so I did some exploring. Though tessellations with squares, triangles, stay perfectly symmetrical as all points of the tessellations, there are other types of tessellations that don't follow these rules or abide by these shapes. These are called wallpaper groups and have translational symmetry. There are 17 types of wallpaper groups, of which have very distinct differences in their pattern, but can vary greatly as they don't follow the boring rules of normal tessellations. Translational symmetry does not maintain a pattern of repeating the same triangle over and over like the tessellations above, but rather gives you a pattern and moves it the same distance over and over to appear unchanged. It creates very pretty patterns that have been used allover the world in things like pots, palaces, and even Egyptian tombs! Here are some cool examples:
ReplyDeletehttp://en.wikipedia.org/wiki/File:Wallpaper_group-p4g-2.jpg
http://en.wikipedia.org/wiki/File:Wallpaper_group-p4m-5.jpg
http://en.wikipedia.org/wiki/File:Wallpaper_group-p3-1.jpg