OliviaGalante

Saturday, April 5, 2014

Tessellates

The first question I had was "what exactly is a tessellate?" After scouring the internet, I found my answer. According to Wikipedia, "a tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps." Tessellations can also be generalized to higher dimensions.

There are four different types of regular polygons tessellations: regular, semi-regular, nonuniform periodic, and nonuniform non-periodic. "A regular tessellation is periodic and uniform and consists of congruent regular polygons." There are only three regular tessellations because only triangles, squares, and hexagons have interior angles that divide 360 degrees evenly. This is important because the tessellate has to be uniform, meaning "the same type of polygon(s) are at each vertex."

Here is a really cool chart I found on the Theory of Regular Polygon Tessellations webpage that explains and shows most of the different combinations a regular tessellate must be (they all have to add up to 360 for the combination to be a regular tessellate):


Combination of Polygons	Combination of Angles
6 triangles	6X 60 = 360
4 squares	4 X 90 = 360
3 hexagons	3 X 120 = 360
3 triangles and 2 squares	(3 X 60) + (2 X 90) = 360
3 triangles and 2 squares - another formation	(3 X 60) + (2 X 90) = 360
2 hexagons and 2 triangles	(2 X 120) + (2 X 60) = 360
2 hexagons and 2 triangles - another formation	(2 X 120) + (2 X 60) = 360
1 hexagon, 2 squares, and 1 triangle	(1 X 120) + (2 X 90) + (1 X 60) = 360
1 hexagon, 2 squares, and 1 triangle - another formation	(1 X 120) + (2 X 90) + (1 X 60) = 360
1 hexagon and 4 triangles	(1 x 120) + (4 X 60) = 360
2 octagons and 1 square	(2 X 135) + (1 X 90) = 360
1 dodecagon, 1 square, and 1 hexagon	(1 X 150) + (1 X 90) + (1 X 120) = 360
2 dodecagons and 1 triangle	(2 X 150) + (1 X 60) = 360
1 dodecagon, 2 triangles, and 1 square	(1 X 150) + (2 X 60) + (1 x 90) = 360

Some common tessellates are in beehives and soccer balls, as well to form:

<--- scales

tiles --->

<--- bark on trees

Sources:
Wikipedia:
http://en.wikipedia.org/wiki/Tessellation

Theory of Regular Polygon Tessellations: http://www.beva.org/math323/asgn5/tess/regpoly.htm

Monday, February 17, 2014

Pascals Triangle

When you first look at the image above, your first reaction is "oh those are just some dumb ol' number strung together in a random order." When you look closer however, you will notice all the cool patterns and tricks of this famous little triangle called Pascal's Triangle.

The first pattern that you will notice is that every new row is made up of sums from the previous row. For example, row 4 is 14,641 because you always start a new row with 1, then 1+3 (from row three) = 4, 3+3 (from row three) = 6, 1+3 (from row three) = 4. Finally, you always end a row with 1.

The second pattern that I noticed was that the blue line (side image) increases by one all the way down the diagonal to 13. And similarly, the green line (side image) increase by adding numbers in numerical order. For example, 1+2 = 3, 3+3 = 6, 6+4 = 10 and so on and so on down the diagonal to 78.

The third pattern is a little harder to discover and is kinda hidden within the triangle. If you take the sum of each row, you will see it will be double the previous rows. I'll show you what I mean:

ROW 0: 1

ROW 1: 1+1=2

ROW 2: 1+2+1=4

ROW 3: 1+3+3+1=8

ROW 4: 1+4+6+4+=16

...

ROW 8: 1+8+28+56+70+56+28+8+1=256

ROW 9: 1+9+36+84+126+126+84+36+9+1=512

...

ROW 12: 1+12+66+220+495+792+924+792+924+792+495+220+66+12+1=4,096

ROW 13: 1+13+78+286+715+1287+1716+1716+1287+715+286+78+13+1=8,192

Of course there are more patterns and tricks in the triangle, but if I mentioned them all then this would be a very long blog. I picked the patterns I thought were interesting and cool to share. But there are more out there so what can you find?