Fun With Parabolas

The parabola results when the plane is parallel to a generating line of the cone. The term parabola comes from Greek para "alongside" and -bola, "to cast, to throw." Overall the word parabola means "thrown parallel". 

While thinking about this blog, I wondered how parabola appear in the "real world". Sure they show up when kicking a soccer ball or shooting a basketball but where else do they appear? I was thinking a lot...and I mean a lot...so I decided to take a "study break" and went straight for my phone. Once I replied to all the text messages and went through all the social media sites, I decided it wasn't time to go back to thinking about parabolas so I played a game. I was scrolling though my abundance of apps when I came across the Angry Bird game. I clicked it, considering I haven't played in a while so why not, and did not realize at the time how that game helped me think for my blog! I soon noticed that as I was hurling the bird across the screen to destroy the green pig's configuration, I was actually creating a parabola!

Math popped up when I least expected it, or wanted it, in the most fun way! So what other fun ways are parabolas included in everyday life? Well, roller coasters that go up and down, up and down have parabolas as well!

Parabolas also show up in architecture! They are in bridges, buildings and arches! 

Speaking of arches, there are two unseen, and very unknown, parabolas in the very well known Golden Arches...Can you see them? 

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

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?