Tessellations

For my blog, I decide to look some stuff up about tessellations - even though they may seem like a very elementary concept, I enjoy their artistic aspects. Thus, here's an answer to your question of "Why are there only three regular polygons that tessellate?" So, a regular tessellation is periodic (repeated translations of polygons) and uniform (same type of polygons at each vertex), composed of congruent regular polygons. Apparently only three regular tessellations exist (triangle, square, and hexagon) and that's because these polygons are the only ones that have interior angles that divide evenly into 360 degrees. To find an interior angle of a regular polygon, you use the formula: a = 180 - 360/n. Also, "of the regular polygons, only triangles, squares, hexagons, octagons, and dodecagons can be used for tiling around a common vertex because of the angel value." (http://www.beva.org/math323/asgn5/tess/regpoly.htm)

Speaking of the artsy side, I think it's pretty beautiful how we can apply math to things such as architectural design, not just architectural structure.   I enjoyed reading this quite a bit! http://www.ysjournal.com/article.asp?issn=0974-6102;year=2009;volume=2;issue=7;spage=35;epage=46;aulast=Khaira Apparently people are really into tessellations and origami, too. Image sources: http://origamiblog.com/origami-tessellations-islamic-design/2009/06/08 http://vi.sualize.us/i_w_a_m_o_o_r_c_h_i_t_e_c_t_u_r_e_voussoir_wooden_art_tessellations_clouds_installation_picture_6rgT.html

Probability Problems

Alice, Mieke, and Grace sat on the floor of Brown Hall doing homework and eating a bag of Skittles candy. For the purposes of this math problem, we will say that there were 55 skittles total in the bag and 11 skittles of each flavor. There are 5 different flavors of skittles: cherry, grape, green apple, lemon, and orange. Alice's favorite flavor is grape, Mieke's favorite flavor is green apple, and Grace's favorite flavor is orange. What is the probability that after picking 9 Skittles out of the bag, there would be at least one grape skittle, one green apple skittle, and one orange skittle?
(A very real life problem - ask them and they'll be witnesses)

A vocabulary test has 30 questions. Twenty questions have choices ABCDE as answer options, five questions are true or false, and five questions are fill in the blank with a word bank that has 10 words, which can only be used once. If you guess at every question, what is the probability of getting at least half of the questions right?

Someone's grandmother really likes to plant flowers in her garden. In a mass package of tulip bulbs, only one out of four bulbs sprout on average. She has twenty packages of tulip bulbs. What is the probability that out of the twenty packages, how many will not sprout?

Seven cards are dealt from a deck of cards that do not have any red hearts.
a) What is the probability of getting all face cards?
b) What is the probability of getting only kings?
c) What is the probability of getting only black cards?

A teacher is pondering if they should give his students a free period or not. He decides to play the silly game where he chooses a number between 29 and 72 and his students have to guess. The number he chose is an even integer. What is the probability that his students will choose the right number if they all get three tries?

Of the 1700 students at a school, 908 have pet dogs, 762 have pet cats, and 45 have both pets (even though dogs are much cuter). If a student is chosen at random, what is the probability that they have either pet?

A coin is bent so that the probability of getting tails is 0.59 instead of 0.50. This coin toss is extremely important because it chooses the sides of high school debaters in their final round. They have to choose a preset combination of heads and tails that consists of two tosses in order to call for the affirmative or the negative side. 
a) Draw a tree diagram showing the probability of 3 tosses of the coins
b) Find P(HH), P(HT), P(TT), and P(TH) for the team that you like more

"Pascal's" Triangle

Blaise Pascal was from France and he was a physicist, mathematician, inventor, Christian philosopher, and writer. I think it's pretty interesting how he was interested in both religious studies and math/science because during his time and earlier there was a separation between the two. He was considered a child prodigy and his earliest work influenced our notions of fluids, pressure, and the vacuum along with defending the scientific method.

The triangle works like this:  On the first row, you only write the number 1. Then to make the next rows you add the number above and to the left with the number above and to the right to find the new value. Lastly, if there isn't a number to the right or left, then you put a zero in its place.
The numbers used in the triangle were originally from studies of binomial numbers combinatorics (how cool)! from India and  the Greek studies of figurate numbers. His triangle was already discovered in China in the early 11th century due to a Chinese mathematician Jia Xian, who lived from 1010 to 1070. Next, Yang Hui in the 13th century formally presented the triangle and how it's still called Yang Hui's triangle in China. I think this brings up a really interesting point because all around the world during earlier times people were constantly making discoveries but in today's world only a few people are actually credited. I think in the US, for instance, we never mention Yang Hui because the education system puts such an emphasis on Western ideals. Back then it would be totally understandable due to a lack of communication systems but at the same time I believe that education overall would be more valuable if American students could learn more about histories of concepts and ideas from other places in the world to be better informed people. I also just find facts like this to be really neat!

Here are visual representations of the

Ancient Chinese patterns:

Number patterns in the triangle:


Odd and even numbers:


Exponents of 11:




Horizontal sums (they double each time!):
.

http://en.wikipedia.org/wiki/Pascal%27s_triangle
http://www.mathsisfun.com/pascals-triangle.html

Math Humor

Here goes nothing:

• mathematicians who can't hear communicate with ease through sin language.

• 
• some mathematicians will be pretty hesitant to cosine loans.
• 
  (at least we have each other, right? 6th period!)
• if you were sin^2x and I was cos^2x, then together we'd make one beautiful couple.
• 
• I like angles, but only to a certain degree. 
•  
• dear Algebra: please stop asking us to find your x. She’s not coming back, don’t ask us y.
•  
• the mathematician worked at home because he only functioned in his domain.
• 
• to people who are bad at math, the equation 2n+2n is always 4n.  
• 
• I used to really dislike decimals until I found out it has really good points.
•  
• what happens after you miss math class too much? the work starts adding up. 
• 
• well, I'm just relieved that there's not too much drama in our class, or else we'd have to work out even more problems. 
•  
• (p + l)(a + n) = pa + pn + la + ln
  guess what? I just foiled your plan.
• 
• sometimes (most of the time) we're still hungry after lunch. please bring us some pi?
• 
• why did the cosine make fun of the sine? it was an odd function.
• 
• should everyone wear glasses during math class to improve our division?
•  
• thanks for being such a great teacher, you really sum things up well. :)