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. :)

"Human Calculator'

It's a really interesting notion to refer to people as "human calculators." My family members have always compared their education system back in China where you had to do mental math all the time compared to the "American" lifestyle were people use their phone calculators to see how much they should tip wait staff at restaurants.

Another human calculator that I found in addition to the two geniuses mentioned in Ms. Mariner's blog would be Scott Flansburg, a bestselling author that has been teaching math for more than 20 years and holds the Guinness World Record for adding the same number to itself more times in 15 seconds than a person can do using a calculator. I think it's hilarious how he earned the branded title of the “Fastest Human Calculator^®.” However, it is somewhat nice that "more important than showing others the skills that he has acquired, he wants to show others that they already have the ability to perform seemingly difficult math problems without a calculator."

Also, on Wikipedia, his occupation is "mental calculator." Wikipedia also talks a bit about his childhood relating to math: "Scott has stated that he was nine years old when he first discovered his mental calculator abilities, after he was able to solve his teacher's math question without needing to write down the calculations. Afterwards he would keep a running tally of his family's groceries at the store, so his father could give the cashier an exact check before the bill had been rung up. In his youth he also began noticing that the shape and number of angles in numbers are clues to their value, and began counting from 0 to 9 on his fingers instead of 1 to 10.^"

On a slightly less math-related note:
I wonder how mathematical skill relates to arrogance and humility. On his website, it states "he would be even faster, but he can’t speak the answers as fast as he can calculate them in his head!" I understand that this man is very intelligent, but I am not amused by his efforts to advertise his "genius." I feel like Mr. Flansburg's attitude is very elitest and slightly offensive. His website states that he "believes everyone has the ability to be good at math and enjoy it, but that most people have not learned to do math in a way that works for them. The Human Calculator^® is dedicated to helping students and adults to overcome their math-related fears." I understand that math is not the easiest subject for every single person, but to state dedication to help people overcome their fears assumes the mindset that he can go in and fix everyone's problems with his own "nontraditional" methods. He also assumes that everyone can enjoy math, but this is completely false because everyone has their own interests and should not be subject to the same forms of joy. Most math teachers that I know are dedicated to encourage learning and understanding of the subject of math, but they don't try to assert or impose their own views as the correct answer to mathematical situations. Maybe this is just the debater in me, but I felt very uneasy reading about Mr. Flansburg on his website, which is why I refuse to blatantly praise any stories that might exist of him.

Many people can possess the skills that he has, but simply don't have the opportunity to show it off to the world. It would be nice if everyone could stop putting labels on talent because talent can be undiscovered or discovered, but what truly matters is whether it exists or not.

Sources:
http://scottflansburg.com/
http://en.wikipedia.org/wiki/Scott_Flansburg

Reponse to Waves in the Pool

After sitting in math class almost everyday, sometimes it's hard to consider the realm of mathematics to extend beyond homework assignments or a test grade simply for the purposes of school. Subconsciously, we all do know that math has real-world applications, but we usually narrow our view to lined-paper and eraser marks. However, after reading Ms. Mariner's blog about the application of sine curves in regards to swimming, I did some research to find other applications of these waves in real life.

The application that I'd like to focus on today would be that of sine waves in music (because who doesn't love music?) It was really interesting to read this little blurb about sine waves and sound by Professor Rogness at UNM. For instance, apparently the A note above middle C's equation is this: , which produces a graph like this:



Then, he states that "if we change the period of the sine wave -- then we change the sound,"e.g. the equations and graphs of C# and E.
 
I always found it interesting that there was such a huge correlation between music and math. One of my favorite Youtubers (ever), Kurt Schneider, was a math graduate from Yale who is incredibly talented at playing various instruments and arranging music. A lot of interviews from his friends and family (and even his math teacher) said that math was the reason of why he understood the way that music works, so well. This makes me think how cool it would be if we could visualize the sine curves of each note as we listened to different songs and sounds in general! Math is pretty cool, isn't it?

Who Uses Trig in His or Her Job and How is it Used?

The answer to this question may seem extremely simple yet extremely complicated at the same time. As students, we often wonder and even criticize when what we learn in school will help us in our future lives. We might think that writing English essays on literature will help improve our literary analysis skills, but what does that say about our possible jobs as musicians or as athletes? Honestly, I think that everything we learn in school will be applicable in our lives in one way or another.

In terms of trigonometry, today I will write about how it is applicable to a very obvious job which requires it: architecture. Without question, one can see that some sort of mathematical skill is needed in order to create buildings/structures that are not prone to collapsing.

According to Ryan Crooks, "Trigonometry is especially important in architecture because it allows the architect to calculate distances and forces related to diagonal elements. Of the six functions in basic trigonometry, the sine, cosine and tangent are the most important to architecture because they allow the architect to easily find the opposite and adjacent values related to an angle or hypotenuse, translating a diagonal vector into horizontal and vertical vectors." Trigonometry allows what architects dream of on paper to be real dreams that can stand before their eyes.

However, the field of architecture isn't limited to structural buildings.

Architecture can also be applied on a smaller scale, such as landscape architecture. Although one would not usually consider this to fall under the category of "trigonometry," many people believe that it actually is - this argument can be justified. For instance, if people required a portion of land for a building, statue, garden, etc. and they needed to clear the space by chopping down trees, then it would probably be best if the height of the tree was known so in the case of it falling no one would get hurt. (Clinometers are great for this!)

Another example of how trigonometry can be used in architecture of roads (transportation infrastructure) is that it is much better for roads and parking lots to be built on a terrain with as little of a "slope" as possible.

Although these examples only provide a brief view into the endless possibilities of trigonometric applications in professions, I think it's great how some of what we learn in school will be very important in our future endeavors.