Communicating Math: Amicable Numbers

Amicable numbers are two numbers whose factors, not including the number itself, add up to the same value. For instance, since the time of Pythagoras, 220 and 284 is a common example of a pair of numbers that are amicable. If we add up the factors of 220 and remember not to include 220 we get

1+2+4+5+10+11+20+22+44+55+110 = 284.

Similarly, if we add up the factors, expect itself, of 284 we yield

1+2+4+71+142 = 220.

To find other pairs of amicable numbers, Thabit ibn Qurra came up with some general formulas to do so. If

T(n)=3*(2^n) - 1 (also known as the 321 number),

T(n - 1)=3*(2^(n - 1)) - 1,

and

9*(2^(2n - 1)) - 1

are prime numbers, then the numbers

(2^n)*T(n)*T(n - 1)

and

(2^n)*(9*(2^(2n - 1)) - 1)

are amicable. 

As of today, the only values of n that give us amicable numbers following Thabit's stipulations are 2, 4, and 7. On the other hand, these formula's don't work for the thousands of amicable pairs that have been found in recent years. One example is the pair of 1184 and 1210. We are not able to get these two numbers from an integer value n by following Thabit's rules, yet they are still amicable. 

It is interesting to think about why someone would first think to add up factors of a number and try to match the sum of factors for another number. Hundreds of thousands of pairs have been found but will there ever be a sure fire formula that provides two amicable numbers? Are there infinitely many pairs of amicable numbers out there? This concept of amicable numbers is kind of an ongoing mystery that keeps mathematicians curious and searching for more answers. 

Reference:
http://mathlair.allfunandgames.ca/amicable.php

Doing Math: Geometric Tessellation

For this week's post I decided to create a geometric tessellation. You might think drawing repeating patterns would be pretty easy, but as each pattern is added the lines start to cause confusion and mistakes are most likely to be made. My eraser was a much needed tool in completing this tessellation. The one shown below took me about and hour and 45 minutes to make and with the help of just a solo cup and a ruler I must say I am proud of it, though it is not very precise.
Step 1:

With not having a protractor or compass I had to use a cup as a starting point. I traced a circle around the base of the cup and then inscribed a hexagon in to that circle using a ruler to try to make the sides as equal in length as possible. I went off of this hexagon for the rest of them and just a ruler to align the overlapping hexagons. I chose to start with a hexagon of this size as it leaves a fair amount of open space to add more details without loosing the contour of each shape added.

Step 2:

Next I added a smaller hexagon at 3 vertices of the each of the larger hexagons. Then I added irregular hexagons inside some of the larger hexagons as shown above, making sure to repeat this pattern. I knew I wanted to center some polygon around some of the vertices of the large hexagons and adding the smaller hexagons was just the thing to do. Once I did this I thought I would continue with the hexagon theme and fill in the space of the larger hexagons. I held back a little bit to get the "every other" pattern going.

Step 3:

For the finishing touch I outlined every line in sharpie for a more pronounced look and to clearly see the repeating pattern.

Tessellations can be just beautiful pieces of art but they also appear in nature and other everyday items. For instance, honeycombs, floor tilings, glass ceiling framework, and brickwork all incorporate patterns that define a tessellation. Since these are such intriguing designs it is not surprising that marketing and advertising companies use them on packaging and products to grab a consumer's attention. It might be a small pattern but I believe that you will always be able find a tessellation in your everyday life.

History/Nature of Math

One of the most influential mathematicians is Euclid. I have searched and read through articles that gave some insight in to what Euclid actually contributed to math. Although, there are many rumor's out there about Euclid as there was another Euclid that was a philosopher just before his time. I tried to find several articles that contained similar information insuring a more concrete perception of him.

Euclid's Legacy:
Euclid has left such an impact on the world today as he is the one who is credited with creating and conceptualizing the rules of Geometry through his 13 volumes of "The Elements." This book comes with one of the oldest algorithms still used to this day, known as Euclid's Algorithm, which allows us to determine the greatest common divisor of two integers.

Common Notions:
Euclid included unproved assumptions in "The Elements" which he called common notions, but to us we are use to the term axioms. Common notions were to be understood as agreed upon statements of science. We define axiom as "a self-evident truth that requires no proof," and "a universally accepted principle or rule," and in the math courses I have taken so far this is exactly how we proceed with the use of axiom's.



Sources:
http://www.britannica.com/EBchecked/topic/194880/Euclid
http://math.about.com/od/mathematicians/a/euclidbio.htm
http://www.thefamouspeople.com/profiles/euclid-436.php
http://dictionary.reference.com/browse/axiom?s=t

What is Math?

So what exactly is math? The term "math" has many interpretations and has different definitions to different people. For some it's the physical act of calculating numbers. On the other hand, others see math as more of the mental game rather than just what a calculator spits out at us. This mind game includes problem solving, logical reasoning, and critical thinking. We cannot not pin point an explicit definition to the term "math" as it is too broad.

Starting from grade school we learn the basics using mathematical operations. Now with the college math courses I have taken, the course material is not as much "math", depending on your definition of this term, as it is logical reasoning.

Top 5 Monumental Moments of Math:
1. The Number System: It's hard to think of where this is not used in our everyday lives!
2. Mathematical Operations:  We see these in functions, computer codes, and in many other instances.
3. Pi: We will never know what pi is exactly equal to, yet we use this in various formula's.
4. The Pythagorean Theorem: This has been used countless times in school and will always be a topic covered in course material.
5. The Quadratic Equation: At some point in our academic careers we have had to learn this equation. The quadratic equation is a great resource that algebra has acquired.