Thursday, June 26, 2014

Skimpression- Second Book Review: Amusements in Mathematics

Amusements in Mathematics: H.E.Dudeney
This book is an entire book dedicated to riddles, puzzles, and games found in Mathematics. Topics such as algebra, geometry, puzzle games, magic square problems, crossing river problems, measuring, weighing, and packing puzzles are included in this book. This book also comes with a complete solution guide for each problem.
I decided to choose a couple of problems throughout the book and attempt to solve them.
#424: The Dovetailed Block
This problem shows the following photo.
 The problem says that the non visible sides of this cube all look exactly the same. Then how were the pieces put together?
Below is an example I found, that is similar to the solution in the back of Dudeney's book.

#246: Visiting the Towns
This problem gave the diagram shown below.
 And asked if the traveler "wishes to visit every town once, and once only, going only by roads indicated by straight lines." You have to determine how many different routes are there to select. The traveler must end his route at town number one from which he started.
Below is my work for this problem. I began by re-arranging the numbers so that there was more of a clear path to travel and finished to conclude that there is only one pathway the traveler can take.
These are just two of the many different types of problems provided in the book. Overall this is a fun book to use if you are looking for a quick puzzle to solve.

Wednesday, June 25, 2014

Weekly #7- Fractals: The Nature of Mathematics

What is a fractal?
Before taking this course I was never formally introduced to the word fractal. It is very interesting that fractals occur frequently within in our daily lives. A fractal is "a never ending pattern." At each step of a fractal there is a repeating pattern. Shown below is an example of a fractal (found on wolfram alpha).
 As you can see each shape follows a pattern within itself and the pattern continues.

Brief History of Fractals
BenoƮt Mandelbrot came up with the term "fractal" as something "that when divided into parts, each part would be a smaller replica of the whole shape."
There are important fractals found including the Mandelbrot Fractals, Julia Fractals, and Newton Fractals. Below are pictorial examples of each fractal.

Mandelbrot Fractals 

Julia Fractals

 Newton Fractals

Fractals Found in Nature

Shown below is a typical fractal found in nature; a leaf is pictured below. From this picture I can see that the leaf pattern continues at each stage of the leaf. This is very interesting to see that such simple things in nature can create such an interesting bridge to mathematics.

Below is a photo of a Roman Cauliflower. As you can see the cauliflower is a fractal that is grown in nature. This pattern is has a very unique look; there are multiple branches that form on this Cauliflower that follow the same repeated pattern. Looking at this I am wondering how this vegetable might taste. Whether it tastes good or not, everyone should know how good mathematics is and the amazing connections that mathematics makes up in our world.

Below is a photo of the Grand Canyon. From the photo you can see that the Grand Canyon is also a fractal found in nature.  This is yet another example of a fractal found with our world.

Here is an example of a lightning fractal. As soon as I found this photo I was very curious whether all lightning strikes were considered a fractal. Looking more into it, I was not able to confirm or deny this idea.

In conclusion, fractals take up many forms throughout our daily lives. It is important to remember that this discovery has led to many other ideas relating back to the fractal. Studying fractals can be fun and can also lead to new discoveries.