Tuesday, February 25, 2014

Book Review

Visions of Infinity

By Ian Stewart

The book Visions of Infinity talked about some of the very important discoveries made my mathematicians in the past. One aspect of this book that I enjoyed was the position that was taken on presenting these discoveries. This book stayed away from long, confusing proofs of mathematical discoveries. Instead of focusing on the actual mathematics of certain people, this book focused more on the thought process behind the math. Throughout the book, many different theorems and conjectures are discussed, as well as how each of these was thought about. Some of the problems that are talked about are the millennial problems. These problems offer a hefty reward of $1,000,000 if they are solved. While there have been numerous attempts to solve these problems, only one of the problems has been solved so far. Visions of Infinity covers a spectrum of topics. These topics range from simple things like prime number and the four color theorem to the Mass Gap Hypothesis.
                This book does a good job of making the topics readable to an audience that is not an expert in mathematics. However, I don’t think that this book is a good read for everyone. Even though Visions of Infinity does a good job putting much of the information in simpler terms, there were a few parts that went clear over my head. I know that I am no master mathematician by any means, but I would like to think that I have more experience and a better understanding of math than the general population.
                Overall I think that this is a great book to read if you are interested in the mathematical discoveries throughout history, and have at least some knowledge of upper level math. I really enjoyed reading about how each of these discoveries has progressed over time and the contributions of the many mathematicians over the years. I also liked seeing that each of these discoveries required the work of so many different people. I would recommend this book to any undergrad math student with the advice to take the time to really this book, as it is not a quick read if you want to really understand it.

Sunday, February 23, 2014

Nature of Math

Does our number system have anything to do with what mathematics is?

I think that this question is can be answered with yes and no depending on how you interpret the question. I would say yes, our number system has everything to do with what mathematics because there is no way to represent what we know about math without using our number system. For example, if we did not adopt our number system, how would we be able to understand just about anything? Would we still understand or discover things like calculus, algebra and physics? How would we quantify anything? How would we make change at a grocery store? How would we add, subtract or multiply?

On the other hand, I would say no, our number system does not have anything to do with mathematics. Our number system doesn’t determine any truth about mathematics. What we know is true about mathematics now will always be true. Our number system is just a way to represent what we know. If we didn’t have our number system, we would more than likely have another representation of what we have discovered about mathematics. To answer the above questions, the concept of calculus, algebra and physics would still remain. Whether it was discovered or represented with our number system would be a different story. We would have found another way to quantify things. Who knows, maybe we would still be using roman numerals. The use of the words “one, two, three… etc.” are essentially just words we agreed to use to represent those numbers. Our number system could go something like “popcorn, carpet, pillow dinner” if that is what we use to represent what we now call “one, two, three, four”. The number system itself does not determine anything about math. It is simply a way for people to represent what they know that is universally accepted so that information can be shared with others.

Sunday, February 9, 2014

Doing Math

Creating a Tessellation

A tessellation is the tiling of a plane using on ore more geometric shapes, called tiles, with no overlaps and no gaps. Tessellations can be simple or intricate and can be categorized into smaller categories which include the following categorizations:

Regular: This type of tessellation is made of the same shape.

Semi-Regular: Tiles of more than one shape.

Aperiodic: Tiling that cannot form a repeated pattern.

Below is an example of a more simple tessellation that I have created.