Tuesday, February 25, 2014

Book Response 1: Journey through Genius

            In Journey through Genius, William Dunham introduced some of the most influential mathematicians in history alongside explanations for one or two of their most profound discoveries in math. Journey through Genius gives a historical context for the mathematicians and their discoveries, and tries to convince the reader of the greatness of the discovery—the worthiness, as it may be, for why that piece of mathematical history has made it into this book. While the content provided in Journey through Genius is accessible to those with only a high-school level mathematics education, the means to coming to the results (via the proofs included in the book) are still so genius that one with a more in-depth understanding can appreciate the proofs given. If the purpose of the book was to inform the reader of some of the great mathematical theorems and those who discovered them, then this book mostly achieved its goal—its main fault is the short-sighted nature of its scope.

            There are twelve chapters in Journey through Genius, each one (more or less) dedicated to a main theorem and to the life of the man who discovered it. Each chapter introduced context of that area of math at the time of the discovery, gave a proof outlining the main theorem, and concluded with an explanation of how the theorem influenced math and sometimes summarized the rest of the discoverer’s life. Not only was each of the twelve chapters focused on someone (or, in Chapter 11’s case, some people) from the Western world, but even introductions and conclusion rarely mention anyone outside of the Western culture. Dunham tries to explain this obvious exclusion in one of his chapters, stating, amongst other things, that “Because the Islamic mathematicians put less emphasis on proving their results in complete generality, no great theorem appears here” (131). This weak excuse for omitting some of the great thinkers of the Islamic world is more consolation than the void of an excuse for why discoveries from China, India, and other realms of the world aren’t explored.

Even if the excuse Dunham supplies is true, it doesn’t explain why more of the Islamic mathematicians’ work isn’t explored in the introduction or conclusion of the chapters. Throughout the introductions and conclusions of the chapters, Dunham does thorough exploration of other theorems that aren’t the chapter’s central topic. He does this time and time again, and in a few of the chapters he goes so far as to mention that there was some progress made in the area by non-Western mathematicians, yet no short proof was supplied, no exploration of any non-Western “discovered” mathematics was ever explained by Dunham. In total, there are less than five pages dedicated to non-Western mathematical discoveries (out of 283), which, in my opinion, is Journey through Genius’s greatest downfall.

The information that Dunham did include about the mathematicians and the discoveries that he did highlight were both intellectually and emotionally satisfying. I appreciated learning how each great mathematician’s discovery was proven and about the background for each great mathematician; hearing how each embodied genius, handled his achievements, and went about his life. The background given for each mathematician increased the impact felt when being led through how each mathematician proved his theorem, giving the already extraordinary proofs another layer of meaning and understanding. I learned much about the state of mathematics from Journey through Genius —from the discipline’s history, to the construction of proofs of some truly great results, to the realities of the Western-centric mindset of some of the mathematical community that still pervade today.

Wednesday, February 19, 2014

History: Brahmagupta

As we learn about history’s great mathematicians, I am frequently struck by how often their pursuits are intertwined with other disciplines. We now have concrete job titles for what a person does—they are a mathematician, an astronomer, an engineer, a poet, a physicist—and it is implied that one will chose a profession and stick to it. I wonder if “mathematician” once encompassed all of these titles, and that to be a mathematician would have implied the various applications, or if we have now forced titles onto the work that people do, thereby corralling their ambitions to one topic.

One individual that was heavily involved in two disciplines—astronomy and mathematics—was Brahmagupta. It has been estimated that Brahmagupta was born in 598 in Ujjain, India, and that he passed in 670. During his lifetime he wrote (at least) two influential books, as well as held the position as the head of the astronomical observatory at Ujjain, “which was the foremost mathematical centre (sic) of ancient India at this time.” (O’Connor and Roberts). A portion of his work has been preserved in his book Brahmasphutasiddhanta, which displayed his insights in both astronomy and mathematics. The first ten chapters of the book described astronomical phenomena. The final fifteen further explored astronomical phenomena, but it also delved into algebra and geometry (O’Connor and Roberts).

Brahmagupta uncovered great mathematical truths in both algebra and geometry, but perhaps even more impressively, he is attributed to defining zero as a number and negative numbers. As early as 200 A.D., a closed circle-symbol or the word “kha” would be used by those who wanted to indicate an absence of a number or an empty place (“Zero as a Number”). Brahmagupta defined zero as a number and dug deeper, thereby unearthing negative numbers. In his book Brahmasphutasiddhanta he outlines rules for math using zero and negative numbers, where he calls an arbitrary negative number a “debt”, an arbitrary positive number a “fortune”, and zero as zero (Mastin).

Not only did Brahmagupta provide us with some of the building blocks of our modern number system, he concluded that quadratic equations could have two possible solutions (and one could be negative), he solved quadratic equations with two unknowns (which wasn’t considered in the West for another 1000 years), and gave a formula for the area of a cyclic quadrilateral as well as a formula for its length in relation to its sides (Mastin; Hayashi). In exploring these concepts, Brahmagupta even began to dabble in how we now view algebra by using the initials of the names of colors to represent the unknowns in his equations (Mastin).

Brahmagupta’s work explored the concretes of our solar system as well as the abstract concepts of mathematics. In describing zero and negative numbers, Brahmagupta unequivocally changed mathematics, and his other contributions further confirm the genius he possessed.


Works Cited
Hayashi, Takao. “Brahmagupta.” Encyclopedia Britannica, n.d. Web. 19 Feb. 2014.
Mastin, Luke. “Indian Mathemtatics-Brahmagupta.” The Story of Mathematics, n.d. Web. 19 Feb. 2014.
O’Connor, JJ, and E. F. Roberts. “Brahmagupta.” School of Mathematics and Statistics, University of
        St. Andrews, Scotland, n.d. Web. 19 Feb. 2014.
“Zero as a Number—Brahmagupta Period.” Wayne State University, n.d. Web. 19 Feb. 2014.

Sunday, February 2, 2014

"Doing Math": Tesselation (s)

Using my novice Geogebra skills, I created the following tesselation:

I also made a few attempts previous to this one for my "base" shape, but the final one I used (above) is my favorite. Here is another "base" that I made, though I didn't find as aesthetically pleasing:
And here is another, more "full" tesselation that I was able to flesh out more.