Reflective Essay
Much of modern day mathematics is either directly from, or is derived from the theories and advancements of Greek culture. This Greek culture slowly spread throughout the Mediterranean and eventually created a unique and distinctive Mediterranean culture with great knowledge and focus towards mathematics. Early on, daily Greek life revolved around mathematics so it was only natural that so many great advancements originated in its culture The Greeks developed “equations” for areas, volumes, and angles. The Greeks also, over a long period of time and input from many mathematicians, developed an extremely accurate value for pi. Greek mathematicians also brought in the advancements from of other cultures, such as those from Indian and the Middle East. This allowed them to apply these advancements to their own developments and over time spread them throughout the Mediterranean creating a culture, of similar foundations, that was always looking to progress mathematics and answer the world’s oldest questions with the help of math.
The Greeks studied every form of mathematics. In doing so, one of their greatest areas of focus was solving and developing the general equations for geometric shapes. Pythagoras, one of the greatest known Greek mathematicians, was fundamental in creating a solid foundation for Greek developments in geometry. Pythagoras was the first to make the distinction between odd and even numbers. He did this by making squares out of pebbles and then dividing them in half. If a number divided in half with no pebbles left over, it was even. Pythagoras’ use of square numbers and his application of the pebble method are seen in Artifact 1 (Chapter 2 homework assignment). Figure 1 below shows a basic example of a Pythagorean number formed with pebbles.
[1] Figure 1: Pythagorean Pebble Method
It is humbling to think that a man as genius as Pythagoras had to start his mathematical career by discovering and proving something that the modern world now takes for granted such as even and odd numbers. It’s even more mind baffling that a man who made so many of advancements in mathematics started by using pebbles and drawings in the sand.
Pythagoras further developed his number theory and taught it to his followers, the Pythagoreans. He then used the idea of putting numbers in “squares” to develop Pythagorean triples. These are numbers that make up the relations of the sides of a right triangle. Pythagoras created tables that contained these ratios and used them to calculate the side relations of many right triangles. Artifact 2 (Cultural Essay 3) describes Pythagoras’ work in more detail. Figure 2 below shows an example of a Pythagorean triple and its corresponding square numbers.
Figure 2: Pythagorean Triple
Growing up and learning the of basic principles mathematics it never really occurred to me how exactly Pythagorean triples worked, I just knew that it was a rule that was discovered a long time ago. Finally seeing and understanding how Pythagoras and his followers came to recognizing Pythagorean triple was eye-opening for me. It showed me that everything I use in mathematics has a origin and proof behind it; nothing in mathematics just come into existence.
As all ancient cultures discovered, the derivations for the many geometric equations were very difficult for one reason, pi. Pi is the ratio for the diameter/radius and area of a circle. Archimedes was the Greek genius who made some of the greatest strides in the determination for the value of pi. In his book, On the Measurements of a Circle, he makes three propositions for the determination of pi. Archimedes three propositions are discussed and analyzed in Artifact 3(Cultural Essay 4). However, it should be noted that Archimedes use of a 96-sided polygons, inscribing, circumscribing, and the method of exhaustion was the first recorded method for actually calculating pi. Archimedes approximation for pi,, from his second proposition has been used as an accurate estimate for pi for centuries. Through Archimedes’ establishment of accurate values for pi, the Greeks were able to develop formulas for and calculate very accurate values for the geometric properties of circles. Figure 3 below depicts an example of how the method of exhaustion, used by Archimedes, started.
[3] Figure 3: Method of Exhaustion
Over the course of the semester it became very clear that pi was the thorn in the side of nearly all cultures. The other cultures had very close value but none so close as Archimedes (up to that point). I had no idea that there were so many different methods of obtaining and approximation for pi. Archimedes had three different propositions alone and one of them involved using a 96-sided polygon, which was arcane to me.
Another great advancement made by Greek mathematic culture and its mathematicians was in the area of conics. One of the grand mathematical questions of ancient times pertained to the doubling of a cube. It was discovered by the Greeks that cones were key to solving this geometric conundrum. Apollonius developed many crucial theories in the area of conics and outlined them in his work, Conics. Using his definition of a cone, Apollonius derived the “equations” for the three basic conics: the ellipse, parabola, and hyperbola. He did this by cutting his cone into different planes and related the measurements of these planes to the base of the cone. Further discussion and the definitions of Apollonius’ conics are presented in Artifact 4 (Cultural Essay 5). Figure 4 shows how the conic shape discovered by Apollonius relate to the cone.
[4] Figure 4: Relation of Conics Shapes to a Cone
One of the most noteworthy mathematicians of all times was Leonardo de Pisa, also known as Leonardo Fibonacci. H Nothing epitomizes how an entire region can be a mathematical dynamo as the work of Fibonacci and his compilation of all of the discoveries of the Mediterranean region. He was the culmination of nearly 1000 years of Greek-based Mediterranean mathematics. Fibonacci’s name is best known through the centuries for his recognition and “discovery” of the Fibonacci sequence. This is the sequence that forms when the number n is formed by adding the two previous terms, n-1 and n-2. However Fibonacci had a far greater effect on Greek mathematics than his sequence; he is the one who introduced the Hindu-Arabic numeral system to the Greeks. Fibonacci’s lifework is outlined in Leonardo de Pisa. Not only did he introduce it, he taught his peers how to apply it. Fibonacci travelled all over the Mediterranean and the Arab lands. He had an Arabic mathematical mentor and in his book Liber Abaci, he defined and demonstrated the application of the Hindu-Arabic numerals. This was groundbreaking because up to this point, the Europeans of the Mediterranean region, predominantly the Greeks and the Romans, were adding using Roman numerals and an abacus. Fibonacci allowed for immeasurable further mathematical development of Mediterranean mathematics. Fibonacci’s mathematical breakthroughs and his major life works are outline and discussed in Artifact 5. Learning about Fibonacci proved to me that one man can take copious amounts of accrued knowledge and put it together to create even more astounding beauty. Fibonacci is the type of person that I know think of when I hear anything about Mediterranean culture. He had so much influence in not only that region but the world and I believe he is one of the most noteworthy of the Mediterranean intellectuals, second only to Pythagoras.
The Mediterranean culture naturally spawned great mathematicians. Mediterranean culture derives most of its distinctions as adaptations and spawns of ancient Greek culture. Everyday life inspired curiosity in the field of mathematics and its stem fields of astronomy and physics. The advancements in the fields of geometry, conics, number theory, and the overall spread of the wealth of knowledge by the Mediterranean culture is nearly unsurpassable by any of the other ancient cultures. So much has been and can be learned from Mediterranean civilizations and their cumulative culture; particularly, their dedication to math.
Works Cited
Image Links:
[1]
http://mediacacheec0.pinimg.com/236x/3d/42/f6/3d42f6b2532764a393ee87300443adf1.jpg
[2] https://encryptedtbn1.gstatic.com/images?q=tbn:ANd9GcSrYQuQfIes9MJNhzwfWutDo8VWOcYc71j12c-DIbCye1NBFeQUzFCnSK8j
[3] http://personal.bgsu.edu/~carother/pi/pigifs/inc-circ-8gonx.gif
[4] http://www.storyofmathematics.com/images2/apollonius_conics.gif
Artifacts:
Artifact 1: Chapter 2 Homework Assignment-
Homework assignment 2 shows how to use dots to prove that any triangular number times eight plus 1 makes a square and that any odd square diminished by one become eight times a triangular number. This is the method used by Pythagoras. I never realized that a man who did so many great and brilliant things had to start somewhere as simple as using pebbles to distinguish odd and even number. It was very humbling to me to think that one culture takes nearly all the credit for most of the basic mathematics that I now take for granted.
Artifact 2: Cultural Essay 3- Greek Advancements in Geometry- Cultural Essay #3
Cultural Essay three talks about the advancements of Greek culture in relation to geometry. It primarily discusses the early discoveries of Pythagoras and how his advancements in geometry affected Greek culture as a whole. Before I just thought Pythagoras was like any other Greek scholar who did something that made him famous. It wasn’t until later that I realized that he was a founding member for an entire culture that dedicated its existence and understanding of the universe to mathematics.
Artifact 3: Cultural Essay 4- Archimedes and Pi- Cultural Essay #4
Cultural Essay 4 discusses the work of Archimedes and his pi discoveries discussed in his work On the Measurements of the Circle. It analyzes how his work on conics affected Greek culture as a whole. Archimedes work n solving for pi greatly influenced my understanding and respect for Greek culture as a whole. His work showed a culture dedicated to mathematics and that a craving for knowledge can envelope an entire culture.
Artifact 4: Cultural Essay 5-Appolonius’ Conics-
Cultural Essay 5 discusses Apollonius’ work on Conics such as Hyperbolas, Parabolas, ellipses, and circles. It discusses the affects of his discoveries and his work Conics on Greek culture and mathematics as a whole. The origin and proofs on conics by Apollonius was eye opening to me as I had never truly understood how they were all connected together (Through Cones-hence the name conics).
Artifact 5: Library Assignment 1-Mathematician Essay on Leonardo de Pisa-
Leonardo de Pisa was the first library assignment and it required a research paper on a mathematician. This paper highlights some of Fibonacci’s greatest mathematical discoveries and life in general. It also analyzes how he influenced the mathematical world as a whole. I learned that mathematics is one of the few things in the world that can connect any and all cultures with no boundary lines or language barriers. Mathematics can spread no matter what conflicts are going on around it.
