2007 Concordia Math May Seminar
 2007 Student Photo Journal
 The Mathematics May Seminar greatly increased my knowledge about math in the world, and it changed my perceptions for the better regarding math. Not only did I read about these ideas and the relevant people, but I was able to see the fruits of their labors in person, from the pyramids to the Greek ruins to art to even the subways, my increased awareness of math helped me see different dimensions of meaning than just a surface view. These ideas will definitely influence my future thoughts and ponderings.
 When I signed up for this class, I anticipated it to be mostly just for fun. While it has been very fun, it has also been a wonderful learning experience for me. By studying the history of math, I have seen math from another perspective. The travels we have undertaken have only enriched what we learned in the classroom.
 The purpose of the seminar, Mathematics in Another Light, is to look at mathematics differently, seeing its farreaching influence in all realms of knowledge: music, art, philosophy, science, and others. For example, one negative impact of mathematics dealt with the rigorization of literature and poetry. Passion and emotion were lost to the cold reason of mathematics. This occurred in the eighteenth century. One positive impact of math was realized in art and paintingin the mode of perspective. We have seen how mathematics has impacted historyfor better and for worse.
 Math is found everywhere, from the ease of everyday occurrences to the intricacies of mindboggling equations. Throughout our travels, it is evident that math is used on a daily basis, whether we realize it or not. Another aspect of math that is not commonly thought of is its use in thought and logic. For example, there are many different branches of religion around the world, but they are all based on the logic of the idea of a higher being. A unique version of this logic is demonstrated by Pascal's Wager. With the probabilities favoring a belief in any sort of religion, this idea of logic shines through. It is natural for humans to maximize pleasure and minimize pain, and through the use of logical thinking they can obtain this.
 As a mathematician, I find some of the pure expressions of math very beautiful. Yet I still like to be grounded, for the physicist in me can be disgusted with some of the highly abstract, seemingly useless ideas in mathematics. However, nonEuclidean geometry seemed pointless at the time, but it has found roots in the reality of Einstein's cosmos. Morris Kline's discussion of geodesics in different geometriesculminating in the discussion of general relativitywas particularly illuminating from a mathematical point of view. In light of these facts, I will try not to brush aside any mathematical idea, for it could find a future physical manifestation. Indeed, mathematics is a storehouse to draw from. Contemplating this ideal realm may be helpful in simplifying problems & finding insight into the nature of the universe. The Seminar has led me to appreciate mathematics more, as well as its effects in the world.
 I am easily frustrated when I have a difficult time with a concept or a problem, and often I do not think about why something is the way it is. I like when I know a process, can follow that process, and can get into the routine of that process. This class has pushed me to leave that routine, and to try to understand math on a deeper level.
 An important notion that struck me as interesting and important and changed my views on math was how mathematics can only grow in certain environments. The best example is in a Greek versus Roman comparison. Since the Greeks were so unconcerned with practical uses, they made significant progress enlarging mathematical knowledge. On the other hand, the practical Romans used the knowledge to build stunning works, but made little progress in discovering new results. I think this idea can be abstracted to almost any field of studyif the only concern is to use the current knowledge body, the field will suffer in the long run. This is great ammunition to use when discussing how important pure math research is and how important the acquisition of knowledge is in general.
 When we got to Rome, I learned that the use of math and art can alter the structure of a building! I leaned this at the church of Sant'Ignazio. I knew that art could be created with a linear perspective (we viewed many of these in the Uffizi and the Louvre), but I had always thought of this as being placed on a flat plane (a canvas or a wall). At Sant'Ignazio, the paintings on the ceilings blew me away. The ceiling was painted to look like the walls were opening up to the sky. So there were painted pillars on the ceiling that seemed to be following the plane of the walls, when in reality they were bending over on top of us. The painter not only had to understand angles for linear perspective, but he also had to have a solid comprehension of spherical geometry. The art didn't stop there, though; Fra Andrea del Pozzo also painted a dome onto the ceiling. When we entered the church, I looked at the dome and it looked fine, but as I walked towards it, it started to twist and lean a bit. Pozzo had used a perspective with the doorway having the vantage point. I had never thought about the fact that math knowledge can be used to create optical illusions. These different sites that we visited really did help me to see new contexts, depths, and uses of math.
 In the last year or two the relations between different branches of math have become very obvious to me, but how math related to other disciplines was not as clear to me. As a future teacher, this concerned me. I knew I could make connections, but many I thought of seemed weak. This class has helped to change that. I can easily discuss how math has influenced art, with advancements in perspective and dimension. Seeing this in person rally solidified the idea for me as well.
 The preMay seminar and the May seminar have really helped me to appreciate the mathematics surrounding me. One thing that really stood out to me was the use of mathematics in art. Previously, I had considered math and the arts to be completely separate fields, but after learning about vanishing points and perspective I see a connection. Even knowing about these two ideas helped me to appreciate and enjoy the art in the museums more because I knew of something to look for in the paintings.
 Overall, this trip and semester class have been great learning experiences. Not only did I see Galileo's theories but I also learned about future math courses like topology at the Science Museum in London. The trip has made me more aware of math's importance and influence in the world and has influenced my mind in understanding the depths of the subject. Thank you for this wonderful opportunity. I have leaned a great deal about the pyramids and beyond. It was a wonderful experience that I would like to repeat again. Thank you.
 Another area that shows mathematical influence is painting: behind the paint are detailed outlines and mathematical calculations. The fulcrum of a piece only works if the painter uses angles and depth perception correctly. For example, Leonardo da Vinci was an expert on angles and contrast. In the Mona Lisa, the eyes and maybe even the body catch the viewer's eye before the background is even seen. Yet the background shows Leonardo's curves and horizon line accurately executed. This shows his thought and understanding of the mathematics behind the art. And in reality it makes all the difference between a masterpiece and another work of art.
 In high school, math teachers tell us of the importance of math. We smile and nod and think math only is used in math class. In college we realize that yes, math does influence other disciplines. We can see that the sphere encompasses the sciences, business, and some other spheres. However, there is still a distinct division between the mathematic area and the art and religion areas. Objective spheres can not combine or blend with subjective spheres. That was what I had originally thought. I was surprised to find out that the golden ratio sometimes determined facial and body proportions in art, and artists used geometric ideas to add depth to pictures. This concept makes sense but I hadn't realized it before. I enjoyed learning about the idea of a vanishing line and also watching how art progressed through the different cultures beginning with the one dimensional Egyptian art going to the geometric art and statues of the Greeks to the paintings of the renaissance. It seemed that as mathematical knowledge increased the paintings became more lifelike. Along similar lines, I didn't think that math and religion could combine. However, the Pythagoreans had an entire religion based on the magical properties of numbers. This practice caused considerable problems when some of the facts they based important religious ideas on proved wrong. For me, mathematics is no longer just numbers on a page. Mathematics now encompasses art, religion, architecture, and people, and can have a reality of its own.
 Looking at the precision with which both the Egyptian and Greek sites were built (minus the years of decay that make them ruins now), one starts to get a better idea of just how much surveying and geometry knowledge and skills they must have had to be able to build such things as the pyramids, Luxor Temple, the Parthenon and other great monuments. Before the class I had never really realized just how far back in history these branches of mathematics must have gone.
 Over the course of this class and throughout the duration of the May seminar trip, I was continuously reminded and amazed by the amount of mathematical knowledge and by how much it influenced and is still influencing history and every facet of human life. If I were to talk about every way in which this has happened or every thing that has enlightened me I would be able to write a book. But there are some things that have really brought themselves to my attention which I will talk about. While we were crawling all over the cities in Egypt and Greece and even Rome, it struck me as to just how old some of our knowledge is. I was also struck by how much knowledge has been lost over the years due to catastrophe, regime change, or other means.
 Everyone that has gone through math in high school has learned about Euclid's findings. Considering the rapidity with which knowledge changes, it is amazing that anything from 100 years ago is still being taught let alone from 2000 years ago. As math progresses through the years it is very interesting to see how everything builds on top of each other. In order to prove a complex problem you have to first prove the most simple parts of the problem and then finally build up to the more complex questions. It seems like this is more true in math than in any other subject.
 Overall, with the combination of the classroom and the actual trip, I truly have seen math in a new light. I have been fortunate to see many famous sites and how math has had an impact on them. I hope to go forward with what I have learned from this experience and incorporate it in other parts of my life.
 This trip opened up a perception that is lost when you are exposed to mathematics only in a traditional class. The size and scale of the sites was not the only thing that made me look at mathematics differently, but also the beauty that mathematics is hidden in. Take the Parthenon on top of the Acropolis in Athens, for example. The design of the pillars so that they do not look thinner in the middle from a distance is a single example of how mathematics can be used to make a building stand out more. Another example is the Klein bottle exhibit at the Science Museum in London. Seeing how mathematics can create unique figures like those would not have the same impact if it came out of a lecture. It is the beauty of mathematics that is lost when you only read textbooks or sit in class taking notes.
 Throughout our journey, no matter if it was in Egypt or London, we continued to see that mathematics surrounds us everywhere and there is no escaping it. Even though the math we see is not exactly what we are studying in our textbooks, we still know that math is used only to be seen in another light. If it is from the construction of the pyramids to the calculation of time, math is always being used throughout the world and I feel that it is one of the most universal subjects that is around.
 During the course of this May seminar and preseminar course I have learned a great amount of information about the use of mathematics in architecture and design. During the precourse I was given my original introduction to the great buildings and monuments I would be seeing on this trip and learned some initial background information. I was very intrigued by the idea that the means of construction for many of the elaborate monuments we have seen is unknown. It also amazes me to know that the math believed to be known at these times was limited and yet the people were still able to create some of the most amazing structures in the world.
 Mathematics, what is it really? Before our May seminar trip, I never really pondered how much mathematics is needed in everyday life. To me, mathematics was just a school subject that everyone had to take, but after this trip, I do see mathematics in a different light. From here on out, math definitely means so much more to me than just a plain old school subject.
 This course has made me realize that the progression of mathematics began and was more advanced much earlier than I had thought. The origin of mathematics is credited to ancient civilizations 3000 years ago and has continued to develop through time.
 The reason for the preMay seminar and the May seminar was to learn to see mathematics in a different way. This I believe was accomplished by first looking at some of the history behind mathematics and learning what has happened through that history. Also by going and seeing what is left of the ancient cultures and seeing how they put their knowledge of math to use. I believe that this May seminar has allowed me to spin the hyperdimensional box of math and see it in a new angle and a new light. It has been a fun trip.
 In addition to being amazed by the Parthenon and the structures at the Acropolis I have found a deeper interest in the history of mathematics. Before I went on this trip math has just been numbers to me. However, I have realized that there is so much more. Math is all around us. There is so much math in things such as architecture and painting. This trip has shown me the simple things that math helps us accomplish. Also I have learned that large things can be accomplished with simple math.
