Author Archives: cbrines

The conchoid of Nicomedes and other engineering geometries

Before I became a Math major, I was an Engineering major, and the things that interested me the most in engineering were ‘non standard’ ways of generating power. Solar power is a classical example of ‘non standard’. Another thing that is really interesting is that ancient geometric principles are being used to generate that power. Though not all of those ways of generating power are considered power generation to most people; some are like the use of the conchoid Nicomedes to develop a heliostat to help generate solar power, and others are like the downtown library in Salt Lake City which uses an understanding of geometry and physics to heat and cool a massive structure.

The conchoid of Nicomedes can be thought of as a curve r=b*secant θ in polar coordinates.  This family of curves was discovered by Nicomedes, who was an ancient Greek mathematician. Wolfram/Alpha describes the conchoid as “…the locus of points a fixed distance away from a line as measured along a line from the focus point (MacTutor Archive). Nicomedes recognized the three distinct forms seen in this family for 0<a/b<1, a/b=1, and a/b>1. (For a=0, it obviously degenerates to a circle.)

ConchoidofNicomedes

ConchoidofNicomedesCurves The Conchoid of Nicomedes ¹

The conchoid of Nicomedes was popular during the 17th century, at least with mathematicians. Currently it is going through a slight revival in popularity, at least with heliostat designers.  For those of you wondering what a heliostat is, heliostat means stationary sun. It is used to project the suns rays wherever you might want them to be pointing, such as a solar energy collector.

An example of the conchoid of Nicomedes being used as a solar concentrator.²

How this works is that there are a series of gimbaled mounts (6) for the mirrors (3) on the poles holding them up and the motion of a single bar (5) is enough to get the whole array to face a new direction. The mirrors will focus the rays of the sun onto the solar collector (14)  and off you go with all that nice fresh solar energy. The solar collector is at the apex of the focus point of the conchoid and all the mirrors are following the path of the line of the conchoid, or put another way each mirror is tangential to a corresponding point on the curve of the conchoid. It really is an interesting way to use this type of math to harness power.

File:Salt Lake City Public Library -IMG 1754.JPG The downtown Library in SLC ³

The Salt Lake City library (downtown) also harnesses the power of the sun but most people don’t seem to realize it. The entire southwest side of the library is shaped like an enormous lens and if you can go in and look at what it is, it really is amazing. That lens captures the warmth of the sun and is used for both heating the entire structure, but also is used for cooling in the summer time. During the winter its operation is pretty basic: capture was much heat as possible. This heating can be calculated by summing the total energy captured by the area of the glass and dividing that total by the total area covered in glass to determine the energy captured by the windows. This equation is true for any window but the shape of the lens really helps out, because of the magnification equation.5 This equation uses the ratio of image and object distance to determine magnification. In this case an image is not being attempted to be made, but the magnification is still able to be used to magnify the available heat.

During the summer the true genius of this piece of geometrical engineering happens.  There are vents that run parallel to the surface of the water (they are dark brown). Those vents open and a corresponding set of vents on the roof line also open. The heat generated by the “lens” then escapes through the roof vents and creates a lower pressure inside the lens that the air from the lower vents rushes to fill. The air from the lower vents has been cooled by passing over the reflecting pool, and more air is pulled from the library creating a natural gentle breeze. This whole set up uses the heat of the sun to cool a huge space, all using ancient geometric principles.

1 http://mathworld.wolfram.com/ConchoidofNicomedes.html

2 http://www.google.com/patents/US7677241

3 http://commons.wikimedia.org/wiki/File:Salt_Lake_City_Public_Library_-IMG_1754.JPG

4 http://www.slcpl.org/branches/view/Main+Library

5 http://www.physicsclassroom.com/class/refrn/Lesson-5/The-Mathematics-of-Lenses

Math Education: Why we are where we are

   I have from time to time read about all the horrors of our current educational system and the ongoing ways that people have come up with to ‘fix’ our broken system. Yes, our system definitely has its faults, and I’ll touch on those in a moment (I think the largest is a dictatorial set up of learning), but have you ever wondered why we are where we are in terms of our education and math education in particular?

Fredrick the Great of Prussia by Anton Graff, via Wikimedia Commons.

   The most common system of education in the United States and the world is based off the Prussian education system. The Prussian education system, which began in 1716, has an odd and strangely juxtapositional beginning. One half of the juxtaposition was a high minded religious ideal and the other half was all about governmental control. Before universal education was brought about by the Prussian education system, education was expensive and you got what your parents could afford. According to  Brendan McGuigan on Wisegeek “The Pietists,(1) among other religious factions of the time, believed that the deepest understanding of God could come only through a personal reading of the Bible; therefore, literacy was important for all people, not just the wealthy.” What seperated the Pietist movement from other creeds that called for literacy was that the Pietists wanted to create schools for all and not just for literacy, math was also considered to be a fundamental part of education. The Pietist movement was started by Johann Kaspar Sehade (2) in 1686 in Dresden, the capital of Saxony in the Germanies.(3) 

    The second half of the juxtaposition was that King Fredrick the Great of Prussia was looking for a way to consolidate power in his newly formed country. King Fredrick took the ideas of Johann Sehade and modified them to cement control of his newly formed dynasty through blind obedience to those in power.(4) (You will obey the teacher…or else.) The Prussian system was not all bad; it introduced the idea of universal education to the world, and some would argue that the advent of universal education heralded the start of the industrial revolution (5) or if not the start then definitely the continuation and acceleration of the revolution into the world we know today.

The Prussian educational system was brought to the United States by Horace Mann (4) in 1843 and has been used for compulsory universal education ever since. I remember that on my first day of college at the University of Arizona I was shocked when at the end of the freshman orientation a questionnaire was handed out by a representative of the state board of education that asked for our opinions on what should or should not be taught to high school students. I don’t remember all of the subjects, but two of them were the swastika and the Holocaust. The reason I was shocked was they wanted to know my opinion on something and not to just a regurgitation of knowledge. This was the first time my opinion was considered of worth at all by any organization, much less the government. The point of this flashback was to point out that the inherent obedience training that was put into the Prussian system is still there in a subtle way; it says without words that you (the student) know nothing, and your opinion is worth even less. At least it had seemed so to me. The Prussian system is very good for rote memorization, and the idea of universal education is unbelievably important, because it is not possible to predict where or when the next super genius world changer will show up.

What does all this have to do with math education? I believe it is best illustrated with a metaphor. Imagine a society in which all of its technology and understanding of the world was based around beautiful and amazing works of art. Now imagine for a moment that this world taught its children art almost exclusively through paint-by-numbers™. Then we for some bizarre reason expect the children to be able to paint masterpieces while they are busy looking for the lines to paint inside of. This is similar to what the Prussian system does for math education. There have been through the years various attempts at reform, some of the latest have been Khan Academy, Google schools, and the new common core standards. These groups and reforms are all attempting to make math a more accessible subject. An aspect to the difficulty of math education that these reforms can not take into account is the societal belief that math is hard and the problem that most of the ‘math’ teachers in primary education are not well versed in mathematics themselves.

   These problems are linked and unfortunately endemic to our society and even though it can not be dealt with until more people really understand math there are some signs of hope. (Quite the catch 22 though, isn’t it?)  Like the recent sea change towards “nerds” and “geeks” in our society.  In recent years it has become more socially acceptable to be smart and, our media is reflecting this. So here is hoping that this societal change of heart really takes hold and we can become much better at math and be proud of it.

1 http://www.wisegeek.com/what-is-the-prussian-education-system.htm

2 http://www.newadvent.org/cathen/12080c.htm

3 it was known as the Germanies at the time because the unification of the German city-states did not happen until 1871 under Emperor Wilhelm I

4 http://www.school.namaya.com/newamericanacademy/images/the-prussian-industrial-history- of-public-schooling1.pdf

5 Me, its argued by me, there may be others but definitely me.

Is Math Culturally Independent?

Is Math culturally independent?   Eleanor Robson asked this question regarding Plimpton 322. She wrote, “We tend to think of mathematics as relatively culture-free; i.e., as something that is out there, waiting to be discovered, rather than a set of socially agreed conventions.  If a simple triangle can vary so much from culture to culture, though, what hope have we in relying on our modern mathematical sensibilities to interpret more complex ancient mathematics?”  And yes, this was a homework question, but for some reason this question stuck with me, and I went looking a bit further.  For those of you who may not know, Plimpton 322 is an ancient Mesopotamian tablet around which there is some controversy. ¹  Scholars have claimed that Plimpton 322 is anything from a set of Pythagorean Triples, to a table of reciprocal numbers, or that it is possibly a trigonometry table.  The truth is, we just don’t know for sure; but whatever it is, it is definitely  more complex than the tax forms or accounting forms we typically expect the Mesopotamians to have left lying around.  (I just put it down a second ago, where did it go?)  Robson’s comment about the triangle mentioned refers to the difference between how we normally picture or represent a triangle and the standard Mesopotamian way of representing a triangle.  We have a tendency to depict a flat side facing down (for example Δ). The Mesopotamians, however, tended to represent their triangles pointing to the right similar to our play symbol. (Emblem unavailable at this time. Please consult your mp3 or video player, sorry for any inconvenience).

This question from Robson brought to my mind the idea of Musica Universalis.  Musica Universalis2 is a philosophical concept that is based on some assumptions made by the Pythagoreans, namely the combination of math and theology.  The Pythagoreans belived that everything had a numerical attribute,³ and they also found an appeal in certain symbols, such as the tetraktys and the Harmony of the Spheres (another name for the Musica Universalis).4  The concept of Music of the Spheres concerns the movements of the Planets, the Stars, the Moon and the Sun. (Remember, the thought at this time was that they all revolved around the Earth.)  One way of interpreting this was that there was some vast Celestial Orrery or Machine that had been set into motion. This Orrery controlled not only the motion of the celestial bodies but also the affairs of men. During these millennia there was no distinction between astronomy and astrology.

An example of an orrery. Image: Sage Ross, via Wikimedia Commons.

An example of an orrery. Image: Sage Ross, via Wikimedia Commons.

Johannes Kepler is a well know and still revered astronomer.  Kepler also believed there to be no distinction (at least it is not recorded) between astronomy and astrology and as an adviser and astronomer to Emperor Rudolph II he made horoscopes for not only the Emperor but also various allies and foreign leaders.  Johannes Kepler believed he had worked out much of the celestial orrery in his Mysterium Cosmographicum.5  The commonly held belief of the time was that all things could be understood by observing natural motions; whether those motions were of the planets, the stars, or in some cases the patterns of other natural phenomena.

Since all patterns can be represented mathematically, math then becomes the language of the universe. This idea can also be traced back to the Pythagoreans.  The concept that everything is a piece of celestial machinery that can be understood through math is still around us to this day, or at least it seems that the repercussions of it are. After all, if everything is patterns, and patterns can be interpreted wonderfully though Math, then Math must therefore be the Language of the universe. (That’s logical, that is.)  This seems to be the idea that Eleanor Robson is arguing against.  (Frankly, I agree with her.) This concept of a pure language of math is rather a strange convention that our society has if you really think about it. After all, the argument could be made that English (or any language really) is some sort of divine language because we can use it to so eloquently describe the world around us. Or perhaps Music is our divine language. It is pattern based, after all. So is this idea of everything being describable through math a belief we have found to be true, is it a truth that we somehow stumbled upon millennia ago, or is it a conceit of our culture?

1 A History of Mathematics by Uta C. Merzbach & Carl B. Boyer

2 Musica Universalis, Wikipedia

3 http://www.math.tamu.edu/~dallen/history/pythag/pythag.html

4 http://www.britannica.com/EBchecked/topic/485235/Pythagoreanism

5 Johannes Keppler Wikipedia

6 http://www.crystalinks.com/musicspheres.html

Archetypes of Wisdom: An Introduction to Philosophy , Douglas J. Soccio