Tag Archives: geometry

Henri Poincaré: A twentieth century polymath

Many of the first scientists would now be considered madly interdisciplinary. Aristotle’s fields of study ranged from mechanics and optics to medicine and the classification of animals, not to mention philosophy and other fields outside the natural sciences. Archimedes not only was fascinated by proving mathematical principles, he also applied them to physics, astronomy, and engineering. Newton invented principles which now are part of calculus while developing his theory of motion. Leonardo da Vinchi and other known Renaissance men were notoriously broad in their fields of knowledge and investigation. Gradually, mathematicians and scientists became more specialized. Darwin focused on biology, Cauchy on mathematics, Einstein on physics, and so on. Now, we recognize some academics as experts in such fields as number theory, particle physics, or Lie groups.

Henri Poincaré was one of the last of the generation of Renaissance men. While he was principally a mathematician, some of his work extended firmly into the world of physics. On the side he was a mining engineer and a philosopher. To see how varied and numerous his contributions were, see this list of things named after him, most of which are mathematical or physical topics.

Henri Poincaré
Image: Connormah via Wikimedia Commons.
Public domain.

Classical physics works very well for large objects with low speeds. In the late 1800s, physicists simultaneously realized that their understanding of the universe utterly failed to explain the behavior of small objects or fast objects. Two theories forever revolutionized our understanding of the universe: relativity, which explains fast moving objects, and quantum mechanics, which explains the behavior of very small objects like electrons. Poincaré contributed mathematically to both of them. Hendrik Antoon Lorentz derived the famous Lorentz transforms which explain relativity on a simple level. Lorentz discovered the Lorentz transforms without collaborating with Poincaré. However, Poincaré did critique Lorentz’ papers and offer additional input, ideas, and encouragement. It was this relationship with Lorentz that would later lead Poincaré into quantum mechanics.

Out of quantum mechanics and relativity, quantum mechanics has by far influenced the world more. It contributed to several major developments, including the understanding of atoms, nuclear power, and semiconductors. Of course, to semiconductors we owe much of our modern society. The development of the transistor would not have been possible without quantum mechanics. Transistors enabled the building of modern computers, cell phones, and the Internet.

For these reasons, Poincaré’s contributions to quantum mechanics are among his most important contributions to math and science. Poincaré was invited to the first Solvay Conference in 1911 on quantum theory by Lorentz. This appears to be the first time Poincaré was exposed to this new theory. In spite of this, his energetic participation in the discussions at the conference were noted by the other participants. In that conference, Max Planck presented a new theory about black body radiation.

Participants in the First Solvay Conference, 1911.
Image: Fastfission via Wikimedia Commons.
Public domain.

Black body radiation simply refers to the light given off by all objects as they cool. By 1911, enough experiments had been done that the wavelengths of light emitted from black bodies of different temperatures were known. However, classical physics failed to explain these results. Plank attempted to explain them by introducing the idea of “resonators” which could produce electromagnetic radiation. Although Planck didn’t consider matter to be made up of these resonators, this is a natural extension of his theory. Poincaré thought of this and questioned how Planck’s theory could explain the transfer of heat within an object. He quickly got to work rederiving Planck’s result and putting it on a more solid theoretical ground. In keeping with quantum theory, his reasoning used probability rather than absolute knowledge about particles. He did arrive at the same result as Planck, although he was more rigorous in doing so:

Unfortunately, just eight months after the First Solvay conference, Henri Poincaré passed away without living to see the impact his research would have on math and physics.

References

McCormmach, Russell (Spring 1967), “Henri Poincaré and the Quantum Theory”, Isis 58 (1): 37-55, doi:10.1086/350182

Plank’s Law on Wikipedia

Henri Poincaré on Wikipedia

Poincaré’s original paper on Planck’s theory (in French) can be seen here.

The Forgotten Mathematician

Pierre Wantzel was born in 1814 on the 5th of June in Paris. His father was a professor of mathematics at École speciale du Commerce after serving in the army. Due to this Pierre started his life with a natural love for mathematics. He attended school at his home town of Ecouen where he demonstrated this love. When he was only 9 years old his teachers would turn to him for help when judging the difficulty of problems. His love and skills for mathematics was realized by his parents when they sent him to École des Arts et Métiers de Châlons. He was surprisingly 12 years old he went there, and this was far younger than most.   His teacher was the well-known Étienne Bobillier, a mathematician known for his works on polar curves and algebraic surfaces. This helped kinder his mathematical skill, but it did not last long because in 1827 the school was reformatted. This was because France itself was facing revolts and other political issues. The school was reformed to become less academic, and this caused Pierre to take his studies elsewhere.

In 1828 he traveled to the Collège Charlemagne to continue his studies and receive language coaching. He later married the daughter of language coach, but before this he accomplished many feats of genius including editing a second edition book by Reynauld, Treatise on arithmetic, at only 15 in 1829. This book featured a method for finding square roots that was never proved. He proved the method, and in doing so he received the first prize for dissertation from his college. Later on he took the entrance exam to École Polytechnique and the science section for École Normale. He placed first in both of these, something never before achieved. Furthering his education he traveled to Ponts et Chaussées, an engineering school, but did not stay long. He remained there for a year until 1835 where he journeyed to the Ardennes. Following a similar pattern he later traveled to Berry after only a year at the Ardennes. After studying engineering he decided that teaching mathematics was his true dedication. In order to achieve this he took a leave from his occupation, and went to become a lecturer for a school from his past, École Polytechnique. He later became a professor of applied mathematics at École des Ponts et Chaussées but not before becoming an engineer in 1841. Continuing with his true interests he began teaching classes on not only mathematics but physics as well. He continued his educational career becoming the entrance exam examiner In 1843. He was not confined by his university, however, as he traveled around Paris to many schools teaching there too.

The tools of Pierre Wantzel. Image: Mcgill via Wikimedia Commons.

Pierre achieved fame when he published what would become his most important works. These were on the subject on radicals, and solving equations and they were dubbed as some of the most famous problems of the time. Publishing them in Liouville’s Journal he was the first to prove that it was impossible to duplicate a cube and trisect an angle with a ruler and compass. Gauss had originally stated that it was impossible but offered no proof. This is what Pierre accomplished in his 1837 paper where he traces the solution back to cube roots, something that proves impossible to do with those tools. This was built off of the work of others, yet it still went beyond what had been previously done. Continuing his works Pierre delved into equations, and from this he created new proofs of algebraic equations deemed impossible. These were solved not by providing a solution but proving they were impossible to solve. He revised a proof of Abel’s theorem in 1845, stating that it was impossible to solve any equations where the exponent n is greater than 5. He also added details to many vague solutions on the subject these solutions were proposed by famous mathematicians such as Ruffini. Pierre published over 20 works throughout the course of his life a few of these branch out into the field of physics, specifically dealing with extreme pressure differences.

Pierre was a strong man who focused on his work so much that he sacrificed his sleep and meals to do so. Pierre Wantzel did not live out his life fully as he overworked himself. He relied on coffee and opium to continue his lifestyle, and this ultimately resulted in his demise. In 1848 at the age of 33 he died and the world lost a great mind. Overall his works were very important yet were not remembered as well as others. This is commonly attributed to the classical nature of the problem he is famous for. Several other mathematicians mentioned the problem yet they have given no proof. Max Simon’s work from 1906 does mention Pierre’s, but it was published as a supplement to another work rather than as its own. Another reason is his early death. Due to his potential yet little time to achieve true greatness he is less known. Sadly he was not elected as a member of the Académie des Sciences. His achievements were great and if he had lived longer he would have achieved much more.

http://www-history.mcs.st-and.ac.uk/Biographies/Wantzel.html

http://fermatslasttheorem.blogspot.com/2008/10/pierre-laurent-wantzel.html

http://www.sciencedirect.com/science/article/pii/S031508600900010X

https://threesixty360.wordpress.com/2008/06/01/mathematician-of-the-week-pierre-wantzel/

The path to Analytic Geometry (Or a few of the many geniuses it took to learn 5th grade math)

Analytic geometry is the study of geometry using a coordinate system. Basically it’s the idea of expressing geometric objects such a as a line or a plane as an algebraic equation, think y=mx+b or ax+by+cz=k. This may be done by use of the more familiar Cartesian coordinates, by something such as polar coordinates or by just about any system for defining coordinates in a Euclidean space. The Common Core has the concept of graphing introduced in 5th grade, and graphing simple functions in the 8th grade. It’s quite interesting that something which took brilliant men so long to develop is now introduced to ten year olds.

The earliest evidence of anything resembling analytic geometry was by the Geek mathematician Menaechmus (380–320 BC), who was a student of Eudoxus and a tutor of Alexander the Great. Proclus and Eutocius both report that Menaechmus discovered the ellipse, hyperbola and parabola and that these were initially called the “Menaechmian triad”. These were used along with something resembling analytic geometry to solve the Delian problem, which is to, given the edge of a cube to construct the edge of a cube with double the volume. Though most of what we know of Menaechmus and his exact solution is second hand as his original work was lost, it appears as though he argued his solution for doubling the cube with proportions of a side length to the area of a side which fairly quickly leads to conics.

Another early manifestation of analytic geometry was by Omar Khayyám, whom we have mentioned in class. He drew a connection between algebra and geometry in his solution of general cubic equations. His idea to do this was to create a geometrical construction of a cubic equation by considering the variable to be the edge of a cube and constructing a set of curves from which a solution could be discerned. While it might seem far flung from Cartesian coordinates it was a significant leap in connecting the separate concepts of algebra and geometry.

Analytic geometry was more or less formalized in the early 17th century independently by René Descartes and Pierre de Fermat. Descartes published first and so he is commonly credited as the sole creator which leads to analytic geometry often being call Cartesian geometry. As Fermat has already been much discussed, I’ll skip his background and instead jump to Descartes. René Descartes was a French mathematician and philosopher who is most well known as the (co-)creator of analytic geometry and as the father of modern philosophy. He is the origin of the well-known quote “Je pense, donc je suis” or “I think, therefore I am” which appeared in in Discours de la methode (Discourse on the Method).

While the Fermat and Descartes constructions are equivalent, they did differ in several ways which primarily stem from which direction their creator worked. Fermat started with the algebraic equation and described the analogous geometric curve while Descartes worked in reverse, starting with the curve and finding the equation. To contrast the methods, the way most of us learn analytic geometry is much more similar to Fermat than to Descartes, where we learn to recognize that a degree 1 polynomial will represent a straight line then we learn how to find that line, next that quadratic function represents a parabola and so on. Whereas if we were to learn as Descartes’ work, we would take a straight line then learn that it represented a degree 1 polynomial which is similar to Fermat.  But then working further in this direction, it doesn’t make sense to jump to parabolas and instead to talk about conics and all degree 2 polynomials with no reason to talk specifically about parabolas.

In 1637, Descartes published his method of connecting arithmetic, algebra, and geometry in the appendix La géométrie (The Geometry) of Discourse on the Method. However, given Descartes’s opaque writing style (to discourage “dabblers”) as well as The Geometry being written in French rather than in the more common (for academic purposes) Latin, the book was not very well received until it was translated into Latin in 1649, by Frans van Schooten, with the addition of commentary clarifying certain arguments. Interestingly, though Descartes is credited with the invention of the coordinate plane, since he describes all necessary concepts, no equations are in fact graphed in The Geometry and his examples used only one axis. It was not until its translation into Latin that the concept of 2 axes was introduced in Schooten’s commentary.

One of the most important early uses for analytic geometry was to help prove the validity of the heliocentric theory of planetary motion, the (then) theory that the planets orbited around the Sun. As analytic geometry was one of the first methods one could use to actually make computations about curves, it was used to model elliptical orbits so as to demonstrate the correctness of this theory. Analytical geometry, and particularly Cartesian coordinates, were instrumental in the creation of calculus. Just consider how you might calculate something like the “area under the curve” without the concept of the curve being described by some algebraic equation. Similarly, the idea of rate of change of as function of time at a particular time becomes much clearer when thought of as the slope of the tangent line, but to do this, we need to think of the function as having some representation in the plane for which we need analytic geometry.

Sources :

https://www1.maths.leeds.ac.uk/~sta6kvm/omar.pdf

http://www.math.wichita.edu/history/Men/descartes.html

http://www.encyclopedia.com/topic/Pierre_de_Fermat.aspx

Mathematics: Its Content, Methods and Meaning (Dover Books on Mathematics) Jul 7, 1999

by A. D. Aleksandrov and A. N. Kolmogorov

http://www-history.mcs.st-and.ac.uk/Biographies/Menaechmus.html

http://academic.sun.ac.za/mathed/Shoma/MATUNIT12_02.htm

http://www.cut-the-knot.org/WhatIs/WhatIsAnalyticGeometry.shtml

http://www.cut-the-knot.org/WhatIs/WhatIsAnalyticGeometry.shtml

http://en.wikipedia.org/wiki/Menaechmus

http://en.wikipedia.org/wiki/Discourse_on_the_Method

http://en.wikipedia.org/wiki/La_G%C3%A9om%C3%A9trie

The Beauty of The Elements

caption

A stone statue by Joseph Durham depicting the famous mathematician Euclid. Image: Garrett Coakley via flickr.

When I was in high school, I eventually learned about the mathematical subject known as geometry. Unlike most schools though, instead of our teacher having us sit down and listen to them talk about the subject, our teacher had each and every one of us go to the library and rent a copy of Euclid’s Elements (Book 1). From that point on till the end of the first semester, each day we would separately read from Euclid’s Elements and then try to prove to our teacher each and every postulate using Euclid’s methods. It wasn’t until recently that I discovered that most children do not learn about geometry in this fashion and how unique of an experience I had. While I can see some of the possible advantages behind the new ways people learn about geometry, I still believe that Euclid’s The Elements has its own advantages that some of these other sources don’t.

One of the most noticeable things about The Elements is that each and every one of Euclid’s postulates build exceptionally well off of each other. While I see proofs building off of each other in most other texts books, there is just something about the way it is done in The Elements that feels much smoother. Perhaps the big advantage with a book like The Elements is that it was never meant to be a “text” book but rather a book for people who are interested in learning about geometry. Because of this, it doesn’t have to continually throw out real world examples or ask the reader to try to use this proof in specific scenarios. Instead, The Elements will just make a statements, go about proving that statement, and then go straight into making another statement and most likely prove it using the previously proven statement.

Another difference between The Elements and other geometry books which I believe makes it far superior is the general way in which it goes about solving proofs. Nowadays, most geometry books will use a popular form of algebra and a number system to solve equations. However, Euclid’s Elements is fully self-contained and takes nothing for granted. Because this book was created in a time where people didn’t necessarily have access to other sources, everything that is necessary to understand what is being stated in this book is there; including its own algebraic system. This self-contained version of algebra within The Elements uses simple comparisons between lines and shapes to each other which replaces constants and variables found in other forms of algebra (which is also explained in the book) to prove that the different statements that are being made are true. These comparisons in combination with previously proven statements allows The Elements to create proofs of all different kinds. While the algebraic like system Euclid’s Elements uses to solve equations may be a little difficult to get one’s mind around sometimes it makes the proofs within its pages much more difficult to refute than other geometry books.

So, why do we not use this book to teach students about geometry today? Perhaps the biggest reason and most obvious is that The Elements is a difficult book to read. Unlike most textbooks today, it doesn’t use numbers and doesn’t give examples. However, just because current day geometry books are easier to teach with and easier for students to understand does not mean that they are better books. Perhaps the final reason that I believe The Elements is such a great geometry book compared to others is that the reader must want to learn about geometry if they wish to get anywhere in Euclid’s Elements.  But, if they are able to get through Euclid’s Elements, they will have a much stronger fundamental idea of geometry than from other textbooks. While it is easy to state the fact that someone who survives being stranded in the wilderness will have a better idea of how to survive in the wild than someone who hasn’t, it doesn’t change the fact that it is true.

Going back to my classroom experience, I thoroughly enjoyed going through the proofs in The Elements and I would spend most of my lunch time going to my teacher and proofing more of Euclid’s Statements. After about 2 weeks of starting the book, I had finished it. After that point, I spent the rest of time in class helping other students understand The Elements. Unfortunately, most of the other students had a hard time getting through that semester and only a few other students were able to understand it in a similar fashion as myself. However, those among us who did understand Euclid’s The Elements had no troubles passing the second semester of class which was going back to the more common form of geometry. In conclusion, I believe that Euclid’s Elements is a fantastic book that does more for geometry than any other book out there and, if someone is really interested in geometry, they should do their best to read through and understand The Elements if they want the best foundation in geometry they can have.

Sources:

(http://en.wikipedia.org/wiki/Euclid%27s_Elements)

(https://archive.org/details/bub_gb_UhgPAAAAIAAJ)

Three Centers of a Triangle

There’s far too little geometry—excluding topology and non-Euclidean stuff—on this blog, so let’s add a little.

Euler Line

Euler line HU. Points H, U, and S are
respectively the circumcenter, centroid,
and orthocenter. Image: Rene Grothmann at the German Language Wikipedia.

Our goal is to get to the Euler line, a line that passes through a triangle’s circumcenter, centroid, and orthocenter. The line is only determined for non-equilateral triangles; the points coincide in the equilateral case. We’ll look at the three points above.

The circumcenter, centroid, and orthocenter are all “centers” of triangle. But what is a center of a triangle? Surely, it’s not a point equidistant to all points on the triangle. Our triangle would be a circle in that case.

The circumcenter of a triangle ABC is the center O of the circle K that triangle ABC is inscribed in.

Circumcenter

Circumcenter O of triangle ABC. Image drawn by me.

The circumcenter is actually the intersection of the three perpendicular bisectors of the triangle: FE, IG, and DH. To see this, first suppose that triangle ABC has a circumscribed circle K with center O. Draw radii AO, BO, and CO to each of the triangles vertices. This creates three smaller triangles AOBBOC, and AOC. In each of these smaller triangles, drop an altitude from O. For example, in triangle AOBaltitude OD would be dropped. This splits AOB into two smaller triangles that are congruent by SAS, Line OD is perpendicular to AB by construction, and AD = DB. Hence OD is indeed a perpendicular bisector of side AB. Repeating this for other sides shows that the center of the circumscribed circle is the intersection of ABC‘s perpendicular bisectors.

Moreover, the intersection of any to perpendicular bisectors is equidistant from each of the triangle’s vertices. The reader can see this by considering triangle AOC. Perpendicular bisector IG splits AOC into triangles that are congruent by SAS. It follows that lengths AO and OC are equal. Repeat for the other sides. We then see that the intersection of the perpendicular bisectors is equidistant from the triangle’s vertices. Thus the perpendicular bisectors of a triangle uniquely determine its circumcenter.

The centroid is the intersection of a triangle’s three medians, lines drawn from a vertex that bisect the opposite side. As said in class, the centroid is the center of mass for a thin, triangular solid with uniformly distributed mass.

Centroid

Centroid O of triangle ABC. Drawn by me.

The reader may suspect whether the three medians of a triangle intersect. Clearly two of the medians intersect; otherwise our triangle ABC would be a line. But the full proof is a little tedious. The proof involves assuming that two medians AF and CE intersect and drawing a parallelogram using the midpoints of the medians. We link to some proofs: http://jwilson.coe.uga.edu/EMAT6680Fa06/Chitsonga/MEDIAN/THE%20MEDIANS%20OF%20A%20TRIANGLE.htm uses classical geometry and http://math.stackexchange.com/questions/432143/prove-analytically-the-medians-of-a-triangle-intersect-in-a-trisection-point-of uses vectors.

Triangles

Four congruent triangles using midpoints. Image drawn by me.

Interestingly, the midpoints of the sides of triangle ABC—the ends of the medians—cut the triangle into four congruent triangles. We will prove this in a roundabout way. Let E be the midpoint of AB. Draw a line EF parallel to AC where F intersects BC. Similarly draw FD parallel to AB. By construction, EFDA and EFCD are parallelograms. Then AD = EF = DC, so D is the midpoint of AC. Similarly, F is the midpoint of BC. The reader can see that the triangles are congruent by repeatedly applying SAS.

Our final center is the orthocenter, the intersection of the three altitudes of a triangle. An altitude is a segment drawn from a vertex that is perpendicular to the opposite side. As with the two previous centers, the intersection of the altitudes at a single point isn’t immediately obvious.

Orthocenter

Orthocenter O of triangle ABC. Drawn by me.

We show that the altitudes of triangle ABC intersect. Construct triangle DEF with triangle ABC inscribed in it by making sides DF, FE, and DE parallel respectively to BC, AB, and AC. Draw altitude BK where intersects DF. Since AC is parallel to DEBK is perpendicular to DE. Moreover, ADBC and BACE are parallelograms, so DB = AC = AE. Hence BK is a perpendicular bisector of DE. We repeat the argument for the other altitudes of triangle ABC. Then the altitudes of ABC intersect because the perpendicular bisectors of DEF intersect.

There are a few other centers of a triangle that are either irrelevant to the Euler line or take too long to construct (i.e. I’m tired of drawing diagrams). The incenter is the center of the circle inscribed within a triangle. The incenter also turns out to be the center of a triangle’s angle bisectors. The Euler line doesn’t pass through the incenter.

The nine-point circle is the circle that passes through the feet of the altitudes (the end that isn’t the vertex) of a triangle.

Nine-Point Circle

Nine-point circle of ABC. Image: Maksim, via Wikimedia Commons.

Strangely, the circle also passes through the midpoints of the sides of its triangle. But that’s not all. The circle passes through the Euler points, the midpoints of the segments joining the triangle’s vertices to the triangle’s orthocenter. Thus the nine-point circle does indeed pass through nine special points of a triangle. The center of the nine-point circle lies on the Euler line.

After all this, we still haven’t proved that the circumcenter, centroid, and orthocenter lie on the same line. We won’t prove this. Here’s a video of the proof by Salman Khan: https://www.youtube.com/watch?v=t_EgAi574sM. The proof uses a few facts about the centers we haven’t discussed, but these facts aren’t too hard to show. Refer back to my four congruent triangles picture. Let O, K, and L respectively be the circumcenter, centroid, and orthocenter of triangle ABC. Then Khan proves that triangle DOK is similar to triangle BLK. This implies angles DKO and CKL are equal, which means O, K, and L lie on the same line.

Sources and cool stuff:

H.S.M. Coxeter and Samuel L. Greitzer’s Geometry Revisited

Paul Zeitz’s The Art and Craft of Problem Solving (Chapter 8 is called “Geometry for Americans”)

Wolfram on the nine-point circle: http://mathworld.wolfram.com/Nine-PointCircle.html

A fun way to play with the Euler line: http://www.mathopenref.com/eulerline.html

Khan’s Euler line video: https://www.youtube.com/watch?v=t_EgAi574sM

Wolfram on the Euler line: http://mathworld.wolfram.com/EulerLine.html

Classical median proof: http://jwilson.coe.uga.edu/EMAT6680Fa06/Chitsonga/MEDIAN/THE%20MEDIANS%20OF%20A%20TRIANGLE.htm

Vector median proof: http://math.stackexchange.com/questions/432143/prove-analytically-the-medians-of-a-triangle-intersect-in-a-trisection-point-of

Math: Is It All In Our Head?

After years of math classes, the crazy truth is finally coming out.  It is all just in our heads.  No way! How can that be? There’s an interesting debate in the world of math.  Are math principles the creation of humanity, or are they universal truths that humans discovered? There are compelling arguments on both sides of the debate and both sides have several different sub-levels of thought.  In this article, I will discuss them both generally.

The realists maintain that mathematical principles would exist even without people.  Humans discovered the principles and brought them into practical use and any intelligent human being could also discover the same principles.  This argument is supported by the fact that many cultures have discovered mathematical principles independent of one another.  Also, mathematical concepts, such as the Fibonacci sequence and some fractals, occur in nature which would suggest that they exist even without people.   Some realists, like the Pythagoreans, believe that the world was created by numbers.  The realist point of view can lead to an almost supernatural view of mathematics.

The challenge with mathematical realism is that there is no physical domain where math entities exist.  We cannot draw a perfect circle or even a line.  We can conceptualize these things in our mind and we can prove them in theory; however, we cannot actually manipulate math entities in the physical world.  Many math concepts exist only in the context of our understanding about them and conceptualizing them.

Another view is the anti-realists.  They maintain that math is the creation of humans in order to make sense of the world.  They recognize that math is an amazing, complex system and that it works as modeled by science.  However, some argue that scientific principles could be explained without math.  One anti-realist, Hartry Field, demonstrated this by explaining Newton Mechanics without referencing numbers or functions.  He explained that, in his opinion, math is fictional and is true only in the context of the story in which it is being told.

So, is it all in our heads?  A fiction that was created to explain properties in our world?  In reality we may never be able to settle the debate and it may not matter.  Math works.  That is the beauty of it.    In his article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Eugene Wigner observes that

the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.  This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

Albert Einstein, 1921. Image: Public domain, via Wikimedia Commons.

Albert Einstein, 1921. Image: Public domain, via Wikimedia Commons.

Perhaps the best thing to commend mathematics as being real, is that it works.  Time and time again, it works.  Its principles, laws and theorems, applied over and over, in different settings produce accurate results and predictions.  Einstein commented in a 1921 address titled Geometry and Experience, “It is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain.”  He explored the question of how math, a product of our mind can be so applicable to the concrete world.  He asked, “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirable appropriate to the object of reality?”  Einstein answers this question with the statement, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”  He looks specifically at the field of geometry and the need humans have to learn about the relationships of real things to one another.  Even though the axioms of geometry are based on “free creations of the human mind”,  he says, “Solid bodies are related, with respect to their possible dispositions, as are bodies on Euclidean geometry of three dimensions.  Then the propositions of Euclid contain affirmations as to the relations of practically-rigid bodies.”  The abstract principles, when applied to “real” world situations prove to be accurate.  Einstein continues to explain how the theory of relativity rests on the concepts of Euclidian and non-Euclidian geometry.  He challenges the mind to conceptualize a universe which is “finite, yet unbounded”.  In the end, it is this ability to use conceptualized principles and apply them to our world that makes mathematics work.  So yes, mathematics may be all in our head and it may be a huge puzzle created by humanity, but it is effective, useful, and even beautiful.

Sources

Einstein, Albert. Geometry and Experience. http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_geometry.html

Wigner, Eugene. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Wikipedia. Philosophy of Mathematics. http://en.wikipedia.org/wiki/Philosophy_of_mathematics

Transition from Euclidean to Non-Euclidean Geometry

Euclidean geometry is the geometry that everyone learns and uses throughout Middle School and High School. In general, geometry is the study of figures, such as points, lines and circles in space. Euclidean geometry is specifically any geometry in which all of Euclid’s postulates and axioms hold. Axioms and postulates are the beginning of reasoning, they are simple statements that are believed to be true without proof. Assuming Euclid’s axioms and postulates found in his book Elements, the rest of Euclid’s classical geometry could be deduced. However, Euclid’s fifth postulate, the parallel postulate, was disconcerting because it was lengthy compared to the rest and not necessarily self evident. Many other ancient mathematicians were dissatisfied with Euclid’s fifth postulate. They thought that it was presumptuous and tried to prove it using lesser axioms or replace it altogether with something they thought to be more intuitive. But their proofs always included an assumption equivalent to the parallel postulate, so for centuries the postulate was assumed to be true.

Centuries passed and the postulate remained unproven; however, development to understand Euclid’s postulate continued into the eighteenth century. Perhaps the most well-known equivalent to the parallel postulate is Playfair’s Axiom, which states “through any point in the plane, there is at most one straight line parallel to a given straight line.” Arguably one of the most influential mathematicians, Carl Friedrich Gauss became interested in proving Euclid’s fifth postulate. After attempting to prove the postulate, he instead took Playfair’s Axiom and altered it, creating a completely new postulate. Gauss’ new postulate stated “Through a given point not on a line, there are at least two lines parallel to the given line through the given point.” With this Gauss had unearthed a completely new space that today is called hyperbolic geometry. However, he chose not to publish any of his results, wishing not to get caught up in any political strife. The work was later published  by Johann Bolyai and Nikolay Lobachevesky, who both had academic ties to Gauss.

Shortly after this discovery another type of Non-Euclidean geometry was discovered by Gauss’ student Georg Friedrich Bernhard Riemann. Riemann looked at what would happen when you altered Playfiar’s Axiom in the opposite direction than Gauss. Riemann’s alternate postulate is stated as follows, “through a given point not on a line, there exist no lines parallel to the line through the given point.” With this, what is known as elliptical or spherical geometry was discovered.

Spherical geometry. Image: Anders Sandberg via Flickr.

Spherical geometry. Image: Anders Sandberg via Flickr.

Spherical geometry provides a somewhat simpler model then hyperbolic geometry. Anyone who has ever attempted to wrap a basketball in paper understands that there are some discrepancies between the two surfaces. For instance, triangles behave quite differently than they do in Euclidean geometry. In hyperbolic space, the angles of a triangle add up to less than 180 degrees, and in spherical space, they add up to more than 180 degrees. Spherical geometry also has other inconsistencies with Euclid’s initial assumptions other than the parallel postulate. In Leonard Mlodinow’s book Euclid’s Window, the author describes how Riemann’s space was inconsistent with other of Euclid’s postulates. For instance, Euclid’s second postulate states that “any line segment can be extended indefinitely in either direction.” In spherical space this is not true; the lines, or great circles, have a limit to their space, namely two pi times the radius of the sphere. Mlodinow describes how Riemann saw this postulate as “only necessary to guarantee that the lines had no bounds, which is true of the great circles.” Also, Euclid’s first postulate became less clear, “Given any two points, a line segment can be drawn with those points as its endpoints.” This postulate can be used to easily describe whether a point is between two other points. However, on the globe, choosing two points on the equator such as Ecuador and Indonesia it is difficult to say whether a third point, Kenya, is “between” them. The problem is that there are two ways to connect the points, one passing over North America and another passing over Africa.

For much of our day to day lives Euclidean geometry works great, because on a local scale we appear to live on a flat world. I can go to a soccer field and trust that it will take four 90 degree turns to walk around the perimeter, or that the Pythagorean theorem will work to describe the path between opposite corners. But looking at a larger scale, the surface that we live on is spherical and has different properties than the flat plane. It is interesting to see how Gauss and Riemann, going against the grain of conventional mathematics, led to new and vast fields of undiscovered mathematics. To me, this shows how mathematics is just as much an experimental science as physics or engineering. These new discoveries of mathematical spaces made possible Einstein’s physical description of the space in which we live. Mlodinow closes his section on Gauss and Riemann saying, “though thoroughly remodeled, geometry continued to be the window to understanding our universe.” Even though the properties of these new geometries differ from classic Euclidean geometry and may have more or less practical use, they are just as important. From Euclid up until Gauss, mathematics was largely pragmatic, but the discovery of these new geometries highlights how math can be appreciated for its own sake.

References:

Case, William A. Euclidean vs. Non Euclidean Geometries. Web. http://www.radford.edu/~wacase/math%20116%20section%207.4%20new%20v1.pdf

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Mlodinow, Leonard. Euclid’s Window. Touchstone New York, 2001. Print

Weisstein, Eric W. “Non-Euclidean Geometry.” http://mathworld.wolfram.com/Non-EuclideanGeometry.html