The Beauty of Mathematics

This semester I am teaching an introductory course in mathematics for students in other departments.  I taught a similar course last year entitled “Basic Ideas in Mathematics,” and this year, following a suggestion of my wife, I changed the name to “The Beauty of Mathematics”. Another change is that starting this year we have a general program in the university, called “Cornerstones“, (initiated by the rector,) whose purpose is to widen the education we offer to our students, and this course is part of the new program.

Talking about beauty rather than about basic ideas, combined with the new Cornerstone program have led many  more students to enlist to the course this year, and subsequently the lectures will use computer presentations.

Of course, the challenge has become  harder. I truly think mathematics is beautiful, but trying to convey its beautiful facets has never been easy. Also, I do not want to sweep under the rug the difficulty of mathematics, and the students will have to learn some basic mathematical skills and some abstract mathematics.   Ideas and suggestions are most welcome.

What do you regard as a great example of the beauty of mathematics?

There will be  is a separate page devoted to the course and the slides for the first lecture (in Hebrew) are linked there (and here). The first lecture was devoted to “numbers.”

Muhammad ibn Mūsā al-Khwārizmī

(From the Wikipedia article on zero.) In 976 Muhammad ibn Musa al-Khwarizmi, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used “to keep the rows.”

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26 thoughts on “The Beauty of Mathematics

  1. Joseph Malkevitch

    Two examples inspired by L. Euler:

    a. A connected graph has a tour starting at vertex v and returning to v which visits each edge once and only once if and only if it is even-valent.

    The proof in each direction is not hard but involves interesting thinking. Furthermore, it is dramatically applicable to solving the Chinese Postman Problem which deals with optimal ways to collect garbage, plow snow, and keep streets clean.

    This is my favorite way to introduce mathematics to “skeptics.”

    b. A connected plane graph satisfies: V (vertices) + F (faces) – E (edges) =2.

    Euler suggested this in the context of 3-dimensional convex polyhedra but did not give a correct proof. In the graph theory form the result is due to Cauchy whose original proof though very appealing is not quite right. He did not prove that a “shelling” exists. Euler’s polyhedral formula is central to many nifty ideas.

    Applications outside of mathematics of this result are not as natural as for a. but there are some.

    I think both of these results are basic, exciting, and beautiful!

  2. arnab

    I think a basic question that needs to be answered right away is whether ‘beauty’ for this class primarily refers to mathematical structures or to solutions to individual problems. There certainly is overlap, but I feel showing gem proofs, like the book proof of the Sylvester-Gallai theorem, or even Euclid’s proof of the infinitude of primes, is a little bit of cheating and perhaps even too intimidating to an outsider. On the other hand, maybe more illuminating and satisfying is showing structure from algebra (like, group structure of multiplication mod primes, Fermat’s little theorem, counting symmetries of polyhedra, field extensions, etc.), from graph theory (like, the aforementioned Euler’s formula, graph genus, quasirandom graphs, Erdos-Stone and other bits of extremal graph theory), from analysis (like, the notion of continuity, differentiability, implications of being a holomorphic complex function, the fourier transform and connecting back to algebra by showing how to define it for abelian groups, specifically F_2^n, etc.), from geometry, from information theory, and so on.

    But perhaps this is just my taste….

  3. Thought

    the proof that sqrt(2) is irrational. Apart from its pure beauty, is its simplicity and that it can help students to understand what in fact an irrational number is and how to determine what one is.

  4. Thought

    How much set theory or discrete math do you think they can handle? I think it’d be great to get a crowd that young in their academic careers to see that math is more than memorizing numbers and formulas. It’d be great if you can get some of them to see the general beauty of a proof.

  5. Luke

    I have always found that the paradoxes and non-intuitive results that mathematics produces are, to me, some of its most beautiful features. My peek movement as far as appreciation of the beauty of mathematics occurred as an undergrad. A professor gave a brief talk on the Banach Tarski paradox. It was intended for math majors (the professor quickly introduced Euclidean motions and the free group on two generators), so it might be above the level of your intended audience. I thought it was amazing because it so stunningly illustrates the interconnectedness of mathematics. Though the group theory was fascinating, I couldn’t see how it would connect to the seemingly geometric act of doubling a sphere. Then, just before the end, I saw what was coming. I still compare all talks to that one.

    It might take some time to introduce the notions of a group, a group action, etc. but I have found that these can be quite nicely illustrated with the simple example of a point group visualized as a finite group of rotations. This example also shows how seemingly abstract mathematics (symmetry groups and group actions) can be very useful, since such discrete symmetry groups are a powerful tool in solid state physics, used to study the structure of crystal latices.

    The Banach-Tarski paradox can be further used to illustrate a non-measurable set and a bizarre consequence of the axiom of choice.

    One other paradox that I have always enjoyed is the classic example that comes from looking at f(x) = 1/x rotated about the x-axis. Taken from 1 to infinity this forms a cone of infinite length. A simple integral (that some of your students may be able to preform on their own) shows that this cone has convergent volume, but divergent surface area; that is, it could be filled with paint, but not painted. This example also leads to an interesting discussion of the potentially paradoxical nature of limits and the concept of infinity.

  6. Wesley Calvert

    The Central Limit Theorem. It states that an enormously wide variety of practical problems abstract (in an intensely useful way) to exactly the same thing. Moreover, the proof calls for some significant mathematics.

  7. Taole Zhu

    proofs without words series
    the philosophical idea of Ramsey theory that complete chaos is impossible.
    fractals are beautiful.

    in regard to chaos, check out the BBC documentary: The Secret Life of Chaos.

  8. N

    Most of the introductory theory of partitions is quite beautiful and accessible: sum-product identities, Ferrer’s diagrams, Euler’s Pentagonal Number Theorem, etc.

  9. Jared Webb

    The countability of the rationals and the uncountability of the reals.

    I suppose that most people don’t care, but whenever I see this first taught to a group that is properly motivated there is vigorous discussion.

    I think it’s a great example of intuition failing and having to rely on rigor to discover what is really true.

  10. Edna

    I know of one student who was very happy after your class, so happy she is convincing all her friends to attend too… Some of them always disliked math and showing them that it’s really not so scary is a great service!

  11. steve uurtamo

    i’m not sure about the level you’re looking for, so i’ll just take a stab at a few things that i think are nice:

    * (for children who have never played with one): give them a rubik’s cube (solved) and let them play with it for awhile, turning the sides until it’s a total mess.

    questions: is it possible for it to be like it once was? i.e. can it have solid colors on all six sides again? if so, why? is it possible to turn the sides in some way so that you can’t ever make it look like it once was? if not, why not? (important not to let one student blurt out the answer).

    * (for high school students): show them (if they haven’t seen it before) sudoku.

    questions: can all possible sudokus be solved? when can they or can’t they? do they all only have one solution? if not, can you invent one that has more than one solution?

    extra credit: how many solutions can a sudoku have, anyway? what does it depend upon?

    * (for college students): tell me if you believe the following claim: what would you say if i told you that for any book, i could add a very specially chosen extra 5 pages to the end of the book (we’re allowed to make these 5 pages depend upon the book in question) so that if something bad happened to the book, destroying up to 4 pages, we could figure out what those destroyed pages used to say using nothing but the remaining pages?

    question: if you believe that, then would you believe me if i told you that we could do the same thing even if up to 4 pages had been subtly modified in such a way that we couldn’t see the modifications? that we could still discover all of the incorrect pages and fix them?

    extra credit: could we do the same thing at a word level instead of a page level? how about at a character level?

  12. tim

    Some of my favorite theorems with practical implications:

    Stone-Weierstrass Theorem: Every continuous function can be uniformly approximated by polynomials in a closed interval (holds as well for more dimensions). Explains why polynomials appear so often in nature.

    Graph Minor Theorem: Every graph family which is closed under contraction can be described via a finite number of minors. Can be used to algorithmically check graph properties.

    Yao’s Principle: The expected cost of any randomized algorithm for solving a given problem, on the worst case input for that algorithm, can be no better than the expected cost, for a worst-case random probability distribution on the inputs, of the deterministic algorithm that performs best against that distribution (from Wikipedia). Simple but extremely useful to obtain lower bounds.

  13. Johan

    Hall’s marriage theorem is one of my personal favorites. Quite simple to explain.

    I also second the recommendation for cardinality of the rationals and reals. The statement of the theorems can seam quite bizarre and explaing what they really mean and how they are proven is an excellent introduction to how mathematicians think. It also has the benefit of making the students feel like they understand some very deep stuff.

  14. Gil Kalai Post author

    Dear all,

    many thanks for the suggestions. Probably it is important for me to try to avoid the temptation of doing too much. The first lectures was about numbers. I gave three proofs: that there are infinitly many primes, that the square root of 2 is not rational, and that the golden ration is irrational. The material was quite a lot for one lecture but we may come back to some of these ideas. There was a small discussion about what proofs are.

    We have more to talk about numbers and we also talk about various riddles. The next topic will be infinity, but perhaps we will also play a little the game of Nim.

    (I am sure there are various presentations of a similar nature in English.)

    Dear Edna, Of course I was very happy to see Einat.

  15. Rajbir Bhattacharjee

    Certainly the most beautiful concept in mathematics is the concept of zero. Told that the concept exists, it is so easy to grasp. and yet, imagine for a while, if you grew up counting sticks, or counting roman numbers, would you be able to realize the concept of nothingness for yourself – I wouldn’t.

  16. Keivan

    The (easy) fact that the procedure for multiplying numbers in decimal basis is the same as multiplication of polynomials is easy, but maybe surprising to non-math majors.


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