[math-fun] beautiful equations and E.T.
Hello, they made a poll asking what are the most beautiful equations to a group of people. http://news.bbc.co.uk/1/hi/magazine/3721406.stm Euler's equation came second : exp(Pi * i) + 1 = 0. Maxwell equation came first. I think that these people are in physics. Nevertheless, I am wondering what if we would make a kind of message to extra-terrestrial intelligence and we would like to show them that we know all these things. What would be the best way to write them down in a universal language because for ET the equation exp(Pi*i)+1=0 means nothing at all if we can't explain what is Pi, e, 1, 0 and the equal sign, isn't? Any clue? ps : E = Mc^2 is only the 6'th. Simon Plouffe
This was a group of physicists, after all! The problem with some beautiful physics equations is that -- properly interpreted -- they can be trivial. They may have taken the whole human race thousands of years to figure them out, but once we get our heads screwed on straight, they are not only obvious, but so necessary that they are built into the foundations of nature. E.g., F=ma is no longer an equation, but a definition. Euler's equation is apparently a favorite of kooks here in Southern California -- last year a nut scrawled this on the windshields of 50 or so brand new SUV's that he trashed. The problem with "imaginary numbers" is that other civilizations might not attach the same importance to them that we do. We like them (as opposed to quaternion, for instance), because they lead to a much richer calculus & analysis. But some other civilization might consider them just one of a whole spectrum of "numbers" that can be derived from matrix representations. The whole "geometric algebra" movement might end up transforming "school math" in a revolution similar to the vector revolution of Gibbs, et al. Although many would claim that "geometric algebra" provides no new theorems, its regularization (i.e., the elimination of so many special cases) of geometric calculations may lead to its quick uptake among engineers and computer software people. So, I think a proper illustration of Euler's equation might be something akin to the "proofs without words" that one finds in some math publications, so that the beauty isn't lost to someone who doesn't understand the notation. I've looked through a number of old books on math -- e.g., translations of Cardano, etc. -- and much of their idea of beauty in these equations is not evident to modern eyes, because they deal with special cases, not the "general" case. Other civilizations looking at Euler's equation may feel the same way, because they have discovered an even more profound equation, of which Euler's is only a special case. At 11:54 AM 10/8/2004, Simon Plouffe wrote:
Hello,
they made a poll asking what are the most beautiful equations to a group of people.
http://news.bbc.co.uk/1/hi/magazine/3721406.stm
Euler's equation came second : exp(Pi * i) + 1 = 0.
Maxwell equation came first. I think that these people are in physics.
Nevertheless, I am wondering what if we would make a kind of message to extra-terrestrial intelligence and we would like to show them that we know all these things.
What would be the best way to write them down in a universal language because for ET the equation exp(Pi*i)+1=0 means nothing at all if we can't explain what is Pi, e, 1, 0 and the equal sign, isn't?
Any clue?
ps : E = Mc^2 is only the 6'th.
Simon Plouffe
For physics, I like Daniel Z Freedman's "Some beautiful equations of mathematical physics": http://arxiv.org/pdf/hep-th/9408175 --Ed Pegg Jr
--- Henry Baker <hbaker1@pipeline.com> wrote: The whole "geometric algebra" movement might end up transforming "school math" in a revolution similar to the vector revolution of Gibbs, et al. Although many would claim that "geometric algebra" provides no new theorems, its regularization (i.e., the elimination of so many special cases) of geometric calculations may lead to its quick uptake among engineers and computer software people. --- Now then, this is much more interesting than how to communicate with alleged E.T.s. Tell us more. Gene _______________________________ Do you Yahoo!? Declare Yourself - Register online to vote today! http://vote.yahoo.com
After thinking about this some more, here are some of the mathematical theories/theorems/equations that I think should make the list: 1. Computability. The fact that there exists a reasonably robust theory of computability, and that this is intimately connected with the fact of the incompleteness of number theory (Goedel's Theorem, et al). It turns out that computability is even easier (in some respects) to present than arithmetic (at least if you know Lisp & EVAL). 2. The fact that the theory of real closed fields is decidable. This means that essentially all of high school geometry is decidable. 3. The FFT algorithm and its generalizations. 4. Fourier theory in general -- astoundingly beautiful. 5. Real numbers -- perhaps the single most important creation of human beings, to date. They are quintessentially human -- using finite means to gain an insight into the (divine) infinite. At 11:54 AM 10/8/2004, Simon Plouffe wrote:
Hello,
they made a poll asking what are the most beautiful equations to a group of people.
The key educational and general intellectual advantage of Euclidean geometry is that so many of its theorems are surprising. For example, it isn't obvious that the angle bisectors of a triangle meet at a point. This is what makes people want to study mathematics or even become mathematicians. 1-1 correspondence of sets as the foundation of elementary arithmetic has no surprises for the school child. Euclid's proof of the infinity of primes is surprising, but it requires an initial interest in primes. The irrationality of sqrt(2) surprised the Greeks, but that was in relation to a prior doctrine of commensurability.
participants (5)
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ed pegg -
Eugene Salamin -
Henry Baker -
John McCarthy -
Simon Plouffe