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G is generated by five elements: x1, x2, x3, x4 and y, subject to the relations
x1^3 = x2^3 = x3^3 = x4^3 = 1
y^12 = 1
(yx1^2)^4 = (yx2^2)^4 = (yx3^2)^4 = (yx4^2)^4 = 1
I'm interested because this group has distinct musical connotations.
We'll have to agree to disagree. During my PhD I only really found one result I considered beautiful, but I just don't like where maths is these days: it is drudgery in the modern world and this is why I don't want a career in it. The art of maths is to find beautiful patterns that avoid the drudgery: drudgery is punishment for not being clever enough.
Haven't read the other comments on this thread but the more interesting question is that if you require that p and q be chosen such that q is not in the extension of Q by p, is it still possible that p^q is rational.
The feeling of stability that you get with a tripod is present with an even permutation but not with an odd one (and this does not depend upon any significant subtleties of your arithmetic or set-theoretic foundations) so the trick to getting a student to intuitively understand the difference between odd and evenness is to lead them somehow to this feeling, and the path is potentially different for every student since no two people start from the same point of view.
Maths is Fun, Maths is enJoyable and Maths is Beautiful. Perhaps we need three websites exemplifying these three distinct properties of maths, but perhaps we can squeeze some of it into this site?
After many years of soul searching as to what keeps me drawn to mathematics, I have traced it to three basic emotions: Fun, Joy and Beauty.
Fundamentally, playing with mathematical structures needs to invoke at least one of these emotions to be properly interesting. Everybody is different, so they will find these emotions in different ways. Thus when thinking of educating a child, I believe that we need to begin solely with the first: Fun. Numbers and patterns need to be fun so that when a child sees them, he or she instinctively and intuitively treats them like a collection of toys to be played with, in any way shape or form possible. Later one needs to develop Joy: that is, a deep enjoyment of the structures and patterns that everybody can see in numbers. Personally I believe that only the times tables should be learned by rote and even then we have yet to find a universal method of making times table learning fun and enjoyable (just as you can do with scales on a musical instrument.) Finally there is beauty, and this for me comes from simple clear proofs such as Euclid's proof of Pythagoras' theorem or from equations such as e^{i\pi}+1=0, which tie what appear to be fundamentally separate concepts together in a stunningly clear way.
I am still thinking through my understanding of how I learned maths and how I think others can or should learn it, so I'll stop here.
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