I can't get the Mango Song out of my head.. Haha.. I will be In the Kitchen preparing food.. & I start Humming... Then singing.. then laughing cause I realize what I'm singing.. :) oh, also I do a little dance. Thxs Dylan :) How are You? Hope you don't have any new injuries.
Hey! It's awesome to find someone who's not only interested, but also knowledgeable about mathematics. Cognitive science is also a fascinating subject!
So you like ruler and compass constructions... This is a deeply interesting subject, dating back to ancient Greece. Did you know that a regular polygon is constructable if its number of sides can be factored as a product of a power of two and a sequence of distinct Fermat primes? This result was proved by Gauss in 1796, when he was only 17 years old! As a corollary of his theorem, he obtained for the first time that the regular heptadecagon is constructable (the first discovery of this kind since ancient Greece!). He even devised an explicit construction for it!
A Fermat prime is a prime number of the form Fn = 2^(2^n) + 1. They have this name because in 1640, in the same letter where he proves his little theorem, he calculated the first five fermat numbers to be:
and conjectured that all numbers of this form were prime numbers. He seemed to be very excited about it, since he wrote letters to several mathematicians asking them to further investigate the matter. However, it turned out that his conjecture was false... In 1732, Euler presents a non-trivial divisor of F5, thus proving it is composite... In fact F0-4 are the only Fermat primes known to this day! So it was not at all a very happy conjecture... Seems to be one of those examples where our inductive intuition fails us... In fact, there are even heuristic arguments that seem to suggest that there are only finitely many Fermat primes; we are not sure about that yet, though, and it's certainly an interesting question of research, though probably very difficult...
In a (rather short) article of 1837, the french mathematician Pierre-Laurent Wantzel (he was 23 years old then and still a student) announced and proved necessary and sufficient conditions for any problem of geometry to be solvable by ruler and compass alone. As a corollary, he obtained necessary conditions for a polygon to be constructable. He also proved as a corollary of his theory that the classical problems of duplication of the cube and trisection of the angle, that had defied the ingenuity of countless mathematicians throughout the ages, where in fact unsolvable...
How many dimensions are there? That depends on what you mean... From a mathematical point of view, it is possible to construct spaces with as many dimensions as you can imagine: finite or infinite (I assume you are aware that there are several kinds of 'infinities' or transfinite numbers, as they are called); there are even spaces with non-integer dimension. The Cantor set, for example; take the unit interval and split it in three, excluding the middle third; repeat this procedure with the remaining two intervals, and so on ad infinitum. The resulting fractal is called the Cantor set. The question you'll be asking right now is: What is the dimension of the Cantor set? Well, it's certainly less than one, since it has zero lenght; but on the other hand, it's clearly greater than zero, since it's not the empty set. So, how to find it?
Well, if you reflect a little on the meaning of the word 'dimension', you'll discover this:
that if you double each side (dimension) of a square (two dimensional object), the area of the square will be multiplied by a factor of 2^2; if you double each side (dimension) of a cube (three dimensional object), the volume of the cube will be multiplied by a factor of 2^3. Now, if you triple the length of the interval from which the Cantor set was constructed, you get two sets identical to the first, which means that the dimension of the Cantor set is log2/log3!!
You can google 'Haussdorf dimension' if you are interested in this.
Now, if you approach the question from a physical point of view, then it's a wholly different matter...
According to M-theory, the physical universe has 11 dimensions... You may like to watch a video called 'Imagining the Tenth Dimension', if you haven't already. It's a bit superficial and you should not take everything there at face value, but it can serve at least as an invitation to investigate the matter in greater depth...
i have always thought that there has to be infinitely many because one dimension cannot exist without lying 'within' the next higher one.
I don't know exactly what you mean by this, but it does seem to be an interesting philosophical argument. Perhaps you could develop it a bit further?
What are your interests in the field of cognitive science?
I wish great success in your journey and in your investigations! Great to be your friend!