the tree

A geometric mean would be [math](a_1 \cdot a_2 \cdot a_3 \cdot a_4 ) ^ {\frac{1}{4}}[/math] But in terms of LibreOffice's notation that'd be along the lines of =POWER( PRODUCT(A1:A4) ; (1/4) ) Or, more neatly =GEOMEAN(A1:A4) I honestly can't see where on Earth you managed to get "power= ((a4/a1),1/5)-1" from.

Well for one, that is a terible estimator for an average. Take the list [0,1,2,6,24] (for a horribly contrived example), the mean is 6.6 but the midpoint is 12 - that's quite a dramatic overestimate.

[imath]f(x)[/imath] is a function, [imath]y=f(x)[/imath] is an equation.

If [imath]x^2 + 2x = 2x^3[/imath] then [imath]2x^3 - x^2 - 2x=0[/imath] so [imath]2x^2 - x -2 =0[/imath]. Which by the Fundamental Theorem of Algebra, is only the case for at most two values of [imath]x[/imath].

I think, an Abelian Grape.

Of course. Considering the amount of Pagan traditions that Christians have adopted (the "Christmas" tree being the biggest example) , there should be no problem with Atheists sharing some of the Christian ones.

Very much this. Even if it looks bad on the government, it looks better when it comes from them than when it comes from someone else.

Surely the first scientist should have been a physicist?

I'm a little confused here: if you're testing your normal using the cross product, then how are you finding it in the first place? A cross product of any two vectors vectors on the plane should give a perfectly good normal vector.

Well it'd only take one non deterministic feature for the whole product to be non deterministic.

Oh heavens, would you look at that? We do try, I swear. Cool, so an extra deterministic input for the deterministic algorithm with a deterministic output. In the Conference Proceedings of the Royal Society of Extraneous Punctuation, no doubt... Yes. The QM folks will doubt be all to enthusiastic to explain indeterminacy and non-causality and how we know there aren't any hidden variables involved. Although in mathematics there are slightly lower standards for randomness, they are to do with information and complexity theory.

So they do, my bad.

It's not from anywhere, what was my error?

It'd have been slightly more cyrptic if you'd given the puzzlee a table and a saw. I recall this being the punchline to a book I had as a kid called Math Curse in which the protagonist was caught in a series of increasingly absurd situations with a numerical bent to them, great way to make kids see both maths and terrible, terrible, terrible puns in everything they do.

I don't really see what opinions have got to do with it. I think that perhaps you're confused between the concepts of nondeterministic and convincingly pseudorandom. Protip: contrary to belief, elipses have no gramatical place at the end of a sentence and don't make you appear any more profound.

Not at all. Clocks are just as deterministic as any other non-random input and from some points of view weaker than a decent PRNG.

"consider a space alpha" is something I'd expect to read in an exam paper, I don't want to be told what to do whilst reading an article.

A great all round book is The Pleasures Of Counting which covers an awful lot of applied mathematics, not in much depth with enough to give you a feel for what is going on and lots of historical context. I think beyond that the question is really dependent on what area of mathematics interests you the most, since there is a hell of a lot of literature to chose from. Can you please, please, please take a second to consider what the point of this post was? Ignoring the grammar and whatnot, you haven't actually named a book.

Or any of the lines, depending on the system in question?

Simply put, I'm not going to believe that without good reason. Deterministic algorithm, deterministic input - the output is only going to be one thing.

Honestly it just looks like a much more convoluted PRNG, using a chaotic system between the initial PRNG and the eventual output would indeed make it much harder to predict but it wouldn't make the output non-deterministic or genuinely random.

Two groups of people can like the same things, y'know. Really good music doesn't need to be shouldn't be age or gender specific.

If you think about it, throwing in the time factor to make an extra projection is really common. We've all seen those animations of MRI slides where we are looking at a 3d object, projected onto a 2d screen but with the 3rd dimension added back in with the aid of calling it time. I'm fairly sure any useful data relating to sound would come in at least two dimensions. No, it's a projection of a 4d cube onto 3d space. No simple shape is going to have more dimensions that the space it occupies. All you've made there is a picture.

For a sufficiently useless metric, it'd be perfectly reasonable to say that 1+1=3 (mod 1) and for that to apply to real spaces. I've never been keen on the mathematical universe, it sounds awfully similar to the 'the universe can be, and so it is' type of ToEs - although the question is entirely philosophical in nature I tend to hold on to the idea that there are ways things could have been but they turned out not be.

It happens, there isn't much that can be done about it beyond being satisfied that you made a worthy effort.