'Infinity Computer' Calculates Area Of Sierpinski Carpet Exactly | Hacker News Reader

'Infinity Computer' Calculates Area Of Sierpinski Carpet Exactly

technologyreview.com

8 pointswglb11 comments

I took a quick look at the arXiv paper behind this article, and it looks suspicious. I didn't take the time to be 100% sure it's vacuous, but it has many red flags, such as being published in "Chaos, Solitons & Fractals," a poorly-regarded journal. Another red flag is the existence of theinfinitycomputer.com.

Now for some real math related to this:

The Cantor Dust and the Sierpinski Carpet really do have area (measure) 0 in the traditional sense. The normal area of the Sierpinski Carpet, during construction, starts at 1 (a full unit square), and is multiplied by 8/9 per iteration. The iterations never stop, so the area goes to zero.

Its 2D measure is 0. There are infinitely many length-1 lines within the final product (any line with equation y=p for any constant p in [0, 1] with no 2's in it's base-3 representation), so it has infinite 1D measure.

So we can look for a dimension between 1 and 2 that measures it, which is where the Hausdorff dimension comes in.

If you double a unit (length-1) line, its measure doubles. If you double a unit square, its measure is multiplied by 4. If you double a unit cube, its measure is multiplied by 8. Here's a pattern we can generalize to non-integer dimensions!

We work from the idea that any measurable set S, when multiplied (scaled) by scalar r, will have its measure (eg length, surface area, volume) multiplied by r^d, where d is the dimension of S. Using m(S) for the measure of S, we can write:

m(S * r) = m(S) * r^d

which means

log [m(S * r) / m(S)] = d * log r

and then

d = log [m(S * r) / m(S)] / log r

This isn't the formal definition of the Hausdorff dimension, but it is the idea behind it, and provides the correct dimension for iterative fractals like the Sierpinski Carpet.

What is d for the Sierpinski Carpet? When we scale it up by 3, we get 8 copies of the original. So m(S * r) / m(S) = 8, and r = 3. So

d = log 8 / log 3 = 1.892789...

for this shape. If you wanted, you could use a factorial-based expression for the size of an n-dimensional sphere, along with the fractional version of !, the gamma function, and use those to get a reasonable number for the exact measure of such a shape.

One more thing. There are numbers of infinite size that you can perform meaningful arithmetic on. My personal favorite are the surreals, championed by John Conway about 40 years ago. This number class basically starts with the ordinals (so including many sizes of infinity) and builds real-number-like operations on those. If omega is the first infinite number, then you can talk about sqrt(omega) in the surreals, and it's a meaningful quantity larger than any finite number and smaller than omega, for example. There are also hyperreal numbers which offer similar freedom of expression.

This article in Technology Review shows a disturbing lack of comprehension of how math works. It is not worth reading. As for the article itself[1], Wikipedia claims[2] that Fractal dimension has been studied since 1975 and Cantor's transfinite numbers[3] since the late 1800's. Perhaps someone else can find insights in this paper, but my own impression is that the authors have not advanced the field, but rather have failed to understand the basics of the field.

[1] http://arxiv.org/pdf/1203.3150v1.pdf [2] http://en.wikipedia.org/wiki/Fractal_dimension [3] http://en.wikipedia.org/wiki/Transfinite_number

The linked paper on the arXiv is pretty bad. The idea as it stands appears to be very mundane -- that we define some new number, which he calls ① and I'll call m, to denote the size of the natural numbers N = {0, 1, 2, ...}. Then m² is the cardinality of N × N, you see. And we could say things like m + 1 > m, surely, and everything would be peachy keen.

Well, not so fast. That sort of mathematics is fraught with difficulty. So, for example, the set of pairs of natural numbers is called N² = {(0, 0), (1, 0), (0, 1), (2, 0), ...}. How many of those are there? The arXiv paper very distinctly says that there are m² of these numbers. In doing so he's apparently unaware of the huge breakthrough that Hilbert made: that in a deep sense, there are only m of them. For every pair, you can associate a number. It's not even hard, the function is:

    def n(x, y):
        return (x + y + 1)*(x + y)/2 + y

It's only a little tougher to invert this mapping, which is not strictly necessary, but is always nice:

    from math import sqrt
    def pair(n):
        row = int(0.5 * (sqrt(1 + 8*n) - 1))
        col = n - row * (row + 1)/2
        return (row - col, col)

Mind you, that's just with examples that he uses. The whole mathematics is poorly defined. Supposing that you remove one element, like the number 0, from the natural numbers, he wants to say that there are m - 1 numbers in the set {1, 2, 3, 4, ...}. And if I take away 2, then presumably there are m - 2. And if I remove 2n for each n in N, I have performed m 'remove' operations, and you would presumably say that the set {1, 3, 5, 7, ...} has m - m elements, no? And even if you say that the resulting set of odd numbers has a size "something like" m/2, how do you know it's not m/2 + 1 or m/2 - 1? Suppose we define the sum of all the odd numbers up to the n'th one as {1, 1+3, 1+3+5, 1+3+5+7, ...} -- this surely has m/2 elements, correct? No -- those sums are {1, 4, 9, 16, ...} and this has sqrt(m) - 1 elements. Wha...? Does sqrt(m) even exist? His other publications suggest that he usually expands calculations in terms of (positive/negative) integer powers of m; there is little discussion about fractional powers, but presumably they must exist.

In short, we must accept that the mathematics of m has no correspondence (or only ad-hoc after-the-fact correspondence) to the natural numbers M. When you do this and uncouple the two, you'll probably discover that his "Infinity computer" is just a normal computer which can represent polynomials, and his ① is just the polynomial's 'x'.

First line:

> Mathematicians have never been comfortable handling infinities...

I stopped reading there, since that is badly false.

Huh. This is not my field, so I am hesitant to comment in detail. However, a few things quickly strike me as troubling about the original paper:

* The work is based on "a new applied point of view on infinite and infinitesimal numbers", the cited source of which is six papers by the same author as the current one. Apparently, nobody else has chosen to work on it.

* The postulates behind this method include the statement that "We shall not tell what are the mathematical objects we deal with", and because of this, "such concepts as bijection, numerable and continuum sets, cardinal and ordinal numbers cannot be used in this paper". In other words, the author is knowingly setting aside most of what I consider to be the essential features of mathematics.

* I have serious doubts that results or concepts that boil down to "infinity minus three" or "infinity times 8/9" would prove to be robust if you considered different infinite sequences with the same limit. I haven't looked up the author's other papers on the subject, but I don't see any evidence that he has addressed this issue.

* This paper (and several of the earlier ones) was apparently published in the journal "Chaos, Solitons, & Fractals" in a 2009 volume. You may recognize that title from recent discussions of Elsevier: a scandal unfolded in 2008-9 where the journal's editor, El Naschie, was found to have published hundreds of his own papers (whose merit has been questioned) and accused of poor editorial standards.[1] It may or may not be meaningful that the current paper's first four references are all to El Naschie's work.

In short, my impression is that this paper is based on decidedly non-mainstream mathematics, and I see no reason to be excited about its claims or conclusions.

[1] http://en.wikipedia.org/wiki/Elsevier#Chaos.2C_Solitons_.26_... http://backreaction.blogspot.com/2008/11/chaos-solitons-and-... http://elnaschiewatch.blogspot.com/2009/02/that-hard-to-find... (The archived posts by Baez here are the point; I know nothing about this site itself.)

The article on Arxiv makes no sense, it uses ill-defined notions, methods used there are contradictory, and most of the references it gives are from "Chaos, solitons and fractals" journal, which many professional mathematicians consider to be a joke.

Mathematicians are great at handling infinities -- it just turns out that Lebesgue's measure is not usefyl when studying fractals. That's why other notions were introduced, like for instance Hausdorff dimension.

so this guy claims he "discovered" Cantor's alef-null?

article is full of wrong statements beginning with the very first one.

He apparently heard about cardinal and ordinal number -- in his paper he states that his grossone is not the same as aleph 0 or omega. What it actually is, he does not say.

It's not just that he doesn't say! He explicitly rejects the notion that he ought to say. As far as I'm concerned, that means he's declared to all the world that he is not actually doing mathematics.

I don't know if this is exactly like aleph-null.

> Because it is governed by the other axioms of real numbers, grossone behaves much like one too. So it's possible to multiply grossone, divide it, add to it and subtract from it, just as is possible with other real numbers

How much is aleph-null minus aleph-null? How much elements have set that is difference of two sets with aleph-null size? Can be anything.

If they solved this somehow, it's huge. But I can't imagine how they did it, and article had not explained it.

there is no way to solve it because it's not a problem. it's a set of not-defined behaviors by defining which in regular ways you get absurd results. if he managed to define them in a way which doesn't produce nonsense and still useful for solving some real problems - it's cool. but i still don't see the problem itself - mathematicians know how to handle fractals and calculate their properties for decades.

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