I am not saying this is wrong. I am just puzzled by the idea.

Assume that the upper most circle in the picture above is “A”. The lower most circle is “B” and the right most circle is “C”.

In the picture above we have the following:

1) A is over B
2) A is under C
3) B is over C
4) B is under A
5) C is over A
6) C is under B

(Once can see that 1 and 4, 2 and 5 & 3 and 6 are merely saying the same thing in the opposite manner.)

I cut three identical circles out of a piece of paper and the cut a slit in each one to allow for interlocking. (One must assume a slit in the circle because one cannot (if not an illusionist, a magician with trick rings or a physicist with quantum mechanical rings) make solid objects pass through each other.) After doing so I was able to construct a BR immediately. No twisting, kinks or deformations were necessary – the feat was accomplished by merely passing the slits through at the appropriate points.

Now perhaps by flat circles it is meant that the circles are always on the same plane such that they would bump into each other and therefore interlocking could not occur. However, if that were the case you would have to say that two flat circles could not interlock. The implication would then be that BRs aren’t that special; therefore, I have to assume that flat circles mean something else (and cause the impossibility of creating BRs) or that I have missed something that is blatantly obvious.

I fear in my naiveté that a “flat circle” in the “physical” world; i.e., that I can cut out of a flat sheet of paper, is not the same as a flat circle in the scientific world.

Thanks!!

http://sprott.physics.wisc.edu/pickover/pc/realitycarnival.html

is an amazing site highly recommended.

As some of you you might know, a normal number is a real number whose digits, in every base, are distributed uniformly. Almost all reals are normal (i.e., non-normals have Lebesgue measure 0). So this is a good example of the kind of base-independent property that makes a “true” mathematical law.

Champernowne’s Constant is a good example of a normal number:

0.12345678910111213141516…

It’s just successive integers in Base-10 here. But the “normalcy” of this constant is invariant across bases, which is fascinating.

On the other hand, Benford’s Law is a kind of statistical property of real-life sets of data; it is not a purely mathematical property in the same way. The continuum of real numbers has just as “many” numbers that start with .1… as .9… (and .123456… for that matter).

Let’s be careful here though, because though the applicability of Benford’s law has to do with *finite* data-sets empirically gleaned, the purely mathematical relations described by the law are as robust and real as you please. For instance, Benford’s law is scale-invariant: it doesn’t matter whether you measure the height of buildings in inches or feet or meters!

Meanwhile, the logarithmic distribution of digits in real-life data-sets, however, can be calculated *across bases too* (though of course the actual probabilities will have to be re-calculated). So in a Base-8 or -16 system the leading digit phenomenon will appear here but the probability will change accordingly. It’s rather simple to calculate, actually. So Benford’s law holds true as long as Base > 2.

So really it’s not as limited as one would think. The Wikipedia page on Benford’s law explains this quite clearly in relation to exponential growth processes. The only real limit is that it is applicable to certain kinds of data-sets.

To be a “Law”, the underlying principle should work in every base of mathematical expression.