Area of the Mandelbrot Set

di | 24 Ottobre 2018

The area of the Mandelbrot Set has been discussed quite a few times during the years since the Mandelbrot Set’s discovery, and particularly starting in the early 1990’s. (See the area history page for details.) An accurate analytical estimate is more difficult than one would think.

Usually the area in question is taken to be the Lebesgue measure of the whole set. Given the definition of membership and the intricacy of the filaments it is a little tough to determine for certain whether the filaments contribute to the area. Therefore, many approaches such as that of Jay Hill measure just the interior.

Since the interior is a union of the interiors of the mu-atoms, which are simply connected and have smooth edges, the area of the interior can be defined as the value of the double integral:

/ 2 / 2
| |
| | M(a+bi) dR dI
| |
/a=-2 /b=-2

where M(x) is the membership function, defined as:

M(x) = 1,      for all x in the Mandelbrot set
M(x) = 0,      otherwise

This can be measured by actually locating the mu-atoms and mearuring them individually (as was done by Jay Hill), or by a statistical method such as pixel counting. Although it might appear that such a method would capture the area (if any) of the boundary, such is not necessarily the case; it depends on the subtle and complicated dynamics of the iterations in Siegel disk Julia sets, whether one’s set theory includes the axiom of determinacy, and other equally complicated things.

Statistical Methods

pixel counting and Monte Carlo are the two statistical methods of measuring the area. The best known estimate so far is 1.506591856 ± 2.54×10-8. See the pixel counting and Monte Carlo pages for a discussion and comparison, and a description of how the error term is computed.

Analytical Methods

Analytical methods use precise mathematical formulas that can give the area, albeit with much calculation. One such method is the Laurent Series method, described in 1992 by Ewing and Schober and discussed fully on its own page.

Another approach that has provided good results is the work of Jay Hill, who has enumerated (located and counted) all of the mu-atoms up to period 15 and all but one of period 16. This includes all the islands up to period 16 and whenever an island is found he also evaluates the areas of the smaller mu-atoms in that island, down to a certain minimum size.

For more information check the links on the Jay Hill page.

Robert P. Munafo, 2003 Oct 21.

The Breakdown of Social Media

di | 13 Settembre 2018

Direttamente da una discussione su 4chan, anche per dimostrare come non sempre quello che viene considerato il peggio del www è realmente solo il peggio. Per chi non se lo ricordasse, quando il world wide web iniziò ad essere utilizzato globalmente era esattamente come 4chan ma meno incattivito dalla presenza dei SJW e di conseguenza meno violento.

Se non riuscite a cogliere la conseguenza logica, siete troppo giovani. O troppo stupidi.

Big ones (social media) are made big by having everyone there.
Small ones are small because all the people using it are friends of friends from an echo chamber.

[…]

All social media turns into echo chambers, There’s no real way to prevent groupthink, and if there was, it would drive away users as people are sheep more interested in having their own opinions validated than the truth.
All publicly available social media is datamining users, whether they want to or not, since they can’t reasonably stop bots from scanning public groups or whatever they might have.
Why on earth you’d want to use any of it is beyond me.
I’d still be interested in hearing your motivations though, so I’ll pose the question. What exactly is it that makes you want to use social media despite the drawbacks? I’m genuinely curious.

[…]

Social media has destroyed so much of the internet. I hate it so much. I’m a huge futurist, technophile, and transhumanist, but we really need to fucking stop using social media before it destroys our species.

I miss the old days where it was easy to find a sense of community and get along with people who were truly different from you.

Marcus e π

di | 18 Agosto 2018

Of course, for any practical application of these numbers, I could probably get away with an approximation which is a fraction. Most engineers are happy to use the estimate 22/7 for π that Archimedes got by approximating a circle with a 96-sided figure. In fact, I need only know 39 digits of π to be able to calculate the circumference ofa circle the size of the observable universe to a precision comparable to the size of a hydrogen atom. There even exists a formula that can tell me what the millionth digit of π is without calculating all the intervening digits.

Not something that I’m desperate to know. But this formula will only ever give me finite knowledge of a number that necessitates the infinite to fully embrace it.

“What we cannot know” Marcus du Sautoy

Ancora Walliman sulla MQ

di | 11 Luglio 2018

Tanto per smentire le solite voci che affermano che “nessuno ci capisce nulla sulla Meccanica Quantistica”…

“…QM is complicated but it does not mean we can’t talk about it at all! […] QM is the description of the smallest things in our universe. How they work when interacting with light. The fundamental rules of the universe. […] We do understand QM. Hard to picture in our head but perfectly described by mathematics…”