The role of Fractals and the Mandelbrot Set in Physics

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Offline JonathanD

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« on: February 01, 2020, 09:41:32 PM »
Hello to all,

After the discovery of a mathematical curiosity in the 80s led to conversations with Benoit Mandelbrot about the significance of M and fractals to cosmology; I have been curious about just how deep this connection goes.  It is a very deep connection indeed.  I took the approach of Feynman to first try to disprove or invalidate what I thought to be true.  When that failed after 10 years; I began to explore the possibility I was right once again, and started exploring other avenues.  And not long after; there was a shake-up in Cosmology (~2k) and then Pietronero and others wrote that astrophysics suggests the universe is a fractal - so I began to wonder.

I got to present my work at the "Crisis in Cosmology" conference in Port Angeles, WA - where I met a bunch of lovable scientists who were willing to explore an alternative view of the Cosmos - including a few well-respected professionals.  Since that time; I have continued to up my game and continue my studies, while attending lectures by and getting to meet some of the top researchers in theoretical Physics alive today - including 7 or 8 Nobel laureates.  I don't know if it counts when you meet someone like Barry Barish before he won the award.

I authored the article on 'Fractal cosmology' at Wikipedia, and continued to research the topic thereafter.  A paper was published at one point in Chaos, Solitons, and Fractals which is almost a snapshot of that entry when it included material favorable to El Naschie, who was the C,S,&F editor at the time.  But I would have thought the book by Baryshev and Teerikopi was a fair indicator that the idea had merit.  It has been 'disproved' a few times, but then they find a yet larger void and I wonder.

In the microcausal realm; their existence is uncontested.  The Physics shows that fractals appear in a vast array of Quantum Gravity theories including Asymptotic Safety (Quantum Einstein Gravity), Causal Dynamical Triangulations (CDT), Loop Quantum Gravity, Horava Lifshitz gravity, and others.  I've had the great privilege to attend lectures by and get to meet the founding researchers for most of those theories.  But I've been one of the few talking abou the Mandelbrot Set.

I've given one or two prestigious lectures, but most of my presentations at top Physics conferences has been on posters.  I am not the only researcher writing about the value of the Mandelbrot Set to Physics.  You can check out Pastor and Romera in Madrid, and Christian Beck in the UK, for starters.  But you might want to see my posters for GR22, which was in Valencia last July.  So I offer these links from viXra.

Massive Gravity Illustrated in the Mandelbrot Set https://www.vixra.org/abs/1906.0559

Theories of Quantum and Analog Gravity Represented in the Mandelbrot Set https://www.vixra.org/abs/1906.0558

Or you can view the introduction on YouTube in the window below.

All the Best,

Jonathan



Linkback: https://fractalforums.org/physics/49/the-role-of-fractals-and-the-mandelbrot-set-in-physics/3299/
Jonathan J. Dickau

Offline gerrit

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« Reply #1 on: February 02, 2020, 10:11:26 PM »
Here's a nice structure of size \( 10^{-308} \) near that (3,1) Misiurewicz point you think has something to do with the cosmos. I computed its location more accurately as
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-1.543689012692076361570855971801747986525203297650983935240804037831168673927973866485157914576059125462120829226367060189278756463322141011522909218905292722992781697157582950422170089518563410700385200128000282366477261779986908996884104614976976927086847969611978920181446128991348585592145309524153482664595734117098522176434389326263543393937637861596321991246778909541806274987275657314799920191066093455727443186409799911554184677317541006847088177067596744933391559037024490072743842093947624299476858021814236680891765387460029177563608220435803992788777820556050367509668431452581000660479923719409341669338344786667998438713931954596704411984984894062625430628197361190271342021144843232805334969761717743968837957977497601033695749567794106261394626976453119061490410209960291581802793650430076138138024025396720461315429830658993371877697398019723803401633422621448328141407053197837142426073136633348667589230259195536140236733894771284836178165007992493871543720417366811166976788846154643698023303756808884

Offline JonathanD

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« Reply #2 on: February 03, 2020, 01:26:48 AM »
I didn't count your digits....

My record for precision is 614 digits long.  That result appears in my FQXi essay, which is currently online in their contest.  I am actually one of 3 or 4 entries already talking about fractals, but the field usually billows out to over 100 participants near the end.  Feel free to join in over there.

But the Julia you have rendered is a different spot entirely, also nice though.  I have lots of images to post, once I collate them.

Thanks again Gerrit,

Jonathan

Offline gerrit

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« Reply #3 on: February 03, 2020, 02:48:09 AM »
But the Julia you have rendered is a different spot entirely
It's at a distance of 2.8E-100 from the M(3,1) Misiurewicz point which incidentally is listed on the wiki page on Misiurewicz points https://en.wikipedia.org/wiki/Misiurewicz_point, but the last 3 digits listed there are wrong.

For this image https://fractalforums.org/fractal-mathematics-and-new-theories/28/another-possible-way-to-accelerate-mb-set-deep-zooming/277/msg13338#msg13338 minibrot location had to be calculated to about a million digits (using Kalles Fractaler's Newton-Raphson zoom option) which is my record.

Offline JonathanD

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« Reply #4 on: February 03, 2020, 04:21:29 PM »
Thanks for the inspiring journey to extremity!

I see that indeed the Wikipedia 'Misiurewicz point' value for M3,1 should be corrected.  I think that was perhaps the value printed in Peitgen and Richter, that was never corrected.  But why stop there?  If you believe the version (of the Misiurewicz points story) according to Julien Sprott; Misiurewicz points are NOT limited to the Mandelbrot Set, but rather are any mathematical structure whose scale factor goes to zero, and then fans out again - which is seen in many figures including the bifurcation diagram or logistic map.

This was pointed out to me by Michel Planat.  If you go back to Grossmann and Thomae, who P and R reference in that section (on M3,1) they show that the bifurcation diagram for the Mandelbrot equation splits everywhere M doubles back on itself, along the real axis, and then all of the prior branches reach to a common point where their trajectories cross - which is M3,1.  So the Wikipedia article has an error right in the lead paragraph.  I'll see if I can verify that statement, and I'll find the Sprott reference in the meanwhile.

All the Best,

Jonathan

Offline JonathanD

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« Reply #5 on: February 03, 2020, 04:38:06 PM »
Here is an image of the two figures overlaid...

This shows how the two figures line up at M3,1 to illustrate that it represents a fundamental behavior of the Mandelbrot Set.  I call the second type (with exactly 2 external arguments) Inflection points.  Part of the miracle of the Butterfly figure is that it makes all of those places easy to spot; they are at the edge of a disc or lozenge.  If you plot the inverse of the Butterfly (where current size > last size > previous size); that figure shows the pinch-off regions for structures of a given period instead.  But the middle type of Misiurewicz point is exquisitely highlighted, in every case.

I'll find some of those images next.

All the Best,

Jonathan

Offline JonathanD

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« Reply #6 on: February 03, 2020, 04:56:14 PM »
FWIW,

Grossmann and Thomae call Misiurewicz point M3,1 a band-merging point and I think the name fits.  What is happening there is a big deal.  It's like the Descent of Innana where all her accoutrements and attributes had to be shed, in order to enter the underworld.

Regards,

Jonathan

Offline JonathanD

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« Reply #7 on: February 03, 2020, 06:14:40 PM »
I have posted images of the inverse Butterfly, which I call the Mandel Shield...

Find that under "Share a Fractal" in the item 'The Butterfly's inverse is also Spectral.'

I'll be posting formula sets for Chaos Pro a bit later today.

Regards,

Jonathan