"$d_ProjMini"'Kaal- Structured Atom Model/240115 emto Steve: Sierpinski [triangles, tertrahedra]: Johannes Kepler, Edo Kaal, Pyramids [Bosnia, Egypt, Mesopotamia, Central America, China (dirt, not stone)].html'

Date : 15Jan2024
To : Stephen Howell. Gas storage consultant. SEBIL Consulting Services Ltd. Calgary <SlHowell.cf@gmail.com>
Subject: Sierpinski [triangles, tetrahedra] versus pyramids: Johannes Kepler, Edo Kaal, Stephen Puetz "Universal Wave Series" (UWS)

Things just seemed to blow up into fireworks this morning while doing a minor update to my draft webPage on Kaal's "Structured Atom Model" (SAM). I was trying to find my notes from circa 2012-2017 giving the name ("Hoh...?" or something), and references to an amateur's presentation to the Natural Philosophy Alliance" (before it became the "Natural Philosophy Society", I think).

His concept was the early stages of an intent to replace Quantum Mechanics (QM). It only had 1 arbitrary parameter, if I remember correctly, compared to something like 20 for QM?
Almost immediately this seemed to tie into many [recent, longstanding] topics of interest to me.
Did most of the Quantum Mechanics (QM)-Sierpinski literature pop up AFTER the NPS presentation, or during the 20+ years that the amateur was developing hus concepts? There is NO indication of any credit given to an amateur who may have figured out the basis [BY HIMSELF, BEFORE] all of the professional scientists listed below??? Or maybe this theme reaches back almost to Boskovitch in the mid 1700s, as a long-established componenet of energy quanta, thence QM?

In any case, here are just a few of MANY references tying Quantum Mechanics, fractals, Sierpinski triangles!!!

	etsy.com second generation Sierpinski tetrahedron VI, beaded art
	Johanna L. Miller 19Nov2018 Quantum mechanics in fractal geometry (<Sierpinski triangle>), Physics Today ...Ingmar Swart, Cristiane Morais Smith, and colleagues at Utrecht University in the Netherlands have taken a step toward experimentally studying quantum physics in a fractional-dimensional system. On a (111) surface of copper they placed carbon monoxide molecules (black indentations in the figure) to corral the surface electrons into a simplified Sierpinski triangle. Howell: it seems to me that Kaal has a fractal-like basis to his "Structured Atom Model" (SAM), albeit with many component parts, and therefore more complex than many well-known fractal patterns? Perhaps the use of fractals in hollywood CGI images is somewhat similar, but taken to artistic extreme?
	Shajesh, Parashar, Cavero-Peláez, Kocik, Brevik 13Nov2017 Casimir energy of Sierpinski triangles, Phys. Rev. D 96, 105010 Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sierpinski triangles and Sierpinski rectangles. The Hausdorff-Besicovitch dimension (fractal dimension) of the Casimir energy is introduced and the Berry-Weyl conjecture is discussed for these geometries. We propose that for a class of fractals, comprising compartmentalized cavities, it is possible to establish a finite value to the Casimir energy even while the Casimir energy of the individual cavities consists of divergent terms. Howell: This does remind me of our recent "pyramids" emails. However, keep in mind that pyramids tend to have a square (or rectangular) base, not triangular like a tetrahedron. For example [Bosnia, Egypt, Mesopotamia, Central America, China (dirt, not stone)]. Even so, the energy-focussing properties of pyramid [material, geometry] may distantly relate to things like the Casimir effect?
	Johannes Kepler Howell: I seem to remember that Johannes Kepler always sought a tetrahedron concept as the proper way to explain planetary motions, even after he formulated Kepler's Laws?
	Great Pyramid of Giza can focus electromagnetic energy through its hidden chambers Andrey Evlyukhin, ... the international research team looked into the relationship between the shape of the Great Pyramid of Giza and its ability to focus electromagnetic energy. To do this, the team led by ITMO University in Saint Petersburg, Russia, created a model of the pyramid, one of the seven wonders of the ancient world, to accurately measure it electromagnetic response. Howell: This does remind me of our recent "pyramids" emails. However, keep in mind that pyramids tend to have a square (or rectangular) base, not triangular like a tetrahedron. For example [Bosnia, Egypt, Mesopotamia, Central America, China (dirt, not stone)]. Even so, the energy-focussing properties of pyramid [material, geometry] may distantly relate to things like the Casimir effect?

Bannink, Buhrman 19Nov2018 "Quantum Pascals Triangle and Sierpinskis carpet"

page 2 of paper: "... Section 2 provides background information on how Pascal’s triangle is related to the Sierpinski triangle when the numbers are taken modulo a prime. ..."
Bannink & Buhrman's "modulo power of 3" immediately reminds me of Stephen J. Puetz's "Universal Wave Series". Puetz and co-author Glenn Borchardt struggled to explain the amazing power of their factor-of-three powers series actross, well, almost all well-known cycles. I haven't done the check yet, but it seems to me that their OR[half, double] series is just due to frequency-beating of their primary factor-of-3 series.

Edo Kaal - frequently mentions the critical role of a tetrahedral structure as one of the key basic building blocks of the nucleus geometry. I also mention carbon, which he describes as the [center-piece, backbone] of complex nuclei.

08********08
Draft work on my webPage "Kaal Structured Atom Model vs Quantum Mechanics.html"

Croatian Roger Boscovich (18May1711 - 13Feb1787) is not mentioned in Edo Kaal's EU2017 presentation?
Boskovitch had very general concept of quanta in [physics, ?biology, etc?] ~150 years prior to Planck
Apparently Neils Borh stated that most colleagues AT THE TIME had assumed that Max Planck's Quantum Mechanics were ?[grounded in, inspired by]? Boskovitch's ideas.
Max Planck was so dissapointed in the "deviation" of his 1901 paper concepts by the scientific community, that he wrote a 1911 paper. Is this [correct, relevant to us now]?
key publication : "... In Vienna in 1758, he published the first edition of his famous work, Philosophiæ naturalis theoria redacta ad unicam legem virium in natura existentium (Theory of Natural philosophy derived to the single Law of forces which exist in Nature), containing his atomic theory and his theory of forces. ..."

Section 2 provides background information on how Pascal’s triangle is related to the Sierpinski triangle when the numbers are taken modulo a prime.
Wikipedia Figure 1 caption: "... The first page of figures from Theoria Philosophiæ Naturalis from 1763. Figure 1 is the force curve which received so much attention from later natural philosophers such as Joseph Priestley, Humphry Davy, and Michael Faraday. The ordinate is force, with positive values being repulsive, and the abscissa is radial distance. Newton's gravitational attractive force is clearly seen at the far right of figure 1. ..."
Howell: perhaps Planck also used Boskovitch's ideas as context for quantum levels of the electron?
1927 Copenhagen convention Quantum Mechanics (QM) vs [1761 Boskovich, 1901 Planck] energy quanta
"... At the end of the 1920s, Bohr, Heisenberg, and Pauli had worked out the Copenhagen interpretation of quantum mechanics, Bannink, Buhrman 19Nov2018 Quantum Pascal’s Triangle and Sierpinski’s carpet

"http://www.BillHowell.ca/ProjMajor/Electric Universe/Kaal SAM nucleus/images/Swart, Smith, UtrechtU 19Nov2018 quantum physics in a fractional-dimensional system, Sierpinski triangle.jpg"
color=green>but it was rejected by Planck, and by Schrödinger, Laue, and Einstein as well. Planck expected that wave mechanics would soon render quantum theory – his own child – unnecessary. ..."
Sierpiński triangles & QM : Presentation in time-period Apr2012-2017 ?Hohohlmann?, an amateur's math-physics project he had spent some years on, but had never found a [person, group] to listen.
- reduces down to only 1 arbitrary constant (vs 10-20? for QM?). I think Bill Lucas had NO QM-specifuc arbitrary constants? - just what came with fundamentals of Universal force, eg [c, h, permittivity, permissivity, permeability, etc].
- Reminds me of Johannes Kepler - apparently always sought a tetrahedron concept to explain planetary motions, even after he formulated Kepler's Laws?

08********08
#] 15Jan2024 search "sierpinski triangles" - long ago NatlPhilAlliance presentation
#] this dateEntry was copied from "$d_web"'My sports & clubs/CNPS/0_JC-NPS notes.txt'
This notes file only goes back to 2018 -> may have to check for other notes files, backups from 2010-2018?

https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle
https://mathigon.org/course/fractals/sierpinski
https://www.mathsisfun.com/sierpinski-triangle.html

08:48$ find "$d_web" -type f -name "*.txt" | grep --invert-match "z_Old" | grep --invert-match "z_Archive" | sort | tr \\n \\0 | xargs -0 -IFILE grep --with-filename --line-number 'pinski' "FILE" | sed "s|$d_web||;s|:.*||" | sort -u >"$d_temp"'find-grep-sed temp.txt'
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bin/0_test/fileops/pinn_getLinseq_povr/230315 13h57m Neural Nets notes public.txt
My sports & clubs/natural- CNPS/0_JC-NPS notes.txt
Neural nets/0_Neural Nets notes public.txt
ProjMajor/Sun pandemics, health/corona virus/Fauci covid emails/Fauci corona virus emails, clean.txt
Qnial/MY_NDFS/dictionaries/combined [Linux, Dictionary.com, vaccine [CDC, Wikipedia]].txt
Qnial/MY_NDFS/dictionaries/dictionary - corona virus.txt
Qnial/MY_NDFS/dictionaries/dictionary Linux american-english noApos.txt
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>> hah! - I failed to find the old NPA presenattion notes, I just caught this entry...

search "Natural Philosophy Alliance and sierpinski"
- nothing

search "sierpinski triangles and Quantum Mechanics"

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https://arxiv.org/pdf/1708.07429.pdf
Bannink, Buhrman 19Nov2018 Quantum Pascal’s Triangle and Sierpinski’s carpet

http://www.BillHowell.ca/ProjMajor/Electric Universe/Kaal SAM nucleus/images/Bannink, Buhrman 19Nov2018 Quantum Pascals Triangle and Sierpinskis carpet: Pascals triangle plotted modulo powers of 3.png"

page 2 of paper: "... Section 2 provides background information on how Pascal’s triangle is related to the Sierpinski triangle when the numbers are taken modulo a prime. ..."

Tom Bannink∗
bannink@cwi.nl
Harry Buhrman∗ ‡
buhrman@cwi.nl
August 2017
Abstract
In this paper we consider a quantum version of Pascal’s triangle. Pascal’s triangle is a well-known triangular array of numbers and when these numbers are plotted modulo 2, a fractal known as the Sierpinski triangle appears. We first prove the appearance of more general fractals when Pascal’s triangle is considered modulo prime powers. The numbers in Pascal’s triangle can be obtained by scaling the probabilities of the simple symmetric random walk on the line. In this paper we consider a quantum version of Pascal’s triangle by replacing the random walk by the quantum walk known as the Hadamard walk. We show that when the amplitudes of the Hadamard walk are scaled to become integers and plotted modulo three, a fractal known as the Sierpinski carpet emerges and we provide a proof of this using Lucas’s theorem. We furthermore give a general class of quantum walks for which this phenomenon occurs.

Physics Today http://www.BillHowell.ca/ProjMajor/Electric Universe/Kaal SAM nucleus/images/

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https://pubs.aip.org/physicstoday/Online/29862/Quantum-mechanics-in-fractal-geometry
Johanna L. Miller 19Nov2018Quantum mechanics in fractal geometry (<Sierpinski triangle>)
Molecules on a metal surface can sculpt the surface electrons into shapes not found in nature.

DOI:https://doi.org/10.1063/PT.6.1.20181119a

What if the number of dimensions could be tuned continuously between 1 and 2? A century ago mathematician Felix Hausdorff developed a new way of measuring dimensionality that could take fractional values. For most familiar shapes—points, lines, planes, and so on—the Hausdorff dimension is equal to the usual topological dimension. But fractals, due to their infinite complexity, have noninteger Hausdorff dimension.

Now Ingmar Swart, Cristiane Morais Smith, and colleagues at Utrecht University in the Netherlands have taken a step toward experimentally studying quantum physics in a fractional-dimensional system. On a (111) surface of copper they placed carbon monoxide molecules (black indentations in the figure) to corral the surface electrons into a simplified Sierpinski triangle.

The full Sierpinski triangle, a fractal made up of infinitely many nested smaller triangles, has a Hausdorff dimension of 1.58. If drawn from line segments, it has infinite length, and if carved from a solid triangle, it has zero area, so intuitively, it’s neither 1D nor 2D but something in between. The Cu(111) surface-electron density inside the triangle is an approximation of the fractal, just as a graphene sheet is an approximation of an infinitely thin plane. But like graphene, the surface-electron system inherits the dimensional properties of its mathematical idealization.

Probing the interesting questions of fractal quantum mechanics will require a more intricate experiment. In particular, ensembles of interacting fermions behave in qualitatively different ways in 1D and 2D (see Physics Today, September 1996, page 19). Their behavior in 1.58 dimensions is an open question—but the surface electrons of Cu(111) interact only weakly. Surfaces of other materials, however, might allow the researchers to study interactions in fractal geometry. (S. N. Kempkes et al., Nat. Phys., 2018, doi:10.1038/s41567-018-0328-0.)

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https://phys.org/news/2018-12-splendid-potential-sierpinski-triangle.html
The splendid generative potential of the Sierpinski triangle
by The Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences
14Dec2018

One transistor can become an oscillator with a surprising richness of behavior. However, even more interesting effects emerge if the structure of connections is fractal and shows some imperfections. Could similar rules explain the diversity and complexity of human brain dynamics?

Intuition suggests that self-similarity appears only in systems as complex as neural networks in the brain, or in fascinating shapes of nature, for example, in fractal Romanesco broccoli buds. At the Institute of Nuclear Physics of the Polish Academy of Sciences (IFJ PAN) in Cracow, researchers made a discovery that in some ways challenges this belief. In close collaboration with colleagues from the University of Catania and the University of Trento in Italy, the researchers constructed an elementary electronic oscillator based on just one transistor. As it turns out, when it contains fractal arrangements of inductors and capacitors, these generate amazingly rich characteristics of the electrical signals.

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https://link.springer.com/article/10.1007/s11128-021-03171-4
Xie, HH., Zeng, GM. 15Jul2021 Quantum walks on Sierpinski gasket and Sierpinski tetrahedron. Quantum Inf Process 20, 240 (2021). https://doi.org/10.1007/s11128-021-03171-4

Abstract
We investigate discrete-time coined quantum walks on the Sierpinski gasket and the Sierpinski tetrahedron which have non-integer dimensions, by concentrating on the probability distribution, return probability and standard deviation. We compare the calculating results with classical random walks on the two fractal structures and quantum walks on the corresponding regular triangle grids. For the quantum walks, we adopt DFT coin and Grover coin, respectively, which exhibit great differences on the above quantities.

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https://journals.aps.org/prd/abstract/10.1103/PhysRevD.96.105010
Shajesh, Parashar, Cavero-Peláez, Kocik, Brevik 13Nov2017 Casimir energy of Sierpinski triangles, Phys. Rev. D 96, 105010
K. V. Shajesh, Prachi Parashar, Inés Cavero-Peláez, Jerzy Kocik, and Iver Brevik
Phys. Rev. D 96, 105010 – Published 13 November 2017
Abstract
Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sierpinski triangles and Sierpinski rectangles. The Hausdorff-Besicovitch dimension (fractal dimension) of the Casimir energy is introduced and the Berry-Weyl conjecture is discussed for these geometries. We propose that for a class of fractals, comprising compartmentalized cavities, it is possible to establish a finite value to the Casimir energy even while the Casimir energy of the individual cavities consists of divergent terms.

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https://arxiv.org/pdf/2003.09604.pdf Krecmar, Zelenayova, Caha, Rapcan, Nishino, Gendiar 24Mar2020 Quantum Potts Models on the Sierpinski Pyramid Roman Krˇcm´ar1, M´aria Zelenayov´a1,2, Libor Caha1, Peter Rapˇcan1, Tomotoshi Nishino3, and Andrej Gendiar Phase transition of the two- and three-state quantum Potts models on the Sierpi´nski pyramid are studied by means of a tensor network framework, the higher-order tensor renormalization group method. Critical values of the transverse magnetic field and the magnetic exponent β are evaluated. Despite the fact that the Hausdorff dimension of the Sierpi´nski pyramid is exactly two (= log2 4), the obtained critical properties show that the effective dimension is lower than two.