Category Archives: Physics

Hofstadter’s Quantum-Mechanical Butterfly

Hofstadter’s Quantum-Mechanical Butterfly relates to the fractional Quantum Hall Effect, which is (IMHO) at the heart of understanding (interpreting) how QM really works!

I modified Wolfram Demonstration code by Enrique Zeleny to produce a short video of the emergence of the Hofstadter’s Quantum-Mechanical Butterfly

I found an interesting pattern. By modifying the integer used in the solution i=12, the Golden Ratio Ф=(1+Sqrt[5])/2=1.618 … emerges within the butterfly! Can you find it (or 1/Ф=.618…)?
Hint: Use the interactive version and mouse-over the red or green dots in the white space of the wings of butterfly.

A snapshot of the code and last frame@n=50:
HofstadterButterfly

If you have the free Mathematica CDF plugin on a non-Chrome browser, you can interactively analyze this with n<11 being presented with symbolic solutions in a mouse-over Tooltip (below).

Visualizing Climate Data

I saw this post on the Wolfram blog.

I’ve enhanced the code to show some error bar indicators and visualized by year w/o the obfuscation of accumulation over time. It also uses parallel CPUs and binary packed arrays to speed up the computation. I also reduced the scale from +2.5C to +1C (since there was no data above it). I wonder if that anomalously larger scale was selected to “suggest” a placeholder for future warming (not appropriate as science – but typical for “influencing” public opinion).

I think my presentation more accurately shows that the recent data from the ’00s (with several El Nino’s), mid 40-60’s ocean temp bias adjustment increases, and more recent changes in (space) technology for collecting data may be a (the?) significant root cause for increases in temperature.

That may not be a politically correct thing to suggest – but I will do more work to substantiate based on the statistical analysis of the various data sets….

If you don’t have the Wolfram CDF Player on a non-Chrome browser for interactive UI, you can watch the YouTube video here or see below.

warming1

warming-1g

Interactive Reimann Zeta Function Zeros Demonstration

This web enabled demonstration shows a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points) along the critical line Zeta(1/2+it) for real values of t running from 0 to 50. The consecutively labeled zeros have 50 red plot points between each, with zeros identified by concentric magenta rings scaled to show the relative distance between their values of t. Gram’s law states that the curve usually crosses the real axis once between zeros.

A downloadable copy is available ZetaZeros-local-art-50.cdf Latest: 05/23/2016.

Note: The interactive CDF plug-in as required below does not currently work on Chrome browsers.

A Snapshot picture for those w/o Wolfram CDF interactivity:
RiemannZeta_Zeros.svg

Selectable example code snippet:
[wlcode]Show[ListPlot[Style[pts2, Red], PlotRange -> {{-2, 4}, {-3, 3}},
AspectRatio -> 1, ImageSize -> imageSize,
AxesStyle ->
Directive[Thick, If[artPrint && ! localize, Large, Medium]],
Graphics[{PointSize@.01,
tttxt := If[artPrint && ! localize, tttxt1, ttxt0];
If[ttxt0 = ToString[# – 1];
Abs@zeroY[[#, 1]] < 10 chop, (* Magenta Critical Line Zeta Zeros *) tttxt1 = Column[{ToString[# - 1], "t=" <> ToString@zeroY[[#, 4]]},
Center];
ttLoc =
zeroY[[#, 4]] If[artPrint && ! localize, 1, 2] imagesize/40000;
(* Flip Point Labels above/below the X axis *)
ttLoc1 = {-1, (-1)^Round[#/2]} ttLoc/Sqrt[2];
{Magenta, Point@ttLoc1,
Circle[zeroY[[#, ;; 2]], ttLoc, {1, 3} \[Pi]/2],
Black,
Tooltip[Text[
Style[tttxt, If[artPrint && ! localize, Large, Medium, Bold]],
(* Shift the Labels off the Point *)
ttLoc1 (1 – (-1)^Round[#/2] .05)], tttxt1]},
(* Orange Critical Line Imaginary zeros w/Real>0 *)
tttxt1 =
Column[{ToString[# – 1], “x=” <> ToString@zeroY[[#, 1]],
“t=” <> ToString@zeroY[[#, 4]]}, Center];
{Tooltip[{Orange, Point@zeroY[[#, ;; 2]], Text[Style[

Column[If[EvenQ[Round[(# – 1)/2]], Prepend,
Append][{“\[UpDownArrow]”}, tttxt], Center,
Frame -> True],
Black, If[artPrint && ! localize, Large, Medium],
Background -> White],
(* Flip Point Labels above/below the X axis *)

zeroY[[#, ;; 2]] + {0, (-1)^Round[(# – 1)/2]} If[
artPrint && ! localize, 1,
If[artPrint, 4/1, 3]] imagesize/5000]},
Column[{ToString[# – 1], “x=” <> ToString@zeroY[[#, 1]],
“t=” <> ToString@zeroY[[#, 4]]}, Center]]}] & /@
Range@Length@zeroY,
Magenta, Disk[{0, 0}, .03]}]][/wlcode]

More plots with various scaling functions and multi-color coding along with Tooltip on mouse-over. Bear in mind the last Smith Chart with a division by Abs@Zeta indicates where the increments go exponential near the 0.

A Snapshot picture for those w/o Wolfram CDF interactivity:
ZetaZeros

A ToE should…

An interesting post re:qualifications for a ToE prompted me to jot down my initial list of requirements. A ToE should inform, expand on, or rationalize, in a mathematically self consistent and rigorous way, the state of the current SM & GR confirmed experimental data:

1) Prescription (aka prediction, retrodiction or specific rationalization) for 3 generations of fundamental fermion and boson particles (and their resulting composite particles), including: charge, spin, color, mass, lifetime, branching ratio, and scattering amplitudes (aka. S-Matrix)
2) Prescription for CKM and PMNS unitary matrices and CPT conservation
3) Framework for the integration of QM and GR, including e/m, weak, strong and gravitational forces
4) Explanation for dark energy and dark matter in proportion to visible matter
5) Solution to the hierarchy problem
6) Provide a realistic computational model based on the above for the evolution of the Universe from BB to present
7) Explain an arrow of time that is consistent with GR and QM CPT conservation symmetries

Non-specific general appeals to the anthropic principle, landscapes, and/or multiverses tend to excuse or avoid prescription and thus become a benign point (or possibly even meta-physical or philosophical), such that they are not considered supportive of an actually verifiable (aka. scientific) theory.

If the theory says “we can’t know” or “we can’t measure” or “it just is that way” – it isn’t science or part of a ToE. again – my opinion and definition of “science”.

There is redundancy in this list, that is expected (even required). Of course, the beauty of the theory would be in conclusively demonstrating that throughout!

Until we can study an actual ToE that is put on the table – the list is only a guide to what might be needed. I am working on a ToE, but it doesn’t yet meet all the criteria (it’s hard work 😉

IF we have a ToE and really understand it, we should, as Feynman suggested, be able to explain it in plain language to anyone. But in the current state of physics, a completed ToE does not yet exist.

IMO, a ToE is about knowing the Universal Laws of Physics (ULPs). It isn’t, in detail, involved in knowing the Universal Initial Conditions UICs).

If you believe that the laws of “climate science” are known (don’t get me started…), then the only problem with predicting the weather is not so much about NOT knowing the laws – it is about not knowing with sufficient accuracy the initial conditions (location & momentum) of enough particles in the system. We’re missing the “butterfly flapping its wings in the Pacific” data points.

The Copenhagen interpretation of QM suggests that is impossible in principle to know any quantum system ICs (vs. the deterministic formulation of QM by DeBroglie-Bohm). Either way, my view is ToE=ULP.s w/o UICs. So prediction of all long term events specifically (like what I will think about next) is NOT the goal.

We just need enough of an idea about the UICs to initiate the computer model so it comes out close enough to get Earth like planets with weather and life forms thinking about this topic.

Easier, but NOT easy!

E8 in E6 Petrie Projection

An article (interview) with John Baez used an E8 projection which I introduced to Wikipedia in Feb of 2010 here. Technically, it is E8 projected to the E6 Coxeter plane.

E8 in E6 Petrie

The projection uses X Y basis vectors of:
X = {-Sqrt[3] + 1, 0, 1, 1, 0, 0, 0, 0};
Y = {0, Sqrt[3] – 1, -1, 1, 0, 0, 0, 0};

Resulting in vertex overlaps of:
24 Yellow with 1 overlap
24 Dark Blue each with 8 overlaps (192 vertices)
1 Light Blue with 24 overlaps (24 vertices)

After doing this for a few example symmetries, Tom took my idea of projecting higher dimensional objects to the 2D (and 3D) symmetries of lower dimensional subgroups – and ran with it in 2D – producing a ton of visualizations across WP. 🙂

It was one of those that was subsequently used that article from the 4_21 E8 WP page.

Here is a representation of E6 in the E6 Coxeter plane:
E6inE6

Resulting in vertex overlaps of:
24 Yellow with 1 overlap
24 Orange each with 2 overlaps (48 vertices)

The Sedenion “Fano Tesseract” Animation

In order to better visualize the sedenion Fano Tesseract, I’ve created an animation that steps through highlighting the vertices and edges of the triads that make up its multiplication table.

I start by showing the simpler octonion version – an 8 dimensional (8D) “Fano Cube” projection. The example octonion is the one (of 480) shown on the octonion Wikipedia article.
[wlcode]{{1, 2, 3}, {1, 4, 5}, {1, 7, 6}, {2, 4, 6}, {2, 5, 7}, {3, 4, 7}, {3, 6, 5}}[/wlcode]
octonionMulitplication-NonFlipped-164

The Fano Cube (as introduced by John Baez here) is created from the ubiquiteous “Fano Plane” by keeping the Real component (e_0) of octonions and requiring that each line of the plane be shown as a hyper-plane through e_0. Each animation step is one of 7 triads representing the triality of 3 vertices in that particular octonion multiplication table:
octonion-Fano_Cube

The 16D sedenion (one of many possible) used in this post uses the Cayley-Dickson doubling procedure on the “parent octonion” (above). The Fano Tesseract is created in the same manner as the octonion visualization, which uses the 16 vertex tesseract to show the sedenon’s 35 triads (with the triads from the “parent octonion” highlighted in red):
sedenion-triads

Wikipedia-sedenion
sedenion-Fano_Tesseract

Introducing the Sedenion Fano Tesseract Mnemonic

The Sedenion “Fano Tesseract” Mnemonic is an extension of the “Fano Cube” idea introduced by John Baez in his much cited blog post. The Fano Cube identifies each valid each triad by a hyper-plane which intersects the e_0 node.

Notice in this VisibLie_E8 output for the pane #3 “Fano Visualization Demonstration”, there are 35 sedenion triads, 7 of which are from the octonion used as an upper left quadrant base for a Cayley-Dickson doubling (highlighted in red).

The 16 vertices of the tesseract are sorted by the same “triad flattening” process used to construct a consistent Fano Plane Mnemonic for all 480 unique octonion multiplication tables.

As in the Fano Cube, the edges are highlighted in Cyan if they are selected in the n1-n3 buttons. Unlike the Fano Plane and Cube, the edges represented by the split octonion multiplication table columns/rows are not highlighted in red.

While there are 32 edges in the formal tesseract, each valid sedenion triad is identified by a hyper-plane which intersects the e_0 node, which are not necessarily those of the formal tesseract.

Here is the computation of the same sedenion table given in the Sedenion Wikipedia article as well as from this website: http://www.derivativesinvesting.net/article/307057068/a-few-hypercomplex-numbers/, which uses the harder to read IJKL style notation.
outFano