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Möbius Strip Latched Left
Using Macsyma - by Doug Renselle
Using parametric equations provided by Paul Bourke:
x = cos(s) - t*cos(s/2) + cos(s) (-t left)
y = sin(s) + t*cos(s/2) + sin(s)
z = t*sin(s/2)

Here's a MAC OS X Grapher example which only took a few minutes to do:


Doug took above parametric equations and made a new graphic using MAC OS X's 10.4 version of (a MAC 'utility') to do above art.

Here are modified parametrics for Grapher which you can retype into a 3D 'New' Grapher file:

Cut and paste piecemeal:

cosv - tcosv + cosv
sinv + tcosv + sinv

t=0...0.25p, v=0...2p

Grapher animates above in a single plane of rotation. Too, you can control rotation rate. And you can grab its cubic frame and move and hold it to any position.

Have some fun playing with omniffering plus and minus signs and p multipliers.

If you like this work, you will love Grapher! It blows other graphic tools to smithereens.

Doug - 11May2007.

"But Doug, that isn't a Möbius strip! It has two sides and two edges!"

OK, OK, try this:

Doug left out 0.5 multipliers on middle column trig arguments.

Note that Grapher doesn't close rotated ends as it should in this case. But it shows us how to assemble a Möbius strip, doesn't it?

Doug - 12May2007.

Erwin Schrödinger's
Double Möbius Tao Helix
Hydrogen Atom Sketch
From his
Notebook N1

See Walter Moore's Schrödinger, p. 193.
CUP USA digital reprint, paperback, 2001.

We want to herald, here, Doug's latest Tao to Möbius to Tao graphic transmutation ontology based upon Schrödinger's hydrogen atom wave function Lissajous Tao shown above.

We have been working on this topological ontology for some time:

We are delighted to be able to offer it here for you to fathom on our popular Möbius Left web page.

Diligent students of Quantonics may notice nexi to not only string theory, but now also to both QED and QCD. You may wish to see how these tie together with Quantonics' latest innovation: fuzzons. Read about it and see comprehensive graphics in our June, 2004 News. Enjoy!

Also, click on graphic to see our more recent fuzzon to fermion ontology AKA "fermionta" which shows how above memeotics may form from more primitive Quantonics' attractor (interrelationships) we call "fuzzons." See QLO and peaqlo.

Please ponder similitude and homomorphism of our two green ellipses above and our two string theory garden hose universe ellipses below, those ones we cut to make a Möbius similar to one shown above. Also ponder sophist, recursive, fractal, self~referent hermaphroditism of Schrödinger's wave and its modeling of a hydrogen atom. See Planaria for a biological exemplar.

What is unique about Möbius strip topology?
A Möbius strip is a quanton.
It can be used to model physical world quantons.
A Möbius strip:

      • has one edge
      • has one surface
      • may be latched left or right (What has this to do with chiralty? I.e., visualize ± ½ fermionic spin. What has this to do with The Riemann Hypothesis? See our Riemann Quanton. Also see our John Nash's Quantum Riemann Hypothesis. And see our 7Jun2002 Möbius 3-Primæ Fermion.)
      • may be formed from a 2-surface, 4-edge plane rectangle
      • as a quanton, it unifies 2 surfaces; if you take a long, narrow strip of paper and write 'particle' ('object') on one side and 'wave' ('subject') on the other; then connect both ends of paper to make a squat cylinder; then rotate one end 180 degrees; then staple or paste both ends together; you have a wave-particle quanton! I.e., quanton(wave,particle)!
      • as a quanton, a Möbius strip offers us transform modeling: try above experiment but instead of 180 degrees, try 360 degrees — what happens? are there other ways to achieve this? what do you see? if you see a bow tie with one lobe facing toward you and other facing away, try morphing your new model, gently — persist! try to get a bow tie both of whose lobes face toward you — now can you show two quasi-cylinders whose edges interfere? this model can produce both a bow tie with one lobe facing away and one facing toward you and a bow tie with both lobes facing toward you — you need latter to achieve next transform to quasi-double-cylinders — think of one cylinder as |0>, other as |1>! what is this quantonic creature modeling? a fermion? or a boson?

        (Yes, we realize our paper model 'physically' IS a fermionic system, but we are trying to show you that it may be able to model aspects of both bosons and fermions! Wouldn't that be handy? Wouldn't that help us and Dr. Stein to develop a new "exegetic and exoteric" quantum ontology?)

        (On exegetics and exoterics: Richard Feynman said "No one understands Quantum Mechanics." Our effort here is to commence a relatively simple Möbius modeling effort which may mitigate Feynman's comment somewhat. We seek a vector toward a simpler way of disclosing (omnisclosing) a new quantum ontology to interested lay folk. Physicists currently perform most experiments at subatomic, atomic, or mesoatomic levels of reality. Our attempt here is to show some much simpler possibilities of tabletop 'scissors and paper' modeling capabilities.)

      • might be capable of modeling aspects of both fermions and bosons!
      • might be capable of modeling aspects of interrelationships with quantum flux!
      • offers us a chance to try to see how a SOMite might perceive a Möbius strip as a sophism
        • four edges of a plane rectangle vis-à-vis unit edge Möbius
        • either/or of a plane rectangle vis-à-vis both/and unit surface Möbius
        • is a 'boson' a kind of quantum sophism? is a fermion? why? why n¤t?
      • try to visualize how a Möbius strip shows quantum c¤mplementarity
      • try to imagine how a Möbius strip unifies SOM opposites
      • notice a quanton is not substance; a quanton is flux!
      • our little Möbius strip models flux creating a 3d 'quantum object'
      • (this whole section, or main bullet, is new as of 6Aug2000) ask yourself, "How might Möbius strips, and transformations of them, model aspects of fermions and bosons?"
        • in our discussion (omniscussion) here, assume no Bose-Einstein Condensation in our Möbius (and transformed) models; we assume however, that we can achieve macroscopic ~bosonic paper models.
        • keep in mind that in our Quantonics' model of quantum reality, we assume all bosons and fermions quantum flux or Vacuum Energy Space as quantons.

          Commence Möbius modeling Q&A session:
        • in actual reality, do you agree a paper model of a Möbius strip is a fermionic system?
        • can aggregate fermionic systems model aspects of bosonic systems?
        • are fermionic systems purely fermionic, i.e. aggregate modular 1/2 integer spin, or do they have some zero spin states?
        • can we use Möbius strips to model both fermions and bosons?
        • do Möbius strips show both rotational symmetry and n¤nsymmetry?
        • does a Möbius strip show 360 degree symmetry?
        • does a Möbius strip show 720 degree symmetry?

          To assist your visualization of what we mean here, take a look at our Quantum Stairs Möbius strip graphic. Can you see how we drew that Möbius strip as a 720 degree n¤nsymmetric spiral? Then we filled it in (rendered it) to make it appear as a 3D Möbius both/and included-middle c¤mplementary stairs modeling of quantum reality. Here is our line artwork for that graphic with start and stop points pulled away to help you see a 720 degree loop (try to imagine a Philippine wine dancer's hand rotation):

          We dragged our inner start point slightly rightward, and our outer stop point slightly leftward. Try drawing a Möbius strip yourself using this 2D artwork technique. It is fun, and opens a whole new realm (many quantum tells) for understanding quantum reality.

          We have found relevant analogies in string theory and want to offer them here. First, permit us to offer a caveat: we are n¤t experts in string theory. We are just now learning both QED and QCD and string theory has entered as another avenue of our quantum research, which we view as our personal learning adventurings, portions of which we share with you here in Quantonics. That said...

          If you are familiar with Brian Greene's The Elegant Universe, 1999, Vintage, you may wish to take a look at his figures in Chapter 10, 'Quantum Geometry.' Brian offers no index on Möbius (though he does offer several index items on rotational nonsymmetry which you may recall relates Feynman's quantum "wobble" and thus 1/2 spin fermionic asymmetrical rotations), so we assume that our work here extends his and may be of Value to researchers in string theory. Also, we are somewhat concerned that string theory is a kind of field theory and suffers from some (both field- and string-theory) problematics which are incompatible quantum reality. As an example, Greene's Ch. 10 title would be an oxymoron in quantum reality. Why? Quantum reality is (appears to be) Bohmian n¤n mechanical. Thus n¤n geometrical. As Henri Louis Bergson might say, "Quantum reality is more qualitative than quantitative." (Of course, many of us suspected that all along...J)

          Here is our rendition of a figure from Greene's book, simplified for our needs:

          We see a cylinder or portion of a string-theoretical "garden hose universe" with two "wrapped" single strings 'on' it and another, intriguing from our perspective, "double wrapped" string. What is that double string? Does it look, somehow, familiar? Do you recognize it?

          One clue is that those single strings are 360° loops vis-à-vis that double loop string is 720°! How do we make that double loop? In a manner very similar to our little double loop above, with its 'ends' pulled apart. What does that tell us? That, in some cases, string theory as shown above, 'creates' Möbius strips! And that is very Good. It says that string theory has a way of representing 1/2 spin fermions. Let's show a graphic of this, without dotted lines, and proceed in evolutionary, quasi-ontic, incremental fecundations:

          First notice how this simple pair of circles appears as an illusion. You can see two circles angled left, and two circles angled right. And you can see quantum reality's included-middle of two ellipses in 2D overlapping one another.


          In our quanton script we show, explicitly — we seldom do this, our two fluxors' included-middle as actual. It's like showing a silhouette line separating Gestalt figure and ground. Most often this included-middle is transparent, cloaked from observation, unseeable. You can observe here an extremely rare trichonic quanton. It has enormous pedagogic Value...

          One example of extreme added pedagogic Value is how we may view our quantum comma-no-space as an ensemble of heterogeneous Quantonic Interrelationships (e.g., a quantum-ensemble of Gestalt silhouettes). See this note in our quantum-subjective Hamiltonian quaternion web page.

          More Value... another nexus we can make here is one directing (omnirecting) us to what quantum physicists call "Bell inequalities." Our illusion expresses quite nicely and graphically "Bell inequalities." Here is another, indirect quantum interrelationship with Möbius strips! See our Bell Theorem Study, especially red text recently added near page top. Too, it offers another quantum epiphany: all quantons, due their quantum-included middlings, are Bell inequalities! Quantum reality is Bell inequalities! Quantum reality is quantonic. From a classical perspective what we just wrote is blatant prevarication and equivocation. But that is quantum reality. Classicists, like Albert Einstein and Richard Feynman, have been calling quantum reality "absurd" for a long time.

          In that simple, yet notably pedantic and quantum telling graphic, if we added dotted lines and two short straight lines top and bottom you would see a cylindrical segment similar to one at left in our previous "garden hose universe" graphic.

          Then you might recall how you constructed your first Möbius strip by making a similar cylinder of a long strip of paper and then rotating one end to form your Möbius. String theory's classical double loop transformation (quantumly, we prefer emerscenture and transemerqancy) accomplishes a similar feat, as we shall show. Instead of one double loop with a single cut, we make two cuts, one on each of our two circles, like this:

          Next, we will pull our left circle's right end to right and connect it where right circle's right end was attached and vice versa for our right circle, like this:

          Voilà! A Möbius strip, unrendered. Notice how it retains its illusory nature. We see here a Möbius strip as a quantum sophism. Left chiralty, included-middle, right chiralty — all in a single line drawing! Students please observe that there are countless ways to draw this strip. Greene's book uses a more isometric perspective which offers a very nice outcome. If you have 3D software, you can do these pixes even better than Doug has. We are using Illustrator 7, which is a very limited 2D application. We can do this in PovRay, but it is not object-oriented at a graphical level.



          I.e., PovRay is object-oriented at a scripting level; it takes a tad more effort to write scripts, but it is really good experience for you to learn how, plus PovRay is freeware and available for both Win$Tel and MAC; we show $ in that one case because acquisition costs are low but far exceeded by installation, operation, usability, compatibility, productivity, maintenance, down time, ubiquitous covert channel (lack of immunity to formal software viruses, worms, prions, phages, bacteria, fungi, even symbionts (a huge threat to formal systems when applied maliciously), etc.), and retirement-recycling costs.

          End aside.

          For more on this quantum Möbius sophism and how it happens from a Quantonics perspective see our Quantum Stairs.

          Before we proceed, let's pause for another avocative aside.



          Quantum reality is energy, energy of abs¤lutely anihmatæ EIMA quantum flux.

          How can we energize our double-wrapped Möbius loops? Try this graphic for a starter.

          We use arrows to show direction (omnirection) of flux in an unrendered quantum Möbius.

          Remember a strange quantum-classical dilemma (omnilemma): classical energy is proportional to amplitude-area, but quantum energy is proportional to flux rate (irrelevant to waveform amplitude).

          So quantum energy doesn't care how big those quasi-circle's Möbius diameters (omniameters) are, only how fast those arrows are whizzing by, and whether their emerqancy is Mobius and thus 1/2 spin, wobbling, fermionic. When those rates are (when quantum energy is) so high that they are exceedingly above our shasbs (standard human accoutrement-assisted sensory bandwidths), they appear substantially, materially objective to our senses. We call it "decoherent" quantum reality. It is posentropic reality.

          J. C. Maxwell thought posentropic reality was all there is to reality and thought posentropy only had one gradient

          Long parenthetical:

          (positive; this is n¤n intuitive; think about it for awhile; increasing posentropy is supposedly increasing classical 'disorder;' any decrease in posentropy is an increase in classical 'order'; of course by observation biological systems are all quantum both-all-while-and-many


          Read Prigogine and Stenger's Order Out of Chaos — note that Ilya Prigogine transitioned mid 2nd quarter 2003; he contributed enormously to n¤vel and innovative sciences and Earth's societies)

          End long parenthetical.

          and thus our universe would eventually suffer an ultimate and final Maxwellian heat death. Wrong!!! Classical scientists refer this 'law' as "J. C. Maxwell's 2nd 'law' of thermodynamics." You may intuit this classically as:

          one_universal_life = dichon(alpha, omega).

          Countless Earth-folk still buy into this fundamentalist Babel.

          Looking at end of our garden hose we can see, perhaps, one circle with arrows all pointing in same loop direction (chiralty matters), even though in this example we know there are two entangled rotationally n¤n symmetric loops

          End aside.

          Rendered (surface filled), it looks a tad like this:

          Now it really looks like a Möbius, right? So we see a strong Möbius nexus (but, potentially due our own ignorance, an ostensibly unrecognized, i..e., Greene doesn't mention it in his TEU index, nexus) with string theory. However, you may recall our (a) problematic (there are many) with string theory. In a genuine Möbius strip its crossing 'point' is n¤t a point. If we were to walk around it to its right we would see our strip's width, behind that crossover, gradually appear. But in string theory, using a model similar to our original string theory graphic above, that crossing IS a point! A real Möbius strip may be thought of as possessing real quantum-arbitrary distribution (omnistribution) in omnispace. If you have progressed well in your studies of Quantonics, you may see how that imposition of a point crossing is a subtle manifestation of classical reality's mechanical and analytic excluded-middle. No self-respecting classical 'point' has arbitrary spatial distribution (omnistribution)! (Big but: Quantonics fuzzons do have arbitrary, spatially probabilistic distributions (we should say "likelihood omnistributions" since quantum reality tends more qualitatively toward a priori vis-à-vis a posteriori; see our 2004 What is Wrong with Probability as Value?)!) Also worry about how string theory itself denies existence of any 'points,' yet it claims any surface may have no thickness, and thus be two dimensional. J

          Here is our Quantonics version of that Möbius as a Bergsonian Durational Möbius:

          Is this a complex quantum analogue of our simple string theoretic 720° wrapped string? Our vote is "Yes!" Notice its intrinsic quantum animacy. Notice its (up to) Planck rate quantization. Notice how 1/2 fermionic spin "wobbles." Notice how "nextings" heterogeneously wobble. So, big one! Decoherent change wobbles! Heterogeneous fermionic timings wobble! Heterogeneous fermionic massings wobble! Heterogeneous fermionic spacings wobble! Heterogeneous fermionic gravityings wobble! Wow! Is that an "And what?" Our glee is going X PON ENtially...HY PER BOLeeee... J

          Let's look at another graphic which explicitly demonstrates a need for our cylinder to offer potential for arbitrary, omnispatial Bergsonian durational distribution:

          We offer students a template to make this quanton. Use an 8.5 x 11 sheet of paper. Draw that bifurcated H on sheet of paper. Use a knife or Xacto to cut along those thin lines.

          You may form a Möbius of that H either before or after you roll up your 8.5 x 11 sheet into a cylinder. Take some timings to ponder quantum philosophical emerscences of this model. They are n¤n shallow. Our purpose, though, is to allow you to see how a genuine Möbius strip emerges from string theory's double-loop wrapped string model. It is a natural byproduct that we also expose string theory's soft underbelly. Have fun. Doug - 6-18Aug2003.
        • do bosons wobble? do systems of bosons wobble?
        • do fermions wobble? do systems of fermions wobble?
        • does a Möbius strip wobble or not? (SOM question!)
          • what can we do to see if Möbius strips wobble or not?
        • does a Möbius strip both wobble and n¤t wobble? (MoQ question!)
          • what can we do to see if Möbius strips both wobble and do n¤t wobble?
        • how might bosonic rotational symmetry and fermionic rotational nonsymmetry be perceived as Möbius or transformed Möbius interrelationships with quantum flux?
        • how might our bar stool/lazy Susan experiment at our wobble link exemplify quantum flux interrelationships?
        • is it worthy of your attention that Möbius strips latch both left and right, and our bar stool experiment demonstrates left (say CCW) and right (say CW) interrelationships?
        • does this have anything to do with chiralty?
        • let's assume by now you know how to recognize a boson modeled using a paper Möbius or Möbius-transformed model:
          • can we locate alternating kets |0> with certainty?
          • can we locate alternating kets |1> with certainty?
          • can we locate them with certainty before transformation?
          • how can we show this with a paper Möbius model or one of its transforms?
          • what does this show us re: Heisenberg's uncertainty interrelationships?
          • can we infer anything general from |0> locations, re: zeroness?
            • is 'zero' physical?
            • philosophical ramifications?
            • mathematical ramifications?
          • can we infer anything general from |1> locations, re: oneness?
            • is 'one' physical?
            • philosophical ramifications?
            • mathematical ramifications?
      • general Möbius transformations are possible; see Eric W. Weisstein's Concise Encyclopedia of Mathematics online; or purchase it online published by CRC Press.


To contact Quantonics write to or call:

Doug Renselle
Quantonics, Inc.
Suite 18, #368 1950 East Greyhound Pass
Carmel, INdiana 46033-7730

©Quantonics, Inc., 1999-2010 — Rev. 28Jul2008  PDR — Created  17Aug1999  PDR
(6Aug2000 rev - Add bullets on Möbius strip and its transformations as fermions and bosons.)
(14Sep2000 rev - Correct link to Eric W. Weisstein's site.)
(5Oct2000 rev - Minor semantic repairs.)
(5Dec2001 rev - Add top of page frame-breaker.)
(14Feb2002 rev - Add 720 degree Möbius line artwork technique. Minor 'o' to '¤' quantum comtextual remediation.)
(5Jun2002 rev - Add link back to our heuristics on Nash's Quantum Riemann Hypothesis.)
(7Jun2002 rev - Add link to our 7Jun2002 Möbius 3-Primæ Fermion.)
(3Jul2002 rev - Minor text edits.)
(21Jul2002 rev - Change QELR links to A-Z pages.)
(6Aug2003 rev - All old red text to black. Add string theory Möbius memes.)
(8-11Aug2003 rev - Repair typos.)
(15Aug2003 rev - Extend list of Win$Tel costs of ownership.)
(16-18Aug2003 rev - Add energy and Bergsonian Durational Möbius graphics.)
(25Nov2003 rev - Repair Nash RH link near page top.)
(14Jan2004 rev - Typo in red text: "See see.")
(4-5Jul2004 rev - Add Schrödinger Möbius to Tao Möbius graphics ontology near page top. Reset legacy red text.)
(15Jul2004 rev - Add 'interrelate' synonyms pull-down menu. Add more red text comments.)
(6Sep2004 rev - Add link and red text box to and describing fermionta re: Tao~Möbius ontologies.)
(18Oct2004 rev - Reset red text and correct a typo.)
(13Jan2006 rev - Update 'interrelates' pull-down menu with 'phase~encodes holographically.')
(14Jun2006 rev - Add 'Mobius Ontology' anchor.)
(11-12May2007 rev - Add a MAC OS X Grapher version of our Möbius. Add fix.)
(5Jun2007 rev - Typo.)
(28Jul2008 rev - Reformat. Reset legacy red text markups.)

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