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| {% include list.liquid all=true %} | ||
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| ## Mobius Strip | ||
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| There are some mathematical shape of this residual objects. [Torus](https://gist.github.com/eq19/e9832026b5b78f694e4ad22c3eb6c3ef#the-crank-of-a-partition) is basically a donut shape, which has the property of of having variable Gaussian curvature. | ||
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| ```note | ||
| The blue parts of the torus above have positive curvature, the red parts negative and the top grey band has zero curvature. If our 3 dimensional space was like the surface areas of a 4 dimensional torus, the parts would have different angle sums. | ||
| ``` | ||
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| [](https://gist.github.com/eq19/e9832026b5b78f694e4ad22c3eb6c3ef#the-crank-of-a-partition) | ||
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| Some parts of the surface has positive curvature, others zero, others negative. | ||
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| If you start anywhere on its surface and follow the curvature round you will eventually return to the same place having travelled on every part of the surface. | ||
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| Mobius strip only has one side, there are two more bizarre shapes with strange properties. | ||
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| ## The Klein bottle | ||
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| The Klein bottleis in someways a 3D version of the Mobius strip and even though it exists in 3 dimensions, to make a true one you need to “fold through” the 4th dimension. | ||
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| ```note | ||
| In [mathematics](https://en.wikipedia.org/wiki/Mathematics), the Klein bottle ([/ˈklaɪn/](https://en.wikipedia.org/wiki/Help:IPA/English)) is an example of a [non-orientable](https://en.wikipedia.org/wiki/Orientability) [surface](https://en.wikipedia.org/wiki/Surface_(topology)); that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. | ||
| - More formally, the Klein bottle is a [two-dimensional](https://en.wikipedia.org/wiki/Two-dimensional) [manifold](https://en.wikipedia.org/wiki/Manifold) on which one cannot define a [normal vector](https://en.wikipedia.org/wiki/Normal_vector) at each point that varies [continuously](https://en.wikipedia.org/wiki/Continuous_function) over the whole manifold. | ||
| - Other related non-orientable surfaces include the [Möbius strip](https://en.wikipedia.org/wiki/M%C3%B6bius_strip) and the [real projective plane](https://en.wikipedia.org/wiki/Real_projective_plane). | ||
| While a Möbius strip is a surface with a [boundary](https://en.wikipedia.org/wiki/Boundary_(topology)), a Klein bottle has no boundary. For comparison, a [sphere](https://en.wikipedia.org/wiki/Sphere) is an orientable surface with no boundary. | ||
| ``` | ||
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| [](https://en.wikipedia.org/wiki/Klein_bottle) | ||
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| [](https://ibmathsresources.com/2014/08/05/non-euclidean-geometry-v-theshapeoftheuniverse/) | ||
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| A sign inversion visualized as a vector pointing along the [Möbius band](https://en.wikipedia.org/wiki/M%C3%B6bius_band) when the circle is continuously rotated through a full turn of 360°. | ||
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| ## The Spinors | ||
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| A spinor associated to the conformal group of the circle, exhibiting a sign inversion on a full rotation of the circle through an angle of 2π. | ||
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| ***(17+13) + (11+19) = (7+11) + (19+23) = 60*** | ||
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| [](https://en.wikipedia.org/wiki/Dirac_spinor#Four-spinor_for_particles) | ||
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| [](https://youtu.be/4NJBvkjpC3E) | ||
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| ```note | ||
| Eigennvalue curves (right) showing a triple eigenvalue at zero for τ = 1 and double eigenvalues at 1 ± √2i for τ = √43. On the left the graph of 1/|Q(λ)| with the same eigenvalue curves plotted in the ground plane. Green stars indicate the eigenvalues of A, blue stars the roots of puv(λ) and triangles the zeroes of Q0 | ||
| (λ) | ||
| ``` | ||
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| ## Global Properties | ||
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| ***7 + 11 + 13 = 31*** | ||
| ***1 + (26+6) + (27+6) = 66*** | ||
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| [](https://www.hexspin.com/0-1-and-negative-numbers/) | ||
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| ```txt | ||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | ||
| ---+---+---+---+---+---+---+---+---+---+----+----+----+----+----+----+----+---- | ||
| - | - | 20| 21| 22| 23| 24| 25| | ||
| ---+---+---+---+---+---+---+---+ | ||
| - | - | - | - | 28| 29| ◄--- missing 26 & 27 ✔️ | ||
| ---+---+---+---+---+---+ | ||
| 30| 31| - | - | ◄--- missing 32 & 33 ✔️ | ||
| ---+---+---+---+ | ||
| 36| | ||
| ``` | ||
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| ```tip | ||
| This behaviour finaly brings us to a suggestion that the dimension in string theory are linked with ***the prime distribution level*** as indicated by the _[self repetition](https://www.eq19.com/exponentiation/#self-repetition)_ on MEC30. | ||
| ``` | ||
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| ***7th spin - 4th spin = (168 - 102)s = 66s = 6 x 11s = 30s + 36s*** | ||
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| ```txt | ||
| $True Prime Pairs: | ||
| (5,7), (11,13), (17,19) | ||
| layer | node | sub | i | f. MEC 30 / 2 | ||
| ------+------+-----+-----+------ ‹------------------------------ 0 {-1/2} | ||
| | | | 1 | -------------------------- | ||
| | | 1 +-----+ | | ||
| | 1 | | 2 | (5) | | ||
| | |-----+-----+ | | ||
| | | | 3 | | | ||
| 1 +------+ 2 +-----+---- | | ||
| | | | 4 | | | ||
| | +-----+-----+ | | ||
| | 2 | | 5 | (7) | | ||
| | | 3 +-----+ | | ||
| | | | 6 | 11s ‹-- ∆28 = (71-43) √ | ||
| ------+------+-----+-----+------ } (36) | | ||
| | | | 7 | | | ||
| | | 4 +-----+ | | ||
| | 3 | | 8 | (11) | | ||
| | +-----+-----+ | | ||
| | | | 9 |‹-- ∆9 = (89-71) / 2 √ | | ||
| 2 +------| 5* +-----+----- | | ||
| | | | 10 | | | ||
| | |-----+-----+ | | ||
| | 4 | | 11 | (13) --------------------- | ||
| | | 6 +-----+ ‹------------------------------ 15 {0} | ||
| | | | 12 |--------------------------- | ||
| ------+------+-----+-----+------------ | | ||
| | | | 13 | | | ||
| | | 7 +-----+ | | ||
| | 5 | | 14 | (17) | | ||
| | |-----+-----+ | | ||
| | | | 15 | 7 x 24 = 168 √ | ||
| 3* +------+ 8 +-----+----- } (36) | | ||
| | | | 16 | | | ||
| | |-----+-----+ | | ||
| | 6 | | 17 | (19) | | ||
| | | 9 +-----+ | | ||
| | | | 18 | -------------------------- | ||
| ------|------|-----+-----+----- ‹----------------------------------- 30 {+1/2} | ||
| ``` | ||
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| This model may explains the newly discovered prime number theorem in relatively simple layman's terms for anyone with a slight background in theoretical physics. | ||
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| ```note | ||
| The property gives an in depth analysis of the not so random distribution of primes by showing how it has solved Goldbach's conjecture and the Ulam spiral. | ||
| ``` | ||
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| The model suggests a possible origin for both charge and half-integer spin and also reconciles the apparently contradictory criteria discussed above. | ||
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| ```note | ||
| ***Arbitrary sequence of three (3) consecutive nucleotides*** along a helical path whose metric distances satisfy the relationship dn,n+3dn,n+2dn,n+1. | ||
| - Sketch showing a characteristic duplex DNA helical standing-wave pattern. | ||
| - The vertical lines depict the cross-section projections of each bp along the helix axis, their length providing a measure of their twist magnitude. | ||
| - Thick lines represent the sugar-phosphate profile. | ||
| Optimally overlapping bps are indicated by the presence of the ovals (m) measures the overlapping resonance correlation length. _([π − π orbital resonance in twisting duplex DNA](https://github.com/eq19/eq19.github.io/files/13790206/prb_Hx2.pdf))_ | ||
| ``` | ||
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| [](https://github.com/eq19/eq19.github.io/files/13790206/prb_Hx2.pdf) | ||
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| Under certain conditions, energy could not take on any indiscriminate value, the energy must be some multiple of a very small quantity (later to be known as a ***quantum***). | ||
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| ```note | ||
| Twisted strip model for one wavelength of a photon with circular polarisation in at space. A similar photon in a closed path in curved space with periodic boundary conditions of length C. | ||
| - The B-fi eld is in the plane of the strip and the E-field is perpendicular to it (a). | ||
| - The E-fi eld vector is radial and directed inwards, and the B-fi eld is vertical (b). | ||
| The magnetic moment ~, angular momentum L~, and direction of propagation with velocity c are also indicated. _([Is the electron a photon with toroidal topology? - pdf](https://github.com/eq19/eq19.github.io/files/13790325/LdBelectoroid.pdf))_ | ||
| ``` | ||
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| [](https://github.com/eq19/eq19.github.io/files/13790325/LdBelectoroid.pdf) | ||
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| A deeper understanding requires a uni cation of the aspects discussed above in terms of an underlying principle. |
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