The triple Klein bottle

Special thanks to @rudimathematici

Felix Klein was a German mathematician best known for a particularsurface thet he introduced in 18882: the Klein bottle, a non-orientable surface without edge, inside, outside and no boundary (for example a sphere is an orientable surface without boundaries).

In 1995 Alan Bennett, a retired glass-blower, became interested in Klein bottles and was in a unique position to satisfy his curiosity. From simple beginnings his researches produced a variety of beautiful and mathematically sophisticated forms. New discoveries have emerged from his work which formed the inspiration for this display.

This is one of a series of glass Klein bottles made by [the artist] for the Science Museum. It consists of three Klein bottles, one inside another. In the series Alan Bennett made Klein bottles analogous to Mobius strips with odd numbers of twists greater than one.

(via archive.org)

JMP 58, 1: magnetic monopoles, spacetime and gravity

Just another selection of papers from the Journal of Mathematical Physics. I would start with the folowing paper:

Fine, D., & Sawin, S. (2017). Path integrals, supersymmetric quantum mechanics, and the Atiyah-Singer index theorem for twisted Dirac Journal of Mathematical Physics, 58 (1) DOI: 10.1063/1.4973368

Feynman’s time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure that these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of $N = 1/2$ supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.

Kováčik, S., & Prešnajder, P. (2017). Magnetic monopoles in noncommutative quantum mechanics Journal of Mathematical Physics, 58 (1) DOI: 10.1063/1.4973503

We discuss a certain generalization of the Hilbert space of states in noncommutative quantum mechanics that, as we show, introduces magnetic monopoles into the theory. Such generalization arises very naturally in the considered model, but can be easily reproduced in ordinary quantum mechanics as well. This approach offers a different viewpoint on the Dirac quantization condition and other important relations for magnetic monopoles. We focus mostly on the kinematic structure of the theory, but investigate also a dynamical problem (with the Coulomb potential).

A flat torus in three dimensional space

Vincent Borrelli, Said Jabrane, Francis Lazarus and Boris Thibert realize for the first time the image of a 3d flat torus. But, what is a flat torus? In a mathematical point of view is a torus without gaussian curvature everywhere. For example: in a 3d space, you can bend a sheet of paper in a cylinder, but you cannot embed it in a torus without stretching the paper itself. But, if you live in a 2d space, a flat torus is a surface like the pacman labirint: you can enter in the right side of the paper and exit behind you from the left side:

We can define a flat torus starting from a couple of real numbers $u$, $v$ such that $u, v \in ]0, 2 \pi]$⁽¹⁾: \[(u, v) \rightarrow \frac{1}{\sqrt{2}} \left ( \cos (u+v), \sin (u+v), \cos (u-v), \sin (u-v) \right )\] The following is the best visualization of a flat torus before Borelli-Jabrane-Lazarus-Thibert:

The mathematical problem around the flat torus was challenged by Nash and Kuiper in 1950s:

Nash and Kuiper proved the existence of a representation that does not perturb the lenghts in the square flat torus. For a long time, this existence remained a challenge for the imagination of mathematicians. But proving and showing should sometimes be clearly distinguished in Mathematics. This is well explained by the thief allegory: Let us assume that a group of people is gathered around a jewel in a closed room. Let us further suppose that the light is turned off for a moment and that the jewel has disappeared when the light is again turned on. We then have the proof that a robber is hiding among the attendance but he can not be exhibited. Although the proofs of Nash and Kuiper are much more than an «existential» trap, their proofs do not provide a sufficiently explicit procedure that would allow for visualization or simply for a mental picture of a square flat torus.⁽²⁾

Between 70s and 80s the Abel Prize Gromov extracted a method from the work of Nash and Kuiper, proposing the so called convex integration, a very useful tool. Indeed it

(...) does not only yield the existence of a solution, it also provides us with an effective construction.⁽²⁾

Starting from this method, Borelli, Jabrane, Lazarus and Thibert realize an algorithm in order to picture the flat torus.

Mathematicians were puzzled by the works of Nash and Kuiper. These works could indeed prove the existence of objects whose regularity was problematic, if not paradoxical. They had to be smooth and rough at the same time... In effect, the mathematical analysis of the images reveals a surface belonging to two antagonist worlds; the smooth surfaces and the fractals,infinitely broken. When zooming in, we invariably observe ripples at smaller and smaller scales. Each ripple - called a corrugation – appears smooth when viewed alone, but the accumulation of those creates an object with a rough and fractal aspect.⁽²⁾

And in conclusion

Demonstrating that convex integration can be implemented open new perspectives in applied mathematics, notably for solving differential systems originating from Physics and Biology.
More specifically, our images reveal a class of objects whose structure lies inbetween smooth surfaces and fractals. Such objects could play a central rôle for shape analysis. They could also resolve some unexplained paradoxes.⁽²⁾

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Read also Wikipedia | phys.org

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(1) The Flat Torus in the Three-Sphere
(2) A flat torus in three dimensional space (pdf) | Hevea Project

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Borrelli, V., Jabrane, S., Lazarus, F., & Thibert, B. (2012). Flat tori in three-dimensional space and convex integration Proceedings of the National Academy of Sciences DOI: 10.1073/pnas.1118478109

Lovecraft's mathematical horrors

Sometimes we can find on the web something of interesting, like the following review of the 4D Man.
We can read on Wikipedia:

Brilliant but irresponsible scientist Tony Nelson (James Congdon) develops an amplifier that allows any object to achieve a 4th dimensional (4D) state. While in this state that object can pass freely through any other object.

Reading these words I immediatly think to Howard Philips Lovecraft and his Cthulhu Mythos, in particular to Dream in the Witch House. In this short story Walter Gilman, a student of mathematics, lives in the house of Keziah Mason, one of the Salem's witches. In the story there are some mathematically interesting quotes:

She had told Judge Hathorne of lines and curves that could be made to point out directions leading through the walls of space to other spaces beyond (...)

We can argue the Lovecraft's use for his purpouse of the non-euclidean geometry, in particular in the following quotation:

[Gilman] wanted to be in the building where some circumstance had more or less suddenly given a mediocre old woman of the Seventeenth Century an insight into mathematical depths perhaps beyond the utmost modern delvings of Planck, Heisenberg, Einstein, and de Sitter.

or in the following point, in which HPL seems refer to Riemann's hypotesys:

He was getting an intuitive knack for solving Riemannian equations, and astonished Professor Upham by his comprehension of fourth-dimensional and other problems (...)

Indeed Gilman was studying

non-Euclidean calculus and quantum physics

Illustration by Greg Nemec

And Walter, dreaming, has experienced the high dimensional space of the limitless abysses:

abysses whose material and gravitational properties, and whose relation to his own entity, he could not even begin to explain. He did not walk or climb, fly or swim, crawl or wriggle; yet always experienced a mode of motion partly voluntary and partly involuntary. Of his own condition he could not well judge, for sight of his arms, legs, and torso seemed always cut off by some odd disarrangement of perspective; (...)

Durign his travel in the fourth-dimension, Gilman seen

risms, labyrinths, clusters of cubes and planes, and Cyclopean buildings

that are characteristic in lovecraftian literature.
Another non-euclidean reference is in The Call of Cthulhu⁽¹⁾:

He said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours.

And Cthulhu itself is a fourth dimensional creature. Cthulhu was one of the Great Old Ones: these creatures

(...) were not composed altogether of flesh and blood. They had shape (...) but that shape was not made of matter.

We can imagine Cthulhu in our world like the projection of a dodecaplex in a three dimensional space, for example: