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A game you're almost certain to lose...

What are the challenges of communicating from the frontiers of mathematical research, and why should we be doing it?

Maths meets politics as early career mathematicians present their work at the Houses of Parliament.

Celebrate this year's International Women's Day with some of the articles and podcasts we have produced with women mathematicians over the last year!

How to sum an infinite series using chocolate.

I love, love. love Mobius loops and Figure-8 Klein Bottles!!! Here are the parts of 3 articles I contributed to http://vixra.org/author/rodney_bartlett which refer to the Mobius and Klein.

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Explaining the “Pioneer anomaly” by refinement of gravitational physics, and related topics ranging from cosmic to human to quantum levels

(1 paragraph)

General Relativity describes gravity and space-time partly by referring to Mercury’s motion. If Einstein’s space-time warping accurately described the motion of the Pioneer probes, we might expect it to conclude that the Pioneers’ reduction in expected distance travelled would be approx. 1 / 30,140 * instead of the actual figure of 1 / 69,032. Rounded to the nearest thousand and inverted, this means the actual warping of space-time has a value of 69 whereas General Relativity gives it a value of 30 (less than half as much – Relativity’s figure is only about 43% of the actual figure, in fact^). This can be explained by warping being based on the Mobius loop. Since one has to travel twice around a Mobius loop to arrive at the same point, the degree of warping is twice as much as it would be in Relativity. This takes us to 86% of the actual value. The foundation of 2-dimensional Mobius loops is then converted into 4-dimensional Figure-8 Klein bottles (see “Mobius loops and Klein Bottles” later in article). Bottles consist of 2 Mobius loops joined on their sides and bottles in motion make up the 3 space dimensions + 1 time dimension of each subuniverse’s space-time. Conversion from Mobius loops into Figure-8 Klein bottles takes energy and could well account for space-time’s warping rising from 86% of its actual value to 100%.

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Intergalactic Plus Time Travel, Hyperspace and Space-Time’s Nature (These Conclusions Address Unusual Things Like Dark Energy by Associating the Work of Einstein, Early Quantum Theorists, Computer Science and Isaac Newton in Unusual Ways.)

(4 separate paragraphs)

1) Discovery.com (March 18, 2010) says: "The universe is not only expanding -- it's being swept along in the direction of constellations Centaurus and Hydra at a steady clip of one million miles per hour, pulled, perhaps, by the gravity of another universe." (this is called “the dark flow”) Could this be describing evidence of an idea suggested by mathematics’ "Poincare conjecture", which has implications for the universe’s shape and says you cannot transform a doughnut shape into a sphere without ripping it. My interpretation follows: This can be viewed as subuniverses shaped like Figure-8 Klein Bottles (above; vaguely similar to doughnuts) gaining rips called wormholes when extended into the spherical spacetime that goes on forever (forming one infinite superuniverse). Picture spacetime existing on the surface of this doughnut which has rips in it. These rips provide shortcuts between points in space and time – and belong in a 5th-dimensional hyperspace. The boundary where subuniverses meet could be called a Cosmic String (the old version of cosmic strings – analogous to cracks that form when water freezes into ice - was first contemplated by the theoretical physicist Tom Kibble in the 1970s). The new cosmic strings are “cracks” in spacetime formed as subuniverses cool, are extremely thin (the diameter of a proton, or smaller), and have immense density (10^19 kg/cm, according to Penguin Encyclopedia, Edited by David Crystal – Penguin Reference Library 2006). This density would vary between any two subuniverses since it depends on the mass and energy content of the boundary regions of the two subuniverses added together, as well as movement of their boundary (the cosmic string) caused by expansion of the subuniverses – because the relativistic motion of each boundary causes enormous quantities of energy to be converted into mass.

2) For simplicity, let’s say the universe is a Mobius loop@ (a Mobius loop can be visualised as a strip of paper which is given a half-twist of 180 degrees before its ends are joined), the twisted nature of a Mobius strip or loop plus the fact that you have to travel around it twice to arrive at your starting point might substitute for the lack of overall spin. Then the cosmos could still function as a time machine. We've seen how it permits travel into the future. We can journey further and further into the future by going farther and farther around the Mobius Universe. We might travel many billions of years ahead - but when we've travelled around the Mobius Universe exactly twice, we'll find ourselves back at our start i.e. we were billions of years in the future … relative to that, we’re now billions of years in the past. (To be strictly accurate, the following references to length/width/height on a Mobius band should refer to those 3 dimensions existing in a Klein bottle, which is an emergent property of combining a couple of 2-dimensional Mobius loops.) The 3 familiar dimensions of length, width and height along, for example, the left side of a loop would have a 4th dimension (time) perpendicular to them (the twisted part at the top). And there would also exist a 5th dimension called hyperspace, at right angles to the 4th and 180 degrees from the length/width/height i.e. on the right. H-space is extended from the side along the loop’s bottom because the WMAP space probe (Wilkinson Microwave Anisotropy Probe) has determined that a very large 72% of the universe is dark energy … and we’ll see later that transmissions of binary digits from hyperspace are an interpretation of dark energy.

@ Actually, each subuniverse in the universe is a Figure-8 Klein bottle: combining 2 Mobius loops in the right way forms a Figure-8 Klein bottle.

(Couldn't insert this diagram at this website)

3) Width a is perpendicular to the length (b or e) which is perpendicular to height c. How can a line be drawn perpendicular to c without retracing b’s path? By positioning it at d, which is then parallel to (or, it could be said, at 180 degrees to) a. d (the spaceship) is already at 90 degrees to length b and height c. To be at right angles to length, width and height simultaneously; it has to also be perpendicular to (not parallel to) a. This is accomplished by a twist, like on the right side of the Mobius loop pictured above, existing in a. Then part of a is indeed at 180 degrees to d, but part of a is at 90 degrees to d. This situation requires a little flexibility or “fuzziness” which allows the numbers to deviate slightly from their precise values of 90 and 180. The fuzziness is represented in nature by past, present, future, space, time, and hyperspace existing everywhere rather than being confined to particular locations. Thus, 90+90 (the degrees between b & c added to the degrees between c & d) can equal 180, making a & d parallel. But 90+90 can also equal 90, making a & d perpendicular. (Saying 90+90=90 sounds ridiculous but it has similarities to the Matrix [of mathematics, not the action-science fiction movie] which is an array of numbers placed in rows and columns. It was worked out in the mid-nineteenth century by British mathematician Arthur Cayley, matrix mechanics is a version of quantum mechanics discovered by Werner Heisenberg in 1925, and matrices say X multiplied by Y does not always equal Y times X. In this paragraph, the first 90 plus the second 90 does not always equal the second 90 plus the first 90 because 90+90 can equal either 180 or 90.) If the infinite universe is composed of subuniverses shaped like figure-8 Klein bottles (diagram at end of paragraph - 2 Mobius loops are joined on their sides to form Bottle), in each subuniverse there would be 2 perpendicularities to the twist (one lot of 90+90, then another 90+90). 180+180 could equal 360 – represented in physics as a subuniverse, a galaxy, or one of the spherical waves above producing quantum entanglement and translating space by 90 degrees. 180+180 could also equal 180 – represented in physics by both of the above spherical waves interacting to produce inversion (translation by 180 degrees) of space which permits the spaceship to enter hyperspace. Since a fuzzily spherical figure-8 Klein bottle is necessary to form (90+90) + (90+90), any spherical or fuzzily spherical thing in this fractal universe (subuniverse, galaxy, black hole, asteroid, subatomic particle, or anything made of either fermions or bosons) would be an example of altered or warped space-time and must include hyperspace in its composition.

4) Now you see why I wrote all that stuff about hyperspace, quantum spin, matrices, Mobius loops and figure-8 Klein bottles: it was necessary to explain travel into the past.

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Whimsical, frivolous title - How ET, or an intelligent robot from the future, might interpret science and maths

More accurate title - reviewing electroweak theory and quarks via Mobius, Figure-8 Klein Bottle, Tegmark's MUH, & revisiting Einstein's G-EM relatedness

(all paragraphs continuous)

Look at the illustration below of a loop (in this case, a Mobius strip). The bottom of it looks like part of a circle while the top has a twist. This particular orientation can be referred to here as “spin 1” – it only looks the same if it’s turned round a complete revolution of 360 degrees, like the Ace of Spades card pictured in “A Brief History of Time”. A photon has spin 1 and when it interacts with a graviton in a wave packet* (gravitons have spin 2 and look the same if turned round 180 degrees or half a revolution, like the double-headed Queen of Spades in “A Brief History of Time”), the particles’ orientations can be the same i.e. they can both have their twist at the top.

* (A wave packet is a short "burst" or "envelope" of wave action that travels as a unit, and is interpreted by quantum mechanics as a probability wave describing the probability that a particle will have a given position and momentum). It acts like 2 hands coming together and catching a ball. Actually, photons are absorbed and emitted just as in laser cooling but instead of a laser beam slowing down atoms, the envelope slows (and traps) photons.

Mobius Loop

If oriented the same way, the electromagnetic and gravity waves forming the wave packets undergo constructive interference and reinforce to produce mass - a massive W+, W- or Z^0 (the carriers of the weak force) that must be turned 360 degrees to look identical i.e. they have spin 1. Slight imperfections in the way the Mobius loops fit together determine the precise nature of the binary-digit currents and therefore of exact mass or charge. If oriented dissimilarly, they undergo destructive interference and partly cancel (there’s little or no twist now – both top and bottom of the new Mobius resemble parts of a circle) to create masslessness - a massless, chargeless gluon (carrier of the strong force) that is identical if turned 360 degrees and similarly possesses spin 1. Quarks – in this interpretation, the gravitational and electromagnetic interference caused by a particular positioning of a Mobius strip - combine into protons, mesons and neutrons but are never found in isolation and cannot be observed directly. (In this explanation, the strong and weak nuclear forces have no existence independently of gravitation and electromagnetism. Since EM is modified gravitation according to this article, it’s perfectly OK to simply say “independently of gravitation”). They could simply be products of graviton-photon interaction: the strong nuclear force - which is 10^38 times gravity’s strength - could be gravity “added to” electromagnetism while the weak nuclear force – 10^25 times gravity’s strength - could be gravity “subtracted from” electromagnetism [identical to the antigravitons of antigravity being added to electromagnetism]. The 2nd example assumes combining with 100 billion antigravitons while the 1st assumes the presence of 100 gravitons per electromagnetic photon, and I believe these “assumptions” are justifiable by photon-graviton oscillation or transmutation …) An antiphoton would be formed by the fitting together of a force-carrying, spin -2 antigraviton with a spin 1 photon. (-2)+(+1) = -1. If it’s correct that “antiparticles are identical in mass to matter particles but opposite in one key property; we would expect the antiparticle of a massless, chargeless photon to have a spin of negative 1.

It has been shown how different spins can be orientations of the Mobius strip. One strip could be called the p (positive) Mobius and another the n (negative) Mobius. Both of these would be purely mathematical^ in nature (after all, a strip is only 2-dimensional and cannot substitute for the physical world). In fractal fashion, they’d be reflected in the world we know as the p-type and n-type semiconductors resulting from the appropriate doping of silicon (adding impurities to alter electrical properties). This is what you’d expect in this electronics-based universe. But as the supplementary matter shows, the strips can be joined on their edges to form a 4-dimensional Figure-8 Klein Bottle. And each Bottle is the 3 space dimensions and 1 time dimension of one of the infinite number of subuniverses making up the cosmos (and also expressing things like 5th-dimensional hyperspace, cosmic wormholes and cosmic strings).

(2 Mobius loops – each one is 2 dimensional - joined along their edges can form a 4 dimensional Klein Bottle)

^ When his paper regarding mathematical formulas creating reality was submitted to a scientific journal and rejected as being too speculative, U.S. cosmologist Max Tegmark showed the rejection letter to his friend John Wheeler (1911-2008), a Princeton theoretical physicist. Wheeler said, “Extremely speculative? Bah!” Then he reminded Tegmark that some of the original papers on quantum mechanics were also considered extremely speculative.

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