In "Who's Afraid of a Big Black Hole?" (BBC program – Dec. 8, 2009), physicist Michio Kaku showed how Einstein divided 1 by 0. The answer in what I call Newtonian mathematics is infinity but the answer in what I call relativistic mathematics would refer to all space-time i.e. the past, present and future + the whole observed and unobserved universe.  (This article is adapted from portions of the Kindle Edition of my book “Tomorrow’s Science Today” on Amazon.com)

Part 1: 1+1=2 but they also equal 1 sometimes

Most people disliked mathematics when they were at school and they were absolutely correct to do so. This is because maths as we know it is severely incomplete. No matter how elaborated and complicated mathematical equations become, in today's world they're based on 1+1=2. This certainly conforms to the world our physical senses perceive and to the world scientific instruments detect. It has been of immeasurable value to all knowledge throughout history and has elevated science to the lofty status it enjoys. Science is now striving towards Unification - where the subatomic realm, all matter, energy, forces, space and time will be seen as entangled parts of one universe. While 1+1=2 has been vital in getting humanity to this point, it's time to suppress our attachments to the past and realize that whereas 1+1 will always equal 2, it's also capable of equalling the 1 which represents unification.

Saying 1+1=1 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.

Science is, in part, in a mess today. It has no understanding at all of more than 90% of the universe. No wonder it's hoping another Isaac Newton or Albert Einstein will come along. But the reality is that it has no need whatever for people such as these. It merely needs to comprehend that a new day has dawned - the times they are a-changing - and 1+1 really does equal 1 as well as 2. Concepts of familiar maths e.g. Mobius loop, figure-8 Klein bottle, E=mc^2 and the fixed-point theorem) are used here but so is "unification maths" (also known as relativistic maths, after Einstein's Theory of Relativity).

Einstein always maintained quantum mechanics was incomplete and could only be properly described using "hidden variables". The "hidden variables" interpretation of quantum mechanics says there is an underlying reality with additional information of the quantum world. My book explains this underlying reality as binary digits generated in 5th-dimensional hyperspace. These allow time travel by making it possible to warp space, simultaneously adding precision and flexibility to the elimination of distances. Similarly to Einstein's idea, the hidden variable behind Earth's incomplete mathematics is Unification.

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Part 2: 1 / 0   =   infinity  or  1

a) Is it possible that mathematical proof of the unified field originated with Filippo Brunelleschi (1377?-1446), the Italian architect who conceived of the vanishing point, the place where parallel lines converge. This allowed the development of perspective in art. We know parallel lines never actually converge – but if we open our eyes and look in the right places, they do e.g. the 2 lines of a railway track seem to meet in the distance. Similarly, division by zero is accepted to be mathematically impossible.  
b) Division by zero is accepted, in Newtonian maths, to be impossible. But we can regard division by zero as division by nothing i.e. division that has no effect. In this case, 1 divided by 0 is 1. However, to a physicist there is no such thing as nothing (even empty space contains energy). What could the something called 0 actually be? It could be a binary digit. If we use the base of ten (for simplicity) and attach one and zero to it as exponents, we get 10^1 divided by 10^0 = 10^1. If we then cancel 10 from each factor in the expression, we get 1 divided by 0 = 1. At the start of the paragraph, this was referred to as division by nothing. Then 0 was called a binary digit and division by nothing became division by something. The 1 that the division equals is the unified field of space-time. Division by 0 is impossible in Newtonian maths because the result can be infinity. But the word “infinity” can, as the last section of this book shows, apply to the unified field of spacetime. So division by zero is not impossible because it results in the universe, which is obviously possible … a possibility that has always been, and always will be, realized.
In "Who's Afraid of a Big Black Hole?" (BBC program – Dec. 8, 2009), physicist Michio Kaku showed how Einstein divided 1 by 0. The answer in what my book calls Newtonian mathematics is infinity but the answer in what I call relativistic mathematics would refer to all space-time i.e. the past, present and future + the whole observed and unobserved universe. Einstein’s relativistic answer could refer to 
a) space and time being infinite in the sense of having no beginning and no end, 
b) space-time being a unified field composed of infinite quantum entanglements a.k.a. localized unified fields (thus, there would be zero separation between all time periods and all particles), 
c) the universe itself being a black hole. Both would be made of the hidden variables of which Einstein is the most famous proponent. My book calls these variables (are they "virtual particles"?) binary digits generated in a 5th-dimensional hyperspace which makes them - as explained in the next sentence - a non-local variety, in agreement with the limits imposed by Bell's theorem. Comparing space-time to an infinite computer screen and the 5th dimension to its relatively small – in this case, so tiny as to be nonexistent in spacetime – Central Processing Unit, the calculations in the “small” CPU would create and influence everything in infinite space and infinite time, and thus permit a distant event to instantly affect another (exemplified by the quantum entanglement of particles separated by light years) or permit effects to influence causes (exemplified by the retrocausality or backward causality promoted by Yakir Aharonov and others). In a universe described by fractal geometry, the 5th dimension wouldn't exist only on a cosmic scale but also as a hyperspace in every fermion and boson. Also, the black hole that is all space-time would manifest as supermassive, stellar, and Stephen Hawking's mini, black holes. From what I've heard of Albert Einstein, I wouldn't be at all surprised if there's a 

d) he mathematically proved all these things by making the "mistake" of dividing 1 by 0.

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Part 3: The Shape of the Universe

In July 2009, electrical engineer Hong Tang and his team at Yale University in the USA demonstrated that, on silicon chip-and transistor-scales, light can attract and repel itself like electric charges/magnets (Discover magazine’s "Top 100 Stories of 2009 #83: Like Magnets, Light Can Attract and Repel Itself" by Stephen Ornes, from the January-February 2010 special issue; published online December 21, 2009). This is the “optical force”, a phenomenon that theorists first predicted in 2005 (this time delay is rather confusing since James Clerk Maxwell showed that light is an electromagnetic disturbance approx. 140 years ago). In the event of the universe having an underlying electronic foundation (hopefully, my summary will make it clear that this must be so), it would be composed of "silicon chip-and transistor-scales” and the Optical Force would not be restricted to microscopic scales but could operate universally. Tang proposes that the optical force could be exploited in telecommunications. For example, switches based on the optical force could be used to speed up the routing of light signals in fibre-optic cables, and optical oscillators could improve cell phone signal processing.

If all forms of EM (electromagnetic) radiation can attract/repel, radio waves will also cause communication revolution e.g. with the Internet and mobile (cell) phones - I anticipate that there may be no more overexposure to ultraviolet or X-rays. In agreement with the wave-particle duality of quantum mechanics, EM waves have particle-like properties (more noticeable at high frequencies) so cosmic rays (actually particles) are sometimes listed on the EM spectrum beyond its highest frequency of gamma rays. If cosmic rays are made to repel, astronauts going to Mars or another star or galaxy would be safe from potentially deadly radiation. And if all particles in the body can be made to attract or repel as necessary, doctors will have new ways of restoring patients to health. 

From 1929 til his death in 1955, Einstein worked on his Unified Field Theory with the aim of uniting electromagnetism (light is one form of this) and gravitation. Future achievement of this means warps of space (gravity, according to General Relativity) between spaceships/stars could be attracted together, thereby eliminating distance. And "warp drive" would not only come to life in future science/technology ... it would be improved tremendously, almost beyond imagination. This reminds me of the 1994 proposal by Mexican physicist Miguel Alcubierre of a method of stretching space in a wave which would in theory cause the fabric of space ahead of a spacecraft to contract and the space behind it to expand. Therefore, the ship would be carried along in a warp bubble like a person being transported on an escalator, reaching its destination faster than a light beam restricted to travelling outside the warp bubble. There are no practical known methods to warp space – however, this extension of the Yale demonstration in electrical engineering may provide one.

                         

                Star Trek’s warp-driven Enterprise

Elimination of diseased matter and/or eliminating the distance in time between a patient and recovery from any adverse medical condition – even death – would be a valuable way of restoring health. With time travel in an electronic universe, people who have long since died could have their minds downloaded into clones of their bodies - a modification of ideas published by robotics/artificial intelligence pioneer Hans Moravec, inventor/futurist Ray Kurzweil and others - allowing them to “recover” from death (establishing colonies throughout space and time would prevent overpopulation). If the distance in time between recovery and a patient is reduced to zero; prevention of any adverse medical condition, including that of a second death for those resurrected, can occur and we can enjoy resurrection to eternal life. 

Since Relativity says space and time can never exist separately, warps in space are actually warps in space-time. Eliminating distances in space also means “distances” between both future and past times are eliminated - and time travel becomes reality. This is “foreseen” by the Enterprise time-travelling back to 20th-century Earth in the 1986 movie "Star Trek IV: The Voyage Home" and by Star Trek's "subspace communications". Doing away with distances in space and time also opens the door to Star Trek-like teleportation. Teleportation wouldn’t involve reproducing the original and there would be no need to destroy the original body – we would “simply” be here one moment, and there the next (wherever and whenever our destination is). 

Can anything more specific about the mechanics of time travel be stated here? If we get into a spaceship and eliminate the distance between us and a planet 700 light-years away, it'll not only be possible to arrive at the planet instantly but we'll instantly be transported 700 years into the future.* On page 247 of "Physics of the Impossible" by physicist Michio Kaku (Penguin Books 2009), it's stated "astronomers today believe that the total spin of the universe is zero". This is bad news for mathematician Kurt Godel, who in 1949 found from Einstein's equations that a spinning universe would be a time machine (p. 223 of "Physics of the Impossible"). Professor Hawking informs us that “all particles in the universe have a property called spin which is related to, but not identical with, the everyday concept of spin” (science is mystified by quantum spin which has mathematical similarities to familiar spin but it does not mean that particles actually rotate like little tops). Everyday spin might be identical to Godel’s hoped-for spinning universe. If 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. * 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 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. Instantly traveling to a planet 700 light years away and instantaneously arriving at a spot in the future which a light beam could only reach by traveling for 7 centuries can be likened to a wave which spreads out from the point of departure. This is because of quantum mechanics’ wave-particle duality which can view the spaceship not as a collection of particles but as a wave, or collection of waves.

At the destination, the convex shape of the spreading wave arrives instantly (meaning the ship and planet are quantum entangled). This situation is equivalent to space being translated (shifted) by 90 degrees so that the ship is perpendicular to length, width and height simultaneously *. What if the spaceship is simultaneously quantum entangled with another wave arriving at the planet from the other side of the universe?  Since the waves are entangled and unified, their motions are instant and this situation is equivalent to space being translated by 180 degrees. It’s inverted and becomes 5th-dimensional hyperspace.

*  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.) If the infinite universe is composed of subuniverses shaped like figure-8 Klein bottles (where 2 Mobius loops are joined on their sides), 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. 

The space-time we live in is described by ordinary (or “real”) numbers which, when multiplied by themselves, result in positive numbers e.g. 2x2=4, and -2x-2 also equals 4. Inverted “positive” space-time becomes negative hyperspace which is described by so-called imaginary numbers that give negative results when multiplied by themselves e.g. i multiplied by itself gives -1.  
(Supporting info from Stephen Hawking’s “A Brief History of Time”, p. 134)

If we encountered an ocean in hyperspace, altitude readings could no longer give positive results like “height of 3 metres above sea level” but would always give negative results like “depth of 3 metres below sea level”. Traversing 700 light years instantly would be meaningless. In hyperspace, time would be traveling backwards for the light beam and we could only ever travel into the past i.e. instantaneously traverse -700 light years.

In 1928 English physicist Paul Dirac (1902-84) proposed that all negative energy states are already occupied by (then) hypothetical antiparticles (particles of antimatter) – “Workings of the Universe”, a book in the series “Voyage Through The Universe”, by Time-Life Books 1992.  Antimatter and antiparticles would therefore be neg(ative)matter and negparticles, described by imaginary numbers. Virtually every modern physicist suspects that antimatter has positive mass and should be affected by gravity just like normal matter, although it is thought that this view has not yet been conclusively empirically observed. (“Negative mass” in Wikipedia) But I agree with the minority and think antimatter has negative mass. In this way, antimatter would be our peek into the mysteries of a hyperspace with 5 dimensions. Isn’t it nice to know that the secret of time travel into the past might be revealed by the antiparticles used in hospitals’ PET (positron emission tomography) scanners, and by the antimatter possibly useful in future space propulsion?

Translating space by 360 degrees would see us turn full circle and arrive back at our start – stationary in space-time. And space, describable by 360 degrees, must be a sphere (actually, a Mobius loop or possibly the closely related Klein bottle which can be constructed by joining the edges of two Mobius strips together). Another way of looking at this is – we’re at point zero (0 degrees) and nowhere. Being in space can’t be said to be nowhere since it’s made of Space-Time Bits and is therefore something. To be a real nowhere man or woman, the space (or space-time-hyperspace) we’re living in must be unified with and inseparable from an N-space (true nothingness – one method of unifying them is, as we’ll see, by thinking of N-space as a potentiality opened by warping of space and called wormholes). This opening in the warping of space is equivalent to forming a wormhole by ripping a doughnut shape and transforming it into the sphere of space (the Poincaré conjecture essentially says that in three dimensions you cannot transform a doughnut shape into a sphere without ripping it). The doughnut and the sphere both resemble, to a degree, the illustration below of a Mobius loop and, to a greater degree, the illustration of a figure-8 Klein bottle. Picture space-time (the 3 dimensions of length, width and height + the 4th dimension of time) existing on the surface of this doughnut/sphere which has rips or Cosmic Wormholes in it. These rips/wormholes provide shortcuts between distant points of space or time - and belong in a hyperspace having a 5th dimension.

If space is translated by 270 degrees? The blue and red waves would meet at right angles and remind us of electromagnetic waves, such as light, which consist of magnetic and electric fields vibrating at right angles to each other. So if spacetime is warped in this way, we might travel at the speed of light (according to p. 203 of Michio Kaku’s “Physics of the Impossible”, “In general relativity, space-time … can stretch faster than light.”) Why can travel be instant at 90 degrees around the cosmic sphere, but restricted to the relatively slow speed of light at 270 degrees? Because the universe gets inverted into hyperspace at the earlier 180 degree mark of circumnavigation and speeds are inverted too. They increase in a smooth linear fashion until 90 degrees and instantaneousness are reached. At 180 degrees inversion occurs and speeds slow to the speed of light at 270 degrees, continuing to slow between 270 and 360. So we can warp space more and more until we fly 700 light years (or many, many more) in literally no time at all. Then we can warp it to the equivalent of 359 degrees, or a fraction of 1 degree, and come in for a landing.

Do all the degrees mentioned above need to be multiplied by two because you have to travel around a Mobius loop twice to arrive at your starting point, and the universe is a Mobius?

                                                 

The universe could be a Mobius strip (above), discovered independently by the German mathematicians August Ferdinand Mobius and Johann Benedict Listing in 1858. It could also be a figure-8 Klein bottle, first described in 1882 by the German mathematician Felix Klein, where 2 Mobius loops are joined (below). The 1st makes it easier to visualize left, top, and right/bottom as space, time, and hyperspace. The 2nd makes it easier to visualize travel into past and future as travel between its red and blue Mobius strips (other colours denote areas of transition between red/blue). Whether strip or bottle, it’s accurate to say we live in a Mobius Universe since the bottle is 2 joined strips. So the term “Mobius strip” (or “loop”) will be used in this book.

                 

If quantum entanglement has existed in the entire universe forever, everything would be everywhere and everywhen. Space, time and 5th-dimensional hyperspace would not be restricted to certain parts of the Mobius Universe but would exist in every particle. Past, present and future would not exist as the distinct periods which everyday life assumes. All instants of all periods would exist eternally, permitting time travel to any point in the past and to any point in the future. Entanglement may be created by simply zipping along at close to the speed of light  - “Quantum entanglement of moving bodies” by Robert M. Gingrich and Christoph Adami in Physical Review Letters 89, 270402 (issue of 30 December 2002) – which might be achieved, according to this book, by warping space so it’s either a fraction of the 90 degrees allowing instantaneous travel or almost at 270 degrees to space as we know it. 

Looking at either the loop or bottle illustrated above, you see spaces and cavities on the inside and outside. These are only potential in reality. The inner hole can be compared to the pleura (the continuous, thin, elastic membrane that covers the outside of the lungs and the inside of the chest).  There is no actual space, except in certain diseases or injuries, between the portion covering the lungs and the portion lining the chest. They’re in constant contact. Much later, I compare hyperspace to the centre of an expanding balloon. Taken literally, this centre would correspond to the inner hole in the above loop/bottle. But just as pleural spaces are only potential, the loop’s/bottle’s inner hole is normally in constant contact with its surfaces and the hyperspace that would exist there is actually part of the loop/bottle. The centre of cosmic expansion is integrated into expanding space-time and there cannot be any one centre to the expansion.

Looking at the illustrations leads to the conclusion that the outside is obviously composed of a true nothingness or N-space (also much later in this book, I take that outlook). But immediately afterwards, I write that infinity (representing the size of the universe in a particular eon) keeps increasing thanks to BITS (binary digits) producing more spacetime – all time is a unification, so a true infinity (where space and time have no beginning and no end) exists right now! Therefore, N-space is likewise only potential and doesn’t actually exist (remember this when you get to that section).

If there is no such thing as N-space then the cosmos must go on and on forever, and the region described by the Mobius loop or figure-8 Klein bottle wouldn’t be the universe as a whole. These shapes do not extend forever. Their finiteness means it’d be accurate to call the region they describe a “subuniverse”. Each subuniverse, with its inner hole that doesn’t actually exist but is merely potential, would have an origin and a Big Bang with binary digits producing more space-time. Thanks to the supreme flexibility of space-time bits, they’d be warped sufficiently to ensure there are no gaps between any loop/bottle and its partners.* This lack of gaps makes space-time (and hyperspace) continuous and the subuniverses combine to form a superuniverse or megauniverse. Such a “network” of subuniverses is comparable to the network of theories which Stephen Hawking’s and Leonard Mlodinow’s "The Grand Design" says may describe our universe).