In physics, physical information refers generally to the information that is contained in a physical system. Its usage in quantum mechanics (ie. quantum information) is important, for example in the concept of quantum entanglement to describe effectively direct or causal relationships between apparently distinct or spatially separated particles.
Information itself may be loosely defined as “that which can distinguish one thing from another”. The information embodied by a thing can thus be said to be the identity of the particular thing itself, that is, all of its properties, all that makes it distinct from other (real or potential) things. It is a complete description of the thing, but in a sense that is divorced from any particular language. We might even consider the sum total of the information in a thing to be the ideal essence of the thing itself, i.e. its form in the sense of Plato’s eidos (The Forms).
http://en.wikipedia.org/wiki/Fifth_dimension_(geometry)
“In 1993 the physicist Gerard ‘t Hooft put forward the holographic principle, which explains that the information about an extra dimension is visible as a curvature in a spacetime with one fewer dimensions. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature of path of a moving infinitesimal (test) particle. Hooft has speculated that the fifth dimension is really the spacetime fabric.”
All dimensions contained in the same space. 0, 1, Infinity
http://en.wikipedia.org/wiki/Digital_physics
Some try to identify single physical particles with simple bits. For example, if one particle, such as an electron, is switching from one quantum state to another, it may be the same as if a bit is changed from one value (0, say) to the other (1). A single bit suffices to describe a single quantum switch of a given particle. As the universe appears to be composed of elementary particles whose behavior can be completely described by the quantum switches they undergo, that implies that the universe as a whole can be described by bits. Every state is information, and every change of state is a change in information (requiring the manipulation of one or more bits). Setting aside dark matter and dark energy, which are poorly understood at present, the known universe consists of about 10^80 protons and the same number of electrons. Hence, the universe could be simulated by a computer capable of storing and manipulating about 10^90 bits.
“Entropy, if considered as information (see information entropy), is measured in bits. The total quantity of bits is related to the total degrees of freedom of matter/energy.
In a given volume, there is an upper limit to the density of information about the whereabouts of all the particles which compose matter in that volume, suggesting that matter itself cannot be subdivided infinitely many times and there must be an ultimate level of fundamental particles. As the degrees of freedom of a particle are the product of all the degrees of freedom of its sub-particles, were a particle to have infinite subdivisions into lower-level particles, then the degrees of freedom of the original particle must be infinite, violating the maximal limit of entropy density. The holographic principle thus implies that the subdivisions must stop at some level, and that the fundamental particle is a bit (1 or 0) of information.”
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“The physical universe is widely seen to be composed of “matter” and “energy”. In his 2003 article published in Scientific American magazine, Jacob Bekenstein summarized a current trend started by John Archibald Wheeler, which suggests scientists may “regard the physical world as made of information, with energy and matter as incidentals.” Bekenstein quotes William Blake and questions whether the Holographic principle implies that seeing “the world in a grain of sand,” could be more than “poetic license”.”
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Energy, matter, and information equivalence
“Shannon’s efforts to find a way to quantify the information contained in, for example, an e-mail message, led him unexpectedly to a formula with the same form as Boltzmann’s. Bekenstein summarizes that “Thermodynamic entropy and Shannon entropy are conceptually equivalent: the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information one would need to implement any particular arrangement…” of matter and energy. The only salient difference between the thermodynamic entropy of physics and the Shannon’s entropy of information is in the units of measure; the former is expressed in units of energy divided by temperature, the latter in essentially dimensionless “bits” of information, and so the difference is merely a matter of convention.
“The holographic principle states that the entropy of ordinary mass (not just black holes) is also proportional to surface area and not volume; that volume itself is illusory and the universe is really a hologram which is isomorphic to the information “inscribed” on the surface of its boundary.”
Richard Feynman on Light
Wheeler’s “it from bit”
Following Jaynes and Weizsäcker, the physicist John Archibald Wheeler wrote the following:
It is not unreasonable to imagine that information sits at the core of physics, just as it sits at the core of a computer.
It from bit. Otherwise put, every ‘it’—every particle, every field of force, even the space-time continuum itself—derives its function, its meaning, its very existence entirely—even if in some contexts indirectly—from the apparatus-elicited answers to yes-or-no questions, binary choices, bits. ‘It from bit’ symbolizes the idea that every item of the physical world has at bottom—a very deep bottom, in most instances—an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes–no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe. (John Archibald Wheeler 1990: 5)
David Chalmers of the Australian National University summarised Wheeler’s views as follows:
Wheeler (1990) has suggested that information is fundamental to the physics of the universe. According to this ‘it from bit’ doctrine, the laws of physics can be cast in terms of information, postulating different states that give rise to different effects without actually saying what those states are. It is only their position in an information space that counts. If so, then information is a natural candidate to also play a role in a fundamental theory of consciousness. We are led to a conception of the world on which information is truly fundamental, and on which it has two basic aspects, corresponding to the physical and the phenomenal features of the world.
Chris Langan also builds upon Wheeler’s views in his epistemological metatheory:
The Future of Reality Theory According to John Wheeler: In 1979, the celebrated physicist John Wheeler, having coined the phrase “black hole”, put it to good philosophical use in the title of an exploratory paper, Beyond the Black Hole, in which he describes the universe as a self-excited circuit. The paper includes an illustration in which one side of an uppercase U, ostensibly standing for Universe, is endowed with a large and rather intelligent-looking eye intently regarding the other side, which it ostensibly acquires through observation as sensory information. By dint of placement, the eye stands for the sensory or cognitive aspect of reality, perhaps even a human spectator within the universe, while the eye’s perceptual target represents the informational aspect of reality. By virtue of these complementary aspects, it seems that the universe can in some sense, but not necessarily that of common usage, be described as “conscious” and “introspective”…perhaps even “infocognitive”.
The first formal presentation of the idea that information might be the fundamental quantity at the core of physics seems to be due to Frederick W. Kantor (a physicist from Columbia University). Kantor’s book Information Mechanics (Wiley-Interscience, 1977) developed this idea in detail, but without mathematical rigor.
The toughest nut to crack in Wheeler’s research program of a digital dissolution of physical being in a unified physics, Wheeler himself says, is time. In a 1986 eulogy to the mathematician, Hermann Weyl, he proclaimed: “Time, among all concepts in the world of physics, puts up the greatest resistance to being dethroned from ideal continuum to the world of the discrete, of information, of bits. … Of all obstacles to a thoroughly penetrating account of existence, none looms up more dismayingly than ‘time.’ Explain time? Not without explaining existence. Explain existence? Not without explaining time. To uncover the deep and hidden connection between time and existence … is a task for the future.”
We Physicists Are the Only Scientists Who Can Say the Word “God” and Not Blush – Michio Kaku
“As you know, I work in something called String Theory which makes the statement that we are reading the mind of God. It’s based on music or little vibrating strings thus giving us particles that we see in nature. The laws of chemistry that we struggled with in high school would be the melodies that you can play on these vibrating strings. The Universe would be a symphony of these vibrating strings and the mind of God that Einstein wrote about at length would be cosmic music resonating through this nirvana… through this 11 dimensional hyperspace—that would be the mind of God. We physicists are the only scientists who can say the word “God” and not blush.
The fact of the matter is that we are dealing with the cosmic questions of existence and meaning. Thomas Huxley, the great biologist of the last century said that the question of all questions for science and religion is to determine our true place and our true role in the Universe. For both science and religion it is the same question.
However, there has essentially been a divorce in the last century or so between that of science and the Humanists and I think that it’s very sad that we don’t speak the same language anymore.”
Alan Watts Animated – Prickles and Goo, I, Appling Peopling, Music and Life
BigThink.com Michio Kaku:
“The “mind of God” is a phrase of Einstein’s, who used it to refer to an ultimate understanding of the universe and its laws, and stated his desire to achieve that understanding through science. The supercollider is a “Genesis Machine” that will re-create, on a microscopic scale, perhaps the most glorious event in the history of the universe–its birth–and take us one step closer to fulfilling Einstein’s goal – and mine.”
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“Today the LHC may have the potential to explain the origin of all four fundamental forces: gravity, electromagnetism, and the strong and weak nuclear forces. Physicists believe that at the beginning of time there was a single superforce that unified these fundamental forces. Finding it could be the crowning achievement in the history of science.
Through the LHC, we hope to finally prove the existence of the Higgs boson, which is the only particle yet to be observed by the Standard Model. There is a hypothetical, ever-present quantum field that is supposedly responsible for giving particles their masses; this field would answer the basic question of why particles have the masses they do or why they have any mass at all. According to CERN, “The answer may be the so-called Higgs mechanism. According to the theory of the Higgs mechanism, the whole of space is filled with a ‘Higgs field,’ and by interacting with this field, particles acquire their masses. Particles that interact intensely with the Higgs field are heavy, while those that have feeble interactions are light. The Higgs field has at least one new particle associated with it, the Higgs boson. If such a particle exists, experiments at the LHC will be able to detect it.”
Ultimately, what we want to get out of the Large Hadron Collider is something so fantastic, it could rewrite science entirely. We want to create something called sparticles (super-particles) that represent the next set of vibrations of the superstring. We are now on the verge of being able to detect signals from the eleventh dimension—signals from hyperspace. In the eleventh dimension, these four forces just melt together into one gorgeous theory that—if understood and proven—will allow us to “read the mind of God.”
From Universe to Multiverse – Micho Kaku
String Theory, M-theory, Parallel Worlds Pt. 1
BigThink.com Michio Kaku:
“The “multiverse” idea—once thought to be so crazy it only belonged on late night television—has now become the dominant theory in all of cosmology. The idea now dominates conversations in science circles and it seems you cannot avoid the theory of the multiverse.
Einstein first gave us the idea that the universe is a soap bubble of some sort and we reside on the skin of this expanding bubble. This observation of an expanding bubble is now one of the greatest experimental achievements of the last century. Now imagine if you run the videotape backwards, the bubble would shrink and eventually become small enough to put in your coat pocket. If this “bang” happened once, it can happen again, again and again. This concept is mind boggling, that idea that entire universes are being created as you are reading this very blog entry.
When speaking about the multiverse, I’m often asked questions about the different kinds of universes that can form as a result of extra dimensions, string theory or even chaotic inflation for example. These are in some sense different kinds of universes but for me personally, it’s very aesthetically pleasing. This all goes back to my childhood with my parents being Buddhist. In Buddhism, you believe in nirvana and timelessness with no beginning and no end. As a child I went to Sunday school where I learned about arks, great floods and the instant of creation when God said, “Let there be Light”. So, all my life I’ve had these two competing paradigms in my head. With the multiverse idea, we have the beautiful melding of these two ideas. The reason being is that we do have this nirvana, this timelessness, this eleven dimensional hyperspace, this arena of string theory. But we also have bubbles that form all the time, almost like a bubble bath. Sometimes the bubbles expand rapidly giving us universes, combine with other bubbles and sometimes even pop. This continual creation, the idea of a multiverse is very pleasing to me because I can now meld Buddhist nirvana with Judeo-Christian epistemology.
We have this arena of eleven-dimensional hyperspace and within it these bubbles start to expand and they vibrate. In string theory we of course have the music of strings which gives us the particles we see in nature. This is also pleasing to me because Einstein spent the last three decades of his life trying to read the mind of God and he asked himself “What are God’s thoughts?” Well, believe it or not, for the first time we now have a candidate for the mind of God. The mind of God, according to this multiverse picture, is cosmic music resonating through eleven dimensional hyperspace. When I say “God,” I’m talking about the God of Spinoza, not necessarily the personal God that answers prayers and feeds the sick. I’m speaking metaphorically about the God of both harmony and beauty. In other words, as I have stated time and time again, it didn’t have to be this way: our Universe could have been random, chaotic and ugly. And I find it absolutely staggering that we can summarize all the laws of physics going back 2,000 years to the Greeks on a single sheet of paper. The goal of string theory is to, of course, have it in an equation no more than an inch long. In the beginning, there was not light but rather there was the one-inch equation which then drives the gears of the entire Universe. This is the Holy Grail.“
“The Golden Ratio, roughly equal to 1.618, was first formally introduced in text by Greek mathematician Pythagoras and later by Euclid in the 5th century BC. In the fourth century BC, Aristotle noted its aesthetic properties.
Aside from interesting mathematical properties, geometric shapes derived from the golden ratio, such as the golden rectangle, the golden triangle, and Kepler’s triangle, were believed to be aesthetically pleasing. As such, many works of ancient art exhibit and incorporate the golden ratio in their design. Various authors can discern the presence of the golden ratio in Egyptian, Sumerian and Greek vases, Chinese pottery, Olmec sculptures, and Cretan and Mycenaean products from as early as the late Bronze Age. The prevalence of this special number in art and architecture even before its formal discovery by Pythagoras is perhaps evidence of an instinctive and primal human cognitive preference for the golden ratio.”
Nature by Numbers
http://en.wikipedia.org/wiki/Mathematics_and_art
It is in De Divina Proportione that the golden ratio is defined as the divine proportion. Pacioli also details the use of the golden ratio as the mathematical definition of beauty when applied to the human face.
“The Ancients, having taken into consideration the rigorous construction of the human body, elaborated all their works, as especially their holy temples, according to these proportions; for they found here the two principal figures without which no project is possible: the perfection of the circle, the principle of all regular bodies, and the equilateral square.” from De Divina Proportione (1509)
Golden ratio discovered in a quantum world
“When applying a magnetic field at right angles to an aligned spin the magnetic chain will transform into a new state called quantum critical, which can be thought of as a quantum version of a fractal pattern. Prof. Alan Tennant, the leader of the Berlin group, explains “The system reaches a quantum uncertain – or a Schrödinger cat state. This is what we did in our experiments with cobalt niobate. We have tuned the system exactly in order to turn it quantum critical.”
By tuning the system and artificially introducing more quantum uncertainty the researchers observed that the chain of atoms acts like a nanoscale guitar string. Dr. Radu Coldea from Oxford University, who is the principal author of the paper and drove the international project from its inception a decade ago until the present, explains: “Here the tension comes from the interaction between spins causing them to magnetically resonate. For these interactions we found a series (scale) of resonant notes: The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618…, which is the golden ratio famous from art and architecture.” Radu Coldea is convinced that this is no coincidence. “It reflects a beautiful property of the quantum system – a hidden symmetry. Actually quite a special one called E8 by mathematicians, and this is its first observation in a material”, he explains.
The observed resonant states in cobalt niobate are a dramatic laboratory illustration of the way in which mathematical theories developed for particle physics may find application in nanoscale science and ultimately in future technology. Prof. Tennant remarks on the perfect harmony found in quantum uncertainty instead of disorder. “Such discoveries are leading physicists to speculate that the quantum, atomic scale world may have its own underlying order. Similar surprises may await researchers in other materials in the quantum critical state.”
geometric psychology, google names for further research
“Subjective geometry” is a term coined by Weintraub and Krantz to describe the distortion imposed upon geometric patterns by the visual system itself—so-called optical illusions. The latter are widely regarded as being generated by misplaced “constancy” effects, i.e., they are regarded as stemming from the invariance of an object’s appearance under wide variations in viewing conditions, such as obliquity, rotations, etc. The invariances represented by these constancies—shape constancy, size constancy, etc.—are spatiotemporal invariants of certain Lie subgroups of P4(R) circled plus CO(1, 3) circled plus GL(4, R) that govern Euclidean and non-Euclidean geometry. That Euclidean subgroups describe a Cyclopean visual world; the non-Euclidean, a binocular (bipolar) world of hyperbolic nature, according to the work of Luneburg, Blank, Indow, and others. The visual field of view is itself a geometric object involving not only “figure” and “ground” but also visual contours (orbits of the Lie groups involved), linear perspective, interposition, and contact and symplectic structures. The retina and “cortical retina” are both covered by a family of “circular-surround” cellular response fields (of a “Mexican hat” nature) which constitute an atlas for the visual manifold S. Upon this manifold are defined certain equivariant vector bundles that account for constancy phenomena and certain jet bundles, arising out of the vector bundles by prolongation, that generate the differential invariants characterizing higher form perception. The resultant theory of perceptual-cognitive processing has been termed “geometric psychology,” in analogy to MacLane’s “geometrical mechanics” and Brockett–Hermann–Mayne’s “geometry of systems,” the mathematical structure being very similar in all three instances. Functorial maps from the category GvFB(S) of equivariant fibre bundles to the simplicial category and the category of simplicial objects complete the theory by extending the perceptual system to cognitive phenomena and information-processing psychology.
Link to post on Infinity – Dangerous Knowledge doc
Link to Circle is absolute intelligence
Link to Science and Art posts
Link to Collective Unconscious
Link to Expansion of Space
Potential Geometry in Quantum Physics and Vision
(In The Character of Physical Law)Richard Feynman:
“It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do?”
“So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the checker board with all its apparent complexities. But this speculation is of the same nature as those other people make—’I like it,’ ‘I don’t like it’—and it is not good to be prejudiced about these things.”
http://en.wikipedia.org/wiki/Completed_infinity
Georg Cantor:
“Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space.”
“Georg Cantor’s grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory.” -Y. Manin
“Thus, exquisite minimalism of expressive means is used by Cantor to achieve a sublime goal: understanding infinity, or rather infinity of infinities.” -Y. Manin
Georg Cantor – Dangerous Knowledge – Infinity of Infinities
The Story of Maths – To Infinity and Beyond
BigThink.com Michio Kaku: “In short, the quantum theory allows us to understand the world of the very small and the fundamental properties of matter.
Our deepest understanding of the atomic world comes from the advent of the quantum theory. Having this deep understanding of the various elements of the theory allows us to do much more than just move atoms around or know exactly why things behave the way they do. The theory itself underlies the entire architecture of the world we see today and beyond. It has ultimately allowed us to develop the most advanced technologies to make our lives easier. The marvels of science that we see and use every single day including the Internet, your cell phone, GPS, your email, HD television—all of it—comes from our deep understanding of this theory. This theory offers a very different way to view the world they we live in—one where the simple laws of conventional physics simply don’t apply at all. Quantum theory is so eccentric and peculiar that even Einstein himself couldn’t wrap his head around it. The great physicist, Richard Feynman once stated that “It is impossible, absolutely impossible to explain it in any classical way”.
Some of what quantum theory predicts and states is almost like something out of science fiction. Matter can essentially be in an infinite number of places at any given time; it is possible that there are many worlds or a multiverse; things disappear and reappear somewhere else; you cannot simultaneously know the exact position and momentum of an object; and even quantum entanglement (Einstein referred to it as spooky action at a distance) where it’s possible for two quantum particles to link together effectively making them part of the same entity or entangled. Even if these particles are separated, a change in one is ultimately and instantly reflected in it’s counterpart. At the end of the day, the world of entanglement caused physicists like Einstein to both dislike the predictions and feel nothing more as if their were serious errors in the calculations. As Einstein once wrote: “I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming house, than a physicist”.”
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“Quantum Computing, which is making direct use of the quantum mechanical phenomena, such as superposition and entanglement to perform operations on data. In contrast with a classical computer which has memory made of bits where each bit represents a one or a zero (binary code), a quantum computer will operate on what is called “qubits.” According to Wikipedia, a single qubit can represent a one, a zero, or, crucially, any quantum superposition of these; moreover, a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8 and so on. Superposition refers to the quantum mechanical property which states that all particles exist in not one state but all possible states at once. In short, a quantum computer will essentially be able to crack any algorithm, solve mathematical problems much more quickly and ultimately operate millions of times faster than conventional computers.”
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“The list goes on an on an on: Quantum Dots; Quantum Wires or Carbon Nanotubes; Metamaterials; Invisibility; Quantum Optics; Teleportation; Communication; Space Elevators; Limitless Quantum Energy; Room Temperature Superconductors; Personal Fabicators; Nanotechnology and even Time Travel. Other applications that will strive are advances in battery technology; solar panels; stealth applications; and even advances in biotechnology and medicine. Needless to say, we have only scratched the surface of some of these technologies and time will perfect them. We’ve got a very interesting future ahead of us….”
Information in the Holographic Universe by Jacob D. Bekenstein [July 14,2003] Scientific American Jakob Bekenstein:
“The proliferation of variations on the holographic motif makes it clear that the subject has not yet reached the status of physical law. But although the holographic way of thinking is not yet fully understood, it seems to be here to stay. And with it comes a realization that the fundamental belief, prevalent for 50 years, that field theory is the ultimate language of physics must give way. Fields, such as the electromagnetic field, vary continuously from point to point, and they thereby describe an infinity of degrees of freedom. Superstring theory also embraces an infinite number of degrees of freedom. Holography restricts the number of degrees of freedom that can be present inside a bounding surface to a finite number; field theory with its infinity cannot be the final story. Furthermore, even if the infinity is tamed, the mysterious dependence of information on surface area must be somehow accommodated.”
Complexity & Chaos – Part 3b (Strange Geometry)
Jakob Bekenstein:
“Thermodynamic entropy is popularly described as the disorder in a physical system. In 1877 Austrian physicist Ludwig Boltzmann characterized it more precisely in terms of the number of distinct microscopic states that the particles composing a chunk of matter could be in while still looking like the same macroscopic chunk of matter. For example, for the air in the room around you, one would count all the ways that the individual gas molecules could be distributed in the room and all the ways they could be moving.”
Singularities and Infinity
Indefinability as ‘failure’ in physics
http://en.wikipedia.org/wiki/Lie_sphere_geometry
“The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.”
http://en.wikipedia.org/wiki/Functional_analysis
In the modern view, functional analysis is seen as the study of vector spaces endowed with a topology, in particular infinite dimensional spaces. In contrast, linear algebra deals mostly with finite dimensional spaces, or does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.
http://en.wikipedia.org/wiki/Real_analysis
http://en.wikipedia.org/wiki/Real_number
“Real numbers can be thought of as points on an infinitely long number line.”
http://en.wikipedia.org/wiki/Fifth_dimension_(geometry)
“In 1993 the physicist Gerard ‘t Hooft put forward the holographic principle, which explains that the information about an extra dimension is visible as a curvature in a spacetime with one fewer dimensions. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature of path of a moving infinitesimal (test) particle. Hooft has speculated that the fifth dimension is really the spacetime fabric.”
http://en.wikipedia.org/wiki/Circle_bundle
In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle.
http://en.wikipedia.org/wiki/Kaluza_Klein
In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and the phenomenon of having a space-time with compact dimensions is referred to as compactification.
In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). Once this geometrical interpretation is understood, it is relatively straightforward to replace U(1) by a general Lie group.
http://en.wikipedia.org/wiki/Point_at_infinity
“The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line. The real projective line is not equivalent to the extended real number line, which has two different points at infinity.
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This construction can be generalized to an arbitrary topological space. The space so obtained is called the one-point compactification or Alexandroff compactification of the original space. Thus the circle is the one-point compactification of the line, and the sphere is the one-point compactification of the plane.
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In hyperbolic geometry, the ideal point is also called the omega point. Given a line l and a point P not on l, right- and left-limiting parallels to l through P meet at a point on the boundary circle of the Poincaré disk model and the Klein model called the omega point. Pasch’s axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.”
http://en.wikipedia.org/wiki/Real_projective_line
The symbol \infty represents the point at infinity, an idealized point that bridges the two “ends” of the real line.
“Sometimes it is convenient in science to describe the state of an object with n degrees of freedom as if it were a point in some n-dimensional space. For example, classical mechanics describes the three-dimensional position and momentum of a point particle as a point in 6-dimensional phase space.”
http://en.wikipedia.org/wiki/Differential_geometry_of_surfaces
Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss (1825-1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.
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“An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections.”
http://www.pbs.org/wnet/hawking/universes/html/bound.html
“A proposal first advanced by Stephen Hawking and Jim Hartle, the no-boundary universe is one in which the universe does not start with a singularity. It uses American physicist Richard Feynman’s proposal to treat quantum mechanics as a “sum over histories,” meaning that a particle does not have one history in space-time but instead follows every possible path to reach its current state. By summing these histories—a difficult process that must be done by treating time as imaginary—you can find the probability that the particle passes through a particular point.”
