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Photographed on the Knersvlakte near Gemsbokrivier (Vanrhynsdorp district, South Africa)
Seeds available from www.capesucculentseeds.com
Trevor and I were on a walk through Cité Soleil, and we went into a church made of tin roofing material. There, on desks made from scrap lumber, were several teens were working on vector calculus problems (that's Calc III). They have no calculus teacher, and are able to do this because they found a textbook in a donation pile and have been teaching themselves.
This picture was not posed, and they didn't know we were coming, this was a totally random interaction.
I'm putting together a laptop to send back to them loaded with learning materials from MIT's OpenCourseWare and other sources. If you'd like to contribute let me know and I'd be happy to include other items in the donation (digital or physical).
For reference, here's what Wikipedia has to say about their home:
en.wikipedia.org/wiki/Cité_Soleil
"Cité Soleil (Kreyol: Site Solèy, English: Sun City) is a very densely populated commune located in the Port-au-Prince metropolitan area in Haiti. The lawless Cite Soleil is regarded by many as one of the most dangerous places on earth. It was developed as a shanty town. Most of its estimated 200,000 to 400,000 residents live in extreme poverty.[1] The area is generally regarded as one of the poorest and most dangerous areas of the Western Hemisphere's poorest country; it is one of the biggest slums in the Northern Hemisphere. There is little to no police presence, no sewers, no stores, and little to no electricity.[2]
The neighborhood, originally designed to house manual laborers for a local Export Processing Zone (EPZ), quickly became home to squatters from around the countryside looking for work in the newly constructed factories. After a 1991 coup d'état deposed President Jean-Bertrand Aristide, a boycott of Haitian products closed the EPZ.[3] Cité Soleil was soon thrust into extreme poverty and persistent unemployment, with high rates of illiteracy.[2]
Armed gangs roam the streets. Murder, rape, kidnapping, looting, and shootings are common as every few blocks is controlled by one of more than 30 armed factions.[4] The area has been called a "microcosm of all the ills in Haitian society: endemic unemployment, illiteracy, non-existent public services, insanitary conditions, rampant crime and armed violence".[5]
After the 2010 Haiti earthquake, it took nearly two weeks for relief aid to arrive in Cité-Soleil.[6] Although the US military have willingly accepted their new role, their relief efforts have been criticized by some as insufficient.[7]"
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Checking out my feet after a long run. More details at my blog, Gay Sex and Advanced Calculus at reflexblue.blogspot.com/
The most famous Belgian comic hero is the brainchild of Herge, the pen name of Brussels author Georges P. Remi. The scene depicted is from The Calculus Affair.
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Be kind to cute creatures.
Professor Calculus knows that.
Little cute hedgehog from Paperdolly.
I bought Professor Calculus.
Gah, The calculus test was brutal. It'll be a freaking miracle if I get a D. Luckily, the class is early on in the day. The rest of my stuff is a joke. History? Already know it. Spanish? Fácil. Language Arts? A joke. And last but not least, Art. Basically, this might as well be recess. The teacher just tells us to draw and we do almost whatever we want. Not only that, I share this class with all my friends and Steph. Though she seems a bit late today. Nevertheless, the other three morons were at our table. These guys, Adam, Mike and Dom...oh boy, where to start? These guys are like a bad sitcom. Hilarious in it's own horrible way. When i walked up and grabbed a chair, they were all like "Heyyyy!" This is gonna be crazy...
Mike: "Been a bit, man, what's goin' on?
"Nothing. The gangwars held up something."
Dom: "Ah yeah. Freeze and his snowmen tearing shit up."
Adam: "About damn time Freeze lost. I was freezing my ass off with that damned arctic weather!"
Mike: "I was making spears from sticks and stones ready to hunt some caribou."
Dom: "You gotta be kidding."
Mike: "No shit I'm kidding. I'm not a caveman."
Adam: Really? You got hair like one."
Mike: "Dude, fuck you!"
Adam: "Lighten up, man. We know I'm not one to talk."
Dom: "Yeah, hence the hat. Seriously, how many detentions do you have because you wear it in class?"
Adam: "I lost count. Still, ain't the worst thing I've done this year."
Mike: "No shit. What was that you just blurted out in the middle of study hall? 'I have the weirdest boner'?"
Adam: "Yep!"
Dom: "Seriously, what the fuck? Why'd you say that?'
Adam: "Just to make everyone laugh."
Mike: "I've met 12-year olds more mature than you, y'know that?"
Adam: "Oh, I'm immature? Hey Tim! Take a look at fartknocker's sketchbook here!"
Oh god. This is gonna be stupid. Adam handed off the book to me and I opened it up, ready for virtually anything. What I got was....rather skillful drawings of boners. How charming. let's see, we've got giant monster boners, boners in knight armor, spaceboners...oh lord, is that a boner dressed like superman? I shouldn't be laughing at this. But yet, I am. I kept skimming what I'll now call the Encyclopedia Bonetannica, until a familiar voice made me shut the thing faster than I could blink.
"Hey guys!"
Adam/Mike/Dom: "Hey Steph..."
"What'cha got there, Tim?"
"Just Dom's book. Here ya go man.."
I quickly handed back the book back to Dom. Man, that was WAY too close. Thank god Steph didn't catch me looking at boners. While I gave a sigh of relief, I saw Adam lean in near Steph, and whispered just loud enough that I could hear what he was saying. His eyes met mine a few times as well.
Adam: "Hey babe, why don'tcha leave this nerd behind and go with a REAL man. I'll be behind the school. Eh?"
We all laughed. Me, him, Steph, everyone. Adam usually does this as a joke. He never means it. he's just here to make everyone laugh. Dom and Mike aren't that different. They all get along with Steph just great too, so it makes our lives much easier. Speaking of Dom and Mike...
Mike: "Boys and...uh...girl, i think we should consult the paper I found in my dad's trash bin last night?"
Adam: "You get your stuff outta your pop's trash? That explains alot..."
Mike: "Here's a rusty pen, go fuck yourself with it. Anyway...."
Mike took out a newspaper and flattened it on the table. It was the frontpage from a few days ago. On it "ROBIN AND SPOILER FOIL CITY BANK ROBBERY". Hehe, I remember that. Me and Steph stopping Roadkill's group from draining out the bank.
Mike: "Hell yeah, Robin doing what he does best!"
Adam: "Kicking ass and chewing bubblegum!"
Dom: "Don't know about the bubblegum, but kicking ass? Hell yeah."
Adam: "Dude, how much of a beast do you have to be, to be a kid, and fight with goddamn Batman?"
Mike: "A huge one. Remember when him and Batman took down the Jokerbot on live TV."
Dom:" Dude I saw that! Bad-fucking-ass!. And then there's his girlfriend, Spoiler."
Mike: "Hey, she can roll with Robin, she's gotta be a badass too, right?"
Adam: "She's kinda cute."
I looked up at Steph. She was looking back at me with a smug smile. Hell, I was smiling too. We kept listening to the three raving over our costumed personas. So, our best friends are also our biggest fans? What more could you ask for?
This program performs the calculation to optimize the work conditions in focus stacking:
MODE 1: for required CoC on the sensor and final image size provides the best pair of Aperture / DoF where the CoC diffraction and CoC defocus contribute equally at total CoC
MODE 2: for a given Aperture and DoF, calculates the total CoC and the contribute of diffracrion and defocus
If anyone wants to try it I'll be happy to send free
Tintin, o filho do Belga Hergé, ou melhor, de Georges Remi, já que era este o seu verdadeiro nome, fez 10 de Janeiro de 1929, anos… 80 anos!
O herói de banda desenhada, de filmes, ou séries animadas, fez e continua a fazer parte da nossa imaginação, acompanhado do seu fiel Milou, o fox terrier que o acompanha nas suas aventuras com o Capitão Haddock, ou os detectives Dupond e Dupont.
Mas porque foi Tintin um sucesso tão grande, que continua a merecer a nossa atenção?
As aventuras de Tintin, com as suas estórias futuristas para a época em que foram criadas, recriam um mundo extremamente parecido com o nosso, que entra facilmente no nosso imaginário. O humor apurado e seco característico, a sátira sofisticada, e os comentários sociais, retratam traços inegáveis das sociedades modernas.
______
The Adventures of Tintin (French: Les Aventures de Tintin) is a series of comic strips created by Belgian artist Hergé, the pen name of Georges Remi (1907–1983). The series first appeared in French in a children's supplement to the Belgian newspaper Le Vingtième Siècle on 10 January 1929. Set in a painstakingly researched world closely mirroring our own, the series has continued as a favourite of readers and critics alike for 80 years.The hero of the series is Tintin, a young Belgian reporter. He is aided in his adventures from the beginning by his faithful fox terrier dog Snowy (Milou in French). Later, popular additions to the cast included the brash, cynical and grumpy Captain Haddock, the bright but hearing-impaired Professor Calculus (Professeur Tournesol) and other colourful supporting characters such as the incompetent detectives Thomson and Thompson (Dupond et Dupont).The success of the series saw the serialised strips collected into a series of albums (24 in all), spun into a successful magazine and adapted for film and theatre. The series is one of the most popular European comics of the 20th century, with translations published in over 50 languages and more than 200 million copies of the books sold to date.
The comic strip series has long been admired for its clean, expressive drawings in Hergé's signature ligne claire style. Engaging, well-researched plots straddle a variety of genres: swashbuckling adventures with elements of fantasy, mysteries, political thrillers, and science fiction. The stories within the Tintin series always feature slapstick humour, offset in later albums by sophisticated satire, and political and cultural commentary.
From Wikipedia, the free encyclopedia
Usually I try to be a good student and pay attention and do my work, but it was Friday, and I just really, really really could not focus on anything that made me feel as stupid as calculus-based physics does.
Cuthbert wasn't participating in a We're Here outing today, but we were watching a comedian on YouTube talking about conspiracies. I hadn't heard of the "Benadryl Hat Man" conspiracy, but we thought we could create a photo depicting such a sighting.
The video: www.youtube.com/watch?v=q3JIRvP-7m8
Info on the conspiracy:
This Royal Bengal Tiger, like all big cats, has an extremely coarse tongue, which they use for self-grooming and stripping meat of their prey. She could literally lick your face off your skull.
You can also see that she is dentally fit with no calculus and healthy intact enamel, indicating that she is not very old.
A five-dimensional space is a space with five dimensions. If interpreted physically, that is one more than the usual three spatial dimensions and the fourth dimension of time used in relativistic physics. It is an abstraction which occurs frequently in mathematics, where it is a legitimate construct. In physics and mathematics, a sequence of N numbers can be understood to represent a location in an N-dimensional space. Whether or not the universe is five-dimensional is a topic of debate.Three Logical Proofs: The Five-Dimensional Reality of Space-Time
West Virginia University at Parkersburg Physics, 300 Campus Drive Parkersburg, West Virginia 26104 e-mail: jebcolst@aol.com
Abstract- A century and a half ago, a revolution in human thought began that has gone largely unrecognized by modern scholars: A system of non-Euclidean geometries was developed that literally changed the way that we view our world. At first, some thought that space itself was non-Euclidean and four-dimensional, but Einstein ended that 'speculation' when he declared that time was the fourth dimension. Yet our commonly perceived space is four-dimensional. Einstein unwittingly circumvented that particular revolution in thought and delayed its completion for a later day, although his work was also necessary for the completion of that revolution. That later day is now approaching. The natural progress of science has brought us back to the point where science again needs to consider the physical reality of a higher-dimensional space. Science must acknowledge the truth that space is four-dimensional and space-time is five- dimensional, as required by accepted physical theories and observations, before it can move forward with a new unified fundamental theory of physical reality.
Keywords: four-dimensional-five-dimensional-space-time-Einstein- Clifford- Kaluza- Kaluza-Klein- magnetic vector potential- electromagnetism- Yukawa potential- xpanding universe- general relativity-unification-superstrings-branes-Randall-Sundmm
Introduction
Individual scientists have been searching for evidence of a fourth dimension of space for more than a century and a half. That search subsided somewhat after Albert Einstein identified time as the fourth dimension and developed the theories of relativity. However, Theodor Kaluza added a fifth dimension to space-time in 1921. Others have contributed to this line of scientific devel- opment, but not to as high an extent. Given the fact the physicists have now developed 10- and 11-dimensional theories of reality, it would seem that the search for a fourth dimension of space would have taken on a new and sig- nificant meaning, but it has not. Yet several generally accepted scientific theories and concepts do imply the existence of a fourth spatial dimension.
On the other hand, a growing number of scientists have acknowledged and embraced the simple fact that physics needs a single fundamental theory to
524 J. E. Beichler
continue its astonishing rate of progress. A complete unification of the funda- mental forces of nature has itself been a long process predating the 1970s, but that unification was made basically from the relativistic point-of-view by Einstein and a few other scientists before the 1960s. Einstein searched for a successful unification of gravity and electromagnetism for the last three decades of his life, hoping that the quantum and quantum effects would emerge from the mathematical formalisms of his unified field theory, but most other scientists shared neither his optimism nor his goal. During the 1970s, quantum physicists finally adopted Einstein's goal, but not his emphasis on a unification based upon general relativity and a continuous view of the ultimate nature of reality. Quantum theorists began their own long search for unification with the discovery of the standard model, then the electroweak force and finally the hope that gravity would eventually submit to quantum analysis. They have utterly failed to achieve this last step toward unification.
All that science can say for certain is that there are presently two theories that can claim to represent the most fundamental nature of reality: Quantum theory and relativity. Unfortunately, these two are mutually incompatible. The near complete dominance of the quantum paradigm over the last century has led most physicists to conclude that any future theory that unifies physics must be based upon a discrete quantum model rather than a continuous relativistic model. The attitude that discreteness can replace continuity at all levels of reality is prob- lematic: It reflects a general disregard for the depth and extreme nature of the major differences between the two theories. This disregard has led scientists to speculate on the structure of reality at as small a level as the Planck length, resulting in the development of quantum loop theories and other attempts to find a quantum gravity theory. Whether the existence of a major conflict between the discrete and continuous is acknowledged or not, the fact that these two models of reality are mutually incompatible is generally minimized or belittled by many theoretical scientists who overwhelmingly assume that discreteness offers the only possible solution to the problem of unification.
Recent attempts to overcome this incompatibility, such as the supergravity, superstring and brane theories, have relied heavily upon the concept of hyper- dimensional spaces. These models have been unsuccessful, yet the overall notion of hyper-dimensionality still offers a way out of the dilemma. Einstein first rendered the notion of a higher-dimensional reality plausible in 1905, but the revolution that Einstein began when he unified three-dimensional space with time to form a four-dimensional space-time continuum has never been fully realized. In the meantime, the opposing quantum concept may have fully run its course and reached its inherent theoretical limits. The modem unification theories based upon the quantum model do not seek to rectify the fundamental differences between the quantum theory and special relativity. Quantum field theories only calculate quantum effects in the relativistic limit; they do not unify the theories at the necessary fundamental level that is often claimed. Many scientists ignore the extent and importance of the differences between continuity
Five Dimensions of Space-Time 525
and the discrete and instead worry about the insignificant problems of inde- terminism and counting bits of information. So the latest attempts at unification have failed utterly even though the quantum theory has been attempting to quantize gravity for several decades.
There are many levels to the hyper-dimensionality problem, many of which have not yet been explored even though the central problem of dimensionality for present day science dates back a century and a half. Science has been misled and has failed to recognize the significance of a far more fundamental revolution that began in the 1850s when Bernhard Riemann developed a generalized system of non-Euclidean geometries (Riemann, 1854). Riemann's work directly implied that space is four-dimensional as well as continuous. His new system of geometry remained relatively unknown for more than a decade and was only popularized within the scientific community in the late 1860s. Simultaneously, James Clerk Maxwell developed Michael Faraday's field concept of electro- magnetism into a complete theory of electromagnetism. Whether the timing of these developments was coincidental or not, and only a careful review of historical documents can determine if the simultaneous development of these theories was truly a coincidence, the two fundamental concepts of the continuity of the electromagnetic field and the four-dimensionality of space are physically related. There are three logical proofs that this fact is true.
The first logical proof derives directly from Maxwell's electromagnetic theory and deals directly with the inability of science to sufficiently explain the nature of the vector or magnetic potential used to explain magnetic induction. The second logical proof deals with the nature of matter itself as represented by the Yukawa potential and the atomic nucleus. The Yukawa potential is normally used to explain how electrical repulsion is overcome to bind particles within the nucleus. However, the mathematical expression for the potential also matches the general shape of space-time curvature within the individual particles that combine to form the nucleus. And finally, the last proof is a more general argu- ment dealing with the simple three-dimensional orientations of spiral galaxies relative to the Riemannian curvature of the universe as a whole. Although these proofs are independent of any particular modern hyper-dimensional theory, they are supported by Kaluza's theory of five-dimensional space-time.
Electromagnetism Speaks Up
The popular concept of a 'force field' is completely erroneous. Even in a classical sense, no force is associated with a field until a material particle or body interacts with it. Force is not a characteristic of the field alone. The interaction of the field and matter results in the force, but the interaction can also be characterized by a potential energy. The energy results from the force acting on the particle in one sense, or from the relative position of the particle in the field in another sense. What exists at any particular position in the field before the interaction takes place is called the potential. So a physical field is char- acterized by the potential of the field, not a force.
526 J. E. Beichler
Gravity presents a good example for the concept of potential. Gravitational field strength decreases radially outward from the center of gravity of a material body like the earth according to the inverse square law. All points that are equidistant from the center of gravity form a surface in three-dimensional space along which the gravitational potential is constant, an equipotential surface. At each point on this surface, the surface is perpendicular to a radial line drawn from the center of gravity. A material body orbiting the earth would have a constant speed along any equipotential surface. Electricity presents another simple example. In this case, the units of potential are 'volts', a common electrical unit with which everyone is familiar. Equipotential surfaces representing specific volt measurements are a commonly accepted fact of electrical fields. The fact that an equipotential surface can be formed and that the surface is perpendicular to the radius of curvature at each and every point where they intersect is a general property of fields. From a theoretical point-of-view, equipotential surfaces must exist for all physical fields. For any field, successive equipotential surfaces form onionskin-like concentric surfaces around point charges or charged bodies.
There is a direct equivalence between electricity and magnetism and that equivalence forms the basis of the electromagnetic theory. Any physical quan- tities or properties of electricity correspond to similar quantities and properties for magnetism. But that equivalence has not yet been fully realized since there is no such thing as magnetic 'volts' or measurable magnetic potential. Magnetic potential has been, is now and will be in the future a mathematical entity alone, given the three-dimensionality of space. Consider a simple magnetic field, per- haps that of a bar magnetic. An equipotential surface cannot be drawn or represented visually as it can for an electric field, although magnetic field lines can still represent the field. A line perpendicular to any field line through a given point on that field line, representing the magnetic vector potential at that point, cannot be connected to neighboring points of equal potential on other field lines to form a continuous surface. In other words, an equipotential surface cannot be formed in the three-dimensional space of the magnetic field represented by the field lines. All equipotential surfaces would go through the same point on a field line in three-dimensional space, which is impossible, but no other conclusion can be reached from the given physical geometry of the magnetic field.
According to Roger Penrose, the magnetic potential is "not uniquely determined by the field F, but is fixed to within the addition of a quantity dO where O is some real scalar field." The scalar field is taken to be a purely mathematical entity, such that the magnetic potential A "is not a locally mea- surable quantity" (Penrose, 2005).The magnetic potential A exists, but no phys- ical experiment can measure or otherwise determine the value of A plus the additional quantity dO, so the value of A alone cannot be uniquely determined. In a sense then, the magnetic potential exists only at the point of intersection, not beyond that point in three-dimensional space. Magnetic potential is purely a point phenomenon in three-dimensional space no matter what its value. It is a mathematical paradox, but the paradox can be solved if a higher dimension to
Five Dimensions of Space-Time 527
space is used. Any connection between a given potential on one field line and neighboring field lines must be in another dimension (orthogonal direction) other than the three normal directions of common space, in order for there to exist an equipotential surface. The 'gauge factor' dO mentioned by Penrose actually represents a minuscule measurement or perturbation in the fourth direction that does not otherwise affect normal three-dimensional field variations in the local environment. This fact can also be seen in the equations that are commonly used to express and model magnetic potential.
Although it cannot be described or measured in a normal three-dimensional space, the magnetic potential can be expressed mathematically, by its rela- tionship to the field, as
and
where B is the magnetic field strength. In this form, the quantity A is known as the magnetic vector potential or just the vector potential. Since the operator
V= (dldxi,dldyj,d/dzk),
taking the curl of A would be the mathematical equivalent of constructing the magnetic field B point-by-point by simultaneously looking at the perpendicular components to A in each of the three dimensions of space. These equations may seem trivial to physicists, but they have far more physical meaning than they have been given in the normally accepted electromagnetic interpretation.
The potential A must be simultaneously perpendicular to all three coordinates used to represent a point in space according to these formulations. However, the only 'thing' that can be perpendicular to all three dimensions of space simulta- neously would be a fourth orthogonal dimension. Therefore, changes in the magnetic potential as well as magnetic potential itself are perpendicular to all three directions at any spatial position in our normally perceived physical space. Different equipotential surfaces would still be expressed by three-dimensional equations even though they are displaced in the fourth direction because they would act like three-dimensional spaces that are parallel to or stacked on top of our common three-dimensional space in the fourth direction. Given the con- tinuity of space, our three-dimensional material world is actually embedded in a four-dimensional space (or manifold). Bernhard Riemann's original develop- ment of the generalized formulations of non-Euclidean geometry posited that an n-dimensional space would be embedded in an n+l-dimensional manifold, which implies that the physical reality of our three-dimensional space (where n= 3) requires the existence of a higher-dimensional manifold. In present theories of higher-dimensional spaces, such as the various superstring theories, several higher embedding dimensions are used, but the Riemannian mathematics used in general relativity only 'requires' one higher embedding dimension.
528 J. E. Beichler
The fact that magnetism implies a fourth dimension is not new. William Kingdom Clifford, a British geometer, tried to express Maxwell's electromag- netic theory using a four-dimensional space model in the 1870s. Clifford is better known for offering the first translation of Riemann's Habilitationsschrift lecture, " On the hypotheses which lie at the bases of geometry" , into English in 1873, among other things. Based on his understanding and interpretation of Riemann's geometry, Clifford claimed that what we sense as matter is nothing more than three-dimensional space curved in a fourth dimension and what we conceive as matter in motion is no more than variations in that curvature (Clifford, 1870). For having stated this, Clifford's geometrical model of space is normally regarded as a precursor to Einstein's model of space-time curvature in the general theory of relativity. Most twentieth century scholars have also concluded that Clifford never developed a theory and had no followers (Eddington, 1921; d'Abro, 1927; Bell, 1940; Jammer, 1954; Hoffman, 1972; Kilmister, 1973; Swenson, 1979)' so his theoretical work is viewed in this regard as a historical footnote and no more. The mathematician and historian E.T. Bell has gone so far as to characterized Clifford's anticipation of Einstein as little more than a case of some lucky person hitting "the side of a barn at forty yards with a charge of buckshot" (Bell, 1937), but this view of history is completely false. While Clifford's physical theories have gone unnoticed, Clifford numbers and his system of bi-quaternions have found new uses in some modern interpretations of quantum theory and relativity (Power, 1970; Gurney, 1983; Chisholm and Common, 1985) even though they were originally developed to describe his four-dimensional space, a fact that should imply new ways of interpreting the quantum.
Many modern scholars have mistakenly interpreted Clifford's theoretical model of a four-dimensional space in physics against a historical mindset biased by an early twentieth century view of general relativity (Beichler, 1996). Clifford's main purpose was not to develop a new theory of gravity, as did Einstein several decades later. Clifford's original theoretical work only dealt with Maxwell's electromagnetic theory even though he planned to add gravity to his theory at a later date (Clifford, 1887), if he had not died. Actually, Clifford was developing what we would today consider a unified field theory or better yet a theory of everything. He was fond of saying that he was " solving the universe" (Pollock in Clifford, 1879),which was his way of describing a single theory that covered all of the natural forces. Clifford attempted first to explain magnetic induction, not gravity, with his four-dimensional geometry (Pearson in Clifford, 1885). Magnetic induction is governed by the equation B = V@A, providing a direct link between the current logical argument for a four-dimensional space and Clifford's interpretation of Maxwell's electromagnetic induction.
Clifford published numerous mathematical papers on the motion of three- dimensional matter in four-dimensional elliptical (single polar Riemannian) spaces. He also published a book that actually presented his first step in building a proper theory, that is, for any of his peers who understood what he was trying to do. Historians and scholars today do not understand what Clifford was
Five Dimensions of Space-Time 529
attempting to accomplish, so they only see the book as a simple introductory trea- tise on kinematics. Anyone looking for a completed gravity theory in Clifford's work simply will not find it. Nearly all modern historians have mistakenly claimed that he never published his theory because they are looking for a nonexistent gravity theory with time as a fourth dimension.
Clifford expressed the opinion that all energies are either potential or kinetic (Clifford, 1880), but he also believed that kinetic energies in three-dimensional space would become potential energies in his four-dimensional spatial frame- work. In other words, forces in three-dimensional space would reduce to constant variations in position along paths in a four-dimensional curved space, an idea that was made current in general relativity. However, the modern concept only deals with gravity as modeled by modem relativity theory while Clifford meant to apply the concept to all forces in his model. Upon this hypothesis, he published the first volume of a series of books titled Elements of Dynamic (Clifford, 1878). His first volume was subtitled Kinematics. Everyone that knew Clifford or his work knew that dynamics in three-dimensional space is just kinematics in Clifford's four-dimensional space, that is why he referred to his explanation of Dynamics as Kinematics in the book title. He was writing about four-dimensional kinematics, which was equivalent to three-dimensional dynamics in his mind and theoretical model. Coincidentally, this same book is recognized by historians as the first published statement by a mathematician that distinguished between the cross and dot products in vector algebra (Crowe, 1967), the same dot and cross products that are used in the vector and scalar representations of magnetic potential given above. It should be clear then that Clifford understood the four- dimensionality of magnetic potential a full century before the modem scientific community took the unification of gravity and electromagnetism seriously.
In developing his theory, Clifford faced the problem that no mathematical formalism existed to express his four-dimensional ideas. So he used a form of quaternions of his own invention (bi-quaternions) to express his four- dimensional model of space (Clifford, 1882). Unfortunately, quaternions lost favor in the late nineteenth century to vectors and their use was largely aban- doned during the first few decades of the twentieth century. So no one today would even recognize that Clifford's mathematics represented his four- dimensional theory of physical reality. Einstein's theoretical work on a theory of gravity used the Levi-Civita tensor formalisms that had developed along a different line of reasoning than Clifford used for his quaternion algebra. The tensor calculus used by Einstein was only developed after Clifford's death.
As stated above, Clifford did not ignore the effect of his four-dimensional model of matter on the Newtonian theory of gravity. Clifford died of consumption in 1879 at the age of 34 and never completed his research, but it is still possible to discover what he planned to eventually accomplish with his four-dimensional model. His colleagues were so impressed with his theoretical ideas that both his published and unpublished works were collected, edited and published within a decade after his death. His followers and colleagues
530 J. E. Beichler
published everything that they could find, including lecture notes of classes that he taught, because they thought that his theoretical work was important enough to save for posterity and the future. Clifford's outline for the second volume of his Elements of Dynamic was among the unfinished works that were published. His student Robert Tucker edited this book. In it, Clifford stated his views on the theory of gravity and outlined how he would change gravity given his new four- dimensional geometry, thus indicating the fact that he was searching for, and may have found but never published, a unified field theory. But we will never know that fact for sure.
Of course, philosophical and mathematical arguments are not as valuable in science as observation and experimental verification. Yet there is some experi- mental evidence supporting the existence of magnetic potential in the Aharonov- Bohm effect (Aharonov & Bohm, 1959). In the Aharonov-Bohm experiment, an electron beam is split in such a manner that the two resulting beams pass on either side of an upright solenoid before coming back together on a screen. The solenoid is oriented in such a way that the twin beams cut across the field lines (perpendicular to B) and thus the net force acting on them is zero. Yet when the beams come together at the screen they interfere with each other. The interference clearly shows that the wave functions associated with the electron beams are out of phase, yet they should not be out of phase by the normal standards of Maxwell's electromagnetic theory. Although the effect is somewhat paradoxical, it is normally interpreted as evidence that the magnetic potential associated with the magnetic field is real even though it cannot be measured or experimentally determined. While the net force is zero, an integration of the potential A in a closed loop around the coil is not zero. The common interpretation of this experiment introduces a quantum solution (Bohm & Hiley, 1993). However, this effect can be simply explained and understood within the four-dimensional framework of electromagnetic induction. In other words, a classical electromagnetic interpretation can be used to explain the results if a physically real four-dimensional space that is associated with the magnetic vector potential is assumed.
While the net force is zero on either of the electron beams, the electrons are moving at a constant speed through different portions of the coil's mag- netic field. So they each follow paths of varying potential (surfaces) in four- dimensional space corresponding to the portions of the magnetic field through which they travel. Since they are following four-dimensional paths of different lengths, they are out of phase when they reach the screen and interfere with each other. The principle is similar to a satellite orbiting the earth at a constant speed. The constant speed holds the satellite to a path along a gravitational equi- potential surface. When the speed changes, the satellite follows a path through different equipotential surfaces. The orbital speed determines the altitude of the orbit and the potential path (surface) along which the satellite travels. The electrons in the beam also follow curved potential paths in the fourth dimension, which are different according to the portions of the magnetic field through which
Five Dimensions of Space-Time 531
they pass in three-dimensional space. The difference in curved paths in four- dimensional space puts them out of phase at the end of the trip even though their paths in three-dimensional space, the projections of their paths in four- dimensional space, are not curved.
And finally, given a real fourth dimension of space that is characterized by magnetic potential, anything that emits a normal transverse electromagnetic wave in three-dimensional space would also cause a corresponding compressive wave of magnetic potential variation in the fourth direction of space. Numerous scientists have claimed to show the mathematical possibility of such longitudinal electromagnetic waves. Edmund T. Whittaker's model of 1903 is perhaps the best known of these attempts (Whittaker 1903, 1904). According to Whittaker,
... thus we have the result, that the general solution of Laplace's equation
wheref is an arbitrary function of the two arguments z+ix cos u+iy sin u and u.
Moreover, it is clear from the proof that no generality is lost by supposing thatf is a periodic function of u (Whittaker, 1903).
The variable u actually represents the fourth dimension of space while V is the magnetic potential. This interpretation renders Whittaker's formulation com- patible with modem advances in the laws of electromagnetism without surren- dering the possibility of a longitudinal electromagnetic wave. The function f is periodical with respect to u, which means that the fourth dimension is closed with respect to the other three dimensions of space. This closure corresponds completely to Kaluza's closure condition for the fifth dimension of space-time, while the factor of du over which the function f is integrated corresponds to Penrose's gauge invariance dO.
In this respect, the fourth dimension of space is independent of the length of the extension in the fourth direction, such that the fifth direction of space-time can be either microscopic or macroscopic in extent. There is no difference between the two in the functionf as long as the fourth dimension of space is closed. Whittaker then analyzed the general form of the differential equations for wave motion
to demonstrate that the mathematical model can account for a longitudinal
532 J. E. Beichler
electromagnetic wave. However, if V is taken to mean the magnetic potential in the fourth direction of space, then the magnetic potential V can be related directly to the concept of proper time in special relativity. Whittaker's concept
I of a longitudinal component of electromagnetic waves can thus be rendered
~
in relativistic terms, which implies that the concept is actually a wave of changing magnetic potential propagating in the fifth direction of a five- dimensional space-time continuum.
Whether or not Maxwell's electromagnetic theory requires a longitudinal wave in its classical three-dimensional interpretation is open to debate, but the existence of a fourth dimension to space would require a corresponding longi- tudinal wave that propagates throughout the fourth dimension relative to the normal three dimensions of space. No one has ever detected a three-dimensional longitudinal wave, but that does not mean the wave cannot be four-dimensional. After all, no one has ever detected or measured a 'magnetic-volt' of potential in three-dimensional space either, even though the potential exists in four- dimensional space.
The Yukawa Field
Modern physics also requires the existence of a fourth spatial dimension, but this time the culprit is the Yukawa potential. The Yukawa potential normally takes the form
The quantity g is real. It represents the coupling constant between the meson field and the fermion with which it interacts, at least in the normal quantum interpretation. The Yukawa potential itself arises from the exchange of a massive scalar field or particle such as the pi meson or pion (Yukawa, 1935). The nega- tive sign guarantees that the force between particles in the nucleus is always attractive.
This potential is associated with the extremely short-range strong nuclear force and it is usually only interpreted as a quantum phenomenon. The potential associated with the Yukawa field decreases exponentially, guaranteeing the short range of the Yukawa field to little more than the outer boundaries of the nucleus. It is simply assumed that the Yukawa field cannot be interpreted within a non-quantum context, yet there is no hard and fast rule that states that the Yukawa potential cannot be interpreted geometrically. Classical fields are nor- mally interpreted geometrically, so it would seem that the Yukawa field should also have a geometrical interpretation. Even the modern view of gravity as resulting from the curvature of space-time is geometrical in nature.
According to a simple interpretation of physical laws, the field strengths of both electric and gravitational fields vary as llr2. Traditionally, this inverse square law has been interpreted as resulting from the three-dimensionality of
Five Dimensions of Space-Time 533
this may seem, the inverse square law has been used in the past to explain the necessity of a three-dimensional space to the laws of physics (Whitrow, 1955; Abramenko, 1958; Biichel, 1963; Freeman, 1969). In other words, the inverse square law is normally thought to imply (if not prove) that space 'must be' three-dimensional. It has also been a common practice in the past to criticize higher-dimensional theories by pointing out that gravity would not work in a higher-dimensioned space because the inverse square law would not apply. However, we commonly accept the notion of a four-dimensional space-time without any alteration to the inverse square law without realizing that we do so. The fourth dimension of time is both qualitatively and quantitatively different from the normal three dimensions of space, so it does not affect the inverse square law. By the same token, there is no hard and fast rule that unequivocally requires that a fourth dimension of space would be both quantitatively and qualitatively the same as our normal three dimensions of space. In fact, given the reality of a fourth dimension of space, nature seems to have ordained that the fourth dimension is different from our normal three dimensions of space and nature rules physics instead of the other way around. So there is no valid or compelling reason to assume that a fourth spatial dimension would have any effect on the inverse square law and gravity. In fact there are reasons to believe that the opposite is true.
Many scientists have long believed that matter is electrically constituted and electricity acts according to the inverse square law. Our perception of space is dependent on the relative positions of matter in that space. So if matter is three- dimensional we sense space as three-dimensional. The three-dimensional surface curvature of a material particle or material body may be sufficient to determine the three-dimensionality of space, but the complete three- dimensionality of the particle is not necessary according to how it outwardly appears. Nor is it complete. The interior portion of a material particle could still be higher dimensional. For instance, the interior of a proton could be a physical singularity stretching into a higher fourth dimension even though the exterior surface of the proton is still curved spherically in three-dimensional space. Space
1 could have any number of dimensions while three-dimensional matter only determines that part of the space or manifold in which the electrical field acts and reacts. Our normal senses evolved in the three-dimensional material world of nature, so they would be limited to detect only the three-dimensionality of matter even given a real fourth dimension. Since gravity acts between material particles, which are three-dimensional due to their electrical nature, it would also act three-dimensionally even if space had four or more dimensions. While it is commonly argued that space is three-dimensional because of the inverse square law, it could also be argued that we only sense three out of a greater number of dimensions because of the inverse square law by which gravity and electricity act as they do in three dimensions.
It seems that the inverse square law only guarantees the three-dimensional actions and interactions of matter, not the other way around. The forces
534 J. E. Beichler
associated with common fields act three-dimensionally and no more. The inverse square law does not guarantee that either space itself or fields in general are three-dimensional or otherwise limited to three dimensions. Fields could be higher-dimensional entities just as space could be higher dimensional even though we only sense three dimensions of space. Matter reacts with fields in three- dimensional space because matter is outwardly three-dimensional, not because fields are three-dimensional. If fields are higher dimensional, there may be field- field interactions that occur only in the higher dimensions of space and thus remain undetected in the three-dimensional material space except by their sec- ondary effects. An effect such as quantum entanglement could be explained in this manner. When all is taken into account, neither physical fields nor space need be limited to three dimensions by either the laws of nature or logic and reason.
On the other hand, the potentials associated with fields vary as llr. So
a physical field associated with a particular potential has one more factor of the
2
variable 'r' than the potential itself because fields vary as l/r . The dimen-
sionality of the space that the field occupies is generally two greater than the exponent of the variable 'r' in the denominator of the formula representing the potential. This logic also follows for the Yukawa potential: The variable 'r' in the denominator reflects the three-dimensionality of the field, but there is another term with an 'r-' factor in the exponent in the numerator of the formula. The variable 'r' in the numerator of the formula could easily represent another dimension, so the Yukawa potential would require that the space occupied by the Yukawa field is four-dimensional, not three-dimensional. The exponential term eKkrrepresents both the geometrical structure of the particle and its associated field as extended into the fourth dimension of space. The extension of a particle in the fourth direction would occur internally relative to three-dimensional space so that the part of the material particle that we sense or detect remains the three- dimensional exterior surface of the particle.
In this model of the Yukawa potential and field, the variable 'r' in the denominator would account for the spherical shape of elementary particles and the nucleus itself. By analogy, this would indicate that the exponential term in the numerator would refer to the geometrical shape of the Yukawa field in the higher fourth dimension. If the Yukawa field conforms to the shape of an exponential curve in the higher dimension, as opposed to the spherical shape in three-dimensional space, then the fourth dimension of space is most certainly different from the other three dimensions of normal space, as noted above.
In fact, elementary particles such as protons and neutrons would be small singularities according to the general theory of relativity; or rather they would be singular at their centers. They would therefore follow curved space-time in a shape similar to a rotated exponential curve, as shown in a normal drawing of the curved metric of a singularity (see Figure 1).
So the Yukawa field would correspond to the shape of a nucleus or elementary particles predicted by relativity theory, if general relativity is taken to depict a real curvature of three-dimensional space in a higher embedding fourth
Five Dimensions of Space-Time 535
Exponential curves define the outer shape of the singularity in
Fig. 1. The internal curvature of an elementary particle.
dimension of space. At this point, there is no need to assume a dimensionality greater than four as used in some recent theories, although there are no re- strictions on space having more than four dimensions. Moreover, the curvature of space-time in general relativity is a function of the mass of a particle or body. The constant k in the Yukawa potential is also related to the mass of the exchange particle between nucleons. In both cases, the mass is related to the curvature explicit in the mathematical model, which indicates that the Yukawa potential could be modeled by the curvature of space-time as expressed by the theory of relativity rather than the particle exchange concept of quantum field theory. In either case, the Yukawa potential logically requires that space is four- dimensional and thus the space-time continuum of relativity is five-dimensional. The relationship between the Yukawa potential and general relativity leads to the third logical proof that space is four-dimensional, only this time the proof deals with the macroscopic world of the greater universe rather than the microscopic world of the quantum.
The Cosmological Connection
In the late 1920s, Edwin Hubble observed that other galaxies were receding from our Milky Way galaxy with increasing speed as the distance to the other galaxies increased. These observations indicated that our universe is expanding. Georges-Henri Lemaitre and others who developed the expansion hypothesis by a theoretical application of general relativity had already predicted the expansion. The marriage of observation and theory in this case produced one of the most spectacular successes for science in the twentieth century. The simple notion of an expanding universe is usually explained by analogy to a two- dimensional surface expanding in a third dimension.
A good example would be a balloon with spirals drawn on its surface to represent galaxies. When the balloon is blown up and expands, the spirals spread
536 J. E. Beichler
apart and move away from each other in the same pattern of motion that the receding galaxies show during astronomical observation. The expanding surface of the balloon is analogous to our expanding universe, the difference being that the balloon is a two-dimensional surface expanding outward in a third direction while the universe is a three-dimensional surface expanding into 'who knows what'. Although the phrase 'who knows what' is not an appropriate phrase for scientific use, it does represent how science views the question of what the universe is expanding into.
Some versions of modern brane theory postulate variously dimensioned branes curved in higher-dimensional bulks, so brane theorists could claim that the universe is expanding into the embedding bulks. However, brane theories have other problems to overcome: There is a discontinuity between the branes and the bulks in which they are embedded, such that the branes and bulks are separate things. As such, they break the continuity of the space-time continuum. The brane theories are based upon Klein's interpretation of Kaluza's five-dimensional theory of space-time, but they violate the basic assumptions upon which Kaluza unified electromagnetism and gravity as expressed by general relativity: Kaluza assumed the continuity of four-dimensional space-time with the fifth and higher dimension. So it would seem that the brane theories as well as the superstring theories upon which they were conslrucled are at odds with their own basic premise.
However, the balloon analogy gives more information about the expansion than ordinarily suspected, which implies an answer to this unanswered question about what the universe is expanding into. The spirals drawn on the balloon's surface are all rotating and expanding relative to a single point, the geometric center of the balloon, rather than any center on the surface of the balloon. This part of the analogy is often used to argue that our universe has no center within its three-dimensional expanse, which is true. The curvature of space-time in general relativity has always been considered an intrinsic property of space-time such that a higher embedding dimension has been unnecessary to explain observed and suspected phenomena. However, a higher embedding dimension, demonstrating that the curvature of space-time is an extrinsic property, is still perfectly compatible with general relativity (Misner et al., 1973). Extrinsic curvature is sufficient to explain the effects of general relativity, but has never been considered necessary as long as the idea of intrinsic curvature was con- sidered more likely. But if the concept of extrinsic curvature and a higher embedding spatial dimension does not represent our true reality, simple rela- tivity will be violated in the case of the expanding universe and other astronomical observations.
In the balloon analogy, as stated above, the plane of rotation of the spirals and the recession of the spirals as the balloon expands are all oriented relative to a single point, the center of curvature of the balloon's surface. In the real three-dimensional spatially extended universe, all of the galaxies rotate and recede from each other at all possible angles or orientations in three-dimensional space. Yet you cannot have a mathematical property true for one configuration
Five Dimensions of Space-Time 537
of spatial dimensions (two dimensions embedded in three-dimensional space) that is not true for another configuration (three dimensions embedded in a four- dimensional space). Such an inconsistency would destroy the validity of the mathematical model. The general geometric properties are the same for all spaces and embedding manifolds for an n-dimensional geometry embedded in an n+l-dimensional manifold. Riemannian geometry is based upon this simple idea. So, there is a logical necessity that the orientation of all of the galaxies in the expanding universe be relative to a single point or center of curvature of the universe. The natural rotations of galaxies in the universe are all relative to the same point, and the planes of galactic rotation are all tangential to the three- dimensional surface that is our space, which is perpendicular to the real extrinsic radii drawn between them and the center of a physically real curvature of our universe in a fourth spatial dimension.
In this case, it is illogical to speak of the overall curvature of the universe and then deny the reality of the higher embedding dimension because of a human sensory and perceptual bias against the possibility of a fourth spatial dimension. Perhaps local spatial curvature can be explained away as an intrinsic charac- teristic of the space-time continuum, but the concept of intrinsic curvature on a global level is untenable. The notion of an intrinsic radius of curvature for the whole of the universe is illogical. The three-dimensional surface of our universe is closed such that it forms a Riemannian sphere, which would require a higher embedding dimension to account for the closure. Once again, the only way to derive a direction perpendicular to all three dimensions of space simultaneously would be to adopt the geometry of a real four-dimensional embedding space. That fourth dimension or direction is orthogonal to the normal three dimensions of space. So the observed three-dimensional orientation of astronomical bodies directly requires the reality of a fourth spatial dimension. In effect, our three- dimensional universe is expanding into a fourth dimension of space. The simple fundamental notions of relative motion and actual observation, rather than any specific theory, logically require that our space is four-dimensional and thus space-time is five-dimensional.
The Kaluza Confirmation
While these logical proofs may not be completely persuasive or even persuasive enough to sway the attitudes of many within the general scientific community, there are other extenuating factors and circumstances that should be persuasive given the validity of the logical proofs. Also, these three logical proofs should be considered independent of any particular hyper-dimensional theory of space-time. They only indicate that some higher-dimensional theory would give a more correct picture of our physical reality without specifying the exact theory to be used. Yet there is already a specific scientific theory that successfully utilizes a five-dimensional space-time geometry to unify general relativity and electromagnetism: Kaluza's 1921 theory. Kaluza's theory has been largely ignored in spite of its successful derivation of Maxwell's electromagnetic
538 J. E. Beichler
theory from the general relativity of a five-dimensional space-time continuum. Most modern scientists are only familiar with Kaluza's theory through its association with the work of Oskar Klein, altering the theory to the Kaluza-Klein model of space-time. Little is known of Kaluza's original theory under these circumstances. Klein's subsequent adaptation of the theory (Klein 1926a, 1926b, 1927) was an attempt to incorporate quantum theory into the geometry of space-
time. But Kaluza's theory can stand alone on its own merits, without considering 7
Klein s extended version of the theory into the realm of the quantum. Kaluza's original theory had nothing to do with the quantum.
According to Kaluza's original theory, two mathematical conditions are necessary to unify general relativity and electromagnetic theory. All points in the four-dimensional space-time continuum are extended orthogonally into the fifth dimension along what Kaluza called A-lines. The A-lines follow circular paths in the fifth direction back to our space-time continuum, so they are closed with respect to the fifth direction. Kaluza's first condition was to close the system in the fifth direction, but the A-lines were also required to be of equal length, giving the second condition. Kaluza also suggested that the A-lines are infinitesimally short to guarantee that we could not detect the fifth dimension, although this suggestion was not a required mathematical condition. The two conditions were necessary to guarantee the mathematical consequences of add- ing the fifth dimension: Deriving the equations of general relativity by applying a four-transformation while obtaining the equations of electromagnetism by applying a cut-transformation.
If either of the initial conditions were to be changed or relaxed in any manner, it is possible and even likely that the results of the change would render electromagnetism and gravity incompatible if not break Kaluza's link between them altogether. But Kaluza also assumed, without so stating, a third condition of continuity in the fifth direction. Continuity was built into the calculus that Kaluza used to develop his geometrical model. So if continuity is forfeited, then Kaluza's theory could still fall apart. Before any of these conditions is changed in new extensions of Kaluza's theory, it must be shown that any of these changes, or any combination of them, does not alter Kaluza's results, the unifi- cation of gravity and electromagnetism. There are no middle roads to take here; it is all either black or white. If Kaluza's initial conditions were altered in any manner that breaks or weakens the link between gravity and electromagnetism, then the extension would be invalid for having destroyed the very foundations upon which the new theory is based. Yet changes in these conditions have been made to expedite the development of modern theories and thus could have a direct bearing on the validity of the supergravity, superstring and brane theories, all of which depend on extended versions of the Kaluza-Klein model.
When Klein adopted Kaluza's theory in an attempt to quantize the unified field, he did not relax or alter Kaluza's conditions. He merely followed Kaluza's suggestion that the extension in the fifth direction must be extremely small since we cannot detect the extra dimension. Klein equated the periodicity in the
Five Dimensions of Space-Time 539
'closed loop' condition to the quantum of action. At the time, Klein's version of the theory was largely ignored by the scientific community, which was mesmer- ized by other developments in quantum theory such as quantum mechanics and wave mechanics. Unfortunately, Klein could not make his theory work. He rejected his first theory and made two later attempts to rectify the errors in his theory, in 1939 and 1947 (Klein 1939, 1947), but eventually rejected his basic hypothesis and gave his theory up as a lost cause.
Klein's adaptation of Kaluza's theory, the Kaluza-Klein theory, was re- discovered in the 1970s and adopted by supergravity theorists as a method to unify gravity with the latest versions of the quantum field theories and the standard model of elementary particles. The superstring theorists adopted the Kaluza-Klein theory a few years later, but both groups of theorists have expanded the number of dimensions to 10,11or more. However, these scientists have never demonstrated that adding the extra dimensions above Kaluza's original five would remain consistent with the original purpose of Kaluza's theory to unify general relativity and electromagnetism. These theories are untenable and speculative and they will remain so until superstring theorists can demonstrate that adding the extra dimensions does not alter the connection between Einstein and Maxwell's theories that Kaluza's five-dimensional structure established.
On the other hand, any extension of the Kaluza-Klein theory that is super- imposed on a quantum field theory should also suffer from fundamental problems because quantum field theories are by their very nature based upon a discrete model that is at odds with the assumed condition of continuity in Kaluza's original theory. Nor have the superstring theorists explained how the curvature of space-time fits into their theories, even though they take general relativity for granted as the basis of their theories. Any Kaluza or Kaluza-Klein theory that retains the infinitesimal (or Planck) extension of length in the fifth direction must deal with the same fundamental problem. The adoption of a real physical five-dimensional space-time structure, instead of a limited purely mathematical model, implies that curvature is an extrinsic characteristic of our common four-dimensional space-time continuum. However, an infinitesimally extended fifth direction seems to retain the intrinsic nature of the four- dimensional space-time by not explaining how the concept of curvature fits into the model, creating a paradox.
The superstring theories have evolved into the more general 'brane' theories. Several 'brane' theorists have speculated on all types of structures including dual three-dimensional branes, five-dimensional branes, colliding branes and curved branes within a bulk, to mention only a few examples. But it seems that they have yet to demonstrate whether these geometrical structures conform to the basic hypotheses upon which their theories depend, Kaluza's initial derivation of the general relativity and electromagnetic formulas from an extremely limited and conditional five-dimensional mathematical model of a continuous space- time. The Randall-Sundrum theory offers a case in point (Randall & Sundrum,
1999a, 1999b). In the Randall-Sundrum model, two branes are separated
1
540 J. E. Beichler
by a higher-dimensional bulk. One of the branes represents our common three-dimensional curved space, while gravitons traveling from our brane to the other brane are the only direct links between the branes. In one model, the second brane is an infinite distance away, effectively limiting our world to the single brane embedded in the bulk and guaranteeing a weak gravitational force. However, this model is in direct violation of Kaluza's condition that our four- dimensional world is closed with respect to the higher fifth dimension. Brane theories of this type must be required to demonstrate that their models do not disrupt the unification of electromagnetism and gravity in the Kaluza model upon which they are based. Yet no one has ever argued or even explored how such changes would affect the basic underlying principles of the original mathematical unification model developed by Kaluza.
The only theoretical research ever conducted to determine the mathematical consequences of changing Kaluza's theory only considered the relaxation of his initial suggestion of an infinitesimal extension, rather than changing any of his initial conditions. Einstein and Peter G. Bergmann completed this change in 1938 (Einstein & Bergmann, 1938). Einstein, Bergmann and Valentine Bargmann again considered it in 1941 (Einstein et al., 1941). They retained the 'closed loop' and 'equal length' conditions and remained within a continuous mathematical model of five-dimensional space-time, but allowed for the possibility of macroscopically extended lengths of the A-lines. Under these conditions, they were still able to derive Maxwell's formulas and thus maintain Kaluza's unification. But Einstein eventually gave up this avenue of research toward his goal of a unified field theory because he could not justify the notion of a normal sized fifth dimension that could not be sensed or detected in any manner. Even so, Einstein listed the five-dimensional approach as one of three possibilities to develop a unified field theory in his last published book before he died (Einstein, 1956). He stipulated that the five-dimensional hypothesis would only be tenable if it could be explained why the fifth dimension cannot be detected.
Conclusion
These three logical proofs, in themselves, will not immediately change the course of science. Science has ignored the implied existence of a real fourth spatial dimension for more than a century, so it will not be so easily compelled to accept it now. However, it is not just the three logical proofs that indicate the existence of a fourth spatial dimension to our universe. It is a preponderance of the evidence that will soon force science to accept the four-dimensional reality of space. The value of these three logical proofs will only become evident over [he lvnger term of scientific advances.
While logically proving the existence of a fourth dimension to space, these proofs also imply the geometric structure of that dimension relative to the other three. First of all, the fourth dimension of space would be different, like time, from the other three common dimensions of space. Otherwise, four- dimensionality would adversely affect the inverse square law and thus conflict
Five Dimensions of Space-Time 541
with normally accepted physical laws. Instead, the fourth dimension should be characterized by changing magnetic potential except inside elementary particles where the space curvature corresponding to matter would assume the shape of an exponential curve. Both of these characteristics imply that the total extension of space in the fourth direction cannot be infinitesimally small or even microscopic as in Klein's version of Kaluza's theory. The exponentially shaped singularity at the center of elementary particles such as protons would require a non- infinitesimal extension of space in the higher dimension.
In other words, if the magnetic potential and Yukawa potential exist in nature as described, then the fourth dimension of space, or the fifth dimension of space- time, cannot be infinitesimally extended. Both logical arguments imply that the extra higher dimension is macroscopically extended as Einstein, Bergmann and Bargmann demonstrated. It is provident that Kaluza's theory has already been developed as the basis for a new unification, but the macroscopic extension in the fourth direction of space means that the present unification theories that are based upon Kaluza's suggestion and Kaluza-Klein models are not valid. The path of unification that science must follow is the path that physics and nature leads us down, not the path that some scientists decide that nature must logically follow, no matter how 'beautiful' or aesthetically pleasing those theories might be. The path that nature has decided for science is the one that leads to the four- dimensionality of space (the Clifford model) and the five-dimensionality of the space-time continuum (the Einstein-Kaluza model).
Much of the early work on five-dimensional space was in an attempt to develop a theory that unifies the four fundamental interactions in nature: strong and weak nuclear forces, gravity and electromagnetism. German mathematician Theodor Kaluza and Swedish physicist Oskar Klein independently developed the Kaluza–Klein theory in 1921, which used the fifth dimension to unify gravity with electromagnetic force. Although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century.
Space-time--time couples Kaluza's five-dimensional geometry with Weyl's conformal space-time geometry to produce an extension that goes beyond what either of those theories can achieve by itself. Kaluza's ``cylinder condition'' is replaced by an ``exponential expansion constraint'' that causes translations along the secondary time dimension to induce both the electromagnetic gauge transformations found in the Kaluza and the Weyl theories and the metrical gauge transformations unique to the Weyl theory, related as Weyl had postulated. A space-time--time geodesic describes a test particle whose rest mass, space-time momentum, and electric charge q, all defined kinematically, evolve in accord with definite dynamical laws. Its motion is governed by four apparent forces: the Einstein gravitational force, the Lorentz electromagnetic force, a force proportional to the electromagnetic potential, and a force proportional to a scalar field's gradient d(ln phi). The test particles exhibit quantum behavior: (1) they appear and disappear in full-blown motion at definite events; (2) all that share an event E of appearance or disappearance do so with the same charge magnitude |q| = phi(E); (3) conservation of space-time--time momentum at such an event entails conservation of electric charge in addition to conservation of space-time momentum, among the participating particles; (4) at such events the d(ln phi) force infinitely dominates the other three --- this strongly biases the appearance and disappearance events to be concentrated deep in the discretely spaced potential wells of ln phi, and sparse elsewhere.
To explain why this dimension would not be directly observable, Klein suggested that the fifth dimension would be rolled up into a tiny, compact loop on the order of 10-33 centimeters. Under his reasoning, he envisioned light as a disturbance caused by rippling in the higher dimension just beyond human perception, similar to how fish in a pond can only see shadows of ripples across the surface of the water caused by raindrops.[2] While not detectable, it would indirectly imply a connection between seemingly unrelated forces. The Kaluza–Klein theory experienced a revival in the 1970s due to the emergence of superstring theory and supergravity: the concept that reality is composed of vibrating strands of energy, a postulate only mathematically viable in ten dimensions or more. Superstring theory then evolved into a more generalized approach known as M-theory. M-theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings. The other 10 dimensions are compacted, or "rolled up", to a size below the subatomic level. The Kaluza–Klein theory today is seen as essentially a gauge theory, with the gauge being the circle group.
The fifth dimension is difficult to directly observe, though the Large Hadron Collider provides an opportunity to record indirect evidence of its existence. Physicists theorize that collisions of subatomic particles in turn produce new particles as a result of the collision, including a graviton that escapes from the fourth dimension, or brane, leaking off into a five-dimensional bulk. M-theory would explain the weakness of gravity relative to the other fundamental forces of nature, as can be seen, for example, when using a magnet to lift a pin off a table — the magnet is able to overcome the gravitational pull of the entire earth with ease.
Mathematical approaches were developed in the early 20th century that viewed the fifth dimension as a theoretical construct. These theories make reference to Hilbert space, a concept that postulates an infinite number of mathematical dimensions to allow for a limitless number of quantum states. Einstein, Bergmann and Bargmann later tried to extend the four-dimensional spacetime of general relativity into an extra physical dimension to incorporate electromagnetism, though they were unsuccessful.[1] In their 1938 paper, Einstein and Bergmann were among the first to introduce the modern viewpoint that a four-dimensional theory, which coincides with Einstein-Maxwell theory at long distances, is derived from a five-dimensional theory with complete symmetry in all five dimensions. They suggested that electromagnetism resulted from a gravitational field that is “polarized” in the fifth dimension.
www.scientificexploration.org/docs/21/jse_21_3_beichler.pdf
The main novelty of Einstein and Bergmann was to seriously consider the fifth dimension as a physical entity, rather than an excuse to combine the metric tensor and electromagnetic potential. But they then reneged, modifying the theory to break its five-dimensional symmetry. Their reasoning, as suggested by Edward Witten, was that the more symmetric version of the theory predicted the existence of a new long range field, one that was both massless and scalar, which would have required a fundamental modification to Einstein's theory of general relativity. Minkowski space and Maxwell's equations in vacuum can be embedded in a five-dimensional Riemann curvature tensor.
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 dimension. 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 path of a moving infinitesimal (test) particle. 'T Hooft has speculated that the fifth dimension is really the spacetime fabric.
<a href="https://en.wikipedia.org/wiki/Five-dimens
Few months ago while I was doing my Calculus homework somehow this word “Evaluate” caught my attention. I knew that it inspired me to write something related to Christianity but I didn’t have a very clear idea what I should write about.
However, a couple of days afterwards while I was waiting at the dentist office and reading Reader’s Digest, the staff of the magazine reported that Toronto is the second honest city in the world among all the cities they have tested. What they did is that in 2007 they travelled the world and put cell phones in public places in specific world famous cities and noted the percentage of the people who were honest enough to return the cell phones in each city. That immediately caught my attention because in 2007 while I was getting on the bus at the subway station (don’t you just love my subway station stories!) I found a cell phone on a seat beside the door. I picked it up and got off the bus trying to find its owner. I couldn’t find the owner because there were over 20 people who got off the bus and were heading in different direction in a hurry. So I gave it to the bus driver, since the bus services in Toronto offer a “lost and found” service. At the time I thought it was kind of odd that nobody found the cell phone while they were existing the bus since the seat was beside the door, or why would a person who is sitting right beside the door exist at the end!
So when I came home and was doing my Calculus homework again I remembered the word “Evaluate” and decided to photograph it. So I set my camera on “super micro” mode and got very close to the paper: the lens was about 0.5 cm away from the paper. And since the sun was setting its light was shining on the paper on a very steep angle so I didn’t have to worry about the lens overshadowing the paper.
After I transferred the image to my PC and viewed it I was amazed by the amount of details in the shot. It seems that under careful and very close observation the smooth surface of the paper looked more like the coarse surface of a rock! And I thought: Perfect…that’s exactly what the world “Evaluate” means to Christians!
You see, because I don’t have any close people to me, nobody knows what I am going through or what I am doing. That’s why I have to always go to my heavenly Father asking Him for guidance and correction. I had not always obeyed Him, but I had never disobeyed Him and not knew it before hand. Sometimes God’s will is not very clear and requires us waiting, but when it comes to sin He always makes His will known. Every time I sinned, sadly, I knew I was doing so. That’s why we feel guilt right after we sin—not because we became perfect right after sinning!—but because we realized that sin did not deliver what it promised as we believed Satan’s lie. In other words, we knew that sin promised something but God said that it was a lie, yet we believed Satan’s false promise—just as the Holy Spirit told us before we sinned.
When I became a Christian I started to realize that my old human standards governed by my conscious were obsolete because the new standards of God governed by the Holy Spirit have taken the definitions of sin and holiness to a whole new level. A level only God’s children living by the guidance and power of the Holy Spirit can live up to.
I didn’t take evaluating my life seriously until I came to Canada and that happened only after few incidents in my late teens.
When I came to Canada in late of 1999 I didn’t speak English, and being enrolled in high school I wanted to excel in studying as I did back home but English was a barrier so I decided to tackle it as best as I could as I went through high school. I started reading books, mainly novels. In the beginning I read those teen suspense novels, you know the ones that always have a weird character and then somehow the good looking guy or girl end up being the bad guy! You know, the ones that are always made up of 120 pages for some reason! My reading was very slow because I am a perfectionist by nature so I looked up every word and phrase, etc. But my ESL (English as a Second Language) teacher told me that when I come across a word that I should first guess its meaning from the contest of the sentence and not bother looked it up! That worked miracles, except that when my mom asks me now what does a certain word mean I have no clue how to explain it in Arabic! I think one semester I read about 38 such novels! Then as my English improved I moved up in my level in reading and started asking my mom what I should read and she told me to read books such as Great Expectations, Wuthering Heights, Jane Eyre, etc, and I loved them…especially Jane Eyre, and I thought that I wouldn’t read a finer novel afterwards! Then in 2002 my mom told me that I should read Gone with the Wind and I thought, “Yea, why not, another novel…” But it was not just another novel: it was THE NOVEL!
I mean, I cannot describe it to you if you haven’t read it! I’ve seen the movie and with all its awards it is nothing comparing with the novel! I remember when I lied in my bed starting to read it thinking to myself that this is different, this is on a whole new level by itself! I read so many wonderful pages in the beginning and they were simply setting up the settings for the first scene! I was so immersed into the story by then that when one evening I saw a Honda Accord passing by the window I kind of got disturb thinking, “What is this! There are no cars in 1860s!” The novel was so well written that it took me to its world rather than it came to mine!
I mean I couldn’t believe a human can write something like this! It was a masterpiece! I can’t describe how I felt when I was reading it. Sometimes I would choke trying to hold back tears. Sometimes I just wanted to scream at the characters especially Scarlet O’Hara saying, “What is wrong with you two! Why are you deliberately making a mess of your life! Why don’t you slow down and think about what you really want!” I was so frustrated at times that I didn’t want the novel to end because I knew it wouldn’t end the way all cheesy novels do! The characters were so real that there was no heroes, no perfect people--just plain ordinary people like me. Then I realized that I was not screaming inside at the characters but I was screaming at myself—it was me that I saw in those characters. I was completely stunned by everything: the reality of the civil war, the settings of the locations, the story, but most of all the characters and how they didn’t improve in a period of 10 years! That was the scariest thing that could’ve ever happen to me: that in 10 years or so I would look back at my life and find that I didn’t become more like Jesus and it was very much possible!
I took me few weeks to finish the novel and by then I was so worn out emotionally that I finished reading it at 5 AM (early in the morning but not sure exactly what time…it was still dark outside), and I broke down crying and crying and crying. I just cried out to God because I knew that I very much could be like those characters in 10 years. I can’t remember for how long I cried but it was a while because I can still remember where I was kneeling on the couch as I was crying. I stopped reading novels after Gone with the Wind.
Then, in 2003 after knowing a lady from work for few weeks she told me that she had been a Christian for 10 years, and she attends church regularly! I was absolutely stunned! I mean, I couldn’t have guessed she was a Christian in a million years! Not that I expect us, Christians, to walk with a halo above our heads, but when someone gets to know us they should not be shocked when they find out we are Christians. The least we could do is let our characters show neutrality toward righteous living and sin, not enjoying sin! Again I had to step back and take a look at my life.
Also around that time I’ve done some things in my teen years that until today I regret—wish hadn’t done—and because I knew that I couldn’t change my past I wanted to control my present so I can change my future. And one day after listening to a sermon by Dr. Stanley I decided that from now on I have to evaluate my daily walk with God and listen to Him, and obey Him in whatever He brings to my mind. I don’t remember what the sermon was about but I remember Dr. Stanley saying that when he makes a decision one of the things he does is he asks himself when he will be 70 years old will he regret making that decision? I think he was talking in that sermon about how to discern God’s will in making decisions, or something like that. After that I started to regularly evaluate my steps in life and making sure they are aligned with His will for us Christians in general and for me personally.
It is very essential that we spend time with God alone and His word to evaluate ourselves, our motives, our actions—our lives. How many of us have seen on TV shows such as Cops a naked man running on the freeway across state lines and when stopped by the police and questioned, he would simply answer, “God told me to do it!” How many honest actors and actress think they are giving God the glory by thanking Him for “enabling” them to act in immortal movies!
(By the way I am listening to: SECRET GARDEN - SOLO VIOLIN www.youtube.com/watch?v=VOhjTpeQhXs&feature=related to help me write.)
It is important to evaluate our faith, that is: why do we do the things we do? What is driving us? What is motivation us? That’s why I often lie down quietly and ask God, “Lord, why do I want to do this?” or, “Why don’t I want to do this?” For example, a lot of relationships we build, the way we act in public, or even dress are motivated by a deep hurt from our past such as a rejection. And the saying “time heals all things” is non-sense to me. Only God heals; time just buries hurts so deep inside that we things are “healed” while they surface from time to time only to make us miserable and ruin our lives.
Let me give you an example:
Sometimes we go and tell people who have low self-image that they are beautiful and lovely and keep on doing so, and then their emotional security becomes wrapped up in us while all along only God’s love can heal them and make them emotionally whole. It is like giving a child chocolate and not expect him or her to become unhealthy! While all along they need healthy food…food only our heavenly Father can give. I am not saying that we should not admire people, but we must be sensitive to the work God wants to do first in other people’s lives. We too need to evaluate our motives.
Sometimes while we think we are doing God’s will, we are in reality simply hindering His work. And sometimes it is hard not to step in and do God’s work for Him, but like a heroin addict who is recovering he is starved from the drug and for a period of few days he may feel miserable without the drug, but once that period is over then he is free. But what we do is step in and give that addict a small quantities of the drug thinking we are helping them, while all along what we are doing is keeping him imprisoned to the addiction. That’s just a small example, and of course there are many.
Another area we need to evaluate ourselves is why do we obey God? Many times we, Christians, hear God’s voice and obey Him expecting something big to happen in our lives. But then when nothing spectacular happens we forget about God’s command and go back to whatever we used to do. Let me give you an example:
I think about 3 years ago after I ate my lunch I went to sit on some boxes in the warehouse I used to work at, to relax and read my Bible a bit. I could simply stretch on the boxes and rest my feet while had a quiet time with God. I opened the Bible and it was the book of James, so I tried to change the page! Then I thought to myself, “Why are you trying to change the page?” Well, the answer was simple: James tells us how to live for God, living for God requires change, change requires sacrifice, sacrifice makes us uncomfortable, and I didn’t want to feel uncomfortable! So I had to make a decision: read the book of James or ignore it? I knew that God often required me to do things that made me uncomfortable (who said carrying our crosses and follow Him is a comfortable thing to do?), so I decided to read the book of James. After some verses I came across this verse, James 4:8:
“Come near to God and he will come near to you. Wash your hands, you sinners, and purify your hearts, you double-minded.”
And I thought to myself, “Phew! Thank God, James can’t disturb my life here because I have never murdered or stolen so I don’t need to wash my hands!” But for some reason I couldn’t move past this verse! And I was thinking “James!!!” And I could clearly hear God asking me through my spirit, “Are your hands really clean? Have you really not stolen anything? Are you sure you don’t have any stolen things in your possession?” I was very confused at first because I had grown in a family that taught me not to steal, cheat, lie, always say “sorry” when I wronged a person, and even taught me not to throw garbage anywhere but in the garbage bin! So again I started to evaluate my possessions to see if I had something that belonged to someone else by may be a mistake!
After some thought I ended up thinking about what I have on my PC. My Windows was an illegal ((from now on I will just say stolen instead of “illegal”) copy, the music I listened to were stolen, and almost all the other software I used were stolen! I calculated it and it was about $5,000! If not counting the 3D software that each costs few thousands! I was absolutely shocked! I couldn’t believe I stole someone else’s hard work, intellectual property and rights, and live hood! So I knew what I had to do. Within a week or so I had removed every stolen piece of software and song from my hard drive and formatted it my hard drive, and broke all the stolen (copy) CDs I had.
You are probably thinking now, “WOW! What happened afterwards? Did you win $10,000 to buy all the software you want because you obeyed God voice?” No, nothing spectacular happened afterwards, even though in the beginning I thought that somehow God will give me the money to get all the software I want in the world! For about two years afterwards I didn’t do any photo editing because I couldn’t afford Photoshop, until one day I prayed asking God to please do something because I had obeyed Him and now I can’t even enjoy photo editing because I can’t afford those expensive software! Within few days I stumbled across Gimp and I absolutely love it :) (That’s why I always say “Thanks Gimp” when I write in regard to my photos.)
For a couple of years afterwards my computer was like a desert! It was so depressing to use it! But you know what? I didn’t get one single virus and I didn’t have any anti-virus or firewall! I learned that I could live without any of the software I had and I was just as happy! I learned that I didn’t need 95% of the software I had! I learned how to do things different ways because the legal software I had now were much older (the ones that came with my PC when I bought it). Now I am in university and so they give me licensed software.
So what I want you to think about is this: let’s say that you obeyed God and that nothing spectacular happened afterwards, would you still obey Him? If not, why not? Ask yourself, ask God, get to know…whatever you do, don’t settle down for what you see and think “I guess this is the Christian life after all.” No, it is not! Open the Bible and read how God worked in the life of the Old Testament prophets, and how He worked in the life of the apostles and He can do the same in your life! Evaluate your steps, your thoughts, your words, your actions, yourself…even evaluate you faith, devotion and obedience to God.
Have God asked you to stop listening to a certain type of music that dishonours Him and when you did so you realized you were still the same person—you didn’t become a saint—and so you decided that may be it was not God’s voice you heard and went back to listen to the same ungodly music? My friend, if every time we expect God to do a miracle because we obeyed Him then we missed the whole point! Christianity is a relationship with God because we love Him because He loved us first! It is not a religion where you input things and you get some outputs! It is a relationship. Here is a passage of scripture for us to think about, Luke 17:7-10:
"Suppose one of you had a servant plowing or looking after the sheep. Would he say to the servant when he comes in from the field, 'Come along now and sit down to eat'? Would he not rather say, 'Prepare my supper, get yourself ready and wait on me while I eat and drink; after that you may eat and drink'? Would he thank the servant because he did what he was told to do? So you also, when you have done everything you were told to do, should say, 'We are unworthy servants; we have only done our duty.' "
That doesn’t mean we should not expect good things, such as blessings and rewards from God, He is our Father after all, but blessings and rewards should not be our motives behind obedience.
That’s it for this topic. I do not make promises, but God does. And I know that I know that I KNOW if you decide today that you will bring everything about you and in your life before God for evaluation, and you commit yourself to listen to His voice, read His word, get to know Him and obey Him that you will not be the same person next year, and the year after you won’t be the same person of next year…as time goes by you will become more like Jesus Christ. But don’t forget it is the work of the Holy Spirit in your life, and many times it is just a matter of letting Him take control of you; submit your tongue, your eyes, your mind, all of you to Him.
I will leave you with those verses:
"Each one of us will give an account of himself to God."
Romans 14:12
"Keep on the alert at all times, praying that you may have strength to escape all these things that are about to take place, and to stand before the Son of Man."
Luke 21:36
"If you are wise, you are wise for yourself, and if you scoff, you alone will bear it."
Proverbs 9:12
"I tell you that every careless word that people speak, they shall give an accounting for it in the day of judgment."
Matthew 12:36
“I have considered my ways
and have turned my steps to your statutes.”
Psalm 119:59
“Let us examine our ways and test them,
and let us return to the LORD.”
Lamentations 3:40
“This is what the LORD Almighty says: "Give careful thought to your ways.”
Haggai 1:7
“Examine yourselves to see whether you are in the faith; test yourselves. Do you not realize that Christ Jesus is in you—unless, of course, you fail the test?”
2 Corinthians 13:5
“If anyone thinks he is something when he is nothing, he deceives himself.”
Galatians 6:3
I just want to say one more thing: if we choose not to evaluate ourselves, we can pretty much live normally, you know, float in life from this to that and nothing will happen. We won’t have a big sign flashing in red telling us to stop and think, or do this or we will bear the consequences. After all we have God’s word and the Holy Spirit, if we choose to ignore their instructions we are pretty much on our own making a mess of our lives.
God will not come down from Heaven to stop you from drinking and driving while God the Holy Spirit lives in you and you are ignoring Him. Common sense still applies to us, Christians, just as much as physical laws such as gravity still apply to us! We break God’s laws, no matter how small they are, we bear the consequences. We steal, even if God forgives us, we are still found guilty under the law. We commit sexual immorality and God forgives us, we still get pregnant or have a sexual transmit disease, not to mention the damage that kind of sin imprint on our minds, and souls.
My point is this: choosing not to evaluate our lives is a very dangerous thing and we will suffer from the consequences and will regret our inactions now and for eternity. We will hurt people we love so much. We will battle against guilt. We will miss on God’s blessings. But most of all we will miss on knowing Him, experiencing His work in our lives, and fulfilling the privilege and honourable task of living for Him.
“You can't teach calculus to a chimpanzee. So just share your banana." -- John Rachel
Poppycock!
For every epsilon banana (ε > 0) there exists a delta banana (δ > 0) such that whenever Ιx - xₒΙ < δ then Ιf(x) - f(xₒ)Ι < ε.
Which is the Limit function, the cornerstone of Calculus. If only John Rachel had considered that two bananas are required, easy-peasy.
Our Daily Challenge - Over Your Head
Canon S.S.C. 50mm f/1.4, taken at f/1.4
i hope the title conveys how stressed and frightened i am. i can't even tell if i'm prepared enough...
oh and sorry for the mess....
wearing:
h&m turtleneck
paisley dress
mum's belt
sweater tights
(pretend i have the brown flats on my feet, hehe)
your necklace
133/365
STROBIST CALCULUS in a little pocket book ....
Hi folks,
have you ever wondered, how to calculate with HSS flash?
How to use gang light?
How to calculate with ND filters?
How to compare continuous light and flash light?
How to adjust an exposure with/without a light meter?
How to calculate with Guide Numbers?
How to benefit from the Inverse Square Law?
>>> Here you go, it is for free: :-)
I made this little pocketbook with RockyNook to say "thank you" to my trusty readers (and to say hi to all the hopefully will-be readers.:-):
fotopraxis.files.wordpress.com/2013/09/light-primer-engl-...
I hope, you enjoy and share this little pocketbook.
Cheers,
~gallo~
--
Tilo ~gallo~ Gockel
54.365
I have a huge Calculus test to study for so I just shot this real quick while studying at Taco Bell. Yes, I study for Calculus at Taco Bell.