A Sphere that is Not so Round

Of course the most basic shape for me would be the sphere, but in our understanding of the earth and the images that we see of earth, our view is shattered by the first time we seen this enormous object, from the eyes of those who had always been earth bound and restricted to the calculation of a abstract world.

Now it is not so round, and the views we recieve of this information help us to understand a few things about the way in which we will look at the earth and its weak field manifestation, as one extreme of the whole gravitational framework we like to understand over this complex perspective of our cosmos from the very strong gravitational foreces, to the very weak.

Gravity is the force that pulls two masses together. Since the earth has varied features such as mountains, valleys, and underground caverns, the mass is not evenly distributed around the globe. The "lumps" observed in the Earth's gravitational field result from an uneven distribution of mass inside the Earth. The GRACE mission will give us a global map of Earth's gravity and how it changes as the mass distribution shifts. The two satellites will provide scientists from all over the world with an efficient and cost-effective way to map the Earth's gravity field.

The primary goal of the GRACE mission is to map the Earth's gravity field more accurately than has ever been done before. You might ask, how will GRACE do this? Two identical spacecraft will fly about 200 kilometers apart. As the two GRACE satellites orbit the Earth they are pulled by areas of higher or lower gravity and will move in relation to each other. The satellites are located by GPS and the distance between them is measured by microwave signals. The two satellites do not just carry science instruments, they become the science instrument. When mass moves from place to place within the Earth's atmosphere, ocean, land or frozen surface (the "cryosphere"), the gravity field changes.

Highlighting in bold, it would not be necessary to go into a full explanation if we considered in context the cosmic string and clumping in our universe? But I have moved to far from the point of reference of our two systems of consideration here so back to the next point.

NASA's Earth Science Enterprise funded this research as part of its mission to understand and protect our home planet by studying the primary causes of climate variability, including trends in solar radiation that may be a factor in global climate change.

One of the interesting ideas in these shapes is understanding what can cause disturbances within their fields and how we might look at these issues if we move our consideration from the normals views of measure stick and straight lines, to variations we have demonstrated on earth(hills and valleys). To how we would percieve "resonance" created in the sun, and use this, to determine the volatile solar coronas that would be ejected into space but of weather systems affected as well. This is a good monitoring tool fore warning system that could affect communication within the sphere of our own influence.



But let me take this one step further, in that we consider both these frames, in context of each other and ask the connection in lagrangian way. How would we see gravitational points of consideration related to each other? Would this help those less inclined to understand the variations in perspective of gravity to comprehend the value Einstein lead us through, to take us to a much more dynamical view of the cosmos?

So in the one sense to take what we know of the formulation at a euclidean level and move it accordingly, to cosmological perspectives. This is a apprehenson of Gr that we are geometrically lead through, to perspectives of the space we would now enjoy of Gauss. Having Reimann views here of spherical consideration, we understand well, the developing roads to mathematical perspective debated and shunned by Peter?:)

So looking at the way in which we entertain the earth view, and how we interpret the sun, we do not limit our views just to the spherical balls one might like to say is self evident, but having the knowledge of what we are lead through geometrically as well as toplogically, it becomes a much more interesting reality to entertain.

A Happy News Years to all those who visit:)

Where to Now?

Once you see parts of the picture, belonging to the whole, then it becomes clear what a nice picture we will have?:) I used it originally for the question of the idea of a royal road to geometry, but have since progressed.

If you look dead center Plato reveals this one thing for us to consider, and to Aristotle, the question contained in the heading of this Blog.

It is beyond me sometimes to wonder how minds who are involved in the approaches of physics and mathematics might have never understood the world Gauss and Reimann revealled to us. The same imaging that moves such a mind for consideration, would have also seen how the dimensional values would have been very discriptive tool for understanding the dynamics at the quantum level?

As part of this process of comprehension for me, was trying to see this evolution of ordering of geometries and the topological integration we are lead too, in our apprehension of the dynamics of high energy considerations. If you follow Gr you understand the evolution too what became inclusive of the geometry developement, to know the physics must be further extended as a basis of our developing comprehension of the small and the large. It is such a easy deduction to understand that if you are facing energy problems in terms of what can be used in terms of our experimentation, that it must be moved to the cosmological pallette for determinations.

As much as we are lead to understand Gr and its cyclical rotation of Taylor and hulse, Mercuries orbits set our mind on how we shall perceive this quantum harmonic oscillator on such a grand scale,that such relevance between the quantum and cosmological world are really never to far apart?

As I have speculated in previous links and bringing to a fruitation, the methods of apprehension in euclidean determinations classically lead the mind into the further dynamcis brought into reality by saccheri was incorporated into Einsteins model of GR. Had Grossman not have shown Einstein of these geoemtrical tendencies would Einstein completed the comprehsive picture that we now see of what is signified as Gravity?

So lets assume then, that brane world is a very dynamcial understanding that hold many visual apparatus for consideration. For instance, how would three sphere might evolve from this?

Proper understanding of three sphere is essential in understanding how this would arise in what I understood of brane considerations.

Spherical considerations to higher dimensions.

Spheres can be generalized to higher dimensions. For any natural number n, an n-sphere is the set of points in n-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is, as before, a positive real number.

a 1-sphere is a pair of points ( - r,r)
a 2-sphere is a circle of radius r
a 3-sphere is an ordinary sphere
a 4-sphere is a sphere in 4-dimensional Euclidean space
However, see the note above about the ambiguity of n-sphere.
Spheres for n ≥ 5 are sometimes called hyperspheres. The n-sphere of unit radius centred at the origin is denoted Sn and is often referred to as "the" n-sphere.

INtegration of geometry with topological consideration then would have found this continuance in how we percieve the road leading to topolgical considerations of this sphere. Thus we would find the definition of sphere extended to higher in dimensions and value in brane world considerations as thus:



In topology, an n-sphere is defined as the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere described above under Geometry, but perhaps lacking its metric. It is denoted Sn and is an n-manifold. A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere.

a 0-sphere is a pair of points with the discrete topology
a 1-sphere is a circle
a 2-sphere is an ordinary sphere
An n-sphere is an example of a compact n-manifold without boundary.

The Heine-Borel theorem is used in a short proof that an n-sphere is compact. The sphere is the inverse image of a one-point set under the continuous function ||x||. Therefore the sphere is closed. Sn is also bounded. Therefore it is compact.

Sometimes it is very hard not to imagine this sphere would have these closed strings that would issue from its poles and expand to its circumference, as in some poincare projection of a radius value seen in 1r. It is troubling to me that the exchange from energy to matter considerations would have seen this topological expression turn itself inside/out only after collapsing, that pre definition of expression would have found the evoltuion to this sphere necessary.

Escher's imaging is very interesting here. The tree structure of these strings going along the length of the cylinder would vary in the structure of its cosmic string length based on this energy determination of the KK tower. The imaging of this closed string is very powerful when seen in the context of how it moves along the length of that cylinder. Along the cosmic string.

To get to this point:) and having shown a Platonic expression of simplices of the sphere, also integration of higher dimension values determined from a monte carlo effect determnation of quantum gravity. John Baez migh have been proud of such a model with such discrete functions?:) But how the heck would you determine the toplogical function of that sphere in higher dimensional vaues other then in nodal point flippings of energy concentration, revealled in that monte carlo model?

Topological consideration would need to be smooth, and without this structure how would you define such collpases in our universe, if you did not consider the blackhole?

So part of the developement here was to understand where I should go with the physics, to point out the evolving consideration in experimentation that would move our minds to consider how such supersymmetrical realities would have been realized in the models of the early universe understanding. How such views would have been revealled in our understanding within that cosmo?

One needed to be able to understand the scale feature of gravity from the very strong to the very weak in order to explain this developing concept of geometry and topological consideration no less then what Einstein did for us, we must do again in some comprehensive model of application.

The Sound of the Landscape

Ashmolean Museum, Oxford, UK

As you know my name is Plato (The School of Athens by Raphael:)I have lived on for many years now, in the ideas that are presented in the ideas of R Buckminister Fuller, and with the helping hands of dyes, have demonstrated, the basis of these sounds in balloon configuration worth wondering, as simplice's of these higher dimensional realizations.

A Chladni plate consist of a flat sheet of metal, usually circular or square, mounted on a central stalk to a sturdy base. When the plate is oscillating in a particular mode of vibration, the nodes and antinodes set up form a complex but symmetrical pattern over its surface. The positions of these nodes and antinodes can be seen by sprinkling sand upon the plates;

Now you know from the previous post, that I have taken the technical aspects of string theory, and the mathematical formulations, and moved them into a encapsulated state of existance, much as brane theory has done.

I look at this point(3 sphere derivation from euclid point line plane), on the brane and I wonder indeed, how 1R radius of this point becomes a circle. Indeed, we find this "idea" leaving the brane into a bulk manifestation of information, that we little specks on earth look for in signs of, through our large interferometers called LIGO's



John Baez:

Ever make a cube out of paper? You draw six square on the paper in a cross-shaped pattern, cut the whole thing out, and then fold it up.... To do this, we take advantage of the fact that the interior angles of 3 squares don't quite add up to 360 degrees: they only add up to 270 degrees. So if we try to tile the plane with squares in such a way that only 3 meet at each vertex, the pattern naturally "curls up" into the 3rd dimension - and becomes a cube!

The same idea applies to all the other Platonic solids. And we can understand the 4d regular polytopes in the same way!

The Hills of M Theory

The hills are alive with the sound of music
With songs they have sung for a thousand years.
The hills fill my heart with the sound of music
My heart wants to sing every song it hears....

It's a wonder indeed that we could talk about the spacetime fabric and the higher dimensions that settle themselves into cohesive structures(my solids) for our satisfaction? What nodal points, do we have to wonder about when a string vibrates, and one does not have to wonder to much about the measure of the Q<->Q distance, as something more then the metric field resonates for us?

This higher dimensional value seen in this distance would speak loudly to its possiblites of shape, but it is not easily accepted that we find lattice structures could have ever settled themselves into mass configurations of my solids.



Lenny Susskind must be very pround of this landscape interpretation, as it is shown in the picture above. But the question is, if the spacetime fabric is the place where all these higher dimensions will reveal themselves, then what structure would have been defined in this expression from it's orignation, to what we see today?

Alas, I am taken to the principles of," Spacetime in String Theory," by Gary T. Horowitz

If one quantizes a free relativistic (super) string in flat spacetimeone finds a infinite tower of modes of increasing mass. Let us assume the string is closed,i.e., topologically a circle

What are Sounds in the New Concept of Theoretics Approach?

I must be true to my word, and follow the tidit of information that I posted on Peter Woit's site. I am pointing to the two positions that both Lubos and Peter declare of themselves, in what they choose to represent of their thinking.

Lubos Said:

As I go towards the present, physics of these topics becomes increasingly difficult, requires higher education, expertise - and I think that something remotely similar exists in any other sufficiently complex field of science, including e.g. number theory, too. Proving the Fermat Last Theorem is a pretty fancy thing that requires some new technology, does not it?

In what Lubos saids, there is no arguing about the prerequsites of insight that follow educational roads to comprehending the world. In a way, that I have mentioned, that few recognize.

So as a commoner and having followed this thinking over the last few years, it so happened that a conceptual frameworld developed that help me look at the physics and approachs that are developing at the the very front lines of theoretical and mathematical developement.

Of course my statements have to be laid in contrast to what is being shown to us on a public scale. To have derived this thinking, gracefully exploding into new phase transitive models of apprehension. What better contrast then to have another mind like Peters Woits to contradict the mathematics that has been developed in string theory?

Peter makes it clear, that the mathematics is in question? If you attack the model of string/M theory you attack it's mathematics. There is no way to avoid this logically. Being the spokesman of why theoretcially this model of strings will never survive? In a innocent enough posting thread of his Peter voices this in a quiet humourous way by point out the logic of his thinking as well? That humour has to be based on some pre existing understanding of math in order to be driven into the jovial states of laughter?:)


Peter saids:

Mathematical Humor

Now for some comic relief:

A new issue of the Notices of the AMS is out. It contains an entertaining article entitled Foolproof: A Sampling of Mathematical Folk Humor with many examples of mathematical humor. Physicists also put in an appearance.

So as if this concept dropped into our views, from the 21 century(the future), we find that these concepts move the common person forward by having our front line physics and theoretcial people explaining how this concept is developing backwards or is it forwards?:)

Yes it is amazing to think, that a whole concept could exist within this reality and that the arguement is being fought on whether to accept this belief or not? That the substance of this reality could mathematical say the same things, from both Peter and Lubos. This will be then the basis of our interpretation, of the way we will derive the physics of approach by elements of structures, that have preceded us in our determnations revealled in the Einsteinian way? This exposition is articulated countless times, on a geoemtrical and topological determination, that rests early in Euclid's developement of postulates, continuing on to the road taken Reimann spherical which determinations lead us in our visions of gravity.

What the hell would any commoner know, if they did not understand that this basis of interpetation did not explode fractorially into the concept we now look at in terms of dimensional attributes above the spacetime we have come to accept and look at, in our everyday lives?



It is nice to have people Like Michio Kaku who can help orientate the common person into the reality that has moved these theoretical positions with clear and concise methods of interpretation. But my start of comprehension in based on the work of Savas Dimopoulos and the conection Nima and others have to a developing view about dimensional interpretation.

Savas Dimopoulos
Here’s an analogy to understand this: imagine that our universe is a two-dimensional pool table, which you look down on from the third spatial dimension. When the billiard balls collide on the table, they scatter into new trajectories across the surface. But we also hear the click of sound as they impact: that’s collision energy being radiated into a third dimension above and beyond the surface. In this picture, the billiard balls are like protons and neutrons, and the sound wave behaves like the graviton.

It is very hard for people to see this third dimension, but if the analogies help, then you should be able to understand the world of this tension, and the harmony involved generated from the musical comparison that is associated?

We can't actually hear gravitational waves, even with the most sophisticated equipment, because the sounds they make are the wrong frequency for our ears to hear. This is similar in principle to the frequency of dog whistles that canines can hear, but are too high for humans. The sounds of gravitional waves are probably too low for us to actually hear. However, the signals that scientists hope to measure with LISA and other gravitational wave detectors are best described as "sounds." If we could hear them, here are some of the possible sounds of a gravitational wave generated by the movement of a small body inspiralling into a black hole.

Have a look here and listen:)Make sure your speakers are on.

This helps one to distinquish the purposes of what might have been driven as a being represented in the manifestation of GR as a spacetime fabric . Holographically, these dimensions consolidate not as a point particles(harmonically driven interpretations) but as a strings on the brane?

The "air," of this particle identification, is density articluated( KK tower on brane thickness?), by energy determinations that are dimensionally related? The only way for us to see this, is to derive some topological feature, that moves into geometrical interpetations of that same energy value determination?



Without these graphs to demonstarte particle movements from collisions, how would you define topologically this energy distribution?

There is more to be added here, to complete this posting, that will show up later.

I would like to tantilize the minds view of this landscape, with a rendition of the Hills are alive with the Sound of Music, and what this looks like, as portrayed by Les Houches.:)

What Cosmologist Wants From String theory

The most surprising difference for the quantum case is the so-called zero-point vibration" of the n=0 ground state. This implies that molecules are not completely at rest, even at absolute zero temperature.

When I looked at the issues strings presented of itself, the very idea of changing the way we percieve the basis of reality came into question. If strings were to exist where did they come from? This assumption, on my part asked me to consider then, that the very basis of this reality was drive by a quantum harmonic oscillator, and that there was never really any zero point in which to consider? Was this a logical assumption about how we would percieve the basis, from some emergent property that had to always exist?



This kind of thinking then forces you to consider what if this universe had always existed? How could it have ever come into being? Could such a realization have been embued into the string cosmology understanding, of the way this universe is operating?

So on a cosmological scale, we have this synopsis of events, as they have been shown in this following Picture taken from Beyond Einstein. Imagine then, how you might encapsulate this whole picture into a simple explanatory feature that trends this quantum beginning to cosmological end, in a cyclcial fashion worth speculating.



From this perspective ,it is with some consideration that we are directed again to Turok, Steinhardt in brane collisions. Or others like Gasperini and Venziano in, "The Pre-Big Bang Scenario in String Cosmology".

The Gravity For Instance Varies with Time

There was some reference made in regards to Lubos and the post Santos quoted from, Superstrings, a Theory of Everything? by P.C.W Davies and J. Brown. The idea of the quote about ole men and how things change, seemed to have been caught in perspective by Santos? That becuase I had this book handy, I went and looked to the reference page that he suggested of 193.

Well, one has to go over the post in question and find the link provided by Peter Woit to understand the context in which this quote is being used. To see that Lubos makes light of his own position, where holding dogmatic to a certain position might be as relevant to where ole men do not accept the new reality easily. It was a innocent enough comment that has certain connatations to it, that moved me through the article santos quotes and brought forth different information for consideration.




For instance before quoting Feynman myself, lets read what Gabriele Veneziano has to say.

String theory suggests that the big bang was not the origin of the universe but simply the outcome of a preexisting state

Now considering the publishing date on this book, "Superstrings, A Theory of Everything," which was 1988, you have to wonder about the progression and way in which we determine dimenisons. I have detailed it in the previous post entitled, "Spheres and their Generalizations in Higher Dimensions." If such a view was to remain consistent then the statement of Richard Feynman to follow, answers today, what he speculated then.

Now it's possible that those kinds of laws in physics may be incomplete. It might be that the laws change absolutely with time; that grvaity for instance varies with time and that this inverse square law has a strength which depends on how long it is since the beginning of time. In other words, it's possible that in the future we'll have more understanding of everything and physics may be completed by some kind of statement of how things started which are external tothe laws of physics.

Pg 206 and 207, of Superstrings, A Theory of Everything, by P.C.W.Davies and J. Brown



So you see, perspective changes because of theoretical positions, concepts form and new insights develope. Because of a personal interest of Feynman's, about his perception of the physics of approach, did you think at first he had a cohesive picture of the toy models he produced? Do you think Andrew Wiles would have ever solved the Fermat Equation?

Such context of the Poincare Conjecture, answers a question we have about how quantum grvaity portrays itself, by the ole soccer ball( my solid), or by how we see conceptual changes allow us to think about the higher dimensional values in regards to gravity? Of tying geometry and topological understandings together.

Do you think Feynman toy models haven't been included?

A Merry Christmas to Everyone, and a Happy New Year.

Andrew Wiles and Fermat

In context of previous post on mathematical problems, it was interesting to carry on, and look at other math issues here. So these links help provide a interesting commentaries, on other math problems, that are interesting and thought provoking.

 

Fermat was a 17th-century mathematician who wrote a note in the margin of his book stating a particular proposition and claiming to have proved it. His proposition was about an equation which is closely related to Pythagoras' equation. Pythagoras' equation gives you:
x² + y² = z²

Update


See Also:

Andrew Wiles: 60th birthday

About Spheres and their Generalizations of Higher Dimensional Spaces

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

— Nikolai Lobachevsky

If a math man were to be left alone, and devoid of the physics, would he understand what the physics world could impart if he were not tied to it in some way?:)

Poincaré Conjecture

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

Now it is of course with some understanding, that few would recognize what this conjecture to mean, and having just read Andre' Weil's last word I do not think it to shabby to say that in this case, the Poincare conjecture has been recognized as a valid conjecture, that has taken some time in it being resolved.




For me being presented with the cosmic string scenario might seem just as complex, when in considertaion of the brane. How our universe could be contained in it. Some might of laughed it off quite easily, being part of some revolution of strings to M-theory, that it could include 11 dimensions.How would you embed these dimensions with these shapes?

What fascinates me, is how we could have found such visualizations of these topological forms within the the brane world and how this may have been described?

So I am looking for traces of literature that would help me in this direction. For example, how a torus would be looked at in a 2 dimensional sheet. Would this be relevant to brane world happenings if we considered, the example of "sound" (make sure speakers are on) in higher dimensions, as viable means of expression of these curvatures of those same shapes?



Of course I know I have to explain myself here, and the intuitive jump I am making. Could be wrong?

Part of this struggle to comprehend what has happened with bringing GR and QM together in one consistent framework, was to undertand that you have altered the perceptions of what dimension will mean? To me, this says that in order now for us to percieve what the poincare conjecture might have implied, that we also understand the framework with which this will show itself, as we look at these forms.

The second panel I showed of the graph, and then the topological form beside it, made sense, in that this higher dimension shown in terms of brane world happenings would have revealled the torus as well in this mapping. This is of course speculation on my part and might fall in response to appropriate knowledge of our mathematical minds. This gives one a flavour of the idea of those extra dimensions.



Well part of this developement of thinking goes to what these three gentlemen have developed for us in our new conceptual frameworks of higher dimenions. If you ask what higher dimension might mean, then I am certain one would have to understand how this concept applies to our thinking world. Without it, these shapes of topological forms would not make sense. It seems this way for me anyway.:)



I hazard to think that John Baez and others, might think they have found the answer to this as well, in what they percieve of the mighty soccerball, and the fifth solid I espouse:)

Hodge Conjecture

Interplay between geometry and topology.

One of the things that I am having difficulty with is if I understood the idea of a cosmic string. The understanding would have to imply that the higher dimensions would reveal themselves within the spacetime curvatures of gravity. So I have been looking to understand how the quantum mechanical nature could have ever reduced itself from those higher dimensions in string theory, and revealed themselves within the context of the cosmos that we know works very well with Gravity.



Part of my attempts at comprehending the abtractness with which this geometry evolved, was raised diffrent times within this blog as to whether or not there was a royal road to geometry?

Throughout, I have shown the processes with which a smooth topological feature would have endowed movements like the donut into the coffee cup and wondered, about this idea of Genus figures and how they to become part of the fixtures of the terrain with which mathematicians like to enjoy themselves over coffee?:)



How would this information in regards to the strings, become a viable subject with regards to non-euclidean realms, to have understood where GR had taken us and where QM had difficult combining with GR(gravity).

Large extra dimensions are an exciting new development … They would imply that we live in a brane world, a four-dimensional surface or brane in a higher dimensional spacetime. Matter and nongravitational forces like the electric force would be confined to the brane … On the other hand, gravity in the form of curved space would permeate the whole bulk of the higher dimensional spacetime …. Because gravity would spread out in the extra dimensions, it would fall off more rapidly with distance than one would expect … If this more rapid falloff of the gravitational force extended to astronomical distances, we would have noticed its effect on the orbits of the planets … they would be unstable… However, this would not happen if the extra dimensions ended on another brane not that far away from the brane on which we live. Then for distances greater than the separation of the branes, gravity would not be able to spread out freely but would be confined to the brane, like the electrical forces, and fall off at the right rate for planetary orbits.

Stephen Hawking, Chapter seven

Mirror Symmetry and Chirality?

What is the mathematical reasoning, for reducing GR down to the quantum world? How would these application be considered in a supersymmetrical world?

Is the figure-eight knot the same knot as its mirror-image? The property of "being the same as your mirror-image" is called chirality by knot theorists. The image sequence below shows the figure-eight knot being transformed into its mirror image -- such knots are called achiral

You have to remember this is the year 2000, article was produced below.

Physicists Finally Find a Way to Test Superstring Theory

As the unification quest has forged ahead, physicists have found it necessary to expand superstring theory to include vibrating membranes -- called branes for short. These are not just two-dimensional surfaces, like the skin of a drum or the world of the Flatlanders. Hard as it may be to picture, there can be branes with three, four, five or more dimensions. These "surfaces" can be tiny like the strings but they can also span across light-years.

by George Johnson

I know this may seem a little slow but if the cosmic string was considered in context of the Fresnel lens and I apologize for my ignorance, how would reverse imaging account for gravitational lensing?

The gravitational lensing has to reveal the warp field as such a possibility? Yet, in that distant time, such alteration of the shapes, amongst it event, would have show two points, but would also have indication that one was the reverse of the other. What signatures would we see of this?



So in keeping with the direction we were given by Lubos in his article, I a am trying to comprehend.

This page is the beginning of a demonstration of strong gravitational lensing about a pseudo-isothermal elliptical mass distribution (PID). You can look at a simple animation or read about the mathematics behind the PID lensing.