Window Phone Concept

Theories about the Origins of Space and Time. 1. Gravity as Thermodynamics Entropic gravity is a theory in modern physics that describes gravity as an entropic force - not a fundamental interaction mediated by a quantum field theory and a gauge particle, but a consequence of physical systems’ tendency to increase their entropy.  2. Loop Quantum Gravity According to Einstein, gravity is not a force – it is a property of space-time itself. Loop quantum gravity is an attempt to develop a quantum theory of gravity based directly on Einstein’s geometrical formulation. The main output of the theory is a physical picture of space where space is granular. More precisely, space can be viewed as an extremely fine fabric or network “woven” of finite loops. These networks of loops are called spin networks. The evolution of a spin network over time is called a spin foam. The predicted size of this structure is the Planck length, which is approximately 10−35 meters. According to the theory, there is no meaning to distance at scales smaller than the Planck scale. Therefore, LQG predicts that not just matter, but space itself, has an atomic structure. 3. Causal Sets Its founding principles are that spacetime is fundamentally discrete and that spacetime events are related by a partial order. The theory postulates that the building blocks of space-time are simple mathematical points that are connected by links, with each link pointing from past to future. Such a link is a bare-bones representation of causality, meaning that an earlier point can affect a later one, but not vice versa. The resulting network is like a growing tree that gradually builds up into space-time. 4. Causal Dynamical Triangulations The idea is to approximate the unknown fundamental constituents with tiny chunks of ordinary space-time caught up in a roiling sea of quantum fluctuations, and to follow how these chunks spontaneously glue themselves together into larger structures. The space-time building blocks were simple hyper-pyramids (four-dimensional counterparts to three-dimensional tetrahedrons) and the simulation’s gluing rules allowed them to combine freely. The result was a series of bizarre ‘universes’ that had far too many dimensions (or too few), and that folded back on themselves or broke into pieces. 5. Holography In this model, the three-dimensional interior of the universe contains strings and black holes governed only by gravity, whereas its two-dimensional boundary contains elementary particles and fields that obey ordinary quantum laws without gravity. Hypothetical residents of the three-dimensional space would never see this boundary, because it would be infinitely far away. But that does not affect the mathematics: anything happening in the three-dimensional universe can be described equally well by equations in the two-dimensional boundary, and vice versa.

We’re used to radiation being invisible. With a Geiger counter, it gets turned into audible clicks. What you see above, though, is radiation’s effects made visible in a cloud chamber. In the center hangs a chunk of radioactive uranium, spitting out alpha and beta particles. The chamber also has a reservoir of alcohol and a floor cooled to -40 degrees Celsius. This generates a supersaturated cloud of alcohol vapor. When the uranium spits out a particle, it zips through the vapor, colliding with atoms and ionizing them. Those now-charged ions serve as nuclei for the vapor, which condenses into droplets that reveal the path of the particle. The characteristics of the trails are distinct to the type of decay particle that created them. In fact, both the positron and muon were first discovered in cloud chambers! (Image credit: Cloudylabs, source)

A Glowing Pool Of Light

"NGC 3132 is a striking example of a planetary nebula. This expanding cloud of gas, surrounding a dying star, is known to amateur astronomers in the southern hemisphere as the "Eight-Burst" or the "Southern Ring" Nebula.

The name “planetary nebula” refers only to the round shape that many of these objects show when examined through a small visual telescope. In reality, these nebulae have little or nothing to do with planets, but are instead huge shells of gas ejected by stars as they near the ends of their lifetimes. NGC 3132 is nearly half a light year in diameter, and at a distance of about 2000 light years is one of the nearer known planetary nebulae. The gases are expanding away from the central star at a speed of 9 miles per second.

This image, captured by NASA’s Hubble Space Telescope, clearly shows two stars near the center of the nebula, a bright white one, and an adjacent, fainter companion to its upper right. (A third, unrelated star lies near the edge of the nebula.) The faint partner is actually the star that has ejected the nebula. This star is now smaller than our own Sun, but extremely hot. The flood of ultraviolet radiation from its surface makes the surrounding gases glow through fluorescence. The brighter star is in an earlier stage of stellar evolution, but in the future it will probably eject its own planetary nebula”

Credit: The Hubble Heritage Team

Mathematics is full of wonderful but “relatively unknown” or “poorly used” theorems. By “relatively unknown” and “poorly used”, I mean they are presented and used in a much more limited scope than they could be, so the theorem may be well known, but only superficially, and its generality and scope can be very understated and underappreciated.

One of my favorites of such theorems was discovered around the 4th century by the Greek mathematician Pappus of Alexandria. It is known as Pappus’s Centroid Theorem.

Pappus’s theorem originally dealt with solids of revolution and their surface areas and volumes. This was all Pappus was able to figure out with the geometric proof methods available to him at the time. Later on, the theorem was rediscovered by Guldin, and studied by Leibniz, Cavalieri and Euler.

The theorem is usually split into two, one for areas and one for volumes. However, the basic principle that makes both work is exactly the same, though it is much more general in the case of areas on a plane, or volumes in space.

The invention/discovery of calculus eventually brought to the table much more general methods that made Pappus’s theorem somewhat limited in comparison.

Still, the theorem is based on a very clever idea that is quite satisfying both conceptually and intuitively, and brilliant in its simplicity. By knowing this key idea one can greatly simplify some problems involving volumes and areas, so the theorem can still be useful, especially when associated with methods from calculus.

Virtually all of the literature that mentions the theorem focus on solids of revolution only, as if the theorem was merely a curiosity, and they never really address why centroids would play a role. This is a big shame. The theorem is much more general, useful and conceptually interesting, and that’s why I’m writing this long post about it. But first, a few words on centroids.

Centroids

Centroids are the purely geometric analogues of centers of mass. They are a single point, not necessarily lying inside a shape, that defines the weighted “average position” of a shape in space. Naturally, the center of mass of any physical object with uniform density coincides with its geometric centroid, since the mass is uniformly distributed along the object’s volume.

Centers of mass are useful because they give you a point of balance: you can balance any object by any point directly under its center of mass (under constant vertical gravity). This comes from the fact the torques acting on the shape exactly cancel out in this configuration, so the object does not tilt either way.

Centroids exist for objects with any dimension you wish. A scattering of points has a centroid that is just their average position. In 1 dimension we have a line segment, whose centroid is always at its center. In 2, 3 or more dimensions, things get a bit trickier. We can always find the centroid by integrating all over the shape and weighting each point by that point’s position. This is generally a complicated enough problem by itself, but for simpler shapes this can be trivial.

Centroids, as well as centers of mass, also have the nice property of being additive: the union of two shapes has a centroid that is the weighted average of both centroids. So if you can decompose a shape into simpler ones, you can find its centroid without any hassle.

Pappus’s Centroid Theorems (for surfaces of revolution)

So, let’s imagine we have ourselves some generic planar curve, which we’ll call the generator. This can be a line segment or some arc of a circle, like in the main animation of this post, above. Anything will do, as long as it lies on a plane.

If we rotate the generator around an axis lying on its plane, and which does not cross the generator, it will “sweep” an area in space. Pappus’s theorem then states:

The surface area swept by a generator curve is equal to the length of the generator c multiplied by the length of the path traced by the geometric centroid of the generator L. That is: A = Lc.

In other words, if you have a generating curve with length c, and its centroid is at a distance a from the axis of rotation, then after a complete turn the generator will have swept an area A = 2πac, where L = 2πa is just the circumference of the circle traced by the centroid. This is what the first animation in this post is showing.

Note that in that animation we are ignoring the circular parts on top and bottom of the cylinder and cone, for simplicity. But this isn’t really a limitation of the theorem at all. If we add line segments for generating those regions we get a new generator with a different centroid, and the theorem still holds. Here’s what that setup looks like:

Try doing the math! It’s nice to see how things work out in the end.

The second theorem is exactly the same statement, but it deals with volumes. So, instead of a planar curve, we now have a closed planar shape with an area A.

If we now rotate it around an axis, the shape will sweep a volume in space. The volume is then just area of the generating shape times the length of the path traced by its centroid, that is, V = LA = 2πRA, where R is the distance the centroid is from the axis of rotation.

Why it works

Now, why does this work? What’s the big deal about the centroid? Here’s an informal, intuitive way to understand it.

You may have noticed that I drew the surfaces in the previous animations in a peculiar translucent way. This was done intentionally, not just to give the surfaces a physically “real” feel, but to illustrate the reason why the theorem works. (It also looks cooler!)

Here are the closed cylinder and cone after they were generated by a rotating curve:

Notice how the center of the top and bottom of the cylinder and the cone are a darker, denser color? This results from the fact that the generator is sweeping more “densely” in those regions. Farther out from the center of rotation, the surface is lighter, indicating a lower “density”.

To better convey this idea, let’s consider a line segment and a curve on a plane.

Below, we see equally spaced line segments of the same length placed along a red curve, perpendicular to it, in two different ways. Under the segments, we see the light blue area that would be swept by the line segment as it moved along the red curve in this way.

In this first case, the segments are placed with one edge on the curve. You can see that when the red curve bends, the spacing between the segments is not constant, since they are not always parallel to each other. This is what results in the different density in the “sweeping” of the surface’s area. Multiplying the length of those line segments by the length of the red curve will NOT give you the total area of the blue strip in this case, because the segments are not placed along the curve centered at their geometric centroids.

What this means is that the differences in areas being swept by the segment as the curve bends don’t cancel out, that is, the bits with more density don’t make up for the ones with a lower density, canceling out the effect of the bend.

However, in this second case, we place the segment so that its centroid is along the curve. In this case, the area is correctly given by Pappus’s theorem, because whenever there’s a bend in the curve one side of the segment is sweeping in a lower density and the other is sweeping at a higher density in a such a way that they both exactly cancel out.

This happens because that “density” is inversely proportional to the “speed” of each point of the line segment as it is moving along the red curve. But this “speed” is directly proportional to the distance to the point of rotation, which lies along the red curve.

Therefore, things only cancel out when you use the centroid as the anchor/pivot on the curve, which allows us to assume a constant density throughout the entire path, which in turn means there’s a constant area/volume being swept per unit of length traversed. The theorem follows directly from this result.

The same argument works in 3D for volumes.

Generalizations and caveats

As mentioned, the theorem is much more general than solids and surfaces of revolution, and in fact works for a lot of tracing curves (open or closed) and generators, as long as certain conditions are met. These are described in detail in the article linked at the end.

First, the tracing curve, the one where the generator moves along (always colored red in these illustrations), needs to be sufficiently smooth, otherwise you can’t properly define the movement of the generator along its extent. Secondly, the generator must be two dimensional, always lying on a plane perpendicular to the tracing curve. Third, in order for the theorem to remain simple, the curve and the plane must intersect at the centroid of the generator.

This means that, in two dimensions, the generator can only be a line segment (or pieces of it), as in the previous images. In this case, both the “2D volume” (the area) and “2D area” (the lateral curves that bound the area, traced by the ends of the segment) can be properly described by the theorem. If the tracing curve has sharp enough bends or crosses itself, then different regions may be covered more than once. This is something that needs to be accounted for via other means.

The immediate extension of this 2D case to 3D is perfectly valid as well, where the line segment gets replaced by a circle as a generator. This gives us a cylindrical tube of constant radius along the tracing curve in space, no longer confined to a plane. Both the lateral surface area of the tube and its volume can be computed directly by the theorem. You can even have knots as the curve!

In 3D, we could also have other shapes instead of a circle as the generator. In this case there are complications, mostly because now the orientation of the shape matters. Say, for example, that we have a square-shaped generator. As it moves along a curve it can also twist around, as seen in the animation below:

In both cases, the volume is exactly the same, and is given by Pappus’ theorem. This can be understood as a generalization of Cavalieri’s principle along the red curve, which also only works because we’re using the centroid as our anchor.

However, the surface areas in this case are NOT the same: the surface area of the twisted version is larger.

But if the tracing curve is planar (2D) itself and the generator does not rotate relative to the curve (that is, it remains “upright” all along the path), like in the case of surfaces of revolution, then the theorem works fine for areas in 3D.

So the theorem holds nicely for “2D volumes” (planar areas) and 3D volumes, but usually breaks down for surface areas in 3D. The theorem only holds for surface areas in 3D in a particular orientation of the generator along the curve (see reference).

In all valid cases, however, the centroid is the only point where you get the direct statement of the theorem as mentioned before.

Further generalizations

Since the theorem holds for 2D and 3D volumes, we can do a lot more with it. So far, we only considered a generator that is constant along the curve, which is why we have the direct expression for the volume. We actually don’t need this restriction, but then we have to use calculus.

For instance, in 3D, given a tracing curve parametrized by 0 ≤ s ≤ L, and a generator as a shape of area A(s), which varies along the curve in such a way that the centroid is always in the curve, then we can compute the total volume simply by evaluating the integral:

V = ∫0LA(s) ds

Which is basically a line integral along the scalar field given by A(s).

This means we can use Pappus theorem to find the volume of all sorts of crazy shapes along a curve in space, which is quite nifty. Think of tentacles, bent pyramids and crazy helices, like this one:

What we have here are five equal equilateral triangles positioned on the vertices of a regular pentagon. Their respective centroids lie on their centers, and since all of them have the same area the overall centroid (the blue dot) is exactly in the middle of the pentagonal shape, which is true no matter how you rotate the pentagon or the individual triangles.

This means we can generate a solid along the red curve by sweeping these triangles while everything rotates in any crazy way we want (like the overall pentagon and each individual triangle separately, as in the animation), and Pappus’s theorem will give us the volume of this shape just the same. (But not the area!)

If this doesn’t convince you this theorem is awesome and underappreciated, I don’t know what will.

So there you go. A nifty theorem that doesn’t get enough love and appreciation.

Reference

For a great, detailed and proper generalization of the theorem (which apparently took centuries to get enough attention of someone) see:

A. W. Goodman and Gary Goodman, The American Mathematical Monthly, Vol. 76, No. 4 (Apr., 1969), pp. 355-366. (You can read it for free online, you just need an account.)

Windswept by Charles Sowers

Though we cannot physically hold wind or see its swirling forms around us, we can definitely feel it.

In order to help visualize wind-currents, artist Charles Sowers created a kinetic installation consisting of 612 aluminum weather vanes called “Windswept” (2011). These were then meticulously placed on the side of the Randall Museum in San Francisco. Through this installation, we are able to see the patterns in the wind; where the currents go, how they turn, and sometimes how wind can abruptly change direction. This gives us a visual representation of the natural, invisible, force which moves around us, and sometimes with enough force, pushes and pulls us.

As the artist states: “Our ordinary experience of wind is as a solitary sample point of a very large invisible phenomenon. Windswept is a kind of large sensor array that samples the wind at its point of interaction with the Randall Museum building and reveals the complexity and structure of that interaction.”

This sort of installation creates a better understanding, and appreciation, of the wind. It is not just one large gust; a single wave can be made up of smaller currents, going in their own directions from the main flow. A dialogue begins to form between the building and the wind, the weather vanes acting as translators.

-Anna Paluch

Life Advice from 50 Beloved Characters in Kid’s Entertainment by AAA State of Play -source-

New Discoveries about Star Formation in the Flame Nebula

Stars are often born in clusters, in giant clouds of gas and dust. Astronomers have studied two star clusters using NASA’s Chandra X-ray Observatory and infrared telescopes and the results show that the simplest ideas for the birth of these clusters cannot work.

This composite image shows one of the clusters, NGC 2024, which is found in the center of the so-called Flame Nebula about 1,400 light years from Earth. In this image, X-rays from Chandra are seen as purple, while infrared data from NASA’s Spitzer Space Telescope are colored red, green, and blue.

A study of NGC 2024 and the Orion Nebula Cluster, another region where many stars are forming, suggest that the stars on the outskirts of these clusters are older than those in the central regions. This is different from what the simplest idea of star formation predicts, where stars are born first in the center of a collapsing cloud of gas and dust when the density is large enough.

Credit: NASA/Spitzer/Chandra

Space Infographicsby Nick Wiinikka

prints/poster/phone cases and more by the artist available here