Saturday, August 22, 2009

Wow so this is cool:

One of the biggest numbers that actually applies to something: Graham's number.

There is no highest number. Given any number, we can easily think of a higher one. But this number is actually higher than any number I've ever thought of before. Using scientific notation, there would not be enough room in the universe to write it. It is higher than the number of particles in the universe, than the number of seconds since our universe began, etc. Despite being inconceivably large, it doesn't end with a string of zeros; the last ten digits of Graham's number are "2464195387"--I almost want to dial that number into my phone... apparently it's in Barbados...

Graham's number can only be written using Knuth's up-arrow notation, which is pretty cool in itself. Graham's number was popularized in the 1980 Guiness Book of World Records, although according to Wikipedia, "specific integers known to be far larger than Graham's number have since appeared in many serious mathematical proofs."

Friday, July 31, 2009

Wind Robots

I found out about this two years ago but I still think Theo Jansen is the man.

video

Wednesday, July 15, 2009

Not math/physics related but still cool:
wiki wiki

Tuesday, July 14, 2009

Mirror Matter

Imagine a universe like ours only all particles are replaced with antiparticles, left and right are reversed, and the direction of motion for all particles is reversed. Don't think too hard; such a universe would be indistinguishable from ours by the laws of physics. The same would be true if we left out the last condition. That is, life would be exactly the same for people who were our mirror images and were made out of antimatter.

If the laws of physics are unchanged under all three laws, and under only the first two, it must also be true that they are unchanged under the third condition only (Perform all three operations, then perform the first two). This means that a universe in which all is the same except for the direction of motion (i.e. the direction of time) is identical to ours. Einstein's special relativity implies such observers, which can best be shown through representation in Minkowski spacetime. Minkowski spacetime is a four dimensional manifold in dimensions, x,y,z, and ct where c is the speed of light. In the image below, as is commonly done, the dimension z is omitted so we can visualize the manifold. The speed of light determines the boundaries.
For an event at (0,0,0), any observer in our universe is represented by a point in the upper cone. For all observers whose spacetime coordinates fall in the lower cone, time is reversed.

Such transformations demonstrate the symmetry of our universe. For anything to be symmetrical, you must be able to exchange the (two) parts without changing anything. Upon the observation that subatomic particles called neutrinos only spin left, a theory of mirror matter was formed, by which every particle has a mirror "twin" that spins in the opposite direction, restoring the symmetry. Such particles would be invisible to us, as our photons (light particles) all spin in one direction, allowing us to see only half of the predicted particles.

Although these particles would be invisible to us and have little to no effect on our universe, they would be affected by gravity (the weakest force, but the only force that is not exchanged through particles, by modern physics). But if that's the case, how could we still not know whether mirror matter actually exists? Many people believe that "dark matter," hypothetical matter that is undetectable by its emitted radiation but effects visible matter via gravity, is in fact mirror matter. Dark matter is postulated to explain missing mass in our galaxies, and may account for more than half of the matter in our universe. Observations of the rotational speeds and temperature distribution of galaxies strongly suggest dark matter's existence.

Just to be clear, there is no "mirror universe," mirror matter would exist in our space-time. In fact, Robert Foot also suggests in his book "Shadowlands: Quest for Mirror Matter in the Universe" that meteorites of mirror matter may collide with our Earth, causing catastrophic events, that seem unexplained. He gives an explosion in Siberia in 1908 as an example, among 6 others. Also, he claims that it is likely that many planets which we perceive as starless, are in fact rotating stars composed of mirror matter.

Perhaps the most interesting thing about mirror matter is that if mirror matter does exist, it is likely that time is reversed for such matter. Meaning left and right would be reversed, and so would time. And although there is no mirror universe, mirror matter should have the same physics as observed matter. This is a result of the fact that our laws are unchanged by the direction of time as long as everything in the system stays consistent. And in fact, reversing time does change the spin from left to right. If a neutrino were coming towards you, it would be spinning left (clockwise), meaning if it were moving away from you, as if the event had been rewound like a VHS, it would appear to be spinning counterclockwise.

Presuming symmetry has led to several discoveries in physics. Consequently several types of matter have been proposed to preserve symmetry:

1. Antimatter
  • A new form of matter necessary to make Einstein's theory of relativity consistent with the quantum mechanical theory of the electron.
  • Predicted by Dirac who noticed that the mathematical description of the electron would only exhibit Lorentz symmetry (the four dimensional rotational symmetry of space-time) if such matter existed.
  • Particles have the same masses as their corresponding anti-particles.
  • Has been experimentally verified.

2. Supersymmetric partner particles
  • Necessary for String theory to accurately describe our universe.
  • Particles have the same mass as their superpartners, although broken symmetry causes the mass of superpartners to be so high that these particles could only be seen at very high energies.
  • Particles also have the same charge as their superpartners.
  • One superpartner is a fermion (a particle with half integer spin, which as far as we know is always a matter particle) and the other is a boson (a particle with integer spin, typically a force carrying particle).
  • Has not been experimentally verified but may be in the next decade using the Large Hadron Collider.

3. Mirror Matter
  • Developed to preserve symmetry despite the fact that most fundamental particles are left-handed.
  • Mirror particles interact with ordinary particles via gravity only.
  • Has not been experimentally verified but poses as an interesting solution to what dark matter is.

My question: Is the theory of mirror matter inconsistent with String theory by which the force of gravity is in fact carried by a particle, the graviton? Can both theories be correct? If not, what is more likely: that gravity is in fact the only force not carried by a particle, or that our universe despite possessing many symmetries is overwhelmingly unbalanced in favor of left-handed particles?

Foot says "The prospect that the most natural symmetry imaginable -- mirror symmetry-- is not a symmetry of nature, while every other obvious symmetry such as rotational symmetry and translational symmetry are indeed symmetries seems rather surprising to say the least." Then again it also seems surprising that all of the fundamental forces besides gravity can be unified.

Foot can't seem to promote his own theory without undermining supersymmetry and String theory. But I like both.

Thursday, July 9, 2009

Hypercubes

I've been planning a "mirror matter" post for a while now but it's so fun to research that I haven't had the discipline to write about it. Hopefully I'll get around to it soon.

In the meantime, I've been reading "Surfing Through Hyperspace" by Clifford Pickover. It might be the strangest book I've ever read, but it does pose some very interesting questions. The book deals with understanding higher dimensions through a second-person narrative science fiction story interspersed with more technical explanations. There is an emphasis on 4-dimensional beings and what it would be like to interact with them. We would only see 3-dimensional blobs changing in size. But a 4-dimensional being looking at us would be able to see everything; they would be able to look at us inside and out all at once. A 4-dimensional being would be much like a god: able to see and hear everything at once; would be incapable of being seen; and it would have miraculous powers, like the ability to remove a tumor without making a single incision. I found it interesting that immediately after, Pickover pointed out that a 4-d being could also impregnate a virgin.

Later on, he discusses hypercubes: 4-dimensional cubes. I've seen pictures of 4-dimensional cubes projected into 2-dimensions. I used to draw them all the time when I was bored in class.
But he also gives beautiful images of 5-d, 6-d, 7-d, and 8-d cubes projected into two dimensions, that I've actually never seen before.

A 5-dimensional cube:
A 6-dimensional cube:
A 7-dimensional cube:
and, an 8-d cube:
What I found even more interesting was a graph he gave of the volume of a radius 2 sphere as a function of dimension.
The x-axis corresponds to the number of dimensions of the sphere, and the y-axis, to the volume of the sphere. The peaks occur at different places depending on the radius of the sphere.

Thursday, July 2, 2009

Wiki Wednesday

Every week I remember to, I plan on sharing a cool wikipedia page, cause there are so many.

Monday, June 22, 2009

Hyperspace

It is particularly challenging but interesting to consider other dimensions of space and time. Physicists became interested in the idea of additional dimensions upon developing string theory, which despite being somewhat controversial, as there are few conceivable ways to test such a theory, is currently our best prospect for a complete physical theory --it would make sense on both small scales and large scales, would unify the fundamental forces, and would actually predict gravity instead of just observing it-- and happens to make the most sense in 10 or 11 dimensions. Regardless of whether string theory proves to be accurate, many physicists see no reason as to why our universe should not have more dimensions than we can perceive. A line exists in one dimensional space, but it also exists in 2-D space, 3-D space, 4-D space, and so on.

Studying extra dimensions has spawned additional theories of wormholes, parallel universes, and the multiverse. Higher dimensions make up what is typically called hyperspace.

Almost any physicist or mathematician working in this field will reference "Flatland," a book written in 1884 by Edwin A. Abbot, when trying to explain their work to the public. I still haven't read Flatland but I've become very familiar with the story. It takes place in a two dimensional world called Flatland and is narrated by a character named "A. Square" who presents to the reader life in a two dimensional world. The story is designed to provoke thought concerning dimensions other than the ones with which we are familiar, and A. Square himself is forced to do the same. First when he visits a one dimensional world in a dream, and then when he is visited by a three dimensional sphere who appears to him as a circle increasing and then decreasing in size.

Flatland is so frequently referenced because the idea extends easily to our own perception. It raises questions of what a 4-dimensional object would look like, and to the true nature of reality. Similar to A. Square's perception of a sphere, a four dimensional object would appear to us in 3-dimensional cross sections. A 4-D sphere, or hypersphere would appear to be a sequence of spheres increasing in size and then decreasing, finally vanishing into a point. We would only be able to see the object in its entirety throughout time (another dimension) but could never imagine its spatial form.

Recently, in trying to understand multiple dimensions and my own perception, I've been trying to deconstruct my experience as an observer. In a single moment, time has no dimension; it is only a point. Space should have three dimensions but without time, I can only perceive two dimensions. Therefore it is my memory and my mind which allow me to assemble these splices to create three spatial dimensions and one time dimension; a succession of moments in space.

I still can't wrap my head around the fact that physical positions are so transient and what it could mean in terms of dimensions. If I move my hand, there is nothing in the space that it occupied. It reminds me again of perceiving only splices of a continuous physical thing. In a video called imagining the tenth dimension which contributed to my early interest in higher dimensions, to help imagine another dimension, the viewer is asked to consider a long "snake" that would correspond to his or her entire life's path. Our experience then consists of cross sections of this 4-dimensional "snake" and if we know we are viewing cross sections, why can we not infer another dimension? If all of our experiences were to persist in physical space, space would become overcrowded. It would require at least another dimension to compensate. If I know that 10 minutes ago I was sitting on my couch and now I am sitting on my bed, I am confident that both are part of reality but the self that was on the couch 10 minutes ago has vanished and will never be again. In every moment time and space are changing and all of experience is piecing together traces. If these moments are not in fact connected, we would not be able to experience them sequentially as we do. Time would never pass. I have read in several books on some of the subjects I have been discussing that we could stack two dimensional surfaces on top of one another to obtain a 3-dimensional object, but that is not true! If the surfaces are in fact two dimensional, they have no thickness and we can stack as many on top of one another as we'd like, we will still have a two dimensional surface. Many theories permit the existence of multiple dimensions by deducing that they must be so small that they are invisible to us. I feel like there must be some "thickness" to our moments; another very small dimension; something that allows time to pass. It has confused me for a while that even though there are an infinite number of numbers between 0 and 1, we can in fact go from 0 to 1. For me it is proof that space and time are not in fact continuous but discrete, but it's something I'm still thinking about.

As I said earlier, these are just ideas. Often when I write things of this nature it is out of confusion. It's possible that in a couple days I will understand why this can't be so. Or come to another understanding of what may be going on. In the meantime it's interesting to think about. I was originally attracted to science and math because of the certainty involved, particularly in math. If you accept very basic and obvious statements to be true you can use these statements to build more complicated ones that eventually build off of each other to create advanced and beautiful theorems whose validity becomes unquestionable. It has always seemed a waste of time to me to learn things that may not be true. Yet I find myself infatuated with theoretical physics, a science which was once undoubtedly a science, but now will more closely resemble philosophy or science fiction at times. Theories by some of the most respected physicists can seem too far out to believe, and it can be hard to know what to take seriously. The idea that so much is unknown leaves a lot of possibility and people are starting to get creative.

Tuesday, June 16, 2009


The very disappointing math section at a local bookstore.

Monday, June 1, 2009

Evolution and Black holes

Is life an accident? Intelligent Design argues that the complexity and efficiency of nature imply a creator. One of the most famous of such arguments is William Paley's "watchmaker analogy" published in 1802. The analogy roughly translates to "design implies designer" using a watch as an example. The argument goes as far back as around 50 B.C., when it was first introduced by Cicero in De Natura Deorum:

"When you see a sundial or a water-clock, you see that it tells the time by design and not by chance. How then can you imagine that the universe as a whole is devoid of purpose and intelligence, when it embraces everything, including these artifacts themselves and their artificers?"

Darwin was fascinated by this idea as a student. However, upon developing the theory of natural selection, he believed he had answered Paley's question. Darwin's theory of evolution has since framed much of science. Additionally computers and modern mathematics have demonstrated complexity resulting from simple rules, over and over again: one example is in dynamical systems, or fractals.

Richard Dawkins, a well-known atheist, author of The Blind Watchmaker, wrote a computer program that, using only a few rules, generated shapes resembling shrimp and insects.

He was surprised at the output of his code, as this was certainly not his intention. He used this result to demonstrate that in fact such things can happen accidentally.

Although science provides some explanation, not everyone is able to drop the idea that life could only exist as a result of a higher power, or creator. It still seems amazing that something so complicated and powerful as the human eye could have come about by chance. A similar example is given in the following video:

http://www.youtube.com/watch?v=y-Fsfou190s

Although I'd expect most atheists find this video amusing instead of a "nightmare," no one can deny that in many ways Earth is perfectly suited for life, as if it were made for us.

Thus far, searches for Extra Terrestrial Intelligence have revealed that complex life is very rare in our universe. Not only is Earth the only planet we know of to support complex life, if conditions were tweaked ever so slightly, Earth may have been yet another barren rock in our solar system. The distance from the sun, lack of meteors, tilt, even the size of the moon are all factors, and in our case, each parameter is just right.

In developing theories of the universe, scientists look at an even bigger picture: how is it that intelligent life was able to develop in our universe? By the anthropic principle, in understanding our universe, we must take into account that we are in it. String theory can generate hundreds of possible universes, and many scientists believe, in some form or another, that hundreds, or in fact an infinite number, of universes can and will exist. It is possible by probability that millions of universes were created, as ours was, before conditions would allow life. The fine-tuned universe refers to an assertion that were our universe only slightly different, that is, had there been a small change in several of the approximately 26 dimensionless physical constants, our universe would be completely different, and in almost all cases, would not be conductive to matter forming. Neuroscientist Larry Abbott wrote:

"the small value of the cosmological constant is telling us that a remarkably precise and totally unexpected relation exists among all the parameters of the Standard Model of particle physics, the bare cosmological constant and unknown physics."


As is the case with Earth, were things only slightly different life may have never formed in the universe at all, which makes it even more amazing that it did.

Just as Darwin's theory replaced bewilderment with theory and reason, theoretical physicist Lee Smolin proposed a theory around 1980 that makes these tiny and unbelievable probabilities a little more reasonable. His theory is sometimes called cosmological natural selection (CNS). He suggested that black holes may be a means by which universes create universes (i.e. universes reproduce). A black hole does contain a singularity at the center and relativity predicts that the initial state of the universe at the beginning of the Big Bang was a singularity. Most importantly, by Smolin's theory, the physics of one universe is passed on to daughter universes created by black holes in that universe, leading to a process analogous to natural selection that favors universes with black holes. Such universes should dominate, and therefore a universe with favorable conditions such as ours is more likely to exist because of such conditions.