12.02.08
Posted in Rendering, Science at 12:45 am by admin
The New Journal of Physics is an open-access physics journal well worth perusing. Unlike many online scientific journals, all of the articles can be read without purchasing a subscription. The titles below are linked directly to the papers’ pages where pdf versions, HTML versions, and movie files and other materials can all be found. NJoP is an impressive resource. The latest issue, Volume 10, Dec 2008, is focused on Visualization in Physics. A wide ranging series of articles explores the application of computer graphics to scientific study; from simulation, to rendering, to analysis, and presentation. Here, I’ve picked a few of my favorite articles from the current issue to pique your interest.
Spiral Metamorphosis, collision between the Milky Way and Andromeda galaxies.
Visualizing astrophysical N-body systems
John Dubinski
Abstract. I begin with a brief history of N-body simulation and visualization and then go on to describe various methods for creating images and animations of modern simulations in cosmology and galactic dynamics. These techniques are incorporated into a specialized particle visualization software library called MYRIAD that is designed to render images within large parallel N-body simulations as they run. I present several case studies that explore the application of these methods to animations in star clusters, interacting galaxies and cosmological structure formation.
An SPH data set rendering a galaxy collision
Colliding galaxies, rotating neutron stars and merging black holes—visualizing high dimensional datasets on arbitrary meshes
Werner Benger
Abstract. Visualization of datasets stemming from diverse sources is challenged by the large variety of substantial differences in topology, geometry and nature of the associated data fields. Since there is no standard on how to formulate and treat data for scientific visualization, algorithms are frequently implemented in a highly domain-specific way. Here, we explore the potential of point-wise rendering as a generic way to represent single or multiple fields instantaneously on arbitrary mesh types.
Electron distribution in silicon
Visualizing a silicon quantum computer
Barry C Sanders, Lloyd C L Hollenberg, Darran Edmundson and Andrew Edmundson
Abstract. Quantum computation is a fast-growing, multi-disciplinary research field. The purpose of a quantum computer is to execute quantum algorithms that efficiently solve computational problems intractable within the existing paradigm of `classical’ computing built on bits and Boolean gates. While collaboration between computer scientists, physicists, chemists, engineers, mathematicians and others is essential to the project’s success, traditional disciplinary boundaries can hinder progress and make communicating the aims of quantum computing and future technologies difficult. We have developed a four minute animation as a tool for representing, understanding and communicating a silicon-based solid-state quantum computer to a variety of audiences, either as a stand-alone animation to be used by expert presenters or embedded into a longer movie as short animated sequences. The paper includes a generally applicable recipe for successful scientific animation production.
A cosmological simulation 10^9 lightyears across
Splotch: visualizing cosmological simulations
K Dolag, M Reinecke, C Gheller and S Imboden
Abstract. We present a light and fast, publicly available, ray-tracer Splotch software tool which supports the effective visualization of cosmological simulations data. We describe the algorithm it relies on, which is designed in order to deal with point-like data, optimizing the ray-tracing calculation by ordering the particles as a function of their `depth’, defined as a function of one of the coordinates or other associated parameters. Realistic three-dimensional impressions are reached through a composition of the final colour in each pixel properly calculating emission and absorption of individual volume elements. We describe several scientific as well as public applications realized with Splotch. We emphasize how different datasets and configurations lead to remarkably different results in terms of the images and animations. A few of these results are available online.
Visualization of an MHD current scroll
Flow visualization and field line advection in computational fluid dynamics: application to magnetic fields and turbulent flows
Pablo Mininni, Ed Lee, Alan Norton and John Clyne
Abstract. Accurately interpreting three dimensional (3D) vector quantities output as solutions to high-resolution computational fluid dynamics (CFD) simulations can be an arduous, time-consuming task. Scientific visualization of these fields can be a powerful aid in their understanding. However, numerous pitfalls present themselves ranging from computational performance to the challenge of generating insightful visual representations of the data. In this paper, we briefly survey current practices for visualizing 3D vector fields, placing particular emphasis on those data arising from CFD simulations of turbulence. We describe the capabilities of a vector field visualization system that we have implemented as part of an open source visual data analysis environment. We also describe a novel algorithm we have developed for illustrating the advection of one vector field by a second flow field. We demonstrate these techniques in the exploration of two sets of runs. The first comprises an ideal and a resistive magnetohydrodynamic (MHD) simulation. This set is used to test the validity of the advection scheme. The second corresponds to a simulation of MHD turbulence. We show the formation of structures in the flows, the evolution of magnetic field lines, and how field line advection can be used effectively to track structures therein.
VisTrails workflow showing output of galaxy cluster simulation
Visualization needs and techniques for astrophysical simulations
W Kapferer and T Riser
Abstract. Numerical simulations have evolved continuously towards being an important field in astrophysics, equivalent to theory and observation. Due to the enormous developments in computer sciences, both hardware- and software-architecture, state-of-the-art simulations produce huge amounts of raw data with increasing complexity. In this paper some aspects of problems in the field of visualization in numerical astrophysics in combination with possible solutions are given. Commonly used visualization packages along with a newly developed approach to real-time visualization, incorporating shader programming to uncover the computational power of modern graphics cards, are presented. With these techniques at hand, real-time visualizations help scientists to understand the coherences in the results of their numerical simulations. Furthermore a fundamental problem in data analysis, i.e. coverage of metadata on how a visualization was created, is highlighted.
Accretion disc around a black hole
How computers can help us in creating an intuitive access to relativity
Hanns Ruder, Daniel Weiskopf, Hans-Peter Nollert and Thomas Müller
Abstract. Computers have added many new possibilities to the tool box used for visualizing science in general and relativity in particular. We present some new results from our own work: (2 + 1) dimensional Minkowski diagrams showing two spatial dimensions, extended wormhole visualization, and the illustration of accretion discs by using the approximation via a rigidly rotating disc of dust. We also discuss some related examples from our earlier work, such as interactive and immersive visualization, or the visualization of the warp drive metric.
Image showing large scale structure of galaxies in the universe
Visualization of large scale structure from the Sloan Digital Sky Survey
M U SubbaRao, M A Aragón-Calvo, H W Chen, J M Quashnock, A S Szalay and D G York
Abstract. We will discuss the challenges of visualizing large cosmological datasets. These include observational issues such as the masks and incomplete nature of the survey volume, cosmological issues such as redshift distortions and the difficulty of visualizing datasets that span cosmological epochs, as well as the inherent visualization challenges in presenting dense three-dimensional (3D) datasets. Two case studies will be presented. The first will feature the identification of filamentary structures in the large scale distribution of galaxies. The second case study will feature visualizations of the correlations between quasar absorption line systems and luminous red galaxies. Finally, we will give an overview of our visualization work-flow which features the use of the open-source 3D modeling program Blender.
Several sources of spiral wave structure
Visualization of spiral and scroll waves in simulated and experimental cardiac tissue
E M Cherry and F H Fenton
Abstract. The heart is a nonlinear biological system that can exhibit complex electrical dynamics, complete with period-doubling bifurcations and spiral and scroll waves that can lead to fibrillatory states that compromise the heart’s ability to contract and pump blood efficiently. Despite the importance of understanding the range of cardiac dynamics, studying how spiral and scroll waves can initiate, evolve, and be terminated is challenging because of the complicated electrophysiology and anatomy of the heart. Nevertheless, over the last two decades advances in experimental techniques have improved access to experimental data and have made it possible to visualize the electrical state of the heart in more detail than ever before.
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12.01.08
Posted in Art, Science at 3:14 am by admin
Photograph from my roof, taken this evening with a 100mm lens on a Canon Xsi (and resized to lower resolution for display on the web). That’s the Moon, Jupiter (upper), and Venus (lower) at about 6pm my time. By this time tomorrow (December 1), the planets will have swung around behind the moon; they reach their closest approach around 3am Pacific time, when they will unfortunately not be visible.
Stellarium, at stellarium.org is an awesome program for working out where to set up your camera; I used it to work out the times to go out on the roof.
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11.29.08
Posted in Science at 2:54 am by admin
Life is a spontaneously autocatalytic process or system that decreases local entropy by converting environmentally available energy to heat.
- Nick Porcino, November 2008
Introduction
How can events in space and time which take place within the spatial boundary of a living organism be accounted for by physics and chemistry?
This was the question with which Schrödinger opened his little treatise, What is Life? (1942). Molecules have a certain stable configuration that cannot change unless the energy necessary to do so is supplied from outside, so he suggests the application of thermodynamics to the study of life. Schrödinger argues that when a non-alive system is left alone, its motion comes to a halt, and the system reaches a state of thermodynamic equilibrium, or maximum entropy. A living system therefore attempts to reach a state far away from entropy, called negative entropy, or negentropy. Schrödinger then derives the notion of a life being a mechanism by which an organism extracts order from the environment. Similarly, in the classic On Growth and Form, Thompson studies the application of mechanical and thermodynamic principles to the expression of life (1917/1944).
The life theorem presented here takes these notions a step forward by taking Schrödinger’s somewhat poetical ideas and grounding them into the mathematical and physical realm of thermodynamic autocatalytic processes, as suggested by Thompson. Following the recent development of Garrett (2008), life is defined in terms of a heat engine, yielding the life theorem.
Autocatalysis
Life begins on its own; it bootstraps itself; when the time is right, it picks up and goes. A seed in the presence of moisture begins to manufacture a plant. A quickened ova divides as factories in the cell’s nucleus begin to run.
In contrast, a mere chemical reaction slows and eventually stops as the amount of substance available for the reaction decreases. A reaction can be accelerated by a catalyser - some substance that by its presence produces or accelerates the reaction. Even in the presence of catalyst, the reaction must inevitably slow down and stop because the amount of energy and material in a system is not altered by the catalyst.
In some chemical reactions, a catalyser is formed as a product, or by-product of the main reaction. As the reaction proceeds, more catalyser is produced, continuously increasing the rate of reaction until some limit of available material or energy is met, or a product is formed that antagonizes the original production. A reaction that produces its own catalyst is called an autocatalytic reaction. (Thompson, 1942, p. 255).
An intriguing experiment by Brailsford Robertson in 1921 compared the growth and reproductive rates of single infusorians in drops of water. Robertson showed that a single infusorian isolated in a drop of water will reproduce faster in a drop of water containing metabolic product from another infusorian, compared to another infusorian in clear water. This clearly indicated that an autocatalytic substance was being given off by infusoria (p. 617). Charmingly, Robertson called this metabolic product the X-Substance.
A plant’s seed is another autocatalytic system. The seed, in the presence of the right resources, will begin a reaction under which catalysts for further reaction are produced, and the plant begins to grow.
Since the formation of a quartz crystal or a snowflake is autocatalytic, might it correspond to life? Crystals are formed in a solution of material around a seed. Ions in the solution are deposited in a growing lattice as energy is given up by the system yielding structure and form. The crystal alters local conditions, particularly the atomic structure of matter at the surface of the crystal, to sustain and accelerate the creation of more crystalline structure. This process is auto-catalytic and might seem life-like. A forest fire is also autocatalytic, once it starts it propagates itself. To determine if the crystal or the fire is alive however, we need also to consider the effect on local entropy, in other words, the thermodynamic aspects. We will return to the crystal and the fire shortly.
Thermodynamics
Erwin Schrödinger (1944) suggested that living systems import and store negative entropy, which he called negentropy. Negentropy is an information theory statistic used as a measure of local normality. As statistical mechanics describes what we can observe of the motion a physical object without describing why the object moves, the storage of negative entropy describes what life is getting up to, but it does not tell what is going on. Nonetheless, Schrödinger and Thompson (1942) provide a starting point for the present theory by grounding our thought in thermodynamics. Here, we consider entropy in the thermodynamic sense - a measure of the ability of a given interval of space-time to do work.
General thermodynamic laws require that all systems, including living systems, convert environmental potential energy into a less available form, such as heat.
A system can be described as an entity separated by a permeable boundary from its environment. The environment is embedded in its surroundings.
The permeable boundary of the entity imposes an energy potential - matter can be transported across the boundary from the environment to the entity in order for the entity to perform work. Work converts the matter to the less available form, heat, which is lost from the system to the surroundings.
Life in this formulation can be considered a heat engine. Unlike a textbook heat engine, in which work is defined in the entity’s contribution to the energy potential of an external system, work is defined here with respect to the internal energy potential of the entity with respect to the environment (Garrett, 2008). Now we can state some simple mathematics to describe the system:
- The temperature of the surroundings is cooler than the temperature of the environment, Ts < Te.
- The energy potential between the entity and its environment is the gradient of temperature and pressure across the permeable boundary, DG(Te, p).
- This boundary is defined to be positive, DG(Te, p) > 0. (Equation 1)
- Work is the rate at which the boundary gradient changes, w = d(DG)/dt (Equation 2)
- The entity converts available energy to work, w, at the rate a (the rate of energy consumption)
- The boundary potential is related to work by a = aDG where a is some entity specific constant. (Equation 3)
- The entity’s efficiency is the ratio of work to the rate of conversion, e = w/a
- Heat is radiated to the surroundings (and is lost to the system) at the rate a - w
- Since Equation 3 relates a to G directly, the system is a positive feedback loop.
We can see that since the boundary gradient is defined to be positive, if w is also positive, the potential at the boundary between the entity and its environment will continuously move to a higher level, and will correspondingly increase energy consumption (Garrett, 2008). Such a system evolves exponentially, and many life processes can be described within this formulation as both Schrödinger and Thompson argued.
And thus the second part of the definition of life is derived. Life, in these thermodynamic terms, is decreasing local entropy by maximizing the conversion of available resources to work, and conserving global entropy by radiating heat to the surroundings.
Our example crystal is therefore not alive by the definition, because the energy gradient between the crystal and its environment does not change as the crystal grows, and so w in equation 2 is zero. As the crystal lattice grows energy is bound into the lattice, heat is radiated to the environment, but these are in equilibrium. In fact, the form of the crystal is governed by a process that minimizes the stored energy (cf. Libbrecht, 2005). The forest fire also is not alive; it transfers all of its energy to the environment and surroundings as heat, the opposite of what an alive thing would do.
Garrett’s thesis states that global society is a heat engine, and that it produces effects and products that increase the efficiency of the conversion of work to heat. This satisfies the condition of autocatalysis, and so global society can be considered to be a life form by the present theory. By extension, we note that distinguishing society from other terrestrial systems and processes is artificial - people eat food, for example. The present theory may therefore provide a grounding for the Gaia hypothesis (Lovelock, 1974). The Gaia hypothesis states that the Earth is complex interacting system that maintains a preferred homeostasis, and is sometimes interpreted to say that the Earth is in fact an organism or life form. One of the principal criticisms of the Gaia hypothesis is that the conventional definition of life does not admit the Earth.
Overturning Convention
The conventional definition of life as found in nearly any textbook is an ad hoc melange of seven rules:
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homeostasis is the regulation of internal state
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organization of one or more cells into an organism
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metabolism is the consumption of energy to create cellular components
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growth means to maintain a higher rate of synthesis than catabolism
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adaptation is the ability to change over a period of time in response to changes in the environment
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response to stimuli, from a rate of change of chemical conversion, to gross motion of an organism
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reproduction by which new organisms can be produced.
One can fiddle away on these rules trying to refine them and catch border cases, but it never really pans out. Some obvious problems with these rules are that they presuppose that life must be composed of cells (rules 2 and 3). A life form must grow (rule 4), yet a senescent organism is clearly still alive. Many organisms such as archaeobacteria have remained unchanged for millions of years, yet they are alive despite rule 5. Worker bees cannot reproduce, but are alive despite rule 7.
Under the present theory, it is possible to show that each of the seven rules are potentially characteristic of life, but none of them are specifically required. Furthermore, the virii can be shown to be life. Although they do not contain the conventional characteristics within themselves, being little more than a strand of DNA in a complex protein shell, they do autocatalytically hijack cells to perform specific work.
Surprising Inductees to the Hall of Life
It would be possible for robots or computers, or even the Internet to be alive, although present day versions are not spontaneously auto-catalytic - today they do not start to self-assemble when conditions are right, humans must do it for them. They do not work to increase the energy potential difference between themselves and the environment as that is fixed by their unchanging structure - for example, a laptop is plugged in to the wall to get energy, and its work does not serve to increase its total energy: thermodynamically it produces only heat. For a computer virus or a bot doing work on the internet to be alive, it would be necessary to equate information with energy. To make such a claim is interesting, but outside of the scope of this presentation.
The present theory welcomes back the worker bee, and the virus. It also allows bacteria that can only survive in host cells. It will make it easier to identify alien life should it present itself some place like Mars or the Internet in some difficult to conventionally pigeonhole fashion. It could provide tests and prognostic models for the Gaia hypothesis. It could also the form of signatures of life processes that could be scanned for in astrophysical observations. Researchers in artificial life, robotics, and nano-technology could use the principles to advance their own work - at a theoretical level, criteria for behavior could be established, and at a practical level, analysis and improvement of efficiency could result.
Conclusion
The Life Theorem, derived from fundamental thermodynamic principles and practical experiment, provides a framework under which predictions about life processes can be made, and a mathematical basis to understand whether or not a system is alive. The conventional definition of life was shown to reflect sufficient but not necessary aspects of systems that are alive. Robots, seeds, viruses, infusoria, and snowflakes were categorized as alive or not-alive. Practical applications were suggested that could give the theorem practical relevance.
Acknowledgements
I would like to thank my daughters, Kevin Kelly, and all the responders on the original posting for asking some hard questions that improved this work.
References
Timothy J. Garrett, Are there basic physical constraints on future anthropogenic emissions of carbon dioxide? arXiv:0811.1855v1 [physics.ao-ph], 12 Nov 2008.
Kenneth G. Libbrecht, The physics of snow crystals, Reports on Progress in Physics, 68 (4). pp. 855-895, 8 March 2005.
T. Brailsford Robertson, Multiplication of Isolated Infusoria, Biochemical Journal, xv, pp. 598-611, 1921, cited in Thompson 1942.
James Lovelock, L. Margulis, Atmospheric homeostasis by and for the biosphere - The Gaia hypothesis, Tellus 26 (1): 2–10, 1974
Erwin Schrödinger, What is Life? The Physical Aspect of the Living Cell, University Press, 1944.
D’Arcy Wentworth Thompson, On Growth and Form, Revised Edition, Cambridge University Press, Cambridge, England, 1942. First published in 1917, reprinted by Dover Publications, Mineola New York, 1992.
Gift Shop
Be careful to get the correct version of D’Arcy Thompson’s wonderful book. The book is 1116 pages long. Many editions are abridged, with “dangerous” and “improper” ideas removed, to the tune of nearly 800 pages of “dangerous” ideas. The Dover reprint linked here is complete and therefore recommended despite being soft-cover and printed on non-archival paper.
Schrödinger’s small book is a great read, and was clearly an inspiration to Feynman when he wrote QED. What is Life? is condensed from a short series of lectures, and engages the reader in thoughtful experiments to clarify tough subjects. It is playful and humble, and anticipates many important discoveries to come. It calls DNA “a variety of contents compressed in a miniature code-script,” a description very appealing to a programmer!
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11.28.08
Posted in Code, Rendering at 11:23 pm by admin
Ziggyware is hosting a fantastic tutorial by Alexander Grafenstein on using geometry clipmaps to display terrain. Geometry clipmapping is a level of detail technique that works by shifting nested regular grids through the terrain data as the viewer moves, by Asirvatham and Hoppe, and published in GPU Gems 2. Grafenstein’s presentation is very clear and comes with working source.
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11.23.08
Posted in Science at 11:53 pm by admin
The wonderful hexagonal structure at Saturn’s north pole has persisted since first viewed by Voyager, 26 years ago. The south pole has an equally unusual structure.
The plasma field surrounding Saturn is interesting, it is precisely aligned with the planet’s rotational axis, due to the fact that the magnetic field is precisely aligned thus. The magnetic field rotates at the same speed as the planet.
Saturn’s moon Enceladus shoots out water that is ionized by the sun. The heavy oxygen ions generated are captured by Saturn’s magnetic field which turns it into a tube ring of energized plasma. As Saturn’s magnetic field rotates, it drags the ions around the planet in a counter-clockwise direction.
My friend Vik Sohal suggested that perhaps the persistent storms were similarly formed by electrodynamic forces. This recent Cassini image showing auroral activity over the hexagon lends credence to that notion.
I noticed that the turbulent structure seems to be quite pronounced near the surface of Saturn’s clouds. That made me think of this paper by Kinney and McWilliams:
The role of coherent structures in magentohydrodynamic turbulence
R. Kinney, and J.C. McWilliams, Small-Scale Structures in Three-Dimensional Hydrodynamic and Magnetohydrodynamic Turbulance, Springer Berlin/Heidelberg, 1995
Long-term self-similar behavior is exhibited in high-resolution numerical solutions of decaying turbulent two-dimensional magnetohydrodynamics. Random initial conditions produce coherent structures which dominate the fluid dynamics. During the self-similar evolution, current monopoles (magnetic vortices) with an accompanying vorticity distribution (which is axisymmetric but not necessarily monotonic) emerge spontaneously. Thin current and vorticity sheets are created as a result of close encounters between the vortices. The sheets are sites of current enstrophy production, conversion into kinetic enstrophy, and dissipation, all of which maintain constant ratios but which produce no net change in the total enstrophy. The chief mechanism for removal of enstrophy is the disappearance of magnetic vortices during coalescence.
I contacted Dr. McWilliams, and he agreed that perhaps Saturn’s atmosphere could be undergoing a process similar to that in his simulations, although he suggested that the prevailing view is that the effect is more likely due to friction. That led me further, to Einstein’s explanation of the tea leaf paradox.
Tempest in a teacup
In the tea leap paradox, tea leafs settle in the center of a cup when stirred, rather than at the edges, as one might expect by analogy to a centrifuge. Stirring the tea makes the water spin about the cup’s central axis, and spiral out from the center. The water below is slowed by friction with the cup’s bottom, and its spin is weakened - this sets up a circulation in the cup where water at the top is spun out, travels down the walls of the cup, and then flow back up the middle, forming a circulating donut.It seems to me that Saturn’s spinning ring current is the spoon stirring the cup, lower layers of atmosphere are the bottom of the cup providing the friction to set up the donut shaped convective flow, and persistent coherent structures in magnetohydrodynamic turbulence provide the fascinating detail observed in the tea leaves.
Nick Porcino, Nov. 2008
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Posted in Science at 11:03 pm by admin
The phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest… The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, … that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body… The introduction of a “luminiferous ether” will prove to be superfluous…
A. Einstein, 1905 (1)
The Lorentz factor describe how time and length scale as measured from two different frames of reference. There are a great many ways to derive the Lorentz factor, each with its particular focus or raison d’etre. For example, some (4) seek to differentiate Galilean from Lorentzian transforms. This derivation, similar to the derivation in Mead 2000 (2), is the simplest I was able to come up with, and easy to work through.
We imagine that we have a rod of length l, in which a signal propagates back and forth at the speed of light, c. Each transit to the other end takes t = l/c, and the round trip takes time t = 2l/c. Einstein compares this scenario to the ticking of a clock. In this experiment, we imagine a second identical rod moving along the x axis at some velocity v (shown blurred along x in the diagram). At t = t’ = 0, a light pulse is emitted; the blue discs in the diagram indicate the position of the pulse sampled regularly in time, as it travels up and down the z axis on both rods.
Since one of the basic postulates of Relativity is that light always propagates with a definite velocity c, we can immediately see in the diagram that the pulse on the second rod appears to be moving slower than that of the first rod (as indicated by the length of the red line between the samples). As the pulse moves the distance l in the z direction, it travels the distance vt’ in the x direction. The observer at the origin will see the pulse on the second rod after a lengthened period of time d indicated on the diagram by the red line extending from 0 to vt. By the Pythagorean theorem:
Equation 1
Since d = ct’ (the length of the red line to vt in the diagram), and l = ct (the length of the rod), substitution into Equation 1 gives
Rearranging and solving for t1 yields
Equation 2
Einstein’s derivation in reference 1 is similar; he shows that if the rod is placed along the x axis, the length of the rod will appear shortened by the Lorentz factor according to the velocity of the observer.
Equation 3
The next step is to show the derivation of the Lorentz transform, given the equation for the contraction of length, equation 3.
References:
1) Albert Einstein, On the Electrodynamics of Moving Bodies, Annalen der Physik, 17:891, 1905. The linked translation was prepared by John Walker.
2) Carver A. Mead, Collective Electrodynamics, Quantum Foundations of Electromagnetism, MIT Press, 2000, p.64
3) Bernhard Rothenstein, A faster than “World’s fastest derivation of the Lorentz transformation”.
4) relativitycalculator.com: The Heart of Special Relativity: Lorentz Transformation Equations (I wasn’t able to determine the author’s name).
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Posted in Robotics, Science at 8:50 pm by admin
My early research in bio-mimetic engineering and autonomous underwater robotics keeps me interested in staying abreast of the latest in robotic sea creatures.
GhostSwimmer is a robotic fish created by researchers at Boston’s Franklin W. Olin College of Engineering and Boston Engineering. It is perhaps the most advanced robotic fish anyone’s built yet. It’s modeled after the tuna, an extremely fast fish. For robotics engineers, it’s particularly attractive because the fish is submarine shaped and has a relatively rigid torso. Aside from the hydrodynamic shape, the major planned improvement of this design is to have a spine formed of electroactive polymers that will flex when voltage is applied. (via The Economist)
This is an idea I had explored, although at the time the technology wasn’t available to build one. Intriguing features of the real-life tuna’s physiology are the fact that when the fish goes fast, all the fins tuck into the slots in the body so the fish becomes a completely streamlined shape with nothing projecting into the flow. In the deep ocean where the tuna lives, getting from place to place quickly is a lot more important than maneuvering.
Earlier work on robotic fish has also focused on propulsion via fins, here’s a more conventional design based on servos.
Gizmodo has an article on work at the University of Washington carried out in the lab of Kristi Morgansen. I am particularly keen on the design of their robot; the pectoral fins are actuated by microlite servos, the rear of the fish is activated by a high speed servo, and the tail fin is driven by a high-torque micro motor.
This sort of design should yield higher maneuverability, and greater mechanical efficiency through reduced cavitation and drag. Of course fish have already proven the superiority of this sort of an arrangement! A useful contribution of their research is that they’ve derived a general mathematical framework for controlling robots that propel themselves by changing their shape.
Their current tests involve trying to make the fish school using very simple communication, and very simple commands, such as “swim in the same direction as me.” The theoretical and practical groundwork for this flocking, schooling, and herding was laid by Craig Reynolds back in 1987. It will be cool to see a physical embodiment of Boids!
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11.22.08
Posted in Rambling at 5:06 pm by admin
According to Typealyizer, I’m an INTP; according to their literature expert, I write sort of like Edgar Allan Poe, or Charles Darwin. That’s all very flattering I suppose. It guessed I’m male. The websites were fun to play with, certainly; you might enjoy pushing some of your work through their analyzers. I wonder if people who read this blog will also come out INTP, and sounding like Poe or Darwin? Post a result!
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11.17.08
Posted in Science at 3:10 am by admin
Version 1.1. Fixed Figures 1 and 2, and accompanying text. If you can see a purple leaf in the diagram, your cache is showing you the old diagrams. Updated the text following Equation 2. Corrected typo in Equation 4. There are more revisions to come, as parts of this derivation are still very arm-wavy.
This post is meant to help create an intuitive feel for how the Lorentz transform works. I work through a number of simple steps, until in the end, an easy formulation of the Lorentz transform using the Clifford algebra is demonstrated.
I think I’ve got everything correct, however if you spot an error, or if something is not clear, or confusing, let me know in the comments!
The Lorentz transformation is used to convert measurements made by two observers into each other’s frames of reference. Here are a couple of moving observers (figure 1); they’re moving in x, y, z, in time, t.
Figure 1
Each observer has an independent frame of reference. We’ll denote the red observer’s frame of reference the original frame, (x, y, z, t). The blue observer’s frame will be (x’, y’, z’, t’).
Figure 2
At the given moment of time, we’ll swing a common reference frame around such that the relative velocity between the two is on one axis, shown by the dark arrow in Figure 2. We can imagine as time goes on, the two observers will become more separated along this axis, commonly chosen to be z. The other two spatial dimensions are not changing. This is called the standard configuration. The Lorentz transformation relates how the two observers see each other. Since we’ve oriented things so that only the z axis is changing, we can use the Lorentz equation. The derivation of these equations are elegantly described in Einstein’s famous 1905 paper (Reference 3), so I won’t describe it here.
Equation 1
The Lorentz transformation describes how to go from one frame of reference to the other. If we graph the time on the vertical axis (it is traditional to do so), and the value we are measuring on the horizontal axis, as velocity increases, the prime frame of reference gets squashed along the diagonal (figure 3). When the relative velocity is zero, the two frames look the same; when the relative velocity is the speed of light, the prime frame collapses on to the diagonal.
Another way to think about this diagram is to realize that events on the vertical axis are events that occur at the same place in both frames but in different times. Events on the horizontal axis are events that occur at the same time in both frames but different places.
Figure 3
If we generate a bunch of frames, and plot the same places and same times for the different relative velocities, we will discover that the shape of the transform is hyperbolic. To give an idea what that means, here is a hyperbolic surface (figure 4). Note that this isn’t a solution of the equations, simply a visualization of what a hyperbolic surface could look like.
Figure 4
Here is the transform visualized another way (figure 5). If we imagine an event occuring at r = 0, and some t, for a given frame, that event will occur at the transformed location which can be found by tracing out the hyperbola until it intersects the frame (figure 6).
Figure 5
Once again, the vertical axis represents time, the horizontal axis represents position. Position divided by time is of course velocity. The diagonals represent the speed of light, which never changes.
Figure 6
The red dots show events as they are transformed into the prime space.
At this point, it is traditional to generate a boost matrix to describe the transformation.
This matrix only gives a boost between two frames in relative motion. A full Lorentz matrix would also include rotations about the three axes. The matrix is already unwieldy, with rotations it becomes more so (see Reference 1).
For a different, and much easier approach, notice that the Lorentz transformation is actually a rotation in hyperbolic geometry! Figure 7 shows the hyperbolic functions. In the diagram, a is defined as the shaded area; it’s analogous to regular trigonometric functions with the circle.
Figure 7
A hyperbolic rotation is therefore another way to think of the Lorentz transform.
Figure 8
Figure 8 demonstrates how the area of the hyberbolic function increases with a, as does the trigonometric function.
The following derivation follows Doran and Lasenby, reference 2. Referring back to Equation 1, we introduce a new identity:
Equation 2
Next, we introduce some basis vectors, e, such that
In the prime frame, these basis vectors will be different; they will describe the transformed frame (recall Figure 3). The correspondence between the original and prime frames can then be expressed as
which we read as the original frame and prime frame are equivalent; the transformed and untransformed vectors are the same, and can be transformed in both directions - the transformation is covariant. Looking back at equation 1, we see that the x and y terms drop out and can be disregarded. Next, we derive the new frame in terms of the old:
Substituting equation 2 through, we can re-express the relations as an exponential. The algebra we will use here is the Clifford algebra.
Now that it is in this form, it can be re-expressed as a rotor:
Equation 3
Note that last e is the natural e, not a basis vector. Finally, let’s work an example (from reference 2). Suppose we observe two objects flying apart, what relative velocity do they see for each other? Recall the definition of a from equation 2, and the rotors of equation 3:
We derive the relative velocities
Equation 4
We read equation 4 as: the relative velocity is the grade 2 components of the differential of the event vs. the proper time, divided by the grade 0 components of the time vs. the proper time as exponentials, which can be recast as hyperbolic functions, and evaluated as follows:
As a validation, replacing the hyperbolic tangents with u = c tanh(a) recovers the familiar equation from Reference 3:
Which shows that addition of colinear velocities in space-time is the same as the addition of the hyperbolic angles - a generalized rotation in hyperbolic space!
References
1) Wikipedia article on the Lorentz Transform
2) Chris Doran and Anthony Lasenby, Geometric Algebra for Physicists, Cambridge University Press, 2003
3)Albert Einstein, On the Electrodynamics of Moving Bodies, Annalen der Physik, 17:891, 1905. The linked translation was prepared by John Walker.
The equations were created using an online LaTeX.
The diagrams were generated using Maxima.
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11.11.08
Posted in Code, Prosody at 1:39 am by admin
- The program is the BEST at what it does
- The program is FAST
- The footprint is SMALL
- The code is CLEAR
- The program is BUG-FREE
- Abstractions must SIMPLIFY
- The unnecessary is ELIMINATED
- The system’s resources are CONSERVED
Derived from Futurist Programming/2.2. I’m dropping the Futurist part as I realized I’m not a Futurist! The Futurists chose to ignore the past, to actively revel in forgetting it, while populating the Earth with explosive, mad, new forms in the spirit of incessant and tumultuous progress.
Ignoring the past means repeating its mistakes as the same roads are trod and the same lessons re-learned, all the while, naif-like, self-congratulatory exclamations of cleverness are exchanged. The redundancy! The waste! The hubris! The Futurists, lemming-like drove themselves off a cliff; after their brief, promiscuous, and prodigious debut, they self-destructed, none surviving their aesthetic extinction on the front lines of the Great War.
I’m renaming the Tenets “Renaissance Programming”, renaissance meaning rebirth. We should take the lessons, arts, and practices of the past; tear them apart, break them down; synthesize them in a new pattern - add new components, new creativity, new idioms, then refine, distill, refactor, remove the dross, iterate! Until at last the product is born, and that product to be input to the next cycle!
Forsooth, ’tis for the win, huzzah!
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11.08.08
Posted in Art at 10:32 pm by admin
A picture I took at the Computer History Museum in Sept. 2008. If you do computer graphics, you know this teapot…
By special request, a wireframe view, and bottom-up view (I couldn’t manage a top down view!)
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Posted in Science, Sustainability at 4:17 pm by admin
Capacity is bound by a finite environment;
Capability is unbounded.
- Nick Porcino
We’re transitioning into a world of ever more constrained resources; and still-increasing demand. Continued growth as a prime societal motivator has created a world of incredible potential, variety, and opportunity. Unfortunately, the model doesn’t scale. The capacity of society to churn through things and resources is finite; there’s only so much capacity for the system to absorb the junk we churn in our wake.
Our ability to maximize production and capacity has created prosperity; with the natural limits to growth becoming all to obvious, we need to find new models for society where prosperity and well-being are not founded on the engine of material and energetic consumption and churn. I argue that although our present capabilities were advanced by prosperity through growth, we are at a tipping point where we can drive our prosperity not through growth, but through increasing capabilities. Our ability to understand the world and shape it can enable us to do it in more and more effective ways; we can develop capabilities where a little can accomplish a lot. Prosperity can be defined in terms of our ability to keep advancing our systems and simultaneously reduce material and energetic consumption and churn.
All graphs of technological progress start low with small change several hundred years ago, and then begin to bend upwards in the last hundred, and then bolt upright to the sky in the last 50.
Progress as a notion did not arrive until the 17th century or so, when it appeared in the West during the Enlightenment. Progress is a child of science and technology. It was born out of the observation that our inventions make life better… Science is sort of like the third wish for the magic lantern. It is the tool that invents new tools… [Upon the invention of science] there is an exponential explosion of both people and progress. But this curious pairing of population and progress has not been examined very much. If we return to the charts of progress we find they fit almost exactly the curves of population. As population rises so does progress and vice versa. The two growths are heavily correlated, but correlated without causation.
From Kevin Kelly’s Techium, The Origins of Progess
Kelly then notes that population is unlikely to continue to expand, and wonders if that presages the end of progress and the prosperity that enabled the population and progress explosion.
Here are my comments on the argument, and my conclusion.
- First, the explosion of population was a necessary precondition for the explosion of knowledge, for all the reasons Kelly elegantly factors (such as the cost/benefit of sharing, and considerations of immediate need).
- Second, we can posit that prosperity is a function of knowledge, supporting the first argument, since prosperity is necessary precondition for the explosion of population, as he argues.
Although total population is countered by death, total knowledge is not dependent on the size of the population. Presuming that we continue to improve storage, search, and analysis methods (rather than burning books and turning computers off), a meta system is being put in place by all of us collectively where knowledge can continue to expand independent of the expansion or absolute size of the population.
I suppose this is a Singularity-like phenomenon. At this point, prosperity needs to be recast in terms of increasing capability instead of increasing capacity.
Capacity is bound by a finite environment; capability is unbounded.
I say that this is the Singularity, and we’re staring it right down the barrel.
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10.18.08
Posted in Code, Games, InsectAI, Robotics at 8:10 pm by admin
Ian Horswill notes that we ascribe character, or intentionality to things that move. His recent work on animation control is summarized in a paper to be presented at AIIDE 2008. The example above is from that paper - two children appear to play with a ball while being observed by an adult.
Despite the apparently complex behavior, under the covers there are simple motivators driving the animation. The children want to approach the ball, but not each other. If they get close to the ball they kick it, and since they have been backing away from each other, they appear to kick it between themselves. If they get too far from the parent, a behavior kicks in where they run to the parent for a moment, then run back to play.
This algorithm can be efficiently implemented in terms of Insect AI (*). The way I translated the description of the behavior to an algorithm was to
- Identify the sensors needed, which are parent, friend, and ball. The output of these sensors has two components, a orientation component, and a scalar distance component.
- Identify the outputs, which in this case are signals to the animation system, move, kick, and hug parent.
- Create transfer functions (the green boxes) that encode the important notions, which for this controller are functions of the scalar distance component. From left to right, output as a function of distance can be read as
- Activate when distant from parent
- Activate when close to friend
- Activate except when close to ball
- Activate when close to ball
- Activate when close to parent
- Enforce a decision hierarchy using thresholding (winner take all) and controlled switches. Kick and Hug Parent take precedence over moving so the controlled switch there switches between the moving hierarchy and a zero input which stops motion completely. Kick or Hug Parent can override moving, so they are summed and thresholded before being fed to the switch.
- Once the child gets a hug, they need to have a chance to get away so the ball chasing behavior can take over again. The triggered delay modulates the close to parent signal to suppress the Hug Parent signal, which will allow the move to ball signal to pull the child out of the parent’s field of influence.
(*)
Porcino, Nick, An Architecture for A-Life, AI Game Programming Wisdom 2,Steve Rabin ed., Charles River Media, 2004, pp. 339-349
Porcino, Nick, Insect AI 2: Implementation Strategies, AI Game Programming Wisdom 3, Steve Rabin ed., Charles River Media, 2006, pp. 189-204
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10.17.08
Posted in Rambling, Sustainability at 1:30 am by admin
”In a world where a movie as incredibly produced as The Dark Knight is raking in gazillions of dollars, Star Trek stands in stark contrast,” Abrams says. ”It was important to me that optimism be cool again.”
Right on! What an awesome position to take.
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10.14.08
Posted in Art, Prosody, Rendering, Science at 12:43 am by admin
If the doors of perception were cleansed
every thing would appear to man as it is, infinite.
For man has closed himself up,
till he sees all things thro’ narrow chinks of his cavern.
William Blake, 1790
This posting is a little parable on the paucity of photorealism as a goal for graphical rendering, and a suggestion that the impressionists and expressionists were on to something.
Deep within the sun, tightly wound and highly compressed space-time gives itself up little by little; neighbouring nexus of hydrogen atoms entrain each other briefly to a higher energetic level in an exchange of quantum state quaintly named “the quantum wave function collapse”. This quantum packet of exchange is the photon - not the ballistic luminous ping-pong ball of Descartes, but the elegant gradient flux of Maxwell.
The energy is exchanged for thousands of years until the wave function collapse couples an atom near the surface of the sun with an atom near you. The entrainment of this atom takes about 500 seconds; it occurs at the speed of light. Uncountably many wave function collapses couple an exchange of energy between the sun and atoms all around you, and so objects are illuminated.
As you bring your eyes to bear on an object, that cascade of energetic couplings begun in the sun entrains molecules making up the receptors in your eyes - the rods and cones are resonant structures tuned to entrain quickly to certain wavelengths of energy.
- At this point, we must briefly interrupt the journey of the photon to note that on the way to perception, this is the last moment when physical reality has any resemblance to a photograph or computer rendering. In fact, it now rapidly diverges; whereas the eye is intermediating an energetic exchange between the environment and the brain, a camera is inducing a chemical change on film (or an electrical exchange on a sensor) for the construction of an image. The constructed image will never have the same energetic response with the environment or the viewer, and will instead only mediate other secondary exchanges between other sources of energy, the paper (or monitor), and your eye. Almost all of the wonderful interplay and change over time of spectra, energy, and the nervous system are lost at this point, flattened to the limited response of silver hallide or a semiconductor’s photo-electric charge.
The rods and cones resonate with the incoming energy, potentiating associated neurons; those neurons quiver at a threshold; when reached, they fire. These patterns of activation spread to neighbouring neurons in the retina, predisposing or inhibiting those neurons’ sensitivities, and likewise, patterns of activation spread through the layers of the retina, where patterns, gradients, and tonic levels are categorized, organized, filtered, and passed through more levels of extraction until finally a highly refined signal train describing the patterns and meta-patterns of energetic interactions of rods and cones with the environment and neurons with each other are passed down the optic nerve.
The optic nerve feeds those through the layers of the portion of the thalamus dedicated to vision where yet more processing takes place, and finally the roving spotlight of attention in the occipital lobe stitches the information into a persistent overall model of the visual field.
The attentional spotlight emphasizes those visual features of interest, retains a model of things that must be noted but are not presently in the field of foveation, and provides signals of note, such as “I think the tiger is over there.” Since the field of high resolution perception in the eye is so small, feedback to the eye is generated to saccade the gaze here and there to fill in the blanks.
Good artists, whether they be painters, directors, animators, or sleight of hand magicians understand the role of attention, and they guide the viewer through the doors of perception to achieve that activation in your mind that most closely makes you see, or most closely makes you to feel. Of course a photograph can be used to have this effect on the viewer, but it is not an intrinsic property of the eponymous photorealism, but rather a property of our perception and the subtle and wonderful fabric we weave between each other through our art.
- Nick Porcino
13 Oct. 2009
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