Rendering Lego Figures

A couple of weeks ago, I described the free software package called LDraw that with MLCad and POV-Ray allows you to design Lego creations and illustrate them with ray tracing software.  The software actually goes far beyond this, as MLCad is a real Computer Aided Design program that allows you to save the steps so that you can automatically generate instructions for your Lego creations.  It also generates a Parts List so that you know which parts you will need beforehand.

I was playing around with it this weekend, and made one of the spaceships described in a good online tutorial.  Then I rendered him with POV-Ray, and then overlaid the Lego image over a lovely picture of Saturn.  The result is: Our Hero Races Past Saturn

Our Hero Races Past Saturn

Go ahead and laugh… that’s the whole point!
Kevin Knuth
Albany NY

Posted under Fun, Lego, Software

This post was written by drknuth on March 21, 2007

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Double-Angle Formulas

To display equations, I recently installed LaTeX into WordPress by using a Latexrender plugin and it works great!  I am going to try it out by blogging about a relatively easy way to derive the double angle formulas for the sine and cosine.  The double angle formulas relate \sin( 2\theta) and  \cos( 2\theta)  to \sin( \theta) and  \cos( \theta)

Let’s be honest, who wants to waste their cortical space remembering formulas like this.  Instead, you can derive them pretty easily by remembering a few more elementary facts.  First, you can write any complex number as:

e^{i\theta}=\cos(\theta)+i\sin(\theta).

Next, you must remember how to work with exponents:

x^{a} x^{b} = x^{a+b}

and

(x^a)^2 = x^{2a}

Now we put it all together.
We start by writing

e^{i\theta}=\cos(\theta)+i\sin(\theta)

We then square both sides

(e^{i\theta})^2=(\cos(\theta)+i\sin(\theta))^2

Now use the rule for exponents on the left-hand side, and multiply out the right-hand side:

(e^{i 2\theta})=\cos^2(\theta)+2i\cos(\theta)\sin(\theta)-\sin^2(\theta)

Now write the left-hand side in terms of sines and cosines again:

\cos(2\theta)+i\sin(2\theta)
      =\cos^2(\theta)+2i\cos(\theta)\sin(\theta)-\sin^2(\theta)

Last, equate the real parts and the imaginary parts and you have your double-angle formulas:

\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)
\sin(2\theta) = 2\cos(\theta)\sin(\theta)

Easy as \pi!
Kevin Knuth
Albany NY

Posted under Mathematics

This post was written by drknuth on March 20, 2007

Celestia

This semester, I am teaching a course on Star Systems, and I have found the free software package Celestia to be excellent.  Celestia performs 3D real-time rendering of the planets and stars with high-quality graphics and allows you to travel through both space and time. 

You can watch a lunar eclipse from Earth, and then go to the Moon and see that the Earth is eclipsing the Sun!  The local stars are all real, and you can actually travel to them, and watch the other stars pass you by in three-dimensions.  It is very interesting to do this with the constellation lines drawn so that you can see that the constellations are meaningless patterns resulting from our particular viewpoint in space.

Animation of Rotating Mars

You can make movies of events, write scripts for tours, and screen capture some very lovely images.  There is a web site called Celestia Motherlode where you can download all sorts of addons that improve the quality of the images (often substantially)  and have various tours already written.

Jupiter-rise from Io (made with Celestia)

Above is an animated gif of Mars that I made from Celestia images, followed by an image I arranged of Jupiter-rise from Io.
I should reiterate that it is free and actually pretty easy to install!

Enjoy,
Kevin Knuth
Albany NY

Posted under Astronomy, Exploration, Fun, Mars, Software

This post was written by drknuth on March 18, 2007

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How to Set the Path in Windows 2000 and Windows XP

Another note to my future self…

The path is now handled by Windows 2000 and Windows XP, so forget about autoexec.bat and the like.  The path is a system environment variable, which can be changed by:

1. Go to the desktop and right-click on My Computer.
2. Click on Properties.  This brings up a System Properties window.
3. Click on the Advanced tab.
4. Way down at the bottom click on the Environment Variables button.
5. Single click on PATH in the upper User variables panel.  This populates a list of path variables in the lower System Variables panel.

Now you will want to do one of two things. 
You will either want to add a New path, Edit an old one, or Delete and old one.

To EDIT:
1. Left-click on the variable to edit to highlight it
2. Click on the Edit button
3. The path can then be edited
4. In the event that you want to have the system look in multiple paths, remember that the are separated by semicolons ; like this:

c:\texmf\miktex\bin;C:\VXIPNP\WinNT\Bin

To add a NEW path:
1. Click New
2. Enter the new system variable name
3. Enter its value

To DELETE an old path:
1. Left-click on the variable to edit to highlight it
2. Click Delete

Kevin Knuth
Albany NY

Posted under Internet, Note to Self, Software, Solutions

This post was written by drknuth on March 17, 2007

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Sonofusion

I am the chair of our physics department’s weekly colloquium, and this week our guest speaker was Dr. Richard T. Lahey Jr., the Edward Hood Jr. Professor of Engineering at Rensselaer Polytechnic Institute.  Dr. Lahey spoke to us about his recent successes in sonofusion.  In short, sonofusion is the act of achieving nuclear fusion by focusing sound waves on bubbles in a liquid to compress them to the density at which nuclear fusion occurs.  The most amazing feature is that this can be performed in a tabletop experimental setup costing about $3000!

Dr. Lahey performs this research in collaboration with  Rusi P. Taleyarkhan, the Ardent Bement Jr. Professor of Nuclear Engineering at Purdue University and Robert Nigmatulin, of the Russian Academy of Sciences.  Below is a photo of Dr. Taleyarkhan with the reactor.

 Dr. Taleyarkhan with the sonofusion experiment

The chamber itself is 65 mm (2.5 inches) in internal diameter and 100 mm (4 inches) tall.  It is filled with a well-degassed deuterated liquid hydrocarbon, in this case D-acetone.  The fact that it is degassed means that there are no air bubbles floating around, and the fact that it is deuterated means that the hydrogens in the acetone molecules have been replaced with hydrogen’s heavier isotope deuterium.  Large amplitude ultrasonic waves are driven into the cylindrical chamber.  These waves come to focus in the center of the cylinder creating pressure changes on the order of 15-40 bar (15-40 times atmospheric pressure), which locally superheats the liquid.

The next step is to get bubbles of acetone vapor in the solution.  They have found that a pulsed neutron source, that shoots neutrons through the fluid leaving a trail of acetone bubbles works very well by creating large bubbles with radii about 10 times larger than those produced by other methods.  The fact that they are using a neutron source makes some researchers a bit suspiscious, since to verify that the reaction is indeed thermonuclear, one needs to detect neutrons of a given energy.  This raises the bar for the researchers; making their experimental results a bit more difficult to demonstrate.  However, the neutrons they are looking for have very specific energies (speeds) from the nuclear reaction (2.45 MeV).  Apparently, forming bubbles in hydrocarbons is extremely difficult and would take extremely low pressures.  For this reason the pulse neutron source works well.

Bubbles near the focus of the ultrasonic waves, are rapidly compressed at rates of 1000s of kilometers per second.  As the bubble compresses a nonlinear shock front forms, which races into the bubble.  As this shock reaches the center of the bubble, the local temperature reaches 100 million degrees Kelvin (180 million degrees F) and a density of 1 trillion bar, which is approximately the density of a neutron star.  This is all occurring on a nanometer scale.  Out of the perhaps 1000 bubbles near the center of the chamber, about 15 of them reach the appropriate conditions to undergo nuclear fusion.  These bubbles then rapidly expand and then bounce back into compression, experiencing fusion again.  This bouncing occurs approximately 50 times!  Each fusion event yields only about 10 neutrons (thus only about 10 pairs of deuterium actually fuse).  But given that there are about 15 bubbles, each bubble will fuse about 50 times, and that they are being driven to implode at 50 times a second gives an overall average neutron production rate of almost 40,000 neutrons per second.

To the outside observer, a fusion event results in a flash of light (similar to that produced by the lower energy phenomenon called sonoluminescence),  and then a shockwave that hits the walls of the cylinder like a hammer!

There are actualy two nuclear processes going on in this reaction:

D + D -> 3He + n (2.45 MeV) + 3.3 MeV (in gamma rays)
D + D -> T + H + 4 MeV

3He is a Helium atom with only one neutron and the normal two protons.  T is the Hydrogen isotope Tritium with one proton and two neutrons.  The reactions have approximately a 1:1 branching ratio, which means that each reaction is equally likely.  Thus 50% of the time, a Tritium is produced, which can also be fused in other types of reactions in the chamber.

The team has experienced many political difficulties.  First, while this is not cold fusion, there is a remaining stigma attached to tabletop physics.  This is reinforced by the fact that researchers who have been working to perfect other fusion techniques, such as laser confinement and magnetic confinement, require enormous budgets.  Certainly, the discovery of a tabletop method is potentially threatening.  However, in reality, each of these methods should be supported as necessary until we finally have successfully created fusion power plants.  Last, these researchers published their initial results in the journal Science, which competes with the journal Nature.  Typically, Science focuses on biology, whereas Nature focuses on other area of science.  Lahey and colleagues, being nuclear engineers, were not aware of these turf boundaries.  The end result of the rivalry between the two journals and the rivalry between scientific competitors was an accusation of scientific misconduct, which have since been cleared

The team continues to perform tests to improve the efficiency and sustainability of the process.  Currently the energy produced by their test chamber is 10,000,000 times less than the energy that they have to put in to get the reaction.  One says that they are 10,000,000 times below breakeven.  The good news, is that they are just starting to refine the process, and they have many parameters to play with, each potentially yielding orders of magnitude increases in efficiency.  To make the process self-sustaining, one idea is to place two cylinders side-by-side and to drive them 180 degrees out of phase.  When one cylinder is fusing, it produces neutrons that make bubbles in the second cylinder, which will then fuse in the next cycle and so on.  A great benefit is that there is absolutely no way to get a runaway reaction.  The second benefit is that for power generation, the cylinders will be designed so that the neutrons are absorbed in the fluid heating it up to eventually run turbines for electricity.  This results in no resdiual radioactivity.

Dr. Lahey and his colleagues are hoping that they can scale the process up to create 100 megawatt generators.  These generators would run on deuterium extracted from seawater, which would last us for about 1 million years at our current rate of energy consumption.  The prospect that we could be on the verge of having an energy source with no CO2 output, no reliance on oil, and no radioactivity is almost too good to be true.  But at this point in history, we could use some good news and from some truly smart people deserving of praise.

Kevin Knuth
Albany NY

Posted under Acoustics, Inventions, Physics, Research

This post was written by drknuth on March 17, 2007