Archive for the ‘Conceptual’ Category

4-dimensional hypercube

July 22, 2008

I’ve been reading Michio Kaku’s Hyperspace, and it’s got me trying to visualize the fourth spacial dimension. It’s not possible to do, but it’s fun to try. Fortunately, the internet has plenty of videos on the matter, a few of which I’ll present here.

As the video I embedded in a previous post about visualizing higher dimensions said, sometimes it’s easier to imagine a higher dimension beyond the three we’re familiar with by thinking of the higher dimension as a dimension you “fold lower dimensions through” to get a desired result. For instance, as shown in the video, folding a 2-dimensional sheet through the third dimension allows the edges of the sheet to touch, so an ant can crawl from one edge to the other. If you lived on the 2-D sheet and could only see in two dimensions, it would appear to you that the ant disappeared from one edge and instantly reappeared on the other. We can’t visualize dimensions higher than three, but we can visualize how actions in these higher dimensions would would look in our 3-dimensional world, analogous to a creature who can only see in two dimensions watching an ant disappear from one place and reappear in another.

A popular 4-dimensional object to try to visualize is a tesseract, which is a 4-dimensional hypercube. We can’t picture it, but we can picture it’s projection, or shadow, in three dimensions. Here is a video of the projection of a 4-D cube rotating:

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More Magnetic Monopoles, With a Summery Hint of Maxwell’s Equations

June 9, 2008

It was pointed out to me that perhaps last time I went off too far into the theoretical setup and didn’t quite wrap up succinctly for you what exactly a magnetic monopole is. In short, a magnetic monopole would be a particle that carries magnetic charge, like how electrons and protons are carriers of electric charge. A bar magnet has two poles, and if you cut it in half, it still has two poles. If you keep cutting it in half and break it down as far as it will go, you will have a spinning electron which still has a “North” and a “South” pole. Whereas, in seeking the most simple possible configuration that produces an electric field, if you broke down a material as far as it would go you would have a single electron radiating a uniform electric field in all directions. This electric field wouldn’t pull objects toward it on one side and push objects away on the other like a dipole; it’s uniform in all directions (pictured here is the electric field of a positive point charge. An electron is a negative charge, so the direction of the field in reversed — pointing in toward the electron — but you get the idea). An electron is an example of an electric monopole. Similarly, a magnetic monopole, which is a magnetic charge, would have a uniform magnetic field radiating uniformly in all directions.

It’s not for the faint of heart, but for those willing to brave some math, I’ve got more for you on Maxwell’s equations, and how they would be symmetrical if a magnetic charge existed. Look at the pretty equations and skip to the summary just above the second set of equations if your eyes start glazing over. These are Maxwell’s equations for charges and electric and magnetic fields in a vacuum:

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Magnetic Monopoles and Magnetism: The Alliteration Choice of Champions

June 8, 2008

Recently, I was reading this article about doomsday, law suits and the Large Hadron Collider. I admit that I’m intrigued and would kind of like to read a rigorous version of the theories behind these doomsday scenarios. It struck me how difficult a problem it is for courts to sort out such things, what with the high level of prerequisite knowledge a person must possess to understand enough about these concerns to make an informed decision about them. I suppose that’s what expert witnesses are for. Still, I think it is important to try to bridge the knowledge gap between scientific experts and folks with only as much science background as their learning institutions required them to learn to graduate in another field. Our society is pretty lacking in those bridges, I think, and I’m going to try to build more.

One of the fears in the article seems to be that the exotic and frighteningly named “magnetic monopoles”, if created, will be malignant and alter all matter they touch in upsetting ways, Ice-Nine style. Having just reinserted the topic of magnetic monopoles into my brain a couple months ago in my class on electricity and magnetism, I’m feeling the spirit to type about them on the internet.

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