While I Was Out...

Well, posting has been sparse recently, and there's a good reason for that: I've been on vacation.

My vacation started on August 19 when I went to a friend's wedding up in Oregon. I was there for the weekend, and it was a good time with lots of food, old friends, and, of course, the wedding. The ceremony was very simple yet elegant, and was followed by a massive reception (apparently the bride's family has a reputation of being very hungry, but not all of them were able to make it, so there ended up being a surplus of food). I helped some in the kitchen and with clean-up, but was mainly there to attend.

Non-ceremony highlights included creating a whirlpool and playing with slot cars. These happened after the wedding when it was just a bunch of guys hanging out. First we went to a small community pool and, wondering if we could actually make a whirlpool, all started running in a circle in the 4-ft. water. There were about 20 of us, and we were able to get a noticeable depression in the middle of the pool. At this point, we attempted to suddenly reverse direction, resulting in mass floundering and hilarity. We managed to repeat this three more times and were all very tired after doing so. Later that night, the father of the bride brought out an old slot car set, and Synk, Hopps, and I had far too much fun seeing just how crazy of a track we could make, with an ever-growing loop-the-loop and a vertical portion of track. Part of our excitement was dulled when we discovered that the cars were magnetic and would stay on the track just about no matter what, but it was still pretty cool.

Right after I got back from Oregon, I drove down to Texas to spend the week at my old dorm, 41, during the floor's "Expended Orientation" for the freshmen. A lot of people immediately think of this as some sort of hazing, but that implies that we're harming the freshmen or lording our seniority over them, when it really consists more of playing late-night games of Capture the Flag and traversing campus en masse to sing "500 Miles" to all the girls dorms. Rather than being some sort of price to join the floor, I found it was the most fun I had my freshman year. And it creates the kind of bond that has people like me using up my vacation days to visit the floor over a year after I graduated.

While the whole EO week was very fun, one thing that was very weird to me was that, except for two days when other alumni showed up, I was the oldest 41er present. There were even some guys who returned as alumni that I had seen come onto the floor as freshmen, making me feel kind of old. Aside from general aging, though, I think much of my disconcertion was caused by the fact that my older brother also stayed on 41. For a long time, I was "Fjord's younger brother," but now there isn't anyone left on the floor who has met Fjord, and I suddenly ended up in the position of being Grey, the eldest.

Weird.



At this point, some of the more astute of you may have realized from the dates given that my vacation ended over a week ago, yet I'm just now getting around to a post about it. This is because after I got back, I lost just about all motivation and my life became very much like this comic. Fortunately, with the long weekend, I'm starting to become bored with being bored, and my motivation cores are coming back online. This means that, with any luck, the Facebook-promised post detailing the construction and workings of my Archimedes Death Ray (oh, yes!) should be coming sometime this week.

Until then, have some steak. I know I will.

Corn Chips Are No Place For Me!

A little over a week ago, I was informed of the existence of the Warrior Dash. This is a 3.5 mile run with 12 obstacles, including an 8-ft. wall, barbed wire, and fire. There just happens to be one going on in Tulsa next month, and a lot of guys from work have been encouraging me to go with them. Aside from the fact that the partying atmosphere isn't really my thing, the race itself does seem pretty awesome and I've decided that it's something I want to do; I'll just decline the beer, enjoy my turkey leg and be on my way.

I have not, however, signed up yet.

This is because I started training last week, and to my disappointment I was unable to even run a mile. Now I know that this is something that my body is capable of (or at least used to be capable of); I was able to run 2 miles without much difficultly a year ago. I suppose I just need to get warmed up to that level of activity after almost a year of sedentary living. I'm just not entirely sure I can get warmed back up in time for the race.

So, I have been following the "I just want to finish" training schedule on their website, with the intent of signing up as soon as I'm able to complete the recommended running distance for that day (I have until the 28th before the price goes up). We'll see how I do tonight.


(I suppose even if I can't run the entire thing, I do know that I can keep up a 4.5 mph walking pace just about indefinitely, so there is that.)

Dnoces a Em Evig

Greetings, Internet.

As you may recall, in my last Project post, I outlined my results concerning the creation of a temporal wormhole in a simple electrical circuit. I also mentioned that these results came with certain assumptions, one of which being that the flow of time was the same on either side of the wormhole. I have since been working to figure out what the results would be if this was not the case.

First, I attempted using differential equations and a similar approach as last time, only instead of having a parallel circuit I had two circuits that influenced each other, but each effect was reversed in time.

Something like this.

After struggling with this for a while, I determined that the math was, in fact, actively malevolent. I therefore decided to try a numerical method. While less exact than an algebraic solution, this would in theory be easier to implement. I chose to use Eulers's method instead of Runge-Kutta, again because it was simpler and fit very nicely with the fact that the equations for the current through a capacitor gave me the slope of the voltage. I also knew that the overall calculations would involve some self-referencing iteration, and I wanted to keep from muddying things up as much as possible.

Since numerical solutions sometimes have problems with accuracy (and also since it was so easy to implement changes) I tested my calculations first with a non-wormhole setup to check for accuracy, then with a common-flow wormhole (the one from last time) to check that the whole self-referencing thing worked out. I had to make several revisions as I apparently made some mistakes in my math somewhere, but I eventually got a working version and was able to run it on a reverse-flow wormhole.

When I was thinking about it before-hand, I pictured that the results from a reverse-flow wormhole would be much "smoother;" that the voltage would always be decreasing, just at different rates. I figured this because a way of thinking about the reverse-flow setup is that the two instances of the wormhole both start (or end) and the middle, then spread out from there. This means that the midpoint where they meet should be decently discontinuous. I also viewed the second instance of the wormhole as a capacitor charging in reverse: a somewhat straight line dropping off faster and faster. These two conceptions formed a mental image something like the tangent of negative pi.

Then I actually ran the numbers.


The result is not a smooth, continually decreasing curve, but rather it forms an even deeper bowl than a common-flow wormhole, with steep sides at either end. The middle area is still smooth, but my idea of the wormhole's second instance looking like a capacitor charging in reverse was wrong; from the perspective of the capacitor, it's still charging in normal, forward time, it's just the voltage source that it charging from is in reverse.

While these results are kind of neat, I am kind of disappointed that they didn't come out how I was hoping. If the voltage was continually decreasing, I was planning on making a graph with the time axis stretched and compressed until the curved looked like a typical capacitor discharge in order to give a better idea of what the wormhole did to the time-line of the event. I had also hoped to make an experiment with an event-dependent wormhole, where the second instance wouldn't start until the voltage reached a certain level or something, but if the voltage is not continually decreasing, there may be issues with opening the wormhole at the "wrong" place. Or maybe not. I'll have to think about it.

Perils of Procrastination

I have learned an important lesson this week: NEVER put off booking a flight.

A good friend of mine is getting married toward the beginning of next month, and both Hopps and I were invited to the wedding. I eagerly responded that I would attend, and proceeded to get the necessary time off from work. I did not, however, book my flight. The reason I did this is because I wanted to coordinate my travel with Hopps, if possible. The actual location of the wedding is about an hour away from the airport, and if we both arrived at the same time it would save on a taxi or be more convenient if someone was picking us up. For his part, Hopps, having just started work, was trying to work out if he could get time off and then waiting for confirmation.

We checked ticket prices about a week ago, and while they were a bit pricey (the airport we would be flying into is not frequented by airlines as often as others) we decided that we could manage to fit them into our respective budgets. Some more time went by without Hopps knowing for sure if he had the day off, and eventually I decided to go ahead and book my flight anyway. So this last weekend I got on the internet to book my flight. I wasn't too concerned about the price; the wedding was still three weeks away, and prices couldn't have gone up that much, right?

WRONG.

In the several days since we had last checked, the price for a round trip had increased to almost 150% of its previous value, putting it way beyond what we had decided was pushing it for our wallets.

And now, I have just gotten of the phone with my friend after explaining to him why I won't be coming even though I had told him many times that I would. Uhhhg.


I have another friend getting married at the end of August. I will be booking that flight tomorrow.

Give Me a Second

Greetings.

So about a week ago, I watched this video on YouTube, and it inspired me to think some more on the topic of time travel. The scope of my current views on the subject is very large and underdeveloped, and I'm trying to get this post up in reasonable time, so I won't go into them here. Suffice it to say that my various thought processes led me to the idea of time travel within a simple RC circuit.

This is because much of the talk surrounding potential time travel deals with wormholes in space-time and the possibility of sending something through one. However, all of the scenarios that I have heard described concerning such wormholes only speak of sending something through in one direction. My understanding of wormholes, on the other hand, is that they would operate more like gates, allowing for bi-directional travel. The model of an electrical circuit works very well for this, since charge will flow from higher to lower potentials whichever side of the wormhole they're on, the quantities being discussed are continuous rather than discrete, and there is the potential for both positive and negative voltages. My thought experiment took the following form:


Under normal conditions, the voltage across the capacitor would decay exponentially according to the time constant tau.


However, I would alter the conditions on this particular circuit by introducing a wormhole between the resistors connecting two points in the discharge time: t1 and t2. The wormhole had to be between the resistors because opening it at ground would give me nothing (always at 0 volts) and opening it at the capacitor would effectively and instantly place the capacitor in parallel with itself at a time when it has a lower voltage, and you can't instantly change the voltage across a capacitor. Also, since an instantaneous spike or drop in voltage within the circuit would have little lasting effect (capacitors are often used specifically to get rid of such spikes), I would have to leave the wormhole open for some duration td.

I assumed that the flow of time on either side of the wormhole to be the same, so that t1 corresponded to t2 and t1+td corresponded to t2+td. Therefore, for the period of time(s) the wormhole was open, the circuit would look like this:


where C1 is the first instance of the capacitor, starting at t1, and C2 is the second instance of the capacitor, starting at t2. There is no need to make a distinction between the instances of the resistors, since their voltages and currents can change instantaneously. The starting voltage of C1 would be the typical RC voltage after decaying for a time of t1, and the starting voltage of C2 would be the voltage of C1 after it had decayed according to the parallel "wormhole" circuit for td then decayed naturally for t2-(t1+td).

My first calculations were performed using the values of R1=R2=1kohm, C=1uF, t1=0.5ms, t2=2ms, and td=0.1ms. Solving the parallel circuit required using some differential equations, but overall it wasn't too difficult, and the result came out looking like this:


As you can see, the voltage across the capacitor dropped slightly once the wormhole was opened at t1, then rose back to about normal when the wormhole was opened at t2. There appeared to be a slight difference between the final voltages of the normal circuit and the wormhole circuit, but this difference was small enough that it could be attributed to rounding errors.

To find out whether there was a net effect to adding a wormhole or if it was indeed just a rounding error, I decided to work through the equations again in general form. This is where things started to get ugly. None of the equations simplified, but instead got more and more complex the further I got. I therefore present the result in the form of many constants that require their own definition.

The discharge cycle of the capacitor is divided into five distinct sections:


With constants defined as follows:


(If you're able to simplify things more, please let me know.)

While the complexity of the equations made it difficult to verify if there was a net effect of the wormhole by algebraic means, I was able to plug them into Excel and tweak parameters until the answer became more obvious. First, I lengthened the duration of the wormhole to 0.5ms:


Then I moved the wormhole "closer" to the capacitor by increasing R1 and decreasing R2. This kept the time constant the same while increasing the influence of the wormhole on the capacitor.


At this point it became obvious that there was a net effect to the creation of the wormhole: the final voltage of the capacitor was greater than it otherwise would have been. Since the rate of decay at the end of both the typical case and the wormhole case was the same, this effectively meant that the voltage decay had been delayed by some amount of time equal to


In order to guard against paradoxes, I calculated the cumulative energy output of the capacitor, just to see if the postponed decay was due to the wormhole creating energy or something. This was relatively easy to calculate by finding the voltage across the resistors, using that to find the instantaneous power, then integrating over time.


The result is that the wormhole capacitor has expended less energy than usual. Since the graph of energy is cumulative, this means that the energy expenditure of the capacitor has been delayed, and quick investigation shows that this delay is the same as that for the voltage decay. This means that the total effect of the wormhole is that, without introducing energy into the system, it delayed the event of capacitor discharge.

In other words, I effectively (and temporarily) slowed the passage of time for the circuit. Pretty cool, huh?


Now, these results are hardly comprehensive. For example, I mentioned earlier that I assumed that time flowed in the same direction on either side of the wormhole, and this is not necessarily valid. The typical picture of a wormhole is fold in space-time, and, at least with the 2-D analogy, if you're traveling in one direction on one side of the wormhole, the fold will reverse your direction by the time you reach the other side.


Applying this would mean recalculating the parallel circuit while one of the capacitors is experiencing time in reverse. This is mind-boggling (though I think the results would fit nicer with the whole slowing-down-time thing), but I hope to give it a try. Whether I get around to it, though, is uncertain, as I have a completely different, physics-based project planned for this weekend.

A Benign Crazy

I don't think there are very many people in the world who would spend their Friday evenings/early Saturday mornings calculating the effects of a temporal wormhole on an RC circuit, but I am one of them.

I'll re-run my calculations and get a "project" post up about it soon, but for now I'll just say that I've always tended to think of time travel in terms of an op-amp feedback circuit, so it just felt natural to run some initial thought experiments with electrical components. Although the circuit just had two resistors and a charged capacitor, the complications introduced by the wormhole were enough that I had to review my old notes from Differential Equations to get a handle on it. I was also aided in my endeavors by Grimholt and Sam, the Homework Gremlins (two whiteboards that I inherited from the Powersuite), who were happy to have something to do.

In other news, Hopps has been moved in for about two weeks now, and things are going fine. We're both somewhat solitary, but it's nice to have another person around. Another one of my friends going to be visiting over the weekend, so we got a grill (and lighter fluid!) and hopefully we won't kill ourselves with it, any July 4 activities, or just the sheer heat outside.

Or rather, hopefully Hopps won't kill himself. I am invincible!

Investigations into Dodecahedral Space

Remember when I warned you that some of these posts may go into extravagant detail about some strange or obscure hobby that I enjoy? Well, this is one of them. So if you're not particularly interested in the interactions of certain polyhedra in three-dimensional space, you should probably leave now.

Today's topic is the interrelation of cubes and dodecahedra and the examination of said relations with regard to the potential of some kind of dodecahedral space.

For those of you who may not know, I really like polyhedra (especially Archimedian and Catlan Solids). I have models of several on my desk at work -- including an intersection of a snub dodecahedron and a pentagonal hexacontahedron, the patterns for which I calculated and designed myself -- and I thoroughly enjoy tinkering around with a Zome kit my parents got me several years ago. My current obsession is with the relation between dodecahedra and cubes. A dodecahedron has twelve regular pentagons for faces and is the shape for a 12-sided die.

I'm pretty sure you all know what a cube looks like.

Now, a dodecahedron doesn't seem to fit into 3D space very neatly: it's faces are pentagons, which aren't easily described in a square coordinate system and aren't as structurally fundamental as triangles, and neither pentagons nor dodecahedra tessellate (pack together without spaces in between). Cubes, on the other hand, represent the very basis of three-dimensional space. It is therefore fascinating to me that, by drawing a line between two vertices on each face, you can inscribe a cube on the surface of a dodecahedron.

(The struts used for this are unfortunately all the same color, so I covered the ones forming the cube in tin foil.)

The edges of the cube all fall on a unique face of the dodecahedron, and every face of the dodecahedron is occupied by an edge of the cube.

This inscribing of a cube on a dodecahedron allows for easier mapping of the latter into cubic space. And while dodecahedra cannot tessellate, they can be arranged without overlap in a repeating pattern along diagonal cubes:


I found this last bit worth further investigation, since the cube formed inside the dodecahedron is not the only one possible; the inscription process can be used to form 5 unique, rotated cubes.

Here are two, shown in black and silver.

So, if I were to arrange several dodecahedra according to the cubic space defined by the silver struts, then generated the black cubic space, would the black cubes intersect the other dodecahedra the same way as the one in which the first cube was generated? If this were true, I could see it giving rise to a new way of defining space according to the arrangement of dodecahedra rather than the cubic space we all know and love. While perhaps not practical, this would be kind of cool. However, preliminary construction showed that, at least one "layer" out from the original dodecahedron, the two cubic spaces did not converge. This didn't mean convergence was impossible, though; it was still possible that the two spaces converged at some more distant point. However, the increased distance would greatly increase the potential complexity of the "dodecahedral space," as every cube spacing the distance to convergence would itself be able to give rise to a dodecahedron and four other cubic spaces, which would in turn be able generate more cubic spaces before they converged, and so on. Further construction only got more and more flimsy the further I got from the original dodecahedron. This would have to be accomplished with math.

Several maths later...

It turns out the two cubic spaces never converge, leading to infinite complexity and the breakdown of "dodecahedral space." I arrived at this conclusion as follows: If I just took the two original cubes,

and then just looked at the top face of the silver cube, particularly the edges originating at the near-left corner,

If I were to figure out the the coordinates of the vertex of the black cube in silver space (given the cubes had sides of length 1), and then find some multiple of these coordinates that was an integer, it would mean that the two spaces would meet at that point. So taking the near-left corner to be (0,0,0),

An extra silver strut has been added to form the coordinate axes.

I calculated the unit vector of the black strut. If the cubes had edges of length 2 (a change from before, I know, but it makes the math easier), then the end of the black strut has coordinates (phi, 1, 1/phi), where phi is the golden ratio and an irrational number (my favorite irrational number, in case you were wondering). Since the coordinates contain irrational numbers, no multiple of them will ever be an integer, and the two spaces will never converge.


If you're disappointed that nothing came of this after such a long post, well... I kind of was, too. However, uniqueness is an important property in and of itself, so perhaps the knowledge that these two coordinate spaces never converge except in this one dodecahedron could be useful. The first thing that comes to mind is cryptography: is there a way to encode a message on the two cubic spaces so it can only be read at the point where they converge? I don't know.


I wish it were faster and easier to get these descriptions prepared and online; my mind moves too fast and sporadically. In the time that I've been preparing these models and writing this, I've been distracted by modeling the tessellation of rhombic dodecahedra


and experimenting with taking the volumes in a dodecahedra not filled by the inscribed cube and mirroring them on the inside of the cube's faces.


Interestingly, the points in the center mark the vertices of an icosahedron (20-sided die), the dual of a dodecahedron. Is this, combined with the planar golden rectangles of an icosahedron, a clue in the relation between cubes and dodecahedra? I must find out!

Exciting Times

Today is an interesting milestone: it is the last day (at least in the near future) that I will be living by myself. Starting tomorrow, this crazy guy will be moving in with me, and provided we don't kill each other, I think it will be a very good arrangement. Still, I can't help but feel a bit nervous about the change. Not that I have anything against him moving in -- it was I who raised the question of sharing the house in the first place -- and having another person under the roof will greatly reduce the rent, utilities, and boredom. Really, aside from me no longer being able to break into renditions of "Be Prepared" from The Lion King whenever I want, there isn't much reason behind my nervousness; just the issue of change.

A similar thing happened about a year ago after I graduated and got a job. Leaving the dorm of 50-ish people and my three roommates, I moved into an apartment with Hopps. He shared the apartment with me for the summer while he was interning, then he went back to college and I was left alone. It was really difficult adjusting from the crowded dorm to the empty apartment and it took a while, but I eventually got used to it. And now, after a year of living on my own, I'm going to switch back to sharing my residence with another. The change won't be nearly as great as before, but it's still there, and while it's not unsettling in the sense of disturbing, it is a bit of a jolt.

I know that this feeling is temporary and I'm actually looking forward to having a house-mate (is that what you call it?), it's just a change, and I tend to be a creature of habit.

Layout Difficulties

I'm somewhat annoyed at Blogger. I had hoped to set up this blog dividing my posts between three distinct tabs: Life, Projects, and Philosophy. This way I could go into great detail in the latter two pages without clogging things up for people who just want to check in and see how I'm doing. Very neat and clean. Unfortunately, while Blogger does have the option for multiple tabs, I haven't found a way to submit multiple posts for anything other than the Home tab. The only way I know of to completely separate posts of different styles is by creating completely separate blogs, which I don't want to do.

So right now it looks like if you want read about my personal happenings, you'll have to wade through posts detailing exactly how I make my chainmaille armor -- both the internet tutorials I learned from and my improvements on them -- should I ever feel like posting that. The only division will be the labels at the bottom.

If you know how to make Blogger sort posts into separate pages under the same blog, please let me know.

On Men and Monsters

These first few posts are a little late in regard to the events that inspired them, but I just now got this blog up and running and I feel that I should put them somewhere, so here they are.



I am convinced that empathy is one of the highest forms of human reasoning. It combines imagination -- one of the highest intellectual achievements -- with selflessness -- one of the highest reaches of morality. Putting ourselves in the shoes of another, particularly someone who appears vastly different from us, keeps us from becoming completely self-serving automata. I have heard it postulated that self-reflection is a form of (and even gives rise to) consciousness. If this is true, then reflecting on the self of another is even greater. Not only do we recognize that we think and have a being, but that others also have a being unto themselves that is very much like our own -- and similar enough that we can imagine how we would feel if we were in their circumstances.

I am also convinced that the greatest atrocities in human history occur due to a loss of empathy. These tragedies occur when a group in power ceases to view people as people and instead sees them as either tools to be used or obstacles to be eliminated. Offenses like this can range from the obvious repression by totalitarian governments who, at best, see the people they expend as tools "for the greater good," to individual bases of "just using someone for their _____." In all cases, there is a fundamental sense that this is wrong, even if it is efficient and utilitarian, and the argument against such actions boils down to the sentiment, "but they're people, too." This can even be applied to the perpetrators; no one ever sets out to be evil (aside, perhaps, from mental illness). They can certainly become that way, but rather than aiming at being the worst possible human they can imagine, they do so by focusing so much on their own interests and ideals that they forget the reality of the selves of others. In thinking (consciously or not) that they are the only true men, they become monsters.

It is very important to remember this in your interactions with others, particularly those you don't like or disagree with. Every person you interact with is human and everything that comes with it. They are not perfect (just like you), but they also have intellectual processes and emotional responses very similar to your own. Aside from environmental conditions, the differences arise in each person's interests and the way in which they pursue and protect those interests. Now, it is perfectly reasonable to find fault in these set of priorities (I just stated that blind pursuit of misplaced priorities can lead to inhumanity), and I'm certainly not against the destruction of erroneous arguments and points of view, but there's a reason ad hominem attacks are considered fallacious. Never forget that the other person is a person.

The whole issue of the monster-izing effect of the lack of empathy is very prevalent in the United States' current War on Terror. I seriously doubt that those who perpetrated the 9/11 attacks thought of the people in the Twin Towers as the same sort of being as their loved ones. I'm fairly certain that those in terrorist cells fit my earlier description of monsters. And I see a war of and against an "us and them" mentality toward Muslims. Several weeks ago, Osama bin Laden was killed in an operation by Seal Team Six. In the United States, this was met with relief, rejoicing, and sharp criticism toward those who were rejoicing. I would tend to side with the third group. Personally, I felt very little emotion at the news other than a very pragmatic acknowledgment of threat mitigation on one hand and the increased threat of retaliation on the other. There are people who are far more emotionally connected with this war than I am and I cannot pretend to know their feelings, but I do not think that bin Laden's death should be celebrated. I'm not trying to detract from the Seals' accomplishment; they very admirably completed their mission and neutralized a threat, but it was not victory, it was the death of a man. The death of a human being who was working to protect his interests. These interests were wrong and they manner in which he protected them was unjust, but in his mind, there was reason behind his actions.

I know there's a very fine line between celebrating a success and a victory (though, again, not ultimate victory), and celebrating the death of the man Osama bin Laden, and so I am not directing any criticism against those who felt glad when they heard the news. I am, however, critical of those who proverbially danced on his grave, who became so focused on their own jingoism that they viewed him as little more than a one-dimensional enemy, an obstacle with a name.

Was I able to fully empathize with Osama? Absolutely not. Personally, I find it very difficult to empathize with anyone who is more than slightly different from me. But it is an ideal toward which I aim. Was bin Laden a monster? Yes, but he was also a man, and if we forget that we become more like him than we would want to admit.