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.

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.

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!