Some Special Relativity exercises with rigid objects

Discussion in 'Physics & Math' started by Halc, Oct 25, 2023.

  1. Halc Registered Senior Member

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    I wanted to open a topic illustrating some concepts in the treatment of rigid objects since there seems to have been much confusion lately in this area. I created exercises labeled P,Q,R below.

    By rigid object, I mean that the object cannot change proper distance with any of its immediate material. This is somewhat more loose than Born rigidity. For a 3D object, the only difference is that I allow the object no not have a rest frame. An accelerating Born-rigid object is already necessarily moving at different speeds at different positions, but I think it is presumed to be simultaneously stationary everywhere in it's (momentary) rest frame.
    As for non-3D object, one cannot stretch the object, similar to above, but one can bend it. So it's ok to send a 1D rigid wire through a pulley for instance. One can fold a 2D sheet, but cannot tear it or stretch it.

    Second note: The only way to apply changes to motion of a rigid object is to apply appropriate force everywhere at once. There is no 'engine in the back' so to speak, since any force applied by the engine at the rear of say a rocket is incapable of affecting the front sooner than a light signal could get to it. Hence a rocket must shorten when the engines fire, which violates rigidity. So no rocket technology. More like rail-gun and such.

    Third note: Kind of keeping things to about 3 digits of precision. Light speed is presumed to be 300,000 m/sec. We have no limits to acceleration rates or speeds (up to c of course).


    P) Finite movement of a 3D object
    This is a 3D object, basically a subway train that is all one piece. It is a light-year long, and stationary relative to the straight track it's on. We want to move it 1.08 e-12 meters (a light hour). The doors are that proper separation apart, and we want to bring the next door to a halt at the stations which are also a light hour apart in the track frame. How long does it take? I get just over 5½ days to do it. I'll be discussing some interesting tricks to attempt to improve on that, but it seems that it cannot be done faster. This is the only one for which I have an easy answer.


    Q) Rotating a 2D disk
    Imagine a flat disk, negligible thickness so we can treat it as 2D if it helps. It has an arrow printed on it pointing 'up' to 12 O Clock (up the y axis). We want to rotate the disk a quarter turn to 3 O Clock (to the +x axis direction). It is a light-hour in radius, and has a hole in it ½ light-hour in radius, so it's kind of a donut disk, like a 45 record.

    The most 'legal' way to accomplish things is to rotate the object about the x axis until the arrow points in the z direction, then rotate along the z axis to bring what was at the 3 O Clock position down to the 6 O Clock place. Finally we rotate about the y axis bringing the arrow down to 3 O Clock. 3 rotations, moving the outermost points π/2 light hours each time, so it takes 3/2 π hours to do it, or about 283 minutes.

    Seems like too much work. Method B: We can fold a 2D object, no? So crumple the whole thing up into a small wad, rotate the wad, and un-crumple it. That takes an hour each way to move all the stuff at the edge to the middle and back, with negligible time to rotate the wad. We've reduced the time to 120 minutes. Can we do better? I think we can.


    R) Rotating a 2D toilet paper roll
    This time we have a tube, a rolled paper in a cylinder, a light-hour in radius again. We want to rotate that a quarter turn jut like above, but this time the Ehrenfest paradox is far less in our way. We still cannot rotate the roll as it is because it will contract, so the idea is to accelerate all the parts inward, and increase the rotation to always match the contracted length of the circumference. We rotate half-way to our destination and then do the same dance in reverse, slowing the rotation to a stop.
    Shouldn't be hard, but I need to write a program to do the integration. The neat thing is that every particle of the object can move at c at all times. It just can't ever move at that speed in a direction tangential to the surface of the material, which would contract it infinitely.


    Comments? Can anybody propose any actual numbers for the above problems? Does anybody deny what I've written above? I've presented not too many numbers (none at all for the 3rd problem so far).
     
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  3. Pinball1970 Valued Senior Member

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    Materials do not behave like this so you are making up your own physics.
    The material, even if it was rigid like a metal rod yet one light hour across would not move as one object if a force was applied even at multiple points.
    1.3 light seconds is from here to the moon.
    The collection of atoms would stop moving as a single object orders of magnitude before this.
     
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  5. Pinball1970 Valued Senior Member

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    What is it you are trying to illustrate exactly? We could be at cross purposes.
     
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  7. Halc Registered Senior Member

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    This is a mathematical exercise, not an engineering one. I've never seen a site dealing with mathematically rigid motion that concerns itself with material properties.
    Any forces not properly distributed over the object will cause stress on the material, and any stress will necessarily cause strain, which would violate rigid motion. So there can be no stress anywhere, and thus any material (silly putty, a cloud of tiny drones, etc) will do. As stated in my OP, the only requirement is that each particle maintain constant proper separation from its immediate neighbors.

    This statement either asserts that there is a limit to the size of a group of particles that can undergo rigid motion, or that you meant something else. If the latter, please clarify, and if the former, you need to back that. Perhaps you're making a reference to the fact that there is a difference in gravitational potential between Earth and moon, which simply puts that environment out of the scope of this topic, which is confined to Minkowskian spacetime and special relativity. There can be no consideration for gravity, energy, mass, or curved spacetime under these confines.

    I actually wanted to see if anybody here is up to discussing this sort of stuff. There have been a lot of threads opened by people seemingly in denial of the properties of say the acceleration of a rigid object, such as the fact that different points in an accelerating object will undergo different proper acceleration, and consequently be moving at different speeds at any given time in inertial frames. Neddy seems to know his SR at least. There are others (Tony say) who just troll, seemingly putting out nonsense deliberately.

    I have some questions myself about the Q case since I want to do seemingly contradictory things. I haven't worked out the answer yet.
    I deliberately put out an answer for the the P case (just over 5½ days) without showing my work, in hopes that somebody would either agree or come up with a different answer. So I didn't want to show my work right away. It's pretty trivial. For the R cases I did, the work is utterly trivial and I showed that work, but I don't think my answers there were optimal.
     
    Last edited: Oct 26, 2023
  8. Pinball1970 Valued Senior Member

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    1,036
    Ok. I think I have come across a paradox similar to what you what you were describing that apparently violated SR. The answer was regarding rigid rods behaving more like a whip than a stick, one the length was very long, I think a light year was mentioned.
    James R is very knowledgeable on SR.
     
  9. Pinball1970 Valued Senior Member

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    1,036
    The upshot is that the forces between the particles move at the speed of sound not light so it would take years to move the rod.
    Less so with your light hour but that is still body about 1/6 th of the way to Pluto.
     
  10. Halc Registered Senior Member

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    350
    Hmm, I think it refers to one dragging a string or a rod behind something that undergoes constant proper acceleration. As I mentioned above, proper acceleration is different along the length, and if one place accelerates at 1g, the acceleration grows at points behind. The Rindler horizon forms about a light year back and the object cannot extend further back than that.
    None of this violates SR in any way, and none of it limits the length of an object. So a 10 light year object is limited to less than 1 m/sec² acceleration at the front. The longer it is, the more time it takes (as measured by a clock at the front) to get it up to any speed. My first problem very much illustrates that, showing a minimum time it takes (far more than an hour) to move the object a light-hour.

    I am beginning to question the answer I got (done in haste with possible missed conversion between meters and km).
    I'll run it again when I'm not in a rush.

    Sound is a form of strain. My object cannot transmit sound without violating rigidity. You need to not worry about such things since they're irrelevant to the mathematics. It doesn't take years to move the rod since all the parts move independently, each being self-propelled so to speak, not controlled (caused) by one signal somewhere. So the motion is choreographed. You can't stop the subway train just because the door closed on somebody's leg. That would require an all-clear signal to the sent the length of the train, and yes, any such signalling would take years.

    All three examples are like this. There is no need for any signals to be sent anywhere. All motion is pre-arranged and initiates at simultaneously relative to the frame in which everything is initially stationary. There is no force here causing an effect elsewhere.

    We cannot make progress on this topic if you cannot dismiss these engineering issues. The topic is a pure mathematical one. There's no leg caught in the door. The door in fact doesn't need to open at all. The train just needs to move a light hour at a time, and be stationary relative to the track at the start and finish.
     
  11. DaveC426913 Valued Senior Member

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    18,959
    It is not merely an engineering issue. Even in a purely mathematical framework, rigid objects cannot exist. SR states there can be no such thing as rigid materials - even on paper. And rigid materials must violate SR - even on paper. In short, you cannot discuss rigid objects and discuss SR in the same breath. They are mutually exclusive.

    (One of the ways to prove this is that it immediately allows you to violate causality. You can send super-luminal messages. In fact, I think you can actually send messages that arrive before you sent them, creating time travel paradoxen.)

    If you want to discuss extended objects, (your silly putty or maglev examples), then why not simply change the scenario to remove reference to rigidity and instead talk about co-moving objects - eg. Bell's Spaceships, attached tail-to-nose with string?


    So my prediction is that you will rapidly arrive at conclusions that are either meaningless, paradoxical, acausal or a combination of the three, and the problem will be easily traced back to the above violation.


    But - since you seem to already be well aware of this, and you plan to push on anyway - I don't see any way I can contribute constructively beyond stating the above problems for your readers.
     
  12. Halc Registered Senior Member

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    350
    I deny this. I was careful in my descriptions to commit none of the violations you list. No message can be send faster than light for instance with my objects.

    Look up web pages on Rindler frames, which describe rigid motion in Minkowskian spacetime. This shows that rigid motion is quite compatible with SR and does not in any way involve locality violations.

    Per Ehrenfest, SR does forbid angular acceleration (but does not forbid angular rotation) of 3D rigid objects. But none of my examples do that.

    This is untrue. If you for instance push only on either end of an object, that end will move and the other end will not. Such an object cannot handle stress, and nowhere in my descriptions am I putting stress on any object. Like I said, you can envision it as simply a cloud of disconnected drones, none of which take signals from any other (since that would take time).
    It is only when you consider actual materials (an engineering issue) do these problems crop up. There can indeed be no infinitely stiff material since yes, its speed of sound would need to be instant.

    So from a pure mathematical standpoint, with each particle being self-propelled so to speak, there are none of the violations you mention.
    Yes, the train can work via something like a maglev, where the entire train is accelerated (not equally) by the track (a sort of glorified railgun) simultaneously, not a force being exerted from one point that needs to take time to be felt elsewhere.

    There is also a way to do continuous acceleration with a force being applied at only one point. It requires the extended object to be pre-stressed, and it fails if the force ever changes. That example is irrelevant here since it is merely an engineering solution.

    Even Bell didn't worry about the rigidity of the string since it was a mathematical exercise. In reality, if a lead ship takes off dragging 100 km of string behind it, the far end of the string remains motionless (if the string wasn't pre-stressed) until the sound wave gets to it.
    I'm not talking about Bell's scenario since I have single objects in all my examples. Still, it was all the nonsense being posted on threads related to that very scenario that prompted this topic. Lots of people find both that and the twins scenario unintuitive, and react by denial rather than by attempting to learn.

    Prediction noted.

    You can back your first statement that rigid motion is incompatible with SR. So far all you've shown is that rigid motion is incompatible with a force being applied only on subsets of the object. In the cloud of drones example, that would alter the path of only one drone, and yes, it would instantly violate the rigidity stipulation that it remain a constant proper separation from its neighbor drone.
     
  13. DaveC426913 Valued Senior Member

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    18,959
    Oh. OK, this is acceptable. This is what I was alluding to with the co-moving spaceships.

    So why not use this setup in your scenarios? That would completely sidestep the problematic "rigid objects" and the objections of any detractors, and the conversation can proceed apace. Otherwise, I suspect you're going to be having this exact clarfication every page or so.
     
  14. Halc Registered Senior Member

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    350
    Well, it is sort of implicit when discussing rigid motion which cannot involve causality in any way. So I don't require it, but if it helps you visualize things, we can always fall back to it.

    There's a couple points to note when using that model in the 2D examples. In that case, we have a 2D grid of drones that are not necessarily planar. I mean, scenario R starts out with a curved rigid plane. While all examples occupy 3D space, the latter two examples are considered 2D. One can for instance spin the rigid cylinder without distorting it. The radius just grows less is all, something which one cannot do with a similar object with nonzero thickness.

    A second point with the drones is that it would sort of allow a sheet to intersect itself. One could have a sheet pass through another part of the sheet, something which cannot be done with a contiguous sheet. So we don't allow that sort of thing.

    The third point has little to do with drones or otherwise. One can roll a 2D sheet and locally it is still entirely flat. Doing so can reduce my light-hour radius disk can be rolled or fan-folded (preferable) into what is effectively a tiny tube approaching the properties of a line. But if we are to further bend that tube, we necessarily have to crease it, meaning there must be a hard fold in it in places. I had suggested crumpling the disk into a wad in the middle. If we disallow creases, we can't go that route. I don't see an obvious reason to ban a hard crease, but it sort of offends my inner topologist, despite me being pretty much an amateur at topology.

    Except for the paper wad bit, all I've posted so far is pretty consistent with what's online concerning rigid motion. I do plan to go off that track a bit. The paper wad is a good example. It not only has creases, but a random paper wad would have regions of surface that are orthogonal to the axis about which I intend to rotate it, and rotating such a surface, even slowly, would run up against the Ehrenfest scenario which forbids it. That's a pretty good counter to the paper wad proposal. Is there a way to crumple it up which avoids such orthogonal surfaces? I think so, especially since I deliberately put a hole in the middle of it.

    While I'm on the subject of restrictions, there are time when I'll be applying impulse (rather than 'force') to the parts of the rigid array of points. There are times when finite rates of acceleration will not work and I need to change the velocity of something (one drone) instantly. Since a point is zero mass, it's not an infinite expenditure of energy, but is rather measured as energy per meter or some such. I'm going to propose such methods of moving my light-year object around. Doing it over finite time doesn't work, as it runs into contradictions quickly.


    More details on case P
    I notice that nobody has suggested that I'm wrong in my posted answer, and no other number has been computed. I initially framed the question in non-natural units, and the math is so much easier in natural units. So for that one (the subway), the natural unit will be a light-day.

    So the train is of length 365. The stations are 1/24 (=0.0417) apart, and we need to move our object (or our cloud of drones if it offends you to think of it otherwise) that far. There are 8761 stations along the train whenever it stops (inclusive of the end points). The thing will never go fast enough that train clocks run significantly slower than station clocks. We're only doing ~3 digits of precision here. In the train frame T, we want to contract the tracks so 8761 become 8760, a Lorentz factor of 8761/8760=1.000114 which happens at about 0.0151 c. Any given point on the object will average half that speed, so we need to travel 0.0417 at an average rate of 0.00755c which takes 5.52 days. That's how I got that number.
     
  15. DaveC426913 Valued Senior Member

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    You don't literally have to be constrained by what drones can and can't do (That is an engineering issue). This is a thought experiment after all.

    Speaking for msyelf at least, it is way easier to accept that a fleet of mini-drones is constrained to certain geometries than to accept the existence of rigid objects in an SR world.

    Besides, you could always just tie the drones together with string. Now they can't make folds that self-intersect. And of course, the drones can be effectively zero size.
     
  16. Halc Registered Senior Member

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    350
    Concerning case Q, the flat disk. I had proposed crunching it into a paper wad, and proceeding from there. It doesn't work since a random crunch of the wad amounts to hand-waving away the problems. It cannot be done without rotating portions of the material along an axis not tangential to the surface. OK, it can be done, but nowhere near as quickly as I proposed.

    So I take the sheet of radius 1 and put a vertical crease line at +/- 0.6 on the x axis. We also make a horizontal crease at the points where those lines meet the edge. I'm forming a sort of 1.2 x 1.6 rectangle now. The vertical creases I drag inward to the x axis, crossing it, and stopping at +/- 0.2. The material outside the rectangle, above and below the horizontal crease gets pulled downward until it runs out, leaving the crease where it is. I can do that because at no point is the action from the sides affecting any point simultaneously. This all takes time 0.8 (moving the sides from 0.6 to -0.2) giving us a rectangle of dimensions 0.4 x 1.6, in the same plane as before, 5+ layers arranged in a sort of fan fold. Second pass is to fold the long ends inward similarly, but this time crunching it accordion style to a narrow rectangle of length 0.4 in the x direction, lying in the x/z plane. This step takes another time 0.8. Now we rotate that thing along the z axis 90 degrees, which takes time π/2 * 0.2 - 0.314. Then just unfold it in reverse of how it was folded. Total time is 0.8 * 4 + 0.314 = 3.514, which (if we're using the light hour natural unit) translates to just under 211 minutes, less than the 283, but far worse than the 120 minutes of the paper wad method, and no, it appears that I cannot do better.

    I realize the above verbal description is hard to follow. I can post pictures if anybody wants to critique this attempt, but it appears that nobody is paying attention to the numbers.
     
  17. Neddy Bate Valued Senior Member

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    I can't follow the crumpling up, or the folding. I am still trying to figure out exercise P. Thanks Halc
     
  18. Halc Registered Senior Member

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    Glad you're reading this at least. I attempted a picture of the folding, and I didn't bother with the hole I initially described. There is no picture of the crumple since it was pretty much what you do to a piece of paper: random, and that will not work. Maybe it does, but this is my attempt at a slower, more formal crumpling that never rotates any surface or moves it in more than one direction at once. I already see issues with my calculations, which nobody caught.

    Please Register or Log in to view the hidden image!



    The circle is diameter 2. The rectangle inscribed is 1.2 x 1.6. The step 1 folds occur simultaneously, folding the top down (linearly, not by rotating).
    To the right and below each figure I show the same object as viewed from the side and/or bottom.
    For step 2 I illustrate 5 folds, but it would probably be many more, maybe 50, which reduces the distance to the sides traveled by each crease.

    Now my problem is that I am moving each piece as if it always had that velocity, I showed in the P example that you can't do that. It needs to be accelerated and then cleanly brought to a halt. This is a big problem in step 1 where everything is moving linearly. In step 2, everything fan-folds to the sides, a very different move than step 1. It still needs time to accelerate, but none to stop at the end since nothing has any extension in the direction of motion at the end. Maybe it would be faster to just do a fan fold in both steps, not just in step 2.

    Anyway, my estimate of 0.8 to do step 1 violates rigidity. We're accelerating an object of length 0.4 and moving it a distance of 0.8 which takes longer than 0.8 to do, as illustrated in exercise P.


    Essentially I am working in the frame of the object. I instantly (as measured at the rear-most point) accelerate it to a speed that contracts the tracks enough to include one more station. Then the front of the object is immediately at the destination we want it. Then I bring the object to a halt instantly (as measured at the leading point) which brings the rear (and all the other points between) forward by the same distance, which is a light hour. The time it takes to do this is the same as the relativity of simultaneity difference between the initial inertial frame and the frame with the ship at max speed, which was 0.0151 c.
    So consider the event of the rear of the object at the beginning (time 0). In the track frame, the front of the object simultaneious with that event is also at (both train and track) time zero. Relative to the frame moving at 0.0151c, the rear event (time 0) is simultaneous with front event t=5.52 days, which is the event where the front accelerates instantly to the same speed as the track.

    This method I use works for slow speeds. For shorter objects, and at higher speeds, the method still works fine, but the time it takes in the original inertial frame is more complicated. At 3 digits of precision, it makes no difference at all in my example. To get the duration, I had taken the max speed and divided it by two since every point on the object accelerates continuously to that max speed and then immediately commences acceleration back to the track frame. But the average speed isn't the max/2 due to relativistic velocity addition, so the divide-by-two step is invalid at higher speeds. So we have to actually do a relativity of simultaneity calculation to get that part right.

    This is straight textbook, although I've never seen any text attempting this particular problem. I haven't gone off the rails yet on exercise P, and to be honest, exercise R is still something I'm trying to work out. I was hoping the solution to R would be applicable to Q, but I've already determined that it will not be
     
  19. Halc Registered Senior Member

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    Case P and Q (with my folding trick) seem quite related, so case Q requires me to move an object of size 0.4 light-hours a distance of 0.8 light hours. At max speed, how long does that take? I cannot do the shortcut calculation here since the speeds are not slow. So we do it the correct way, and also do case P the correct way to show that it yields the same answer as before.

    Distance as measured in S, the inertial frame, from rear event and time 0 and front event at time X is 1.2, three times the size of our object, so we want a Lorentz factor of γ=3, which happens at v=0.943
    Relativity of simultaneity time change: t'=Lv where L is the distance in S between the two events (1.2) and v is the velocity change (0.943) so elapsed time to move the object is 1.2*0.943 = 1.132, not 0.8 like I mistakenly used before.

    So the time to do my folding plan is (1.132 + 0.8) * 2 + 0.314 = 4.177 hours or over 250 minutes, still better than 283, but barely.

    Using the same method on case P: t'=Lv where L=(8761/8760) * 365 = 365.0417 and v = 0.0151
    365.04 * 0.0151 = 5.51 days, not 5.52 like I computed the approximation way. I take back my assertion that it makes no difference with 3 digits of precision. There was a slight difference.
     
  20. DaveC426913 Valued Senior Member

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    18,959
    I'm still not managing to contribute meaningfully; it's all I can do to define the problem stated. Mostly because the premise is not given.

    Take your first example, above.
    The object, extended or no, takes as long as it takes to travel one light hour, based on its acceleration, coasting and deceleration, none of which are defined in the problem.
    The fact that
    • it is a subway train,
    • a light year long,
    • stationary relative to the track it's on*,
    • it has doors,
    are all irrelevant to the problem as-stated.

    * I guess only initially? Since the problem wants the subway doors to come to rest elsewhere on the track, it's not going to stay at-rest relative to the track. Maybe I'm just over-thinking it.



    Length/distance contraction - as exampled by Bell's Spaceship's, strung together with string - has already been dismissed by the OP as not relevant to the scenario, as far as I understand (post 9).

    So: all parts of the extended object are free to accelerate (presumably, all at the same rate), coast and decelerate to a stop.

    What am I missing?



    (The generous side of me thinks that whatever "gotcha" there is to the problem is inadvertently left out, say, as part of an earlier, but not referenced, discussion. The cynical side of me can't help but wonder if this may have been deliberate, for rather cynical reasons. I am open to clarification.)
     
    Last edited: Oct 28, 2023
  21. Neddy Bate Valued Senior Member

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    2,548
    Exercise P:

    I didn't realize that was your approach. I thought we would have to work with realistic finite accelerations, and calculate carefully to ensure no matter exceeds the speed of light, (as measured from the inertial frame).

    Allow me to elaborate. Consider your approach from the track frame. If the train were to accelerate instantaneously, and if the proper length of the train is to be held constant, then that means the train's length becomes length-contracted instantaneously in the track frame. This would cause parts of the train to exceed the speed of light, surely?
     
    Last edited: Oct 29, 2023
  22. Neddy Bate Valued Senior Member

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    2,548
    Wait, I could be wrong. Maybe you are saying that the rear of the train accelerates instantaneously in the track frame, remaining in its location, and that the rest of the train does not accelerate until RoS propagates up the train from the rear to the front. Hmmm...
     
  23. Halc Registered Senior Member

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    350
    Good point. I didn't make it clear that in all three scenarios, the object is to accomplish the task in as little time as possible. I read the OP and it never comes out and explicitly says that.

    Clarification questions are encouraged. I'll do my best.

    Max possible acceleration. No coasting (unless it's better than not coasting). It has to be stopped at the end in all three cases, so no blasting through the station.
    And 'deceleration' is such a non-physics term, but it is sort of applicable here since very little of my analysis involves frame changes.

    The length is very much relevant, as is the spacing of the doors/stations which defines how far it needs to move. It being stationary relative to the track at the beginning and end of the exercise is a hard requirement. The track defines frame 'S', our main inertial frame.

    So the problem is: What is the minimum time to move a light-year length object from and to a stationary state, a distance of a light hour, without violating rigidity (as defined)? In the initial solution of 5.5 days I gave, the train is always everywhere stationary in its own frame. That will not be the case in later methods, hence my not including that criteria in my definition of rigid motion.

    The train becomes length contracted in S whenever it isn't stationary in S. In that sense, it is very much relevant. There is not string, and no separate object at each end that measures the same proper acceleration. In that sense, Bell's scenario simply is inapplicable.

    Free to accelerate, yes. There is no 'coast' phase. Accelerate at the same rate, no, since that would 'break' the train, per Bell's scenario. The proper acceleration measured depends on where you are in the train. A human riding it would live (in discomfort) if he was in the middle, but not if near either end. We're going for absolute minimum duration here, so maximum brutal acceleration is called for.

    Not trying to be deliberately obscure or anything, and if you're reading the posts, I don't know the answers myself, at least not to the latter 2 problems. I'm very familiar with the linear object thing and confident that the 5.5 day duration cannot be improved upon. The mathematics is quite trivial and doesn't require me to write code or anything. Code is going to be needed for the other tasks.

    There is no prior discussion, but it was quite an oversight not to explicitly call out that we're trying whatever it takes to minimize the time needed to do each of the tasks.


    I took a long walk today and I am convinced that I can get scenario 2 down to 200 minutes, or close at least. Too complicated to do in my head, but I got a mental estimate near 200 using a different folding strategy. If anybody feels up to the task, I encourage attempts to do better than my methods, each of which hasn't been so good so far, getting it only down to about 250.
     

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