< Earlier Kibitzing · PAGE 289 OF 289 ·
|Oct-14-16|| ||beatgiant: <john barleycorn>
Better sharpen your crayon again; I posted some more data to your forum.|
|Oct-14-16|| ||sea otter: <Circle packing in a square>|
This might be a fun visual challenge. The aim is to pack n unit circles into the smallest possible square. For example, packing five circles in a square would look like: https://en.wikipedia.org/wiki/Circl...
Just change the number in the link for more examples. It only goes to 20. Here's a link to all 20 solutions: https://en.wikipedia.org/wiki/Circl...
I've never tried it myself, but it looks fun. Try to get all the way to 25.
|Oct-17-16|| ||al wazir: <sea otter>: Thanks for that link. The solution for 13 is really cool. It reminds me of some Islamic tiling patterns.|
It's hard to believe that the solution given for 7 is really optimum.You can use coins of the same denomination or pawns from a chess set to test different arrangements. The obvious first try is to surround one circle with six others. (They fit snugly.) This can be enclosed in a square of size 6.0, which is certainly bigger than 5.732... But is that the smallest square that will enclose them? Suppose you rotate the square slightly. Does that allow it to shrink?
I'm having trouble calculating the dimensions of the square as a function of the rotation angle. Can someone help me?
|Oct-17-16|| ||al wazir: Here's a somewhat related topic.
The flag of the United States has always had 13 stripes alternating red and white, with a blue rectangular field in an upper corner (conventionally portrayed on the left) on which n stars are placed, where n is the number of states in the union at a given time. The 39 historical flags are shown here: https://en.wikipedia.org/wiki/Flag_...
In most cases the stars were arranged in highly symmetrical patterns. For example, the current 50-star pattern is symmetric with respect to 180-degree rotation and inversion about either the vertical or horizontal axis, but not with respect to 90-degree rotations.
Some of the earlier star patterns were more symmetric, but some (e.g., those with 21 and 23 stars) were less symmetric. Can you think of improved designs? Of course those flags were not in use very long, and your designs will never be adopted in the future unless secession takes place, which seems unlikely, so this is not a practical exercise.
But the most symmetric arrangement for 51 stars, in case Puerto Rico or the District of Columbia becomes a state does have a potential application. That could actually happen. (Both have more inhabitants than Vermont and Wyoming.) What about 52, in case both join? Or 53, 54, etc., if some of our other island possessions do.
The idea is to arrange n identical objects in a planar pattern with the highest possible symmetry, for arbitrary n.
|Oct-17-16|| ||al wazir: <sea otter>: I've thought some more about packing seven identical circles in a square box and found the answer to the question I asked.|
Start with the arrangement I suggested (one circle in contact with six others surrounding it), enclosed by a square of size 6.0 (three times the diameter of a unit circle), with three of the circles aligned either horizontally or vertically. The square *can* be shrunk by rotating it. The minimum size is reached with a 15-degree rotation. The dimension of the square then is 6.0 cos(15) = 5.79555.... That, however, is bigger than the solution given at the website you linked to, 5.73205...
The arrangement I suggested is indeed the most compact one. It minimizes the space between circles. But the wasted space *outside the circles*, i.e., between the circles and the walls of the box, increases.
|Oct-17-16|| ||sea otter: <al wazir>, you may find the solution to N=23 on this link interesting: http://hydra.nat.uni-magdeburg.de/p...|
It's pretty fun to observe the different grain boundaries when N is large, for example around N=100. The site also has circle packing in a Circle, when hexagonal lattices come into play pretty early sometimes.
|Oct-17-16|| ||sea otter: The 51 star problem is especially appealing to me, since 17 is a very nice number imo, and three way symmetry is often very pretty. I'm going to try some of those out later today.|
|Oct-17-16|| ||al wazir: <sea otter: you may find the solution to N=23 on this link interesting>. Yes! That's the pattern that should have been used for the flag! But flag manufacturers of the 19th century would have been unhappy with it.|
I noticed that the circle-packing solution for N=13 given at the Magdeburg site is *not* isomorphic to the one given at the Wiki site; but in both the size of the square is 2[2+sqrt(3)] = 7.4641... .
|Nov-23-16|| ||WannaBe: OMG, we have a picture for Mr Stumpers!!! Yepee.|
In other news, http://www.forbes.com/sites/startsw...
What was actually detected was M. Carlsen pounding his head against a wall in anger after his loss.
|Nov-24-16|| ||Sneaky: What's the deal with the "EM Drive" anyhow? I still have to remain skeptical that we've found some wrinkle in the conversation of momentum, but gigantic tenets of physics have been overthrown before.|
|Nov-25-16|| ||al wazir: <Sneaky: What's the deal with the "EM Drive" anyhow?> First I've heard of it. But maybe you're referring to this WaPo story (which I missed): https://www.washingtonpost.com/news...|
I'm not going to look for the journal publication the WaPo piece refers to, because it says that <their machine generated 1.2 millinewtons of thrust per kilowatt of electricity pumped in.> That's not enough to propel a model airplane, let alone drive a spacecraft that will explore strange new worlds, seek out new life and new civilizations, and boldly go where no man has gone before.
The article says that <its thrust seems to come from the impact of photons on the walls of the copper cavity.> That's not very perspicuous. But it also says, more helpfully, that its operation is based on <Newton's third law of motion. It's [analogous to] how jet engines work: As hot gases are expelled out the back of the plane, they produce a thrusting force that moves the plane forward.>
So here's my picture of what's going on: The device works like a microwave oven that somehow functions with the door open, which your kitchen microwave oven does not. Microwaves (the photons) bounce around inside and some of them fly out though the open door. They're analogous to the exhaust jet from a rocket. The "push" is exerted against the closed front side of the oven, just as the thrust produced by a rocket engine is applied to the closed front surface of the combustion chamber. (In the turbojets and fanjets that power airliners the thrust is applied mostly to the blades of the compressors located at the front of the engine.)
In my conception the photons that come out through the back end (the open door) act like an exhaust. But the WaPo piece says that it <generates a small amount of thrust simply by bouncing microwaves around a cone-shaped copper chamber. No propellant goes in, no exhaust comes out, and yet, somehow, the engine can make things move.>.
So maybe they're saying that *nothing* come out the back end, not even photons. Then I don't understand how it can work, unless there's a clue in the "cone-shaped" chamber. If photons are absorbed preferentially at one end because of the conical geometry, then that end heats up, while the photons bouncing off the opposite end cause a bigger force to be applied to that end, because a photon that is reflected transfers twice as much momentum to the surface it hits as one that gets absorbed. That would create a net thrust.
If I were you, I wouldn't call my stockbroker just yet.
|Nov-25-16|| ||beatgiant: <Sneaky>, <al wazir>
Wikipedia has a decent article on the phenomenon: https://en.wikipedia.org/wiki/RF_re...|
|Nov-25-16|| ||al wazir: <beatgiant: Wikipedia has a decent article on the phenomenon>. The article mentions plasma thrusters and gravity assists, which are already in use, as well as two familiar, scientifically uncontroversial but not very practical "propellantless" mechanisms: solar sails and propulsion by radiation pressure from ground-based beams.|
It also says of the "Cannae drive" that <its thrust is derived from a reduced reflection coefficient at one end plate">. That sounds like the second version of the device I described.
The idea underlying its operation is very easy to understand. Imagine that you are an ice skater, standing on the ice. Somebody throws a heavy medicine ball to you and you catch it. That imparts a certain amount of momentum to you and you begin to move in the direction in which the ball was thrown. If you throw the ball back where it came from at the speed with which it arrived, that imparts the same amount of momentum to you and you move twice as fast.
The momentum transferred by photons reflected from the far end of the microwave oven I described is analogous to the momentum transfer resulting from catching and throwing back the medicine ball. The photons that are absorbed in the surface of the opposite side transfer only *half* as much momentum to that surface, because they aren't "thrown back." Force is equal to momentum transferred per unit time, so the force on the far end of the box is twice the force applied to the opposite side.
The calculation is a little trickier than that, however. If photons are merely fired from a source at one end of the "oven" (the chamber) and absorbed at the other without bouncing around at all, there is no net momentum transfer. It's as if a pair of skaters are linked by a rope and one throws a medicine ball to the other. So you have to look closely at how the photons are produced.
Suppose the source is two "photon guns" situated in the middle of the chamber. One fires photons in the forward direction, the other in the aft direction, at the same rate. Each photon carries momentum μ.
No net momentum transfer so far.
Each photon that hits the forward end transfers momentum 2μ. Then it reflects back and hits the aft end and transfers momentum μ in the rearward direction. Meanwhile, each photon fired in the aft direction hits the rear plate and is absorbed, transferring momentum μ. Since the guns fire at the same rate, the momentum transferred in the rearward direction is also 2μ.
So there is still no net momentum transfer.
But what happens to the *energy* carried by those photons that are absorbed? It turns into heat. The absorbing surface gets hotter. When it does, it radiates... photons! Not the same photons that were absorbed, but photons of longer wavelength. If they emerge preferentially from the aft (external) surface of the chamber, then *that* will result in a net momentum transfer to the chamber, and this will create a force that will propel the craft forward. It's not perpetual motion, of course, because a source of energy was needed to create the photons that the photon guns fired.
Can this work? I don't see why not. If the external and internal sides of the absorbing wall are painted different colors, they will radiate at different rates.
There is another sf idea for a form of space propulsion that really is bonkers: the "ram jet." (Not to be confused with air-breathing ram jets, which really work, though only at supersonic -- better, hypersonic -- speeds.)
The idea is that most of the interstellar matter is hydrogen, and some fraction of that is the isotope deuterium, which in principle can produce useful energy through fusion: D + D --> He + energy. (In principle. It works in the Sun, but so far not in the laboratory.) So the space ship carries huge vanes that scoop up interstellar hydrogen atoms, funnel them to the craft, where the deuterium is winnowed out and burned in a controlled fusion reactor.
The trouble with this concept is that it neglects the drag due to all the interstellar matter that gets scooped up that *isn't* deuterium. At near-relativistic speeds the energy needed to scoop up a mass m is close to mc^2, which is what you would get if you converted *all* of that mass *entirely* to energy. But only a tiny fraction of the mass scooped up is deuterium, and the D-D reaction converts only a tiny fraction of it to energy.
|Dec-01-16|| ||WannaBe: So, in theory, water can freeze in hell.
|Dec-01-16|| ||diceman: <WannaBe: So, in theory, water can freeze in hell.>|
I'm sure the Clintons have ics cubes.
|Dec-08-16|| ||zanzibar: Here's one that's got me stumped...
Why does Shing-Tung Yau have a photograph of Hamilton in his book <The Shape of Inner Space>?
After all, the guy was in the very next office to him for years... and he's got pictures of just about everybody else.
|Dec-09-16|| ||Shams: <WannaBe> <So, in theory, water can freeze in hell.>|
You only need to hop two planets over to find ice on the surface of an 800 Fahrenheit planet. :)
|Dec-09-16|| ||zanzibar: Heck, I do that all the time... it should read
<Why <doesn't> Shing-Tung Yau have a photograph of <Richard Hamilton> in his book
<The Shape of Inner Space>
|Dec-09-16|| ||zanzibar: BTW- a pretty good review of the book (or so it appears to me), is found here:|
|Jan-02-17|| ||TheFocus: Where does it say that Humpty Dumpty is an egg?|
|Jan-02-17|| ||TheFocus: The worst thing about getting hit in the face with pi is that it never ends.|
|Feb-16-17|| ||TheFocus: A computer lets you make more mistakes faster than any invention in human history, with the possible exceptions of handguns and tequila.|
|Mar-15-17|| ||alexmagnus: Talking about unsolved mathematical problems with a statement looking elementary, here is another one:|
Consider the sequence 1, 12, 123, 1234,...12345678910,...
<Unsolved problem>: Will the sequence ever produce a prime number?
It is conjectured that there are infinitely many of them, yet so far none is found - and they checked it till 344869th number so far...
|Mar-15-17|| ||alexmagnus: <<Unsolved problem>: Will the sequence ever produce a prime number?>|
But looking at the factorizations of the first 30 numbers (starting with n=2), there is a funny pattern in the smallest factor:
|Mar-17-17|| ||alexmagnus: It's actually quite easy to prove that the 2,3,2,3,2 pattern will repeat forever. So if there is a prime there it will be on a position 1 mod 6. But the numbers at those positions show no apparent pattern in the factorization at all.|
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