I'm keen to try explaining something to a younger person, so The Idea was to run it past, and see what happens. So I postulate that there's this space, the domain of frequency, and that time--in the domain--is a function of frequency. Simultaneous events occur in the same frame of reference, but this frame can be the difference in frequency between two sources, or oscillating systems. In fact in order to 'count the days, hours, minutes' etc a requirement is that a choice of oscillating system is made. Choosing for instance, the sun and the shadow made by an upright object, you have a sundial with a period that begins and ends with sunrises and sunsets. When the sun sets, so the clock stops. So the whole synchronization argument is down to a choice of oscillators with the same frequency. But then a third oscillator with a different frequency will be a phase-difference, which is the frame of reference for all three frequencies. So is it ok to say that this third frequency is choosing a gauge? Including the start-stop paradigm should then form a gauge group of "clocks that can be reset and run at different frequencies"?
What you are talking about (badly) is a Fourier transform. If you have something that's a function of time, and you do a Fourier transform on it then you'll be left with something that's a function of frequency.
No, what I'm talking about is the difference in frequency. What you're talking about is how to analyse say, a mixed frequency. If you treat, as I asked about, the choice of a frequency as equivalent to choosing a clock, then is choosing a different frequency, which will be out of phase with the first, equivalent to choosing a gauge; or since two identical frequencies can be out of phase, is that a better choice? The choice being the difference between the phases of identical or non-identical frequencies, of oscillating systems. Given that you have "the set of non-identical frequencies" and the operation of synchronization, don't you have a group?
Suppose that you assume the selection of a different frequency is like choosing an angle, like a dihedral (or digonal) element. Does it follow that choosing coordinates is like a tiling, and choosing a frequency is like coloring the tile? Or that the clocking mechanism of choice in a frame of reference corresponds to an identification, the frame has a past and a future, in other words, once it gets 'colored'?
If Fourier analysis doesn't cover your bases then I don't understand what you're asking. I'm not completely sure that you understand what you're asking either.
I'm asking if a map coloring corresponds to choosing coordinates. Is a stationary frame equivalent to say, coloring a face of a regular solid? I'm not asking if there is a way to decompose the output of an oscillating system with a Fourier analysis.
Hmm. The theory I'm trying to flesh out, even if it's quite a modest attempt, is that relativity is really about clocks when space and time are both 'flat', and when they aren't (i.e. there is curvature to explain). First of all, explain what a clock is in basic terms--regular periodic motion, a frequency which is stabilised, modular arithmetic that counts a number of periods and divides the output of an oscillator (earth clocks use mod 12 and the earth is divided into timezones). Time, when you use clocks, is essentially a frequency mod some fixed number of equal periods. In permutation puzzles, time is part of the problem. More exactly the time it takes to solve a puzzle corresponds to 'real time', you can measure it with an external clock. Is there an internal kind of clock though--does 'cube time' exist? To explore the idea I think you have to see if a Rubik's cube corresponds to a system with fixed frequency and a way to count periods. My theory says that the cube puzzles (and their variations, all Platonic solids) tell us something fundamental about the nature of space and time. "Analysis" like Fourier, is of course the shortest word problem or God's algorithm, and the 'sub-problems' of finding the shortest paths between each step. I already know that the words that get the "cube-clock" from the origin to the first few steps away have redundancies (for instance, words with length 6 corresponding to three steps have a total of 12 combinations/permutations, but 3 of the 12 are redundant, writing out all 12 doesn't seem to yield many clues about how to pair the redundant words). I'm also fairly sure that, despite its shape, the Rubik's cube is actually a composition of all five of the regular solids. The message these puzzles convey is a matter of personal experience on the one hand, and also finding out there is a deeper message--there is a hidden structure 'living' in the 3-space, which is 'expressed' in 2-space on the surface. Time is then just a universal resource--the 4th dimension--which allows this inner structure to evolve. Why are the Rubik's cubes all deeply-sliced and what is a 'deep' slice? What about puzzles that have shallow slices? And, are there any glaring problems with "the model" anyone can see?
Aha. I think I've seen through to the next 'level' in my somewhat detached view of these little buggers. Using the familiar labeling and I suppose what would have to be dubbed the operator algebra: X, X', XX, where X is an element of {URFDBL} the set of 'face moves', you have that the forms (A,n), where A is the ' word', or algorithm, and n is the number of alternate faces, all correspond to the following: Since n is the number of sides of a polyhedron, you have successively: one-sided, two-sided and so on 'figures' which since they have a 4th dimension (the rotation of coordinates in time) are polychora (not polyhedra). So you have 'trivial' polychora until n = 4, the tetrachoron (or tetrahedral group), and at n = 6, the hexachoron (hexahedral group), etc. The problem with this is deciding what odd n means--is there a "monochoron", a "dichoron" etc, or is it ok to see these as just odd to even compositions? I know that when n =4 and A is length 8, there are 24 words, however the 2x2x2 "sliced 3-ball" at (8,4) has only 5 representatives and so 19 redundancies. In other words the tetrachoron has the symmetry of a pentagon in this (8,4) representation, or the 5 'faces' of the tesseract have a pentagonal relation to each other as a diagram of sorts. The 2x2x2 is a direct representation of the quaternions which is the group of rotations in 3 dimensions (that's SO(3)). And no, I don't know all that much about group theory, but I do know a bit about knot theory, braids and links, and I've assumed the set of puzzles is like a set of knots (initially unknotted), or braids embedded in R. Could be up the creek a little with that, but the fun is in finding out if you are or not.
I had a think last night about the explanation. Which will be about what a strand is and where the sporadic group comes into it. If you want to braid (or plait) n strands n has to be at least 3, like the sides of a polygon do in order to be a closed figure (in fact, in order for the idea of closure to be 'rigorous' in the sense of a closed cycle). Plaiting three strands is an iteration of: "Cross the leftmost strand over the centre strand, then cross the rightmost strand over the (new) centre strand." As the 'instruction' is repeated, you get a braided structure, as the centre strand alternates. With n = 2, you can only twist them around repeatedly--unbraiding is a matter of untwisting a 2-braid, or unplaiting a 3-braid is a matter of inverting the instruction the same number of times, which is "the unlink" or "unknot". But these are really simple examples of braids and knots. The 3-braid is labeled, for example each strand can be given a symbol: a,b,c and so you can write sequences of letters that specify a particular 'knotting". The strands in the Rubik's cube are labeled with colored sections. In the 2x2x2 representation they are identified by the two colors adjacent to each "missing" edge piece--the edges are contracted to a point on the 2-sphere. I suppose you could call these "imaginary" edge faces. You can "invent" to a certain extent your own interpretation of where or what a braid group is, but here we have one on the boundary of a 3-ball. We can show that it's an immersion or embedding in a higher dimension which is the one the imaginary pieces are in.
The idea of contracting a circle or a sphere--a piece of the puzzle 'fits' inside a circumsphere--to a point is important. A point has no interior so the interior is the empty ball. A 1-ball is a line, or an interval like (0,1) with two points projected 'over' each other. In other words you rotate two points to 'see' a strand. The 0-sphere is the two-point boundary of the 1-ball. On the cube the 1-balls are the edges, you make 8 of them disappear by rotating the cube so you can only see one face in perspective. This is the projection of the interior of the face, which is a disk with a 1-sphere boundary. So because you can rotate four of the adjacent faces, much like rotating the sides of a box, you can 'project' the interior of the disk onto an imaginary 4-sided figure. This is just a number that looks like a 4x4 matrix of square sections, with connections between each row and column, and these are where the strands are found in so-called projective space. Rotating the outer perspective is a form of composition that essentially yields alternating projections of opposite faces. Here the 'divisor' is the group of rotational symmetries of a cube. So you're seeing a projection of one half of the edgewise connection--an abstraction--to the sporadic group on each disk, which is in a 'sublattice' of points all the braids can project over or 'pass through' like holes in a surface, as strings of abstract words in a kind of knotting machine. These words have a natural class, which is their length, like the length of a list. In the plaiting example, the instructions can be made more 'machine' like by recognising that you need one that can identify which is the centre strand--a sequence composed from a,b,c as "address of the central strand", and which is the left or right strand. So what you have is two strings of address-instruction "data" like bacbacbac and acbacbacb, which mean respectively "b is the centre, ..." and "cross a over the centre", ...". Anyway, this 'imaginary' space is like another boundary (the one of the sporadic group, or at least some of the points), and the one I have is the boundary that the real pieces reach, a bit like the boundary of a chess board, but the pieces keep changing. There are 0-sphere boundaries for these half-spaces of connections--on the disk-- but the algebra is kind of geometric. The thing here is, the lattice is too, and it has a left hand and a right hand boundary, the first is kind of stepped or, 'more tangled' and the second is vertical--these are projections onto the 1-sphere boundaries of 2-spheres. And getting back to frequency, I'm looking for the frequency of letters in these words, and the letters are colors that get braided together along the boundaries.
Something else that's occured to me is, the idea of a link and "unlinking" is connected to the physical problem of momentum transport in fluids, like the problem in another thread. How it is is in the following explanation--again aimed at a young person. There are three theories which are germane to the problem: the theory of molecular transport or diffusion; the theory of heat conduction; and the theory of viscous flow. These theories are linked together by the fact they all have terms in the formulas that are the product of velocity and mean free path. i.e. vl, where v is the average velocity per particle; there are n particles transporting heat as vibrational and rotational momentum, and the viscosity or 'delay' is a function of the momentum times the mean free path. The idea here is to unlink these--make the mfp, the average velocity, number of particles and momentum vanish alternately to define boundary conditions. The bulk flow is tied to the viscosity of the flow itself, so if possible look for 'diffusionless' and 'constant temperature' solutions to reduce the number of links. The other link is the number of particles, where the velocity is low these will be 'cooler' and the density will be higher, regions with high velocity particles will be less dense; in both cases they can have the same momentum product which is their pressure. Pressure (including gravitation) is a momentum product which you can decompose into 'layers' of flow, which has three coefficients to determine. In other words, you 'slice' the flow into a rectangular section and compute momentum products--their direction is aligned along a flow gradient and I believe the solution will require a formulation of one at least. The slice gives you a perspective of the knotted momentum product. This is why knots and braids are so cool.
I would like to discuss some terminology. First of all what is "the unlink" and why is it different to a link? I know the unlink or unknot is a closed loop. Does this mean any closed loop is 'cobordant' to the unlink? In the permutation puzzles you get cycles--the fundamental group is a composition of 4-cycles and 2-cycles, the latter are essentially "two letters on the same face", so that all the cycles are loops and all cobordant to the unlink? So with (1,1), (2,1) and (2,2) you have respectively single letters, identical pairs, and pairs which are different a kind of classification or ordering. The first class corresponds to vectors because any X has a direction and magnitude, the second two classes are compositions of vectors which are formally rank 1 tensors? With compositions of 2 different letters you also get loops (cycles) which are unknots? Then odd cycles have a centre, even cycles don't? Further, is a 3-ball smoothly embedded in 3+1 spacetime? One more: is it ok to say the boundary, or part of the boundary, of an embedded 3-ball is the base of a number? That is, to conjecture that a number in base Lp exists, where p is a group of points on the boundary of a lattice?
Ah well. And so, young person, if the question is "does the lattice of points exist because someone wrote a program that "filtered" all the permutations of a Pocket Cube recently so it sorted them into minimum word lengths, and because the Rubik's Cube was invented?" the answer is no, the cube is an artifact of mathematics and architecture, the lattice existed before Erno Rubik did. Anyone could have, since the Greeks at least, place eight identical blocks in a stack and colored the external faces, then thought about rotating groups of them along the inner plane surfaces, and then rotating adjacent sides in steps, and so on. The lattice and the shape it has have always existed--it's a universal lattice which has been around since the universe began. This is definitely a more interesting idea.
One other idea I've had is about history, and what I call "mirroring". In physics there is the CPT theory, where different kinds of abstract mirror are reflecting charge, parity, and time and an asymmetry is found which is connected to the history of the universe, which is roughly speaking the difference between the amount of matter and antimatter today. The lattice is a kind of abstract mirror because it has boundaries, with compositions of letters and 'grouping' into sets of word-length per number of face rotations, the rotations are what places the limit on the word lengths. You have a maximum span of the lattice which is 11 if you count "XX" as a single span. Time reversal--the T mirror--is equivalent to running an algorithm backwards, so then X' is the reverse or inverse of X in notation that uses the quote character to signify an inversion of the "time direction"--if you abstract time to be the direction of rotation. But (XX)' is the same span as XX, so here the time mirror reflects XX as itself--its algorithmic form is unchanged in either direction. So part of finding the "skeleton key" is understanding the difference between spatial mirroring and inverting an algorithm. Also, because any composition of letters is an algorithm it's also a cycle and contains its own reflection (at a boundary). There is more to the existence of this structure, I believe, than in the philosophy of men, Horatio. It spans the history of the universe too (even if you only go back to 1979, or to Plato's time or Euclid's), but does it go all the way back, to the initial conditions??
What the connections to fluid dynamics and flow and transport are, is in the same guise as imagining the 2x2x2 structure as a kind of pump, then you replace X with pumping forward or up a slope of 1, and X' with pumping backward or down a slope of 1, then when either "acts on itself", you get respectively lifting a unit of fluid, and 'dropping' a unit of fluid--that is, XX is either a capillary unit, or a unit of free fall. This is the idea of a flow of fluid in a lattice of tubes. In a continuous fluid the flow is everywhere a product of particle (molecule) momentum. Since momentum is transported you have free-fall of the momentum as a limit velocity, the average velocity of diffusion, in the algebra of lifting or dropping a unit of fluid the "momentum product" vanishes, or it's an algorithmic flow. The connection is the one of gravitational free-fall of a particle through a viscous fluid. If viscosity is zero it will accelerate, if the momentum is transferred, say with the walls of a tube, it will change velocity--perhaps come to a state of rest.
Please provide evidence that this is all gibberish: Please Register or Log in to view the hidden image! And if you can, some evidence that this lattice of points doesn't exist, it's all gibberish, please.
A peculiar question. Can evidence of gibberish be provided? Take this example: PT tor amoughly is there CPT the differ tory of mirroughly speaking tory is rount mirroughly speakind today. In theory, whics the hich is theor and where re rory speaking the is therent mirror the asymmetweent king ch istract kind asymme abstry of abstry speakind and ant of asymme differen an the ant kind abstoday. In there and and whis connected arity, abstry, whe his ch is rouniverse, whis foughly is re rory speaking connectimatter the arge, whis the histry speakind there differeflect matted the Gibberish? Not? What evidence is there either way?
