Another mental model Turing left for the kids of the worlds who also like math is that of the ice cream cone. With it, one conquers another topic that typical American kids, taught by just awful math teaching methods, struggle to even conceive properly: hyberbolics. In particular, so what does that damn cosh() key do on my calculator?
The good news is… that we don’t bother answering that question. Its just not necessary. What matters is the shape of an ice cream cone (before you eat it). What matters is that what’s in the cone (the ice cream) is not the same as the cone itself. And later, we will worry about the side of the code vs. the outside since the outside is something we can imagine shining a lamp and then putting something between lamp and cone ( and then watching how the shadows dance on the surface of the code, before we eat it…)
Concerning the cone and its ice cream, realize that this is fundamental to quantum mechanics. So study QM, since there is lots of ice cream cones, each lecture! Note that the cone has an orientation, which is typically up (so the ice cream doesn’t fall out). If we call the up direction Z (out of X, Y and Z), we are saying that of the three coordinates for a point of ice cream within the cone we know its up-ness. I cannot tell you any more (except that its Y and X fall somewhere within the boundary of the cone.) I always imagine a string within the ice cream, connected to the lower tip at one end with the other in my (free) hand, and able to move a taught string freely within the ice cream thereby “stirring it”.
Concerning phase space, and the last memo, this is our space (and we have vectors of 3 components, only one of which we know anything about certainly. The other two are inherently uncertain (but fall within bounds of the cone).
now comes the fun part, since the seller of the cone probably demands money before handing it over. So she or he puts the cone’s tip into a plastic holder – a horizontal with a hole, with two legs – before doing the money thing. Later you lift it up , in the direction, and no longer does the top of the plastic touch the cones outer surface. Of course, if the hole of the device was too small, not much of the cone tip would have passed through (since the expanding diameter of the cone would soon preclude passage into the hole. And it would probably tip over once the guiding hand is removed (since the mass of ice cream well above the center of gravity of the dynamics would cause the cone’s tip to break off, making both a literal and a math mess) as “collapse” occurs. quite where the ice cream would end up is rather random (though predictably within a few feet of the disaster zone). The likelihood of it ending up in the kids mouth is small, though – which was perhaps what the kid was “expecting” the outcome to be.
anyways, enough of collapses and onto puppet shows. lets imagine that the cone is properly mounted in its holder. Now take that torch you always carry around with you and darken the ice cream selling room. Assuming the holder is not transparent, so positing the torch light so it illuminate the cone, but the lamp is positioned BELOW the holder, with some light pass around and some being blocked.
Now look at the shadow on the cone (From where you are standing). Here you are in hyperbolics. Go now look from the top, and note how the shadow changes (as you look a it). Here you are in hyperbolics, again. One shadow, lots of different lengths of shadows. And, if we have different lengths, we have difference angles (between two lines of different lengths). Which means we have different cosines.
Anyways, back to crypto, since Turing is showing us how. HE imagines two cones, tip to tip, in a line They are filled with air (rather than ice cream, for obvious reasons that one is now oriented downwards. One points up, Z, one points down Z. IN the middle there is still the “horizontal of the holder which can now move upwards (till it hits a limit against the upper cones expanding diameter) or it can move downloads (till it hits a limit of its opening against the lower cone’s expanding diameter).
In Turing’s view, we have the concept now of history and future. Relative to the present (the horizontal ‘hyperplane’ we can move say up as future of the systems evolution or backwards in time (down).
What matters is that the system of cones, their diameters, and the hole sizes in hyperplanes are set such that “feedback” occurs, in which there ARE dynamics relating history to future.
Its even more fun when you tip the hyperplane a little and THEN move it up and down, where one side of the hyperplane hole intersects with the cone “at an hyperbolic angle” – before the lower edge of the planes hole. No you don’t need the torch, since you can create all the possible shadows from considering these “innate, or internal coordinates” – meaning you have a self-describing system – that doesn’t need the torch as a reference system.
Then you go to Tunny cryptanalysis, and leverage arctanh() – since when reasoning with probability measures its useful to calculate in these spaces since differentials in these space converge in ways that are “more interesting” than in the math you got taught at school – designed to keep you dumb about crypto design.
Now you can study http://en.wikipedia.org/wiki/Hyperbolic_cosine. Its awful. But its easy when you realize that the string between origin is elastic. It’s the elasticity (and its stress curve) that ensure that in some sense limits the ball on the other end attempting to go round in a circle. The pull of the elastic, in my mental model, deforms the original circular path, and indeed prevents the ball from even continuing on its circular path beyond a certain point (since one reaches the maximum elasticity of the stretchy connector). Then you read about cosh(), being accompanies by sinh(), which are the two parameters for the position of the point as it wander (or tries to wander) around the circle, or phase space.
Its interesting to crypto when the circle has unit volume, and thus the space being carved out or the space through which paths follows are probabilities, and convergence of sequence of probabilities densities. Then one can reason about space allowing such convergence (and which do not).Then you get to talk about “embeddings and extensions” – which gets you go higher ciphering math – since in “stochastic” systems we get to define lots more types of differentials – beyond the gearbox in your car and beyond the “Gradient” of the curve taught to most 14 year olds (or 19 year olds in the US) as “calculus” (one hears legions of USA exceptionals shuddering and crashing to go find the dope fix to cover-up their non-exceptionalism, at the mere mention of the word “calculus”).