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A question such as where the ball comes to rest will depend on the pattern of bumps on the surface. … One can roll this ball like a die, and then look to see which color is on top when the ball comes to rest. … And what one sees is that it takes only a small change in the initial speed to make the ball come to rest in a completely different orientation.
(The three dots in the representation of each rule stand for the rest of the elements in the sequence.)
Starting with an ordinary base 2 digit sequence, one prepends a unary specification of its length, then a specification of that length specification, and so on: (Flatten[{Sign[-Range[1 - Length[#], 0]], #}] &)[ Map[Rest, IntegerDigits[Rest[Reverse[NestWhileList[ Floor[Log[2, #] &, n + 1, # > 1 &]]],2]]] (d) Binary-coded base 3. … Apply[Take, RealDigits[(N[#, N[Log[10, #] + 3]] &)[ n √ 5 /GoldenRatio 2 + 1/2], GoldenRatio]] The representations of all the first Fibonacci[n] - 1 numbers can be obtained from (the version in the main text has Rest[RotateLeft[Join[#, {0, 1}]]] & applied) Apply[Join, Map[Last, NestList[{# 〚 2 〛 ], Join[Map[Join[{1, 0}, Rest[#]] & , # 〚 2 〛 ], Map[Join[{1, 0}, #] &, # 〚 1 〛 ]]} &, {{}, {{1}}}, n-3]]]
A fixed interval of time for the clock—as indicated by the length of the darker gray regions—corresponds to a progressively longer interval of time at rest. … It leads to the so-called twin paradox in which less time will pass for a member of a twin going at high speed in a spacecraft than one staying at rest.
For assuming that around the particle there is some kind of uniformity in the causal network—and thus in the apparent structure of space—taking slices through the causal network at an appropriate angle will always make any particle appear to be at rest. … What will the events associated with the second particle look like if one takes slices through the causal network so that the first particle appears to be at rest?
To make the patterns of flow easier to see, the velocities shown are transformed so that the fluid is on average at rest, and the plate is moving.
A typical example is time dilation, in which a fixed time interval for a system moving at some speed seems to correspond to a longer time interval for a system at rest. … But as the pictures indicate, it is then essentially just a matter of geometry to see that this dark gray region will correspond to progressively larger amounts of time for a system at rest—in just the way predicted by the standard formula of relativistic time dilation.
And from this it is often concluded that there can be nothing like an ether that one can consider as defining an absolute state of rest in the universe. … For in standard cosmological models it fills the universe, but is everywhere at rest relative to the global center of mass of the universe. … In particle physics standard models also in effect introduce things that are assumed to be at rest relative to the center of mass of the universe.
Picture (d) is obtained by transforming to a reference frame in which the fluid is on average at rest.
If I am correct that there is a simple underlying program for the universe, then this means that theoretical physics must at some level have only a very small amount of true physical input—and the rest must in a sense all just be mathematics.
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