Speed of light is not constant!

[knightlife]

The speed of light is one of two different values used in various discussions. There is an orthogonal speed of light
and then there is a diagonal speed of light. The two speeds are related by the squareroot of 2. No one questions
this because there are no knightships to talk about currently. The reason I am bringing this up is that I discovered
a reaction that travels at an oblique angle and would like to calculate its speed as a single number, not (3,5)c/28.

[H. V. McIntosh]

The speed of light is one of two different values used in various discussions. There is an orthogonal speed of light and then there is a diagonal speed of light. The two speeds are related by the squareroot of 2. No one questions this because there are no knightships to talk about currently. The reason I am bringing this up is that I discovered a reaction that travels at an oblique angle and would like to calculate its speed as a single number, not (3,5)c/28.

Might one recall that John Horton Conway was a master punster, and that his followers have jubilantly followed the tradition? The Life vocabulary bears this out: Glide to crystallographers was a translation combined with a reflection in a mirror containing the glide axis. Conway knew this when he named his little bug so because that is what it did. Since then glider tends to mean any (small) thing that moves, whether it reflects or not, and even if it does, the name has already been given to that first object. And then there are space ships, even though there is no Weltraum. Which brings us to "the velocity of light," even though there is no light, G'd not yet having issued His famous dedazo and Michaelson, Morely, and Einstein all having been preoccupied elsewhere.

Does the Universe in which Life exist have a name? Anyway, over there velocity is measured in cells per generation rather than centimeters per second; the topology is L1 rather than L2, and oblique velocities might best be reckoned by connecting the center of a cell with the center of the cell a generation later, counting the number of cells traversed, and calling that the slowness (reciprocal of the speed) of the motion. So light speed gets you to a neighboring cell, diagonal, lateral or transversal, in one generation. Half light speed gets you over there in two generations, third light speed in three, and so on whatever the zigzag path "actually" followed. What a shame if this turns out to be inconsistent, as it surely would have been if one had insisted on imposing a continuous space-time on a discrete space.

[calcyman]

The speed of light (maximum propogation through any medium) is defined as one cell per generation using the L-infinity distance. Interestingly, the maximum propogation speed through vacuum, or empty space, is one cell per two generations using the Manhattan distance. This is unusual that the former should use one metric whilst the latter should use the other.

The two speeds are related by the squareroot of 2.

In our universe, which is not anisotropic, there is no differentiation between orthogonal and diagonal. In fact, you can't define the terms 'orthogonal' and 'diagonal' in our universe, without setting an arbitrary coordinate system. For this reason, it is best to use the Euclidian distance in our universe.

However, the Life universe is not anisotropic - different directions are treated differently. In Life, Euclidian distance and Pythagoras' theorem become meaningless. That's why no B*/S* rule can exhibit perfect circular growth. You can't apply these principles to the space in which Life operates.

The reason I am bringing this up is that I discovered a reaction that travels at an oblique angle and would like to calculate its speed as a single number, not (3,5)c/28.

I'm sorry, but there is not much of an option; 'c/4 diagonal', for example, is simply shorthand for '(1,1)c/4'.

[Phantom Hoover]

Actually, effects can propagate into the vacuum at c, just not forever.

[Macbi]

However, the Life universe is not anisotropic - different directions are treated differently. In Life, Euclidian distance and Pythagoras' theorem become meaningless. That's why no B*/S* rule can exhibit perfect circular growth. You can't apply these principles to the space in which Life operates.

Although a lot of them try quite hard to end up circular. I wonder why that is?

[doccolinni]

That's why no B*/S* rule can exhibit perfect circular growth.

Well "circle", if you define it as "set of cells equally distant from a certain (centre) cell" and use the L-infinity form of distance then "circle" in the Life universe is actually a square whose sides are diagonal.

[calcyman]

... and use the L-infinity form of distance then "circle" in the Life universe is actually a square whose sides are diagonal. ...

In the L-infinity metric, the set of equidistant points form a square whose sides are orthogonal (rectilinear). Manhattan distance has a diagonal square as a 'circle'.

Although a lot of them try quite hare to end up circular. I wonder why that is?

Probably for the same reason that certain lattice gases (e.g. FHP, PI-LGA) are isotropic at the macroscopic level.

[doccolinni]

... and use the L-infinity form of distance then "circle" in the Life universe is actually a square whose sides are diagonal. ...

In the L-infinity metric, the set of equidistant points form a square whose sides are orthogonal (rectilinear). Manhattan distance has a diagonal square as a 'circle'.

Oh yeah, that's what I meant. But why would we use L-infinity metric instead of Manhattan metric in the Life universe?

[Phantom Hoover]

Because Manhattan is not appropriate for Life: the distance between a cell and a cell diagonally adjacent to it should be 1, not 2.

[calcyman]

But why would we use L-infinity metric instead of Manhattan metric in the Life universe?

Because Life uses the Moore neighbourhood (surrounding eight cells), rather than the von Neumann neighbourhood (surrounding four cells).


Just to make sure everyone knows what these different metrics look like, here are the cells within 5 units of the centre in each:



Metrics are Manhattan, L-infinity, and Euclidean, respectively.

[doccolinni]

But why would we use L-infinity metric instead of Manhattan metric in the Life universe?

Because Life uses the Moore neighbourhood (surrounding eight cells), rather than the von Neumann neighbourhood (surrounding four cells).

Ah, yes, that makes sense.

[doccolinni]

If cells were hexagonal, however, then the metric would be more Euclidean-like. Because when the cells are squares the Euclidean distance between the centre of a cell and centres of the cells in its Moore neighbourhood is not constant (it's 1 for the cells which are also in the von Neumann neighbourhood and √2 for the other four cells), but for hexagonal cells, if we were to analogously define Moore neighbourhood (which would in this case be equivalent to the analogous definition of von Neumann neighbourhood), the Euclidean distance between the centre of a cell and centres of the cells in its Moore neighbourhood would be constant. Indeed, it still wouldn't be a perfectly Euclidean metric but it would definitely be much closer to it than L-infinity.

[H. V. McIntosh]

It seems to be an everlasting discussion, how to measure the speed of light. Insisting on metric topology, isn't there a whole sequence of l<sup>p's matching the L<sup>p's, the difference being that the former is based on integers, the latter on real numbers? Insisting on literalcy, "speed of light" belongs to Minkowski spaces, related to experiments of Gallileo, Michaelson, Morely, and so on. Dynamical systems theory and cellular automata practicioners picked up the name rather than inventing their own in part because the term was familiar and in part for the effort that would have been required to invent a new name and then to have to explain it.

Then, physicists like to distinguish "speed" and "velocity," the latter being more appropriate when a direction is involved and/or there is anisotropy.

It is also possible to base a protracted discussion on whether or not the velocity of light can be exceeded.

[doccolinni]

It is also possible to base a protracted discussion on whether or not the velocity of light can be exceeded.

Well in the Life universe I'd say definitely no, unless the person running the simulation interferes.

[137ben]

It makes the most sense to use the L-infinity metric because we end up with a simpler definition of the speed of light. Defining the maximum speed of a signal in terms of the Manhattan metric IS possible, it would just require taking direction into account (e.g., the maxiumum speed diagonally under the Manhattan metric is 2 cells per generation, while orthogonally it is 1 cell per generation.)
The L-infinity metric, however, gives the same max signal speed regardless of direction.

While it seems strange that the maximum speed for spaceships uses a different metric, it should be noted that that limitation is specific to B3/S23, while under the L-infinity metric the speed of "light" is the same in ALL life-like cellular automatons.