Her eyelids grew heavy, and a sense of peace and reassurance suffused her thoughts. There was no need for words now. They were sharing light, and the light carried Nino’s promise to protect what she would become.
APPENDIX 1
UNITS AND MEASUREMENTS
APPENDIX 2
LIGHT AND COLORS
The names of colors are translated so that the progression from “red” to “violet” implies shorter wavelengths. In the Orthogonal universe this progression is accompanied by a decrease in the light’s frequency in time. In our own universe the opposite holds: shorter wavelengths correspond to higher frequencies.
The smallest possible wavelength of light, λmin, is about 231 piccolo-scants; this is for light with an infinite velocity, at the “ultraviolet limit”. The highest possible time frequency of light, νmax, is about 49 generoso-cycles per pause; this is for stationary light, at the “infrared limit”.
All the colors of light arise from the same pattern of wavefronts, rotated into different orientations in four-space.
In the diagram above, AB is the separation between the wavefronts in four-space, which is fixed regardless of the light’s color. AD is the light’s wavelength (the distance between wavefronts at a given moment) and BE is the light’s period (the time between wavefronts at a fixed location).
The right triangles ACB and ABD are similar triangles, because the angles at A are the same. It follows that AC/AB = AB/AD, and:
AC = (AB)2/AD
Also, the right triangles ACB and EAB are similar, because the angles at B are the same. It follows that BC/AB = AB/BE, and:
BC = (AB)2/BE
Pythagoras’s Theorem, applied to the right triangle ACB, gives us:
(AC)2 + (BC)2 = (AB)2
Combining these three results yields:
(AB)4/(AD)2 + (AB)4/(BE)2 = (AB)2
If we divide through by (AB)4 we have:
1/(AD)2 + 1/(BE)2 = 1/(AB)2
Since AD is the light’s wavelength, 1/AD is its spatial frequency, κ, the number of waves in a unit distance. Since BE is the light’s period, 1/BE is its time frequency, ν, the number of cycles in a unit time. And since AB is the fixed separation between wavefronts, 1/AB is the maximum frequency of light, νmax, the frequency we get in the infrared limit when the period is AB.
So what we’ve established is that the sum of the squares of the light’s frequencies in space and in time is a constant:
κ2 + ν2 = νmax2
This result assumes that we’re measuring time and space in identical units. But in the table above we’re using traditional units that pre-date Yalda’s rotational physics. The data Yalda gathered on Mount Peerless showed that if we treat intervals of time as being equivalent to the distance blue light would travel in that time, the relationship between the spatial and time frequencies takes the simple form derived above. So the appropriate conversion factor from traditional units to “geometrical units” is the speed of blue light, vblue, and we have:
(vblue × κ)2 + ν2 = νmax2
The values in the table are expressed in a variety of units that have been chosen so that the figures all have just two or three digits. When we include a factor to harmonise the units, the relationship becomes:
(78/144 × κ)2 + ν2 = νmax2
Now, the velocity, v, of light of a particular color is simply the ratio between the distance the light travels and the time in which it does so. If we take the pulses of light in our first diagram, they travel a distance AC in a time BC, giving v = AC/BC. If we then use the relationships we’ve found between AC and AB and the spatial frequency κ, and between BC and BE and the time frequency ν, we have:
v = κ/ν
Again, we can only use this formula with traditional units after applying the appropriate conversion factor:
v = (vblue × κ)/ν
which, if we’re taking frequencies from the table above, becomes:
v = (78/144 × κ)/ν
The velocity we’ve been describing so far is a dimensionless quantity, related to the slope of a line tracing out the history of the light pulse on a space-time diagram. (The way we draw our diagrams, with the time axis vertical and the space axis horizontal, it’s actually the inverse of the slope.) Multiplying the dimensionless velocity by a further factor of 78, the speed of blue light in severances per pause, gives us the values in traditional units that appear in the table.
AFTERWORD
Much of what we know about the physics of our universe can be understood in terms of the fundamental symmetries of space-time. If you imagine any experiment that can be fully contained on a floating platform out in space, then orienting the platform in different directions or setting it in motion with different velocities will have no bearing on the outcome of the experiment. The particular directions in space and in time with which the platform is aligned make no difference.
However, in our universe the laws of physics distinguish very clearly between directions in space and directions in time. While you’re free to travel through space precisely due north if you wish, as you do so you will also be moving forward in time (as measured, say, by GMT). Expecting to be able to depart from Accra at 1:00:00.000 GMT and arrive at Greenwich to see the clocks showing exactly the same time—because you’d arranged to move “purely northwards” without any of that annoying progress through other people’s idea of time—is not just a tad optimistic, it’s physically impossible. “North” is a “space-like” direction (whatever else might be merely conventional about it), while “the future” is a “time-like” direction (however much it might differ from person to person traveling at relativistic speeds). No amount of relative motion can transform space-like into time-like or vice versa.