Well, it has turned out to be an interesting week. Somehow I found enough time to do some meta-mathematics of my own.
Or to stumble across some by someone else. I was sorting through a pile of books that had been sitting on the floor of my library for too long. One of the books I found was called “Cognition” by Dodd & White. In one section it talks about why some syllogisms seem harder than others for people to get right. I haven’t taken notes on this yet, but I’m looking forward to it. It should serve as spice when I get to talking about syllogisms. Real Soon Now. Probably this weekend.
In color, I spent too much time trying to do something that was impossible. At least, I’m pretty sure the fundamental geometries of two particular descriptions of color are just too different from each other. I have shown you how to get and to lay out tints, tones, and shades from HSB. The resulting layout is rectangular or square. In particular, it has a line which is black.
On the other hand, the original representation of tint-tone-shade uses a triangle whose three vertices are labeled color, white, and black. (I think I have never displayed one of those, but here’s something. Not exactly a triangle, but use your imagination.) In particular, let me emphasize that it has only a vertex — not a line — which is black.
I do not believe these can be reconciled. To be specific, I do not believe that HSB can be reconciled with color-white-black specifications of color. Incidentally, I didn’t set out to reconcile them — I was just trying to get unique color-white-black specifications for Birren’s color triangles.
If it’s any consolation, let me quote Fairchild:
“It is not possible to say that one is better than the other [speaking of the Munsell, rather than HSB, and NCS color order systems — but NCS is a specification in terms of white-black-color], it can only be stated that the two are different. This was recently reaffirmed in the report of CIE TC1-31…, which was requested by ISO to recommend a single color order system as an international standard along with techniques to convert from one to another. This international committee of experts concluded that such a task is impossible.”
In color, I also learned something new… and I will probably be adding a comment to the post about from tristimulus to spectrum.
Here it is. We know how to take an illumination spectrum and a reflectance spectrum, and compute tristimulus values. What happens if we divide the illumination spectrum by two?
That is, what happens if we reduce the absolute level of illumination?
Answer: we will get exactly the same tristimulus values (because Y is scaled by the illuminant).
This means that unless we have an absolute measure of the illumination level — or at least a relative measure of “full illumination”, and I was assuming we did — then there are still an infinite number of reflectance spectra (hence illumination spectra) which will reproduce the same tristimulus values.
We cannot recover the absolute intensity from XYZ alone.
Meanwhile, my kid (me doing whatever I want) has been playing with the simplest models of projectile motion (constant gravity and no air resistance), and it has been fascinating to look back at something I understand.
It emphasizes, once again, that it is far, far easier for me to supply my own meta-mathematical commentary for mathematics I understand than for mathematics I’m trying to figure out.
I set out to do as many calculations as I could from first principles — more precisely, perhaps second principles. That is, I didn’t start from the differential equations (first principles) over and over again — but my initial approach was simply to use the general solutions (second principles? I’m being foolishly pedantic.), i.e. four equations: horizontal and vertical position, and horizontal and vertical velocity, as functions of time.
I’ve already discovered — been reminded of — three significant caveats… that is, it is extremely worthwhile to use additional information.
For one thing, it is extremely worthwhile to use magnitude and direction for the initial velocity — much as I hate to use magnitude and direction for vectors! You see, the angle of launch completely dictates the form, the shape, of the trajectory; while the initial speed — the magnitude of the initial velocity — dictates the scale of the trajectory. That correspondence can be very useful; more importantly, the independence of the form and the scale can be very useful.
For another thing, conservation of energy makes it trivial to compute the maximum altitude as a function of the initial velocity vector (strictly, of the initial vertical component of velocity).
Third, once we have the maximum altitude, it is almost trivial to get the time at which it occurs… and by symmetry, twice that time is the time to impact for equal initial and final altitudes… and then it is trivial to get the range. And having the range, even for a problem which does not have equal initial and final altitudes, can be a useful sanity check on the answer to the given, albeit different, problem.
To put that another way: the sequence of calculations
initial vertical speed -> maximum altitude -> time to maximum altitude -> time to impact (at final altitude = initial altitude) -> distance to impact (i.e. range)
is easy and informative. (It also might answer the given question, but that’s another issue.)
Fourth, in the modern world the sensible way to check your answer is to graph the entire solution… and if your solution was algebraic, the sensible way to check it is to graph it for a selection of numerical values of the parameters.
Part of what I’m saying there is that even though a question may ask for one specific characteristic of some trajectory, a sensible real world solution would end up with a complete description, in order to draw at least one picture, if not more.