I think Poology is a super interesting system, and not that it matters - but its also definitely the coolest name of any of the aiming systems!
I'm one who has played with it (I have original rev of your book, 2014) and while it's fascinating to see how you have mapped out a relatively simple pool/diamond math system for fractional aiming, I'm one who still finds it overly complex. Obviously the math itself is 1st grade level arithmetic, but what I found hard to keep straight was the differences in how OB position values are used among the 3 different A/B/C table zones. This is also not overly complicated per se - just requires memorization and lots of practice.
Overall, what I found is that both basic CTE and TOI's style of using CTC/CTE references & inside tip fractions just sort of makes your math superfluous. In no way do I mean any disrespect to Poology!
Huge caveat, what I'm about to say here is certainly NOT endorsed by the official CTE heads, and I really have no idea what CJ thinks of my interpretation of his TOI aiming concepts, but I have spent serious time learning & testing these systems, including all vids & books. I’ve benefited greatly from them all. Basically, here's my thinking for how simple it all can get.
As I've mentioned in previous discussions, I think all these fractional, CTE, TOI, and shaft aiming systems are very closely related and actually all boil down to essentially different ways to visualize 1/8b fractions.
It's very easy & quick for anyone to recognize & classify all pool shots into a small number of "categories"; thick, thin, very thin, razor thin. Much to what I assume will be the chagrin of the vocal CTE guys, Hal Houle himself said this in his infamous original 3-angle
document. Directly quoting Mr. Houle, speaking of CTE's primary 3 aiming points (15, 30, 45 deg, aka 1/4b fractions): "Your only requirement is to recognize whether your shot is a 15, 30, or 45° angle. Recognizing those 3 angles can be accomplished in an instant by aiming the edge of the cue ball to one of the spots on the object ball. It will be obvious which object ball spot is correct.
This is literally the very definition of so called "feel". Its super bizarre that for some reason most CTE guys (at least the ones around here) are so violently allergic to this concept, when the founder himself described a subjective assessment process, albeit a very easy one. This is a feature, not a bug!
I completely concur with Hal's statement, and as we know CTE really actually has 4 primary aimpoints, Thick = A (15°), Thin = B (30°), Very thin = C (45°), and the Razor Thin (1/8b!?) cut ref for 60+° angles (For some reason they don't name it "D", never understood that…) Over time, the concept of inside & outside pivots was introduced, so essentially they have 4 primary aiming refs, each with 2 modifiers, meaning 4x2 = 8 effective aimpoints or angles are actually used. The mapping to 1/8b fractions is rather obvious, when seen from this perspective, yes? Again, all of this is decided subjectively by the shooter, and most of us understand that all this happens via feel & progressive experience. [insert violent apoplectic response & attack from Spiderboy or Cookieman here...]
From a mental processing perspective, in the CTE system you essentially have what may be considered in mathematical terms as a 4x2 matrix representing a subjective visual shot selection decision tree, ie a narrowing down of options. First, you decide whether the shot is an A, B, C or 60°. As Hal said - this actually obvious and a great feature of the system. Then you decide whether it needs thinned or thickened by using inside or outside pivots. No diamond or angle math required, and a very simple 2 tier decision matrix; 1) choose among 4 obvious choices, 2) choose inside vs outside, done. All based on one's perception of how steep the cut angle is. I think this is why CTE guys find CTE to be far more efficient than Poology, they never really think of angles, fractions, or do math
In my interpretation of CJ's TOI aiming system, it's actually a very similar mental process to CTE but is essentially a 2x3 matrix that uses only 2 super obvious primary refs which only use true centers & edges of the balls (no fractions, no mid ball aimpoints), and can reuse the same 3 secondary refs which is our own cue tip totally under our control. This reduces choices and is much simpler & more efficient, among other things. Really no selection needed, if needed, one can simply step through all 3 tip fractions and pick one that looks best, it will be obvious and one calibrates to it since its our own cue tip, always in our consistent near field vision.
In this TOI system, first you decide whether the shot is CTC (thick) or CTE (thin) - again, super, super obvious, much more so than CTE's first aimpoint selection process. Then you visualize & decide roughly how much of an inside tip fraction you need to make the shot; 1/4, 1/2, or full tip. Done.
All of this is done during PSR in less than 1-3 seconds. It's worth pointing out that the vast majority of shots we shoot on the table including all banks are dealing with angles < 40°. Of course we have no hard data on the true distribution of angles that all pool players see on average, but I'd hazard a guess that for anyone except total noobs, for most of us with any modicum of CB control, probably 80-90% of all our shots are < 40°, which narrows down the mental shotline selection process process considerably.
I know you Brian know all this stuff and probably most others will think we're nutjob nerds with a math fetish, but below I've included a table that roughly illustrates the points I've tried to make above.
One of the reasons why the TOI system gets simpler, is that we recognize that basic CTC covers 0-3.5° cuts and are so obvious that any child can see & shoot them purely intuitively. Also extreme cuts > 60° are rare & also pretty obvious and effectively such a thin slice that we just use ETE and a bunch of feel to make them (I completely agree with Brian’s comments on those issues above). That really narrows down our 1/8b system to 6 angles, vs 8. You'll further notice that I've also omitted the 5/8 fraction from the table. This is because I've personally found this angle between 3/4 and 1/2 ball to be almost indiscernible due to 3D curvature distortion from most angles & distances, and thus can also be ignored (ie a very slightly thickened 1/2b CTE shot does it).
There is a lot more that could be said about how to see & implement TOI's tip fractions at the CB, or vanilla 1/8b fractions. I believe this is essentially also a form of "parallel shaft aiming". In the table, I use the terms .25, .50 and 1 full inside tip fractions, as seen by looking at the center or edge of one's tip (1/2 tip), or 1/2 way between the two (1/4 tip) - set against either the center or edge of the CB/OB visual, when looking at CTC or CTE alignment. Essentially these are micro pivots, but since they have distinct shaft visuals on both CB & OB - I think they are much more effective than thinking about pivoting, which I think opens the door to all sort of messy & unnecessary stuff with stance & stroke. I'm definitely a proponent of fully visualizing the shot in PSR and then coming down directly onto it - this is another area where I completely depart from mainstream CTE. The TOI tip offsets can be thought of as just a few minor "wedge" offsets from basic CTC/CTE alignments - in practice these offsets are tiny shifts away from CTC/CTE in stance & alignment. The beauty is that they are the same 3-4 basic cueing positions for CTC and CTE, and in practice one is always just hovering within 1/2 a tip of inside for 80% of most shots. I think this is really hard to describe, and I'm butchering it, and I'm certainly not any blessed, certified TOI master, this is just how I've digested and integrated CJ's concepts into my own game.
The point of this post is not to bash any of these aiming systems, but rather to better explain a point I've made in several of these chats - which is that all these systems can essentially be thought of as derivatives of 1/8b fractions, and of course our visual-proprioceptive systems fine tune the rest, including all the standard micro adjustments required for speed & spin.
I think the main topic that Brian introduced is really "how much mental processing" is required for Poology's approach to fractional aiming? I suggest that over time, a student of the game using any one of Poology, CTE, TOI, Shaft Aiming or vanilla fractions - will develop a natural, very efficient method of classifying & down selecting all the possible shot angles into a much smaller pool of just a few basic choices, then our natural cognitive systems make them work. Over time this all becomes very intuitive, pressed down into our subconscious like breathing & blinking, and is essentially just short cutting the path to where some folks get to with HAMB. The advantage of course over 100% feel - is that this creates a fairly objective, consistent framework to lean into under external & internal adverse physical & mental conditions.
By the way, in computer science & control system theory, highly efficient, tiered, progressive down-selection algorithms are used to achieve first level solutions, such as coarse/initial alignment using known references, then work from there to achieve fined tuned accuracy. The simplest one that is widely used is called binary search, and it's essentially what I've described above, a progressive bifurcation of value ranges until the answer is reached - its widely used because its super efficient & super easy to implement. I think this logic applies to our aiming process in pool.
I'm open to the idea that all this might just be pure gibberish from a total nutjob. Flame away!
Peace & love