Previously, I had (mistakenly) decided that Sq' could NOT be described by cos(t)+sin(t)=(radius). BUT: If you plot Ci, As, and Sq' on a (t,r) periodic/ harmonic plane, you will note f(t)=r is always 1 for Ci, which doesn't coincide with Sq' and its functional graph cos(t)+sin(t), which has a variable radius.
I'm going to quadruple check this (again), but the (t,r) periodic/harmonic functional graph of Sq' gives a wave-function whose values coincide with Sq' and its radius with respect to an origin when represented as a 1-manifold. So, for now, I'm sticking with Sq'=cos(/star)+sin(/star) as the general formula for the square rotated by pi/4.
*I'm also working on formalizing my previous papers, since they feel too intuitive/conversational to me (as I constantly note in other posts).
Update 8/29/18 I'm reposting the original errata from 3/18/17, since the above doesn't hold and the original does hold. So, for all previous papers, Sq' does NOT equal cos(\star)+sin(\star). Original "major errata alert 2" is posted below:
I don't know why I've been expressing the square rotated by (pi/4) as (cos(t)+sin(t)) this whole time. in all of these papers. Total bonehead move.
One (actual, existent, and extraordinarily non-circular...) expression for the square in parametric form would be:
(1+tan(t)) on t=((-pi/4) to (pi/4))
(cot(t), 1) on t=((pi/4) to (3pi/4))
(-1, -tan(t)) on t=(3pi/4 to5pi/4)
(-cot(t), -1) on t=(5pi/4 to 7pi/4).
Then rotate those coordinates by pi/4 to obtain Sq', which must then be scaled down to intersect at e_{i}. I haven't found the proper scale yet to attain Sq' as described in the papers. That's part of my "Winter Break project".
(note changing the scale of the square and it's rotation doesn't change the overall results or conjectures. Sq' as described in the papers still describes a surface with Euclidean flatness (except at e_{i} as described in paper 5).
Currently, I'm looking for a better equation to denote that in terms of being both a geometric surface AND a manifold. Something easier to manipulate (in the same way cos^2(t)+sin^2(t) is easier to manipulate & work with)..
Anyway, I had been thinking about this not seeming right since after the Second paper, but before the Second talk I gave.
Every other aspect of the papers I've written (in terms of building Geometric Lenses to study surfaces and conjectures about solids implied by the quermassintegral conjecture I made in paper 2 or 3) should hold, but this was obviously a HUGE "equational representation" error that had to be addressed.
I will say that I'm not 100% sure if paper 4 still fits into the whole "Geometric Lens" line of study given my mistake, but (in my opinion) it still has some neat results in terms periodic equations of the type discussed in that paper.
I obviously feel really dumb about that right now, and also retroactively....
It would have been nice to have someone point that error out to me. I guess that says a lot about a few things (and I'm not NOT pointing a finger or two at myself when I make that statement)...
I'm just glad I (FINALLY!!) caught that error.
Anyway....OOOOOPS!!!
Aside/Non-related realization: Whether I've noted it or not, I've mostly been approaching these manifolds from a Reimannian point of view. It occurs to me that these manifolds might also be expressible as Kahler manfiolds, since they're all Reimannian, they're all expressible as complex manifolds, and (these are the parts I'm less sure about) they're all (PROBABLY) symplectic and they're POSSIBLY all "projective varieties" (in quotes because I'm not 100% sure if that's the most accepted terminology). The hyperbolic octahedron is the only manifold I'm not 100% sure about, but from what I've seen/worked on, it should be. It's something I'm going to look into more. Symplectic manifolds/geometry is still extremely new to me, so I'm not ready to go back and say "these are all definitely Kahler manifolds" yet, but I suspect they are.
edit (2/1/18): I'd like to add that one of the on-going questions behind this research is "Can this methodology be used to create smooth corners in geometric shapes and/or manifolds". It occurs to me that I never really explicitly stated this, even though it's been an underlying question since the 2nd paper I posted. In fact, this question was at the heart of the 5th math paper I posted on here: "If a circle can encompass a manifold with sharp corners and intersects said manifold at it's corners, can those corners be interchanged such that those corners can be considered smooth and differentiable?"
This is to say, at can we express As(\star) as cos^3(\star)+sin^3(\star)-cos^3(0)-sin^3(0)+cos^2(0)+sin^2(0) (using 0 as an example), so that the smoother curvature of Ci(\star) at \theta={0,pi/2,pi,3pi/2} can replace the sharper values of As at those points (since those are points of intersection).
edit (2/11/18): One BIG mistake I noticed in going back through papers & lecture notes is the way I defined "reparametrization". While it's correct to say that all properties hold under reparametrization AND changes of variable sets, I was intermixing the definitions (and representations) of reparametrization and changes of variable sets. (eg saying cos(x)+sin(y)---->cos(t)+sin(t) was a reparametrization). OOPS! In a separate errata for an in
edit(8/6/19): So, while working on a new gL-related paper, I noticed another HUGE error on my part as I read through my older work. The error? I implied the Laplacian/2nd derivative is exactly the (unit) normal vector. Obviously, this is not true. This is another case where I messed up terminology/methodology (and ran with it for WAY too long), but the heart of the conjectures I've made are true. It's true that the vectors pointwise-orthogonal to the surface (or "surface" in the 1-manifold case) can be compared in the manner implied by my papers. But, from what I can tell, the use of Laplacian/2nd derivative is absolutely and completely wrong. (since it's not always true that f''(\star)=T'/||T'||. In fact, it's almost NEVER true). I'm really hoping to eventually grow out of this tendency to make massive nomenclature/computational mistakes while making otherwise true (and hopefully interesting) conjectures.
update 8/8/19: NEVERMIND! There are actually points that require the normal to NOT be unit, as I had originally stated. There are the points on As(\star) that are close to the points associated with the elementary vectors. The length of these particular vectors lie in the inequality sqrt(2)>|As''(\star)-Ci(\star')|>1. Thus, the Laplacian/2nd deriv actually DOES quality as the NON-unit normal, which is required here here. So yeah...nevermind the above edit from 8/6. Maybe some day I'll drop the annoying habit of questioning and doubting things I actually got right the first time around.