Just finished typing & submitting (for a grade) the paper on which my NUMS'15 talk was based. This one was a ton of fun to write, and I'm looking forward to continuing this line of research (amongst others) for times to come. Enjoy, constructively critique, and as always: I own this and am more than happy to share the work so long as I'm properly cited.
As a post-script: yes, errata is forthcoming. And yes, my style is possibly too conversational, so I plan on implementing a more "sequential and consequential logical structure" to this and subsequent work. One thing that became clear to me in re-reading my own work was how much easier it would have been to read and digest if I had shown "theorem leads to lemma leads to..is a result of..etc.". It wasn't hard to see that everything comes down to the context of your initial definitions and the subsequent logical approaches are what allow certain formulas to be relevant or not. I think what I was aiming for here was something more akin to Zong's "The Cube.." in that I was trying to use this composite manifold (and the derived composite 2-mainifolds) as a model to study different elements of mathematics. It's interesting how working on a project like this really makes the whole puzzle start to come together.
ERRATA:
1) As discussed with Dr. Guo, -(e{i+1}-e{i}) doesn't always travel ccw. I realized I was confusing this with the directional derivative, which would give Sq' its ccw "motion". See any book or pdf on differential geometry for the definition of a directional derivative.
2) One thing I feel I failed to (at least explicitly) state was why this is important or relevant as a course of study. The over-all goal of my research is to create 3-manifolds of constant curvature (tesseract, and hyperbolic octahedron and sphere 3-manifold analogues) and play with the idea of using geometric lenses created by analogous intersection properties from these and seeing if the properties hold for higher dimensions, odd numbered dimensions, and their graphs and the heat kernels of those graphs. and other interesting (to me) phenomena.
3) I think the importance of the geometric lens approach isn't just the ability to apply linear sweeps and linear point-wise association of spaces, but also in analyzing densities and non-linear point-wise associations of the complementary spaces and seeing if deformation of those associations in the "mixed volumes" of the complementary spaces can be utilized in methods of creating mixed spaces. This would be the reason for introduction of quermassintegrals, as that is a method of analyzing the densities and associations in mixed volume spaces. One thing I realized in re-reads was that I just kind of wrote "quermassintegral" without really explaining why. So there's the "why".
4) page 12, the distance from ci to sq' is (in fact!) equal to itself. I meant to say that the distance from ci to sq' is not equal to the distance from as to sq'. (this error is located in the last few lines above the equations on pg 12)
5) As discussed w/ Dr. Guo (also during quasi-defense of this paper): WHAT ABOUT ELLIPTIC GEOMETRIES? I gotcha!
The spherical elliptic analogue is most obvious: the oval (2-manifold analogue is the oval sphere). For the aster, the elliptic form is simple the corners on the horizontal axis contracting in to (+/- 1/2,0) (the 2-mani analogue is the astroidal ellipsoid). The square is elliptically represented as an elliptic octahedron, which would be dual to the rectangle/rectangular prism as the octahedron is to the cube.
6) I noticed in a few spots I wrote "dimensional" instead of manifold. For example, on page 4 I write "1 dimensional surface" when I meant a surface in the 2nd dimension (ie; a 1-manifold).
7) Throughout: the interval [0,2pi] should be amended to [0,2pi) (open at 2pi, so as to not count it twice). Though one of the benefits of looking at intervals in terms of these three structures is that they can be expressed both on open and closed intervals (clopen sets)* without rendering the usefulness of this type of analysis irrelevant, we don't want to count values at 2pi twice!
8) One thing not explicitly stated in this paper (and probably subsequent papers and posted lecture notes) is the idea that all of these functions can be expressed in any variable set. For example, As(x,y)=x^(2/3)+y^(2/3) and As(r,theta)=rcos^(3)(theta)+rsin^(3)(theta) remain the same comparable distances and angles between orthonormal vectors of Ci(star) and As(star) regardless of variable sets. The same holds true in the complex plane. .
*The main advantage of this will be more apparent in a subsequent work, as yet unwritten as of 1/4/18. The basic idea being that your can swap , for say As(x,y) and Ci(x,y) and Sq'(x,y), the values at the points of intersection (e_{i} for all i in natural numbers). This holds in all variable sets covered so far (as well as complex space and possible hyperbolic space).