Modular is a concept with wide applicability in a variety of domains. And was lucky that weirdly enough after Gauss, development of modular with mathematics apparently lagged. Which I know because was able to introduce first true modular algebra, which then not surprisingly, revolutionized much.
However modular algebra was not my first step into modular with mathematics, as was kind of a relief to notice was key to an earlier puzzling to me result, as SO simple. Where considered packing of spheres in a modular way. Then it makes more sense that I could easily prove that distorting that one piece from hexagonal close packing would lead to a lower density. Since full space is filled with same modules, where any distortion leads to lower density, had an easy proof of optimum packing.
Wrote up a two page paper. Having that rejected by an editor for the Proceedings of the AMS as too simple is just kind of telling to me now.
Would go on to develop a full modular algebra where like to give as easy for context:
x+y+vz = 0(mod x+y+vz)
Which is the simplest form I found with some research, and I call, a tautological space. Have created terminology over more advanced forms.
And have noted that remarkably, can use to study algebraic expressions where can give things which are algebraic manipulations, done by the modular algebra. As the modular algebra is doing all the work, is perfect and can reveal things humans never found before.
Who knew? How could anyone? Watching modular algebra do algebra? Is surreal.
My best example is what I call the BQD Iterator lots. A very simple relation which I've used quite a bit, which the modular algebra gave me, with my method for generally reducing binary quadratic Diophantine equations. My method remarkably is better than that from Gauss.
BUT can now also note that Gauss was am sure doing something I don't need to do any more--complex algebraic manipulations.
Letting the math do the work is more effective and complete. Is faster, and is of course easier.
The math easily checks infinite possibilities in a modular way. The math is complete intelligence.
Those are the kinds of things that tend to show you where human beings will go, in time.
Why do we not notice certain things? Probably is about how we are built.
Of course, how do we know what we do not know?
With the math performing better in a way that humans have tried with our intelligence, can sense are watching an intelligence, but is an infinite one which is perfect, makes no mistakes, and gets to the best answer.
Which then lets us look at what we humans did not find on our own, and wonder.
Could go on and on about it. And have deferred on what I call the social problem as to acceptance until 2028, which is not that big of a deal for me. Figured out my modular approach to packing sphere back in 1996. And had finally the evidence that the method I found around what I decided to call tautological spaces, was doing algebraic manipulations back in 2008.
And that was September 2008, with: Quadratic Diophantine Result
Figure that is decent matter-of-fact naming for that post. At THAT time did not say to myself, have a true modular algebra which can do algebraic manipulations. Didn't even realize had actually handled three variables as only focused on two. That is wild, eh? Recently talked the three variable.
So much did not know then, where would learn in time. At the time, was more like, wow, what's this thing? That's SOMETHING when you first realize that the mod can just go away. Where thank God that was years ago, and have settled down much since then.
Takes years to properly process discovering such things have learned with myself, as slowly you adjust, to what changed for you so quickly. So much a sense of things, of what I thought I knew got shattered, so quickly.
There was often elation as well, but learned to be suspicious of it. Like to say that math and emotion do not mix well. But that effort is more to maintain an even keel. As can end up with massive high's and intense low's which to me? Is weird.
Took YEARS as slowly rebuilt a firm sense of reality. And noticed was better than before, but so much work to get there. The world seems more crisp to me now, often. And appreciate so much more as well.
And web has helped me greatly and lucky was here evolving as I found these things. As web allowed me a more gradual sense of certain things, while being aware, yeah, people around the globe...what have you all been doing? Web analytics only tell me so much. Guess you would be the people reading this post though.
Where is strange yes, can have simple but powerful ideas, but one reason became fascinated with modular elsewhere. Like would stare at cargo ships loaded down with shipping containers when lived in San Francisco. And would ponder.
Noted was lucky for me that further development of modular seemed to have waited until now. Still there has been some resistance, as now over 18 years since I introduced what I call tautological spaces. With over 21 since my first paper with modular ideas and packing of spheres.
Regardless the ideas are flowing, which web metrics lets me know. So I guess something has changed for humanity allowing the possibility and simply live in the right time.
My view is the web has made the difference as notice, how are you reading these words?