The model presented below, in the following post, defines a geometric function. In the .gif animation, the small numbers in the rectangular box change as a function of the small numbers at the top of the model. It should be possible to express the function as an equality, α = f(E).
The expression is deceptively complex. This problem has been posted on several other math and science discussion sites and has received thousands of views and hundreds of comments. To date, no one, not even myself, has been able to come up with an expression for the function. Again, it is deceptively complex.
The origins of the model date back to ten years ago. We were trying to find a (trigonometric) function that would return the slope of a tangent line on a small circle on a sphere. We were a little taken aback when we discovered that it appears that there is currently no way to do this. It seems that very little (nothing) has been done with regard to spherical geometry since Napier. Modern techniques seem to treat the sphere as a surface or manifold and tend to ignore the gross placement of the object in 3space.
The model presented has nothing to do with a sphere (a 2-dimensional surface) and instead describes the 3-dimensional thing that the sphere occupies (not a ball, more like all points adjacent to the sphere center.) This is a new equality, and there’s much more going on than it looks like at first glance.
Discussion of the new equality among mathematicians and physicists has become a little contentious, the consensus being that it is too simplistic and straightforward of a problem for anyone to want to waste any time on it. I heartily disagree.
The new equality accomplishes some rather sophisticated calculus by performing certain functions in three dimensions, simultaneously. Although each operation is straightforward, the way that they are combined with one another is complicated.
The history of the new equality isn’t all that interesting. We were not looking for this; we were trying to find something else. Recently, I was going over our notes from ten years ago and I found a couple of pages that were some sort of cryptic message that I still don’t quite understand. I think that what I was trying to say is that my attempts to compose a formula for α = f(E) led to the equation that cot(α) = 0, or some such nonsense. At least it makes no sense to me now. And it is incorrect, according to Mathematica.
In any event, several months ago I had my brother prepare some animations to accompany a written description of the function. They are:https://www.youtube.com/watch?v=ho8XCHI ... e=youtu.behttps://www.youtube.com/watch?v=xwjIeHC3Nb0https://www.youtube.com/watch?v=6OnSZki ... e=youtu.be
I posed the problem on Wolfram Community and a member there, Hans Milton, prepared a mathematical model in Mathematica from the animations. Then, after some persuasion, a member of another forum, strange at http://www.scienceforums.net
, actually graphed the function from the mathematical model.
Although we have a partial understanding of what the function does, the model is only a partial model. It must have another degree of freedom added to it in order to complete it. What are described as the ordinal and cardinal axes are shown at 45 degrees to one another in the model. These axes can be any angle to one another. As the angle between the cardinal and ordinal axes changes, the curve in the graph of the function will also change. The curve should vary from (possibly) a sine curve at 0 degrees to (possibly) a right angle at 90 degrees.
Composing this function has been called an “interesting math problem.” The following post will define the function mathematically. Understand that the function has been partially modeled in a math modeling software package, and it has been partially graphed. Much of the math has already been accomplished. Help is needed in completing the project.
There will probably be several more conversations about this new function in order to discuss the implications for mathematics and physics and philosophy and so forth. Hopefully the moderators here will allow this thread to stand alone, and will not merge it into any of the other discussions. Experience has proven that insisting on concatenating these separate discussions only leads to confusion and lots of questions and answers about topics that are unrelated to solving the remaining issues with this function.
I would ask the moderation team to leave this thread as a stand-alone “interesting math problem.”