FAQ
It is currently Tue May 23, 2017 6:58 am


Author Message
steve upson
Post  Post subject: Expressing a New Equality  |  Posted: Sun May 29, 2016 2:33 pm
User avatar

Joined: Tue May 24, 2016 12:25 am
Posts: 14

Offline
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.be

https://www.youtube.com/watch?v=xwjIeHC3Nb0

https://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.”


Top
steve upson
Post  Post subject: Re: Expressing a New Equality  |  Posted: Sun May 29, 2016 2:36 pm
User avatar

Joined: Tue May 24, 2016 12:25 am
Posts: 14

Offline
Gratitude and credit go to Hans Milton for creating this model in Mathematica.

Image



The first angle that we are concerned with is the elevation angle E. It is the angle that is formed between the equatorial plane (beige) and the elevation plane (yellow).

The second angle is more complicated. It is formed between a plane of longitude (green) and a "tangent" plane (blue) which lies in a conical orbit. The construction of the longitude plane is not too complicated, but its orientation is very specific.

The orientation of the longitude plane is such that it intersects a point on a small circle which is constructed on the surface of the sphere at a 45 degree angle. This small circle can be defined as a circle at the 45 degree latitude which has been tilted 45 degrees such that it intersects both the pole and the equator.

The "tangent" plane lies along the surface of a cone formed by the sphere center and the 45 degree small circle. As the elevation angle E is varied, the position of this tangent plane changes such that it remains coincident with the intersecting point on the circumference of the small circle.

If we call the angle that is made between the longitude plane and the tangent plane the angle α, then the object is defined by the function which expresses the relationship between the two angles, E and α. This function is the one that is troublesome, for some reason or other. How is this function expressed?

Thanks to a member of another forum, strange, for graphing the results from the model:


Image


blue: α = f(E)
red: sine



It should be mentioned that, although a sphere is used for construction of the animation that is shown here, there actually is no sphere involved in the function itself. In other words, there is no two-dimensional surface involving spherical excess, or anything like that. The sphere is simply used as an aid in visualizing how the object is constructed.

Also, the model isn’t really complete, as it should have another degree of freedom added to it. The following image shows how the model should work when it is complete.



Image



The illustration at a is the example shown in the model, where the relationship between the ordinal and cardinal directions is 45 degrees. Example b shows an angle less than 45 while c shows an angle greater than 45. Example d is provided to show that the model is not based on any particular size. In other words, there is no metric.

It’s unfortunate that the model includes the variables E for elevation and alpha for the tangent angle, but we were dealing with spherical geometry at the time those variables were chosen and we weren’t even considering that those variables are reserved for specific usage in physics.


Top
steve upson
Post  Post subject: Re: Expressing a New Equality  |  Posted: Sun May 29, 2016 3:25 pm
User avatar

Joined: Tue May 24, 2016 12:25 am
Posts: 14

Offline
The Mathematica model is contained in a .cdf file. I'm not sure of how to go about posting the file here. If anyone is interested in viewing the model (rather than a .gif animation) then let me know how to publish it and I will put it in this post. The actual model allows you to turn off certain objects such that the image is much less cluttered. Also, it lets you move the slider manually, again, for more clarity.


Top
Display posts from previous:  Sort by  
Print view

Who is online
Users browsing this forum: No registered users and 0 guests
You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
Jump to:   
cron

Delete all board cookies | The team | All times are UTC


This free forum is proudly hosted by ProphpBB | phpBB software | Report Abuse | Privacy