20081113—from beginning of logarithmic spiral theory 2007 The latest “Hunch” is just an essay. I have no mathematical prof for the following idea. It is just that, an idea. Previously it was theorized that a logarithmic spiral could be used to find an equation to the series of Prime numbers. This idea can be applied to all series. We found a parabola that described Prime numbers occurring every Pi radians. The pattern is not easily seen. But we can use a logarithmic spiral to describe that parabola. That is based on the fact that all parabolas can be described by a logarithmic spiral. So there is an increase in the spiral with each turn. But the question remains: Can the pattern of Prime number be described by two geometric figures that increase proportionally? In other words, does the parabola actually reveal the pattern? And can that parabola be described by a logarithmic spiral? It is well known that a logarithmic spiral can be shifted by a line. But building on to this idea we may be able to find a geometric figure that can be used to find a pattern in any series. The idea is: What would happen if we use a periodic function (such as a sine curve) and use it to shift a logarithmic spiral? That is the shift can alternate “bending” towards and away from the center of the spiral. Also note what is already known about periodic functions such as the sine curve. They can be manipulated or transformed to fit unknown curves. Such transformation has now just become that much powerful. The purpose of the periodic function would be to show how much the logarithmic spiral had been shifted from the center reference point. Just as a line can refract and bend a logarithmic spiral in a specified direction, a periodic or any function could be used to show the change the pattern takes from an ordinary logarithmic spiral equation. In other words if the pattern (Prime numbers) could not be described by a logarithmic spiral or parabola by them selves, the pattern may be described by a function that changes the shape of a logarithmic spiral. Challenges to this theory: First we have to describe how the periodic function changes or transforms the logarithmic spiral. For example a sine curve could start at the center of the logarithmic spiral and “cycle” (where the function occurs) at an angle perpendicular to the horizontal. There may also be a problem with the math being too complicated. That might be solved by the fact that usually the numbers of the series are known. That means a approximate logarithmic spiral can be “fitted” to the numbers. Perhaps over a small portion of this makeshift logarithmic spiral a pattern will occur that can describe the function that transforms the logarithmic spiral. And of course the whole theory behind the logarithmic spiral being able to solve series needs to be proved. But remember, the golden spiral, involutes, are types of logarithmic spirals and themselves are formed by series. Review: Prime Summation May the Creative Force be with You
------- Hopefully it is clear what is being attempted to be solved here. I will post updates to better explain and hopefully solve this problem. This is a good group project. If you have read this and want to work on a problem email: trurlthe_constructor@hotmail.com . Also more math can be found in the math_hunches section of Constructor’s Corner. |