Modeling
In the general sense building a model means to render a scaled down or enlarged replica of a structure. In this sense models are dilations proportional to the objects they model, and thus the modeling process relies on spatial thinking.
A model is analogous to the modeled object in appearance, but usually it doesn't posses all the functions of the original.
In this Module, I wanted to somehow bring together two separately represented patterns explored in Module 3: prime waves and factor-dots. What sort of a model could juxtapose the periodicity of prime waves with the deeper fractal pattern involved in composite numbers' prime factorization? And, could I use this model to gain new insights into primes' pattern?
Look for instance at Two's wave and its dots, how could both patterns be brought into a single structure?
A model is analogous to the modeled object in appearance, but usually it doesn't posses all the functions of the original.
In this Module, I wanted to somehow bring together two separately represented patterns explored in Module 3: prime waves and factor-dots. What sort of a model could juxtapose the periodicity of prime waves with the deeper fractal pattern involved in composite numbers' prime factorization? And, could I use this model to gain new insights into primes' pattern?
Look for instance at Two's wave and its dots, how could both patterns be brought into a single structure?
To begin to solve this problem I had to visualize the new construct. Since prime waves are two dimensional, the fractal property must add a dimension to the wave-structure making it 3D; with this as a basic requirement I took Two's wave and I tried visualizing spirals of different shapes. Some were circular and had periodic widenings and narrowings, others were flattened and constantly widening. I tried to picture how the triangular peaks of the fractal image would enhance with depth the wave's periodicity. This visualization was a purely mental exercise and proved to be more difficult than I expected. My thoughts often shifted from one shape to another before I could behold a structure's main characteristics over a long enough interval. This thought-exercise was also challenging because as my imaging improved lots of questions arose about the new construct's shape. I could imagine fairly well what the curve did at notable points of a given interval, but I was at odds about the curve's behavior between those points. This initial processes of imaging was heavily reliant of embodied thinking as the best way of seeing the spiral in question was to see it from the inside. I also noticed myself using hand gestures as I moved from one point to another in three dimensions. The embodied thinking that eluded me has finally showed itself! Still, the exact nature of the shape puzzled my imagination. Moving forward, I tried to use a 3D scatter plot to aid my visualization. I knew where many of the points were located, but the mathematics of the z-dimension proved to be way too complicated, and the individually calculated points in space were to few and far in between to reveal the overall design on the 3D plot. With this my work came to a halt. I could not entirely visualize, not even with placing myself into the spiral structure; I couldn't describe the z-dimension of the shape by way of a mathematical function, and didn't have enough known data points to "plot my way out" from this conundrum. Coincidentally however, this turned to be a problem perfectly suited for modeling (which made me wonder what would I have done in this module if I could have solved this problem by other means !?). So, I set out to build a model for the structure whose entire shape I couldn't quite yet conceive. In the following images I'll show you the process of building the model that I finally accepted as reasonable (many others I discarded).
Circles in the Y and Z dimensions help to maintain the image's depth. The circles' heights match the exponents of even numbers' factors of base two. For instance at x = 2 the circle's height is 1 (2 = 2^ 1 ), and at x = 16 (the greatest circle below) the circle's height is 4 (16 =2^4).
Next, I overlaid Two's wave in the X and Y dimensions, the scaffolding was at this point complete.
Finally, I stretched the new curve over the scaffolding.
See how g(x) combines the wave and dot properties. Notice that contrary to my initial guess g(x)'s graph is not a spiral at all because it doesn't turn around itself; modeling has paid off! While at some of the peaks g(x)'s z-coordinate is log2(x), most of the other points' z-dimension is conjectured.
What makes this a model; and how is the model building process different from the one used to create the prime waves shown in Module 3? The structure is a model because its shape as shown is somewhat speculative; known points are connected with unknown but plausibly shaped connective elements. In other words, this is a model because not all points on the graph are mathematically computed, rather they are stretched over a scaffolding of known elements with the hope that enough of the scaffolding existed to produce a coherent and aesthetic 3D structure. The "graph" was actually not graphed at all but was drawn which makes the design process distinct, and less mathematically rigorous, than Module 3's graphing, all the while the drawing had to be obey mathematics and required a constant mindfulness to bringing together Two's sinus-wave with the fractal-peak's added dimension.
What did I learn about primes' pattern by developing this model? We can see that f(x) and g(x) are concurrent at points where z = 0 and x is a prime number. (correction:: f(x) and g(x) are concurrent at points z=0 and x = odd). Beyond this initial insight for now I can't make other conclusions about primes' pattern using this model. I'll be revisiting this question through this model in the coming weeks!
The next question would be: How to mathematically confirm the location the conjectured bits?
Go to module 7!
What did I learn about primes' pattern by developing this model? We can see that f(x) and g(x) are concurrent at points where z = 0 and x is a prime number. (correction:: f(x) and g(x) are concurrent at points z=0 and x = odd). Beyond this initial insight for now I can't make other conclusions about primes' pattern using this model. I'll be revisiting this question through this model in the coming weeks!
The next question would be: How to mathematically confirm the location the conjectured bits?
Go to module 7!