Patterning
Patterning is the process of perceiving order in the arrangement of objects. Based on the perceived order a logical representation of the pattern is created in the form of an image, sound, shape, mathematical formula, etc. An informative pattern either entirely captures the order in the arrangement of the observed objects, or at least creates some sort of analogy between the objects' arrangement and their representation.
Patterning is the process of perceiving order in the arrangement of objects. Based on the perceived order a logical representation of the pattern is created in the form of an image, sound, shape, mathematical formula, etc. An informative pattern either entirely captures the order in the arrangement of the observed objects, or at least creates some sort of analogy between the objects' arrangement and their representation.
In Module 2 I have shown a representation of primes perceived as waves undulating through the real number line. In this Module I’ll show a few applications of the wave representation of primes, I will also point out the limitations of this representation. All evidence points to the belief that it’s impossible to describe the overall pattern involved in primes, however by restricting the interval of analysis to anything less than infinite, a myriad of definite and complete patterns emerge. In this module I’ll venture into describing a few of such patterns by building on the wave representation shown in Module 2. The two conceptions each miss some important element of the whole prime conundrum, superimposed however they fill in some of conceptual gaps inherent in each other's design.
Here is a slightly modified version of the prime wave image:
Here is a slightly modified version of the prime wave image:
In this representation the viewer is asked to shift focus from the overall balance of the design and concentrate on its parts, on specific waves. To facilitate this shift in vision, the waves are color coded and the function rule describing each is given. Notice that each wave is defined by a sine function. To maintain the harmony of the image, and thereby its usefulness, waves alternate starting their paths with the decreasing or increasing interval of their cycle. Waves’ periods have twice the length of the prime they represent, and their amplitudes are equal to their primes’ order. For instance Two’s wave has a period of four and the amplitude of one because Two is the first prime; and Three’s wave has a period of six and the amplitude of two because Three is the second prime.
The first application of this design is an exercise in analyzing graphs of simple trigonometric functions. The concepts of periodicity, wave height, amplitude, period, cycle, wave length and frequency can be discussed in a classroom activity by simple use or recreation of this image. A good question in an 11th grade classroom could be: What’s the role of π in obtaining the “proper” period for each wave?
Another application is more closely related to the original perception of prime waves. Here it is, all so obvious:
The first application of this design is an exercise in analyzing graphs of simple trigonometric functions. The concepts of periodicity, wave height, amplitude, period, cycle, wave length and frequency can be discussed in a classroom activity by simple use or recreation of this image. A good question in an 11th grade classroom could be: What’s the role of π in obtaining the “proper” period for each wave?
Another application is more closely related to the original perception of prime waves. Here it is, all so obvious:
Okay, I was just joking!
By zooming onto a specific segment of the color-coded waves, we can use the design to give the partial prime factorization of numbers. Let’s zoom in!
By zooming onto a specific segment of the color-coded waves, we can use the design to give the partial prime factorization of numbers. Let’s zoom in!
On this image we see prime waves from Two to Ninety-Seven on the interval of 9390 to 9420. What can we tell about these numbers using the image? We can tell for instance that 9391 has no factors other than 1 and itself. That is, we can visually confirm that 9391 is a prime, and so is 9403, because waves don’t mark them. (Note that beyond 97² = 9409 we can’t make such predictions about primeness because the last wave is that of Ninety-Seven, beyond that point primes may sneak into this design unnoticed.) We can also see on the image that 9394 has prime factors of 2, 7, 11, 61; these are the waves passing through 9394 (use the color codes and amplitudes to tell which wave is which in the tangle). However this isn’t a complete prime factorization; the image doesn’t show what powers of 2, 7, 11, and 61 make up 9394 (coincidentally in this case they are all raised to the first power! 2 x 7 x 11 x 61 = 9394). Look at for instance 9408, apparently it’s made of waves of Two, Three, and Seven, but raised to what power? There is an incredible amount of patterns in this design; however, a marked limitation of it is its inconvenience in determining complete prime factorizations.
Observing the not so obvious patterns hidden in this design, then re-imaging them in the mind’s eye (a whole new cycle of perception), a new representation of the patterns emerges.
Observing the not so obvious patterns hidden in this design, then re-imaging them in the mind’s eye (a whole new cycle of perception), a new representation of the patterns emerges.
On the newly perceived image we see prime dots instead of prime waves, apparently less aesthetic, less balanced. This is however the missing element of the prime wave conception, because on this representation we can visualize full prime factorizations of numbers. Take for instance 36. It has two green and two pink dots 2² x 3² the complete fingerprint of 36. Or look at 60, blue, pink, pink, green; 5 x 2 x 2 x 3. Now let’s see the two simultaneously:
The two designs complement each other; waves together with dots offer a fuller story of primes and composite numbers' prime factorizations. But we pay a high price for the added information: the image is contrived. This cannot be Module 3’s new pattern!
Taking away some layers of complexity and looking again at the dot image a more pleasing pattern emerges:
Taking away some layers of complexity and looking again at the dot image a more pleasing pattern emerges:
Note: 56 is missing a pink dot! It should have 3.
What we see here is the symmetricity hidden in prime factorization. A “full cycle” of the bases 2, 3, and 5 is shown. It is a full cycle because reading it from front to back or back to front the distribution of the dots (look at 2’s for instance) is the same, making it analogous to an ambigram. Moreover the overall shape of this distribution would not change at all even if we’d look at a much larger interval. In other words this is a self-same image or a fractal. The cycles of 2, 3, and 5 do overlap on a large enough interval, however by then other prime bases sneak in and blemish the cycles. Grasping this idea is another way understanding why an overall pattern in primes is so problematic. In other words, this new representation of the cyclicality of the bases in prime factorization allows for a new insight into why only “temporary rules” are obtainable when dealing with primes. The appearance of a new wave or a new dot makes the rule incorrect thereof. The last look at the new pattern is Module 3’s true new pattern. Related to Module 2’s prime waves, but taken a step further. The image is about the fractal cyclicality of the bases of 2 in composite numbers’ prime fingerprints. Perceiving the triangular pattern in the dots’ arrangement leads to the idea of a self-same fractal image that is very similar to the Sierpinski Triangle. (I didn't connect all the dots into triangles letting you to imagine the rest.) The function rules describing the triangles’ sides through the properly spread dots reveal a wealth of further hidden patterns in this fractal (I won’t describe them here, but take a look at the functions to see for yourself).
The number of patterns in prime waves and in prime factorization is virtually unlimited. I hope that shifting from waves to dots and then to fractals was an interesting journey on the surface facts of this great mathematical landscape!
Visit the next module!
Visit the next module!