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more Julia sets?


Zelly

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So we know the overall map landscape of Roshar is related to Julia sets, making it all......spiral-ish. 

I'm wondering if that has carried over to other aspects of Roshar, such as marbled skin patterns on the Parshendi/Fused or the ever spiraling symmetry of Cryptic heads.  The Cryptics in particular I could understand, with their interest in math principles.

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But they're different things, I think? (math, music, and/or physics are so not my skill set).

I don't even know what questions to ask that make sense....

Are cymatics and sets related?  Waves?  Functions? (wave function is a thing, right?)

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4 hours ago, Zelly said:

But they're different things, I think? (math, music, and/or physics are so not my skill set).

I don't even know what questions to ask that make sense....

Are cymatics and sets related?  Waves?  Functions? (wave function is a thing, right?)

I am not an expert on these things - I'm only an Engineer, not a Mathematician, though I took my fair number of maths courses, and I am interested in the subject :-P - but to give a simplified overview:

  • A function is basically a relation between two (or more) values - an input (or more than one input), and an output. So, for example, f(x) = x^2 + 3 means that the value of f(x) is equal to x squared and then whatever value you get from that you add 3 to - so x = 1 means f(1) = 4 (that is, (1)^2+3 = 1 + 3 = 4), and x = 4 means f(4) = 19 (as (4)^2 + 3 = 16 + 3 = 19). If you have a graph and set f(x) as the y value (so the vertical line), and x as the horizontal line, then for each value of x there is a value of y, which you can then use to draw a line that connects those points, so if your x and y axes are labelled you would see that if you drew a line straight up from the number 3 on the x axis, if would touch the graph of the function f(x) = x^2 + 3 at the same point a horizontal line from 19 on the y axis would touch it. Different equations produce different graphs, and each of these graphs is a function.
  • A wave is a repeating, periodic pattern - that is, something which cycles. A wave function is a function whose output is a wave, so f(x) = sin(3*x) is a wave function which produces a pattern that repeats every time 3*x = 2*PI*n where n is an integer (and assuming we are using radians, but don't worry about that, its just an alternative to using degrees to describe angles but which is much nicer mathematically). For sin, cos, etc., the PI shows up because you are basically looking at the x and y values when moving around a circle, and 2*PI is the equivalent of going fully around that circle, so getting back to where you started.
  • A set is a group of numbers, or group of any mathematical objects. I know there is more to it than that, but that is the long and the short of it. A set can have various properties (such as having a certain cardinality - a fancy way of saying having a certain number of elements, such as the set of all multiples of five greater than 12 and less than 99 having a cardinality of 17, consisting of {15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95}), be constructed in various ways (such as, for example, talking about the set of all even numbers), and have various relations to other sets, such as both having the same cardinality or one being contained in the other (like the set of all powers of 2 being contained in the set of all integers), etc. (I apologise if I left anything else important about sets out, but that is what I understand of them)
  • Cymatics is about how - when waves interact - they produce regions where there is either greater movement than the individual waves would create, or far less. This happens because a wave can have two points that are both moving in the same direction interact - so adding together to produce a larger change - or two areas which are moving in opposite directions interact - so they subtract and produce a lower value. The shape of the surface vibrating changes the length of the waves that can "fully fit" on the plate, as only waves equal in length or who have a length that can divide into it (that is, have a frequency that is an integer multiple of the fundamental frequency) can fit. A great way to think about this is with a piece of string - tie one end to something so it can't move, and then shake the other end - at a low frequency you can make the middle of the string go up and down while the end you are holding - and the end tied to the object that can't move - is basically stationary. Wave it a little faster and you can get two points on it to go up and down, only now the two ends AND THE MIDDLE are stationary. The two ends and the middle would be - if this were a plate - where sand would pool due to those areas not disturbing the sand, while the sand would be fully cleared away from the two points on the string that are moving. Wave it even faster, and you can change where the stationary and moving points are. Cymatics is basically applying that principle across a two dimensional surface, where the shape of the plate and the frequencies involved determine the pattern.
  • ([Edit] Note that for cymatics and the string example the wave is actually interacting with itself, the wave bouncing off of the sides and coming back at itself again. The patterns are stationary despite the waves moving because the interactions form what is called a standing wave, the patterns of movement fully balanced due to the wave length matching the size of the string or plate.)

 

If that was a little high level then here is a summary: functions are ways of mapping two values to each other so for any given input you have a specific output. A wave is a repeating periodic pattern that repeats over a given period or distance so you will see copies of it if you go far enough along or wait long enough. A wave function is a function that takes an input and produces a wave. A set is a collection of numbers, etc., which have been grouped together. Cyamtics is the result of waves interacting based on the plate shape and frequencies applied to produce regions where there is a lot of movement or very little, and so sand moves from where there is a lot of movement and settles where there is little movement.

 

I would rather like to see more mathematical functions in the series :-)

 

As I said, I'm not an expert but this is an area I am interested in. I do still do remember the non-Mathematician lecturers when they used maths in a more informal and fudged way reminding us not to show the maths department how they were doing it :-P If any Mathematicians would care to correct me on any of my mistakes here, I would appreciate that :-)

Edited by Ixthos
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