Roshar's Map and the Julia Set

[Pagerunner he/him]

I'm wondering about the implications of the origin of Roshar's shape and its connection to something called the Julia Set:

Quote

Q: Has anyone figured out what the secret in the map was, in Words of Radiance?

A: Yeah, they have. That it's modeled after the Julia Set. Which is meant to indicate that Roshar was designed specifically.

Q: Did it happen through crem buildup?

A: No.

Now, I'm not entirely sure of the meaning of the Julia Set, and I'm having a hard time researching it because my mathematics vocabulary isn't up to the task. I understand that the input is a function, and the output is a set of points. I think it's roughly analogous to the set of points that are valid for a mathematical 'cosmic number' riddle, but that's not all that important.

The specific map of Roshar can be developed from a particular instance of the Julia Set. There's an example on Wikipedia that corresponds exactly to the map of Roshar. If I understand it correctly, the function behind it uses an advanced form of complex numbers, that isn't just a + bi, but a + bi + cj + dk. Or, in other words, the set of points that is an output of the Julia set is in four-dimensional space.

Roshar is obviously three-dimensional. The animation from Wikipedia contains three-dimensional 'slices' of the four-dimensional set of points. It's analogous to an MRI, which shows two-dimensional slices of a three-dimensional object, like a brain. Roshar's three-dimensional geography matches all points with a particular value in the fourth dimension, like taking all points where a = 0 and then using b, c, and d to define our three-dimensional coordinates.

That's all the groundwork, my understanding of the situation. My questions are these:

• Do I have any errors in my math explanation above?
• Why the Julia Set to begin with? Did Brandon stumble across the animation on Wikipedia and decide it looked cool as a map? Or was he already looking at the Julia Set specifically for Roshar's map for some other reason? If the latter, what sort of thematic connection can be drawn from the nature of the Julia Set?
• What is the fourth dimension, that is being used to define a three-dimensional slice? Is it time, and the continent of Roshar has a definitive lifetime? Since the geography hasn't changed significantly since the time of the Silver Kingdoms (4500 years), that would mean it was a very slow change, and Adonalsium would have created the continent hundreds thousands of years ago, or even more. (Actually, has anyone done an in-depth comparison of the Silver Kingdoms map and the modern map? I've looked at a few things, like how Rishir changed into Herdaz, but I haven't checked to see if its the exact same slice of the Julia Set.) Alternatively, does this fourth dimension have to do with three Realms? (Although those appear a little too quantized to match a four-dimensional set.) Do we suspect Adonalsium has created any other slices of this Julia Set anywhere else?
• What do we think the significance of the function is that's used as an input? The function itself must utilize a point in four-dimensional space as both an input and output, correct? (Since the Julia Set involves iteration?)

So, this is predicated that Brandon did not merely find that image and say "That image looks like it would make a neat map. Since Roshar is an artificial continent, I'll make it look like that." I recognize that it's possible there are no meaningful interpretations that can be made from the the similarities, that they may be merely superficial, especially since the original Julia Set slice was not created by Team Sanderson. But I figure I'd check if anyone who's more experienced in some of these advanced mathematics has put any thought into whether a deeper analysis could produce any insights.

I recognize that most of these will not have definitive answers. This might have worked better in one of the theory forums. But since I don't actually assert or deduce anything, I figured I'd put it here.

[skaa he/him]

It's probably not going to have any interesting implications, I'm afraid. From here:

Quote

Let me remind you this is not going to be a mind blowing revelation. It is going to be a nifty thing.

 

[+Extesian he/him]

3 hours ago, Pagerunner said:

I'm wondering about the implications of the origin of Roshar's shape and its connection to something called the Julia Set:

Now, I'm not entirely sure of the meaning of the Julia Set, and I'm having a hard time researching it because my mathematics vocabulary isn't up to the task. I understand that the input is a function, and the output is a set of points. I think it's roughly analogous to the set of points that are valid for a mathematical 'cosmic number' riddle, but that's not all that important.

The specific map of Roshar can be developed from a particular instance of the Julia Set. There's an example on Wikipedia that corresponds exactly to the map of Roshar. If I understand it correctly, the function behind it uses an advanced form of complex numbers, that isn't just a + bi, but a + bi + cj + dk. Or, in other words, the set of points that is an output of the Julia set is in four-dimensional space.

Roshar is obviously three-dimensional. The animation from Wikipedia contains three-dimensional 'slices' of the four-dimensional set of points. It's analogous to an MRI, which shows two-dimensional slices of a three-dimensional object, like a brain. Roshar's three-dimensional geography matches all points with a particular value in the fourth dimension, like taking all points where a = 0 and then using b, c, and d to define our three-dimensional coordinates.

That's all the groundwork, my understanding of the situation. My questions are these:

• Do I have any errors in my math explanation above?
• Why the Julia Set to begin with? Did Brandon stumble across the animation on Wikipedia and decide it looked cool as a map? Or was he already looking at the Julia Set specifically for Roshar's map for some other reason? If the latter, what sort of thematic connection can be drawn from the nature of the Julia Set?
• What is the fourth dimension, that is being used to define a three-dimensional slice? Is it time, and the continent of Roshar has a definitive lifetime? Since the geography hasn't changed significantly since the time of the Silver Kingdoms (4500 years), that would mean it was a very slow change, and Adonalsium would have created the continent hundreds thousands of years ago, or even more. (Actually, has anyone done an in-depth comparison of the Silver Kingdoms map and the modern map? I've looked at a few things, like how Rishir changed into Herdaz, but I haven't checked to see if its the exact same slice of the Julia Set.) Alternatively, does this fourth dimension have to do with three Realms? (Although those appear a little too quantized to match a four-dimensional set.) Do we suspect Adonalsium has created any other slices of this Julia Set anywhere else?
• What do we think the significance of the function is that's used as an input? The function itself must utilize a point in four-dimensional space as both an input and output, correct? (Since the Julia Set involves iteration?)

So, this is predicated that Brandon did not merely find that image and say "That image looks like it would make a neat map. Since Roshar is an artificial continent, I'll make it look like that." I recognize that it's possible there are no meaningful interpretations that can be made from the the similarities, that they may be merely superficial, especially since the original Julia Set slice was not created by Team Sanderson. But I figure I'd check if anyone who's more experienced in some of these advanced mathematics has put any thought into whether a deeper analysis could produce any insights.

I recognize that most of these will not have definitive answers. This might have worked better in one of the theory forums. But since I don't actually assert or deduce anything, I figured I'd put it here.

@Pagerunner I assume you have seen the megathread where this was originally discussed, theorized and confirmed?

There's also this handy breakdown of the maths by @Argent

Quote

Here's what I consider a pretty simple explanation of the Julia set.

 

Take a mathematical function f(x). Choose a random number x₀, from the domain of that function. Find out what f(x₀) is (let's say f(x₀) = x₁), then feed it back into f. Find out what f(x₁) is. Feed that back into f to get x₂. Repeat to infinity for all possible initial numbers x₀. All the original numbers, or seeds, that give you a sequence x₀, x₁, x₂, x₃, ... of fairly similar numbers are something we'll call "prisoner set of f;" all the seeds that give you erratic sequences x₀, x₁, x₂, x₃, ... will be the "escapee set of f." Well, the Julia set of the function f is the set of those seeds that form the border between the prisoner set and the escapee set.

 

Here, let me edit with an example.

 

 

---

John Carroll University has a really nice vignette on Julia sets, so I am going to use their example - it's clear enough and it saves me some work.

 

Let's take our function to be f(x) = x² - 0.5. We now need to look at all the possible x₀ we could feed into this function - which, in this case, means all (real and imaginary!) numbers. To illustrate the prisoner and escapee sets, however, we'll only look at a couple of numbers.

 

If we take x₀ = 0 (0 is always nice, makes math easy), we get the following sequence x₀, x₁, x₂, x₃, ... :

x₀ = 0

x₁ = f(x₀) = f(0) = - 0.5

x₂ = f(x₁) = f(-0.5) = - 0.25

x₃ = f(x₂) = f(-0.75) = - 0.3086

...

[May 2, 2016] Edit: My numbers were off, I was using two different formulas to compute x. The final result was still qualitatively the same, but the numbers are now correct.

 

If we continue this, we'll see that the numbers we get never move too far away from the original x₀ = 0. This means 0 is part of our prisoner set.

 

Now, we take a different x₀, let's say 2. Do the same thing:

x₀ = 2

x₁ = f(x₀) = f(2) = 3.5

x₂ = f(x₁) = f(3.5) = 11.75

x₃ = f(x₂) = f(11.75) = 137.563

...

Unlike the case where x₀ = 0, here it's easy to see that the numbers will continue growing (exponentially). This kind of behavior means x₀ = 2 is a part of the escapee set of f. It's also easy to see that any number greater than 2 will also be a part of the escapee set, by the way.

 

Now all you need to do is repeat the same thing for all the numbers (keep in mind, the example I gave doesn't even touch the imaginary numbers / components), plot them, see which one(s) form a boundary between prisoner and escapee sets, and voilà - Julia set!

 

If you are curious, here's how the Julia set of our example function looks like:

 

I've spent a lot of time researching this, trying to understand the maths (I still don't really) and trying to figure out any symbolic or practical significance. To the point where my profile pic is a slice of the Mandelbrot Set, a related mathematical fractal. But I've come up with nothing.

However I at least feel that it's also related to the cymatics that the cities are based on (presumably as the geography was formed by them). I'm no mathemagician but I do know they're quite separate mathematical functions. But a rough, uneducated feeling I have is around Adonalsium building Roshar through mathematical functions in the form of vibrations (have a swirling ball of molten rock, set to cool, apply massive vibrational force to cooling lava-ball, resulting continent matches applied function). The cymatic patterns are then (deliberately) smaller-scale vibrational patterns within the wider one. No idea if that makes mathematical sense. But it's the directing my mind went before I too realized I didn't have enough to post a theory from it.

I don't think it has some meaningful purpose, that he specifically used a slice of that particular function. He mostly indicated that it was because he liked the look of it. But he did say that a lot of his planning is around mathematical functions and fractals, and that that does have some significance for understanding creation history, so that's why I feel it's about Adonalsium using maths to form planets. And the only way I could think of that 'working' is by vibrational force. Choosing the Julia set itself, I have my doubts that it's too meaningful.

[ZenBossanova]

The Julia set is related to the Mandelbrot set, and is a classical example of a fractal. Most coastlines are very fractal, in the real world, but only in the general geometric sense, not in the sense of being built from a single function. For instance, one of the initial problems in fractal geometry, is the Length of the Coastline in Britain problem. 

It suggests to me, that Roshar was constructed very specifically by a vast intelligence. The gas giants and moons suggest that as well. 

 

[Landis963 he/him]

1 hour ago, skaa said:

It's probably not going to have any interesting implications, I'm afraid. From here:

Quote

Let me remind you this is not going to be a mind blowing revelation. It is going to be a nifty thing.

In addition, the rest of that quote mentions that it has to do with "the history of the world."  What that quote neglects to mention is that it apparently has something to do with the prehistory of the world.  

[Argent he/him]

On 4/8/2017 at 8:59 PM, ZenBossanova said:

The Julia set is related to the Mandelbrot set, and is a classical example of a fractal. Most coastlines are very fractal, in the real world, but only in the general geometric sense, not in the sense of being built from a single function. For instance, one of the initial problems in fractal geometry, is the Length of the Coastline in Britain problem. 

It suggests to me, that Roshar was constructed very specifically by a vast intelligence. The gas giants and moons suggest that as well. 

 

I am not sure if we have a WoB on this, but there has been a long-standing assumption that Roshar was created manually by Adonalsium - and the mathematical nature of its geography is a hint to that.

[skaa he/him]

9 hours ago, Argent said:

I am not sure if we have a WoB on this, but there has been a long-standing assumption that Roshar was created manually by Adonalsium - and the mathematical nature of its geography is a hint to that.

Yeah, that's my interpretation of the "Roshar was designed specifically" part of the WoB that @Pagerunner included in his question. This intentional formation of Roshar by Adonalsium is likely what Brandon referred to when he said "it is a fun easter egg that will tell you more about the history of the world" in the WoB I linked above.

So, again, it's just a nifty thing. If I were to let my imagination run wild (as I sometimes do), I'd speculate that Roshar's geographic history looks like this animation (where each frame is a three-dimensional slice of the four-dimensional Julia set from which the current shape of Roshar was taken), meaning that at some point in the past there was a land bridge between Shin Kak Nish and Aimia, and that at some point in the far future the whole continent will disappear. But I don't really find that likely.