the mandelbrot set

Regarding Fractals and Non-Integral Dimensionality

Alright, I know it’s past midnight (at least it is where I am), but let’s talk about fractal geometry.

Fractals

If you don’t know what fractals are, they’re essentially just any shape that gets rougher (or has more detail) as you zoom in, rather than getting smoother. Non-fractals include easy geometric shapes like squares, circles, and triangles, while fractals include more complex or natural shapes like the coast of Great Britain, Sierpinski’s Triangle, or a Koch Snowflake.

Fractals, in turn, can be broken down further. Some fractals are the product of an iterative process and repeat smaller versions of themselves throughout them. Others are more natural and just happen to be more jagged.

Fractals and Non-Integral Dimensionality

Now that we’ve gotten the actual explanation of what fractals are out of the way, let’s talk about their most interesting property: non-integral dimensionality. The idea that fractals do not actually have an integral dimension was originally thought up by this guy, Benoit Mandelbrot.

He studied fractals a lot, even finding one of his own: the Mandelbrot Set. The important thing about this guy is that he realized that fractals are interesting when it comes to defining their dimension. Most regular shapes can have their dimension found easily: lines with their finite length but no width or height; squares with their finite length and width but no height; and cubes with their finite length, width, and height. Take note that each dimension has its own measure. The deal with many fractals is that they can’t be measured very easily at all using these terms. Take Sierpinski’s triangle as an example.

Is this shape one- or two-dimensional? Many would say two-dimensional from first glance, but the same shape can be created using a line rather than a triangle.

So now it seems a bit more tricky. Is it one-dimensional since it can be made out of a line, or is it two-dimensional since it can be made out of a triangle? The answer is neither. The problem is that, if we were to treat it like a two-dimensional object, the measure of its dimension (area) would be zero. This is because we’ve technically taken away all of its area by taking out smaller and smaller triangles in every available space. On the other hand, if we were to treat it like a one-dimensional object, the measure of its dimension (length) would be infinity. This is because the line keeps getting longer and longer to stretch around each and every hole, of which there are an infinite number. So now we run into a problem: if it’s neither one- nor two-dimensional, then what is its dimensionality? To find out, we can use non-fractals

Measuring Integral Dimensions and Applying to Fractals

Let’s start with a one-dimensional line. The measure for a one-dimensional object is length. If we were to scale the line down by one-half, what is the fraction of the new length compared to the original length?

The new length of each line is one-half the original length.

Now let’s try the same thing for squares. The measure for a two-dimensional object is area. If we were to scale down a square by one-half (that is to say, if we were to divide the square’s length in half and divide its width in half), what is the fraction of the new area compared to the original area?

The new area of each square is one-quarter the original area.

If we were to try the same with cubes, the volume of each new cube would be one-eighth the original volume of a cube. These fractions provide us with a pattern we can work with.

In one dimension, the new length (one-half) is equal to the scaling factor (one-half) put to the first power (given by it being one-dimensional).

In two dimensions, the new area (one-quarter) is equal to the scaling factor (one-half) put to the second power (given by it being two-dimensional).

In three dimensions, the same pattern follows suit, in which the new volume (one-eighth) is equivalent to the scaling factor (one-half) put to the third power.

We can infer from this trend that the dimension of an object could be (not is) defined as the exponent fixed to the scaling factor of an object that determines the new measure of the object. To put it in mathematical terms:

Examples of this equation would include the one-dimensional line, the two-dimensional square, and the three-dimensional cube:

½ = ½^1

¼ = ½^2

1/8 = ½^3

Now this equation can be used to define the dimensionality of a given fractal. Let’s try Sierpinski’s Triangle again.

Here we can see that the triangle as a whole is made from three smaller versions of itself, each of which is scaled down by half of the original (this is proven by each side of the smaller triangles being half the length of the side of the whole triangle). So now we can just plug in the numbers to our equation and leave the dimension slot blank.

1/3 = ½^D

To solve for D, we need to know what power ½ must be put to in order to get 1/3. To do this, we can use logarithms (quick note: in this case, we can replace ½ with 2 and 1/3 with 3).

log_2(3) = roughly 1.585

So we can conclude that Sierpinski’s triangle is 1.585-dimensional. Now we can repeat this process with many other fractals. For example, this Sierpinski-esque square:

It’s made up of eight smaller versions of itself, each of which is scaled down by one-third. Plugging this into the equation, we get

1/8 = 1/3^D

log_3(8) = roughly 1.893

So we can conclude that this square fractal is 1.893-dimensional.

We can do this on this cubic version of it, too:

This cube is made up of 20 smaller versions of itself, each of which is scaled down by 1/3.

1/20 = 1/3^D

log_3(20) = roughly 2.727

So we can conclude that this fractal is 2.727-dimensional.

Some S02 E03 thoughts

I have a whole list that isn’t even complete yet, but I wanted to share a couple thoughts. Honestly, they’re mostly questions and they’re mostly Amanda/Vogel/Blackwing/Priest related.

- Dirk’s “I know this is a strange ask”. Why do Blackwing subjects talk so oddly sometimes? Vogel talks about “truthing” and asks Amanda what her “malfunction” is later and they talk fairly normally most of the time so what’s going on with these very odd gaps in their language? Should we be stringing together the missing words to find a clue? Were they raised by badly programmed robots?

- Amanda tells Vogel “I really appreciate you stealing this food for me”, so does that mean for sure that Vogel doesn’t (need to) eat, besides the energy feeding he does?

- Vogel has no idea how family relations work.. which makes sense. Also Amanda apparently knows how old Vogel is. Did they have a conversation about this? Does Vogel know his own age? Did she guess based off of how long he says he’s been in Blackwing? I wouldn’t be able to tell from appearance at least. 

- Vogel is thoroughly grossed out about the thought of being a boyfriend. He doesn’t know what a profile or poetry or a problem is but he knows what a boyfriend is. Who had this talk with him? What exactly did they tell him? 

- What does the scarecrow mean? Anything other than just something in Amanda’s vision that’s tied to Priest? In Dutch a scarecrow is vogelverschrikker or bird(”vogel”)-scarer. Which is definitely what Priest is. This has got to be on purpose, right? 

- Priest says “They’re traveling in a car with a 3 on it. It’s not exactly geometry.” What is his obsession with geometry? Later he says that things are about to go “oblong” instead of South.

- Why were the drives from the first Blackwing not a priority? If bringing in the subjects is a priority, why not, you know, all the stuff they know about them?

- Seriously the theme of this season has something to do with geometry. The shapes that Suzie shoots at Bart. The odd things Priest says. Amanda’s connection to the Mandelbrot Set (which is fractal geometry).

Prediction for “Nice jacket”


Amanda will slay Suzie, because Suzie’s (and Mage’s) magic manifests as simple geometrical forms, while Amanda is something like an universal fractal - mathematical set of complex numbers, which can be represented geometrically. (If I got it right; but anyway, this is something MUCH MORE COMPLICATED than just circles and squares)

By the way, the Mandelbrot set consists of cardioids (though not only?) which is


I feel this kinda describes the relation “blackwing projects” @ universe …idk

Bonus:

I’m sure someone had already screamed about it here but


( 1.Maaaaaaaax????!!! 2. Was Mike Howell a Blackwing subject?)

Amanda Brotzman, season 2 wardrobe

Episodes 1 & 2:  WHO. IS. THIS. GIRL. Look at her! Go back and look at season 1. Now look at this. Then go look at season 1. THEN LOOK. AT. THIS. Gone are the plain, oversize clothes! She’s wearing makeup, and her skin is flush with life and color! The dark circles under her eyes are GONE.  She’s rocking a fitted olive & grey striped tee, a black leather jacket that fits, with a blue Mandelbrot set painted on the back, black jeans, black boots, black hair, shaved sides (which, you know she probably had to face down a pararibulitis attack over the razor but she did it anyway!) She looks GOOD.

Episodes 3-5: She’s got her trusty jacket, a grey tee, and striped pants (which look alternately grey or blue in different scenes), black boots. The shirt gets increasingly ripped at the collar as each episode passes, possibly a mirror of her stress levels, that she’s coming apart a bit after the escape from Priest and ending up in CrazyMoonLand–similarly, her makeup, though punkishly smoky the first couple eps, starts to get increasingly raccoon-faced as shit gets Super Weird. Obviously going for a dive in Portal Lake is partly responsible, but it defies the laws of physics as it keeps getting more exaggerated as crazyass shit keeps happening, until she looks like an extra from the chimney sweep dance in Mary Poppins

Episodes 6-10: Look at our little witchakookoo. She’s embraced Wendimoor, and what it means for her. Although she’s still our little punk, she’s now a punk ~witch~ and historically witches have a connection to nature, and we also have Amanda connecting to herself, and becoming more in tune with the interconnectedness of the universe, so we see a look on her here that’s very rooted (heh) in nature with earthy tones and natural looking fabrics, her little woven hood dotted with bright color like a field of flowers, or a bird’s nest with tufty down feathers woven into it. Also, note the fresh, clean face (though rosy and full of color, not the sickly face from season 1): the Rowdies are back, she connected with her powers and saved them, hence saving the village…she’s feeling stronger, more confident, pulled together.

Episode 10: Grey lady!  Grey leather jacket with black mandelbrot on the back under black leather straps, pink and maroon striped v-neck tee, grey jeans, grey boots, 3 arrow-shaped enamel pins on her lapel, pointing in different directions. Now, grey is OFTEN used as a color signifying negative things: depression, sadness, gloom, blandness, and so on. But in this series, as we see in Amanda’s season 1 wardrobe, they use *faded, worn, washed-out black* for that purpose.  Amanda declares herself Miss Duality here; everyone has light and dark in them, good and bad. It also indicates that their mission isn’t about the triumph of Good and the abolition of Evil, it’s about restoring Balance. They’re not out to make the world a Utopia, just set it back on its tracks, as it’s supposed to be, with still some good in the world, and some bad. They are meant to be a morally Neutral force. Interestingly, because, in the color world, grey is a neutral color, so it works well with all, or most, other colors, and I think we can see the ‘colors’ as being the Tools that Amanda is supposed to collect, but also a nod to Dirk, yes? Good. Or Amanda’s just being a pretentious art student, who the fuck knows? ;)

Notes

I love that both Hobbs and Tina know Farah is a better cop than any of them and defer to her in police work. And apologise when failing at it. And cheer when Farah is awesome (like that fight when she took Panto out). It was very moving when Hobbs gave her what she always wanted and made her his deputy. This helped her later realise that it isn’t what she needs any more but she’s still great at it.

I too suspected Arnold Cardenas was The Boy when he was first mentioned but now I think maybe the “Infant, Male, Pollock, Francis” is The Boy. Not the boat - the boy it’s named after.

I really like the look of Wakti Wapnasi and other monsters. The make up was amazing. Still many people think she and Mona might be the same and she did have a version of Friedkin’s stress toy with shining eyes and that’s how Mona started her shapeshifting journey to free Dirk.

Vogel looked very good with flowers in his hair. And both his and Amanda’s reunion with the rest of Rowdy 3 was the happiest part of the episode.

7,775,000 seconds is about 90 days. That’s a long time to be strapped immovable in one position. But at least project Incubus have special powers helping them survive until their Drummer pulls them out. Ken was in not much better situation and he is just a human. This shows that psychic vampires are better people than anyone in Blackwing. They try to feed only on those it doesn’t hurt or on those who attack them and others. Friedkin and Priest were harming them just to feel in control.

Those patterns in Wakti Wapnasi’s pool were fractals - Mandelbrot set to be precise. And then they morph into kaleidoscope images when Amanda is using it. these are all self repeating patterns.

Amanda’s vision in S02E05: GIRL POWER

Decided to take a closer look at these! There’s tons of stuff to unpack.

OK, first bit. We’ve got Amanda at the lake when they first arrive in Wendimoor. Then Amanda(??) with Mandelbrot face and oddly colorful eyes - though that could just be the vision. Then we get Priest with what looks like a split face who goes all kaleidoscope on us. Then there’s Panto laying on the floor surrounded by yellow flower petals.

Second bit, there’s Dirk wearing a noose around his neck and apparently in the room with yellow flowers everywhere. Dirk looks around, as if he’s confusedly recognizing someone. Then there’s the familiar Mandelbrot set but with red eyes flashing in it. Then what looks like a woman with dark eyes and eyebrows and absolutely covered in blood, which moves into alien blue Amanda that we know from previous visions.

We get the Mage in an actual black and red Mage outfit, with a circle, a square and a star on the front of his robe. He looks to be projected onto the mural or emerging from it - a flashback to when the Mage came into being somehow? Next up is Suzie in full Evil Queen getup in an overgrown room absolutely smashing an electric guitar to bits and having pink smoke erupt from it. There’s a second of a kaleidoscope view of a forest, and then it goes over into a kaleidoscope view of the Kellum Knights.

this vision is basically just Kellum knights.They look to be under an arch with the Dengdamor sigil on it.

The entire vision ends with the Project Incubus symbol that sort of refracts into the pattern of the water of the vision pool.