- Gale: ?
- Gale: What are you drawing?
- Scarlet: A function that’s continuous everywhere yet differentiable nowhere.
- Gale: Huh?
- Scarlet: As you can see, the function has no break points. But it never becomes linear, no matter how far you zoom in. It oscillates at every zoom level.
- Scarlet: However, I haven’t found an exact formula for such a function yet. But it must be an infinite series with a cosine function.
- Gale: I’m sorry, but you’ve lost me there...
- Scarlet: What would you like me to explain?
- Gale: What is a function?
- Scarlet: … Do you have some time?
A Sky Full of Stars 071
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21 thoughts on “A Sky Full of Stars 071”
Frith Ra
I recognize myself in this picture, and I do not approve.
Sekhmet
I recognize myself in this picture, and I find this extremely cute.
Cyndayn
I do not recognize myself in this picture, and I approve.
ToBeFree
https://en.wikipedia.org/wiki/Weierstrass_function
bh
“Becomes linear” is wrong. When you zoom in on a parabola, for example, the result is congruent to the original; there’s only one shape of parabola. To be differentiable at a point, a function just has to have a tangent line at that point. But if Scarlet is self-taught, I guess we can forgive her for missing that subtlety.
HKMaly
Well … the term used on wikipedia is “straight” but the idea is same: When you zoom in on a parabola, it won’t BECOME straight but it will get closer to it the more you zoom. That’s not true for Weierstrass function: the more you zoom, it stays exactly same.
Tauben
Well, one can define differentiability in a point as the property that a function can be approximated arbitrarily well by a linear function around that point, as long as one chooses a sufficiently small neighborhood of the point to look at. (I.e. going with the definition of the Frechet derivative, which matches the regular differentiability term in finite dimensions)
So while there of course is a difference between being linear and being arbitrarily close to being linear, I think that might be too much rigor to expect for a comic.
Adge
Yes – the correct technical term is “smooth”
Alex
bh: It means a bit different thing. If you draw a tangent line to a parabola at some point, then that line will be pretty close to the parabola if you don’t move away from that point too far.
It can be restated more formally with the help of limits: for any value of epsilon there exists an area near the tangent point, where the distance between the points of a parabola and the line is less than epsilon.
For the Weirstrass function there are _no_ such points.
Alice
I came here for the story.
A stayed for the comment section 😉
Graybeard
I’m having a hard time understanding, or imagining, what this function has to do with the plot here. Yes, it shows that Scarlet is extremely bright, and Gale’s reaction shows that she is highly charismatic, or if you prefer, cute. But we knew both of those already. One possible path forward is that she “discovers” fractals, which then have something or other to do with the Machine — key, maybe?
bh
She’s also too nice! At her age, I didn’t think everyone else was as smart as I was; I thought I was the smartest person in the world. (I didn’t learn how wrong I was until college.)
bh
(Addendum after the next page was posted) Okay, I was wrong. She’s just like young me. Only maybe she really is that smart. 🙂
Intrebute
I’m loving the little math tidbits, even if they’re a little handwavy. I’m liking Scarlet more and more as time goes on.
MegaJar
Of course, this whole exchange is ignoring the most vital question of all: why did she feel the need to draw this highly advanced mathematical function in the dirt with a stick? Does she not have access to paper and pencils?
MegaJar
Actually, I can answer the second question myself: of course she does, she used them to write her short list of books to borrow from Gale, and then again to fill out the “orderly do-gooder vs. free-spirited scoundrel” personality quiz. Which makes her choice of a dirt-based medium to perform higher mathematics all the more perplexing.
Clevershitter
Advanced mathematics works better on a large surface.
Source: Trust me bro.
Or trust Archimedes.
Cirom
Probably because she’s currently on guard duty? I imagine it’s easier to keep an eye on things if your focus isn’t entirely on a book.
Also, she’s like 12 at this point, I think? You saying you didn’t want to doodle in the dirt or especially sand when you were a kid?
Hegel-Marx
To find the Machine of Eternal Summer, Scarlet will have to be an action heroine. She will need followers, so she must be a spiritual leader. To turn the Machine on, she will have to be a Sage. Boudica, Joan of Arc and Hypathia rolled into one. (If you’re interested in Hypathia, the movie “Agora” is available online.)
Vicious Sand
I’m reminded of genetic colonists and that Scarlet is the end product or even just middle range product of them. You’re going to want to breed for hearty people- strength, endurance, great immune system, etc. But you’ve also got the chance to breed for smart people as well. And starting off from a space faring civilization, that’s a strong foundation of intelligence rankers. If everyone’s an astronaut, then their children, or other distant descendants, should be about as smart as well, theoretically speaking. Math Doodles aren’t even her final proof.
Hilmar Zonneveld
The function being discussed (and the function drawn) is the Weierstrass function.
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