#quadratic scaling

[iridescentholly]

X spells in magic the gathering are where the fact it was designed by a maths guy really pops. Majority of the maths in magic is "can I pay cost" or "is damage dealt enough to kill a creature/player", so playing around with that number being variable rather than fixed makes some sense, and maths has X as a default variable, so you get fireball

right there in the first set - variable cost for variable damage, 1:1. This spell is great for ending games, which makes up for it's poor rate for the time (see lightning bolt, where you pay 1 mana for 3 damage)

So now you can use X for the cost to give a linear scaling effect, such as additional counters on creatures

Where we get really mathsy next is the relatively few examples of effectively "do something X times, X times" for example:

These two cards are roughly equivalent in what they do to the previous two - but appear to scale at a much worse rate, having to pay two or three times the cost for each time you do "X" but the trick here is these aren't actually scaling linearly, they're scaling quadratically, making them much more effective than they seem.

Anyway all this to say that sometimes evaluating how good a magic card can require understanding quadratic scaling!*

*but almost never will because these effects are almost certainly intentionally costed too highly and attached to traditionally uncompetitive effects, likely so that players don't actually have to think about quadratic scaling. This is probably for the best. Also there's one card that scales exponentially (2 to the power of X) rather than quadratically (X to the power of 2). It's literally called exponential growth.

[meinkatz]

SPIEGEL BY VERPAN

SPIEGEL was made famous for its role in Verner Panton’s complete interior installation of Der Spiegel Publishing House in Hamburg in 1969. The SPIEGEL lamp is an avant-garde statement piece for the modern contemporary home. Typically for a Panton design, the lamp can be mounted either individually or as part of a system - on the wall or in the ceiling - enabling the user to create a stunning single-piece statement or a dramatic large-scale installation, depending on mood, personality and space. The large quadratic lamp has a circular indentation in the center, which holds the light source, which is then concealed under a hemispheric shade that breaks and softens the light and banish glare. What Panton created with SPIEGEL was an architectural and sculptural art piece that doubles as a lamp providing a beautiful and soft ambient light.

You can connect them together (No cheats needed)

DOWNLOAD (FREE)

[itsapmseymour]

Hehehee

On a scale of 1-10, what is the quadratic equation to Einstein's Theory of Gravity?

Markiplier

[argumate]

scaling up the context window by eliminating quadratic overhead of the transformer architecture?

[is-the-snake-video-cute]

this isn’t a husbandry question, but do you know why snakes are considered only a suborder taxonomically rather than a full order? It seems like the order squamates is diverse enough that I’d expect it be split into one order for snakes and another for lizards

The general consensus among herpetologists is that squamates, while a very diverse group, have similarities strong enough that it's more useful to consider them all part of the same order rather than split it up. There are actually three suborders in Squamata: snakes, lizards, and amphisbaenians. They all have things that tell us they're members of the same lineage:

Scales. Squamates are also called "scaled reptiles" becuase they all have scaly skin that must be shed every so often. This makes squamates distinct from other reptiles, like crocodilians, who have scutes instead.

Kinetic quadrate bones - they all have bones in their skulls that make it possible to move the upper jaw independently from the rest of the skull. Not every squamate can, but their skull structure makes it possible.

The potential for venom. Not every squamate is venomous (not by a long shot, only about 15% of snakes are venomous and that number is even smaller for lizards), but something like 60% of all squamates belong to clades with venomous members somewhere in their lineage.

Taken all together, these three things that all squamates share tell us that they are extremely closely related and considering them as one order altogether can tell us a great deal about how all squamates evolved. If we considered snakes as their own order instead of a suborder, then we'd lose some perspective on how interrelated all three suborders are.

If we take a look at the single sister order to Squamata, the rhynchocephalians, we can see why no squamates are distinct enough to warrant becoming their own order. Rhynchocephalians, unlike all squamates, have a distinct tooth type, have fused skull bones, do not have the potential for venom, and have a unique skeletal structure including hourglass-shaped vertebrae and gastralia. Snakes are a little different from other squamates, but they're not that different!

[prkwook]

CUPID 💌🏹

IN WHICH our cast plays cupid for their loser (endearingly) friends who can’t seem to take their own relationship advice.

CHAPTER 002. sneaky little tax evader (smau + written) 🎧

“HI!” you greet the boy that you’ve been not-so-secretly eyeing for the past hour and a half.

“Hi!” he smiles. you internally melt and almost don’t notice the hands belonging to the boy in front of you that are giving you a book to check out. He doesn’t break eye contact once. Usually, that would be a little odd and would kind of sketch you out but there’s just something about his big brown eyes that makes you feel safe.

“Will that be all for you today?” you ask. You’re the one to finally break eye contact, but only to quickly look down so you can locate the barcode and scan it. Your eyes graze over the cover, “The Crisis of the 3rd Century? Interesting choice…” You must have a quizzical look on your face because he laughs and it’s the most beautiful sound you think you’ve ever heard. 

“I know. It's one of my favorite things to whip out at trivia night. No one ever sees it coming! First, it’s the Quadratic Formula … easy peasy … and then Boom! It’s the near collapse of Ancient Rome! Well, that and the fact that Henry David Thoreau committed tax evasion. Multiple times.” 

“Oh my gosh, no way… no one ever knows about that! Every time I reference it, my friends think I'm some sort of nerd. Not that you’re a nerd or anything.” you could’ve rambled on and on but a thought pops up and stops you in your tracks. “I'm sorry, I just have to ask. Exactly how often do you think about the Roman Empire, and I’m not talking about The Gladiator (2000) with Russell Crowe all oiled up Roman Empire. I’m talking like actual Roman Empire?”

“Trust me … you don’t want to know. “ the pretty, brown haired boy replies while trying his best to look serious. It doesn’t work though because a few seconds later, he bursts into the same laughter as before and you swear your world stops for a second. All you can do is pray that he doesn’t hear your heart about to beat right out of your chest. In an attempt to hide your rosy cheeks, you look away from him and towards the register which now displays his total. 

“Your total is going to be $15.89 today, cash or card?”

He pulls out a $20 from what seems to be an off-brand Lightning McQueen wallet and hands it to you. You take it, put it in the register, and give him back his change. He takes it from you with a smile that you’re convinced could bring world peace and drops all $4.11 into the tip jar sitting on the counter in front of you. He says “Have a good evening.” to which you reply “you too.” and watch him walk out the door, bag and cup in hand.

“Woah, your cheeks are redder than a tomato. Be honest, on a scale of 1 to 10, how down bad are you?” Taerae, your best friend, questions as he walks over from his post at Bluebird, the in-store cafe that sits right by the checkout and also happens to give him the perfect view of what just went down.

You sigh and dramatically place a hand to your forehead as if you’re swooning. “I fear I’m in love and I don’t even know his name.”

🔭 ★ mlist. previous. next.

☆★ TAGLIST:  @annoyingbitch83 @vernonburger @doiedecimal @taekwondoes @imthisclosetokms @kaynunu (bold can't be tagged ; pls check your settings + make sure u can be tagged ... ty ^_^)

want to join? -> taglist form . 🫀

[lonely-vault-boy]

Tag yourself Fatui cabbage addition!!

Goodest boy cabbage

- good cook

- very wholesome

- likes to stab things

- has never done anything wrong ever

- orange

Greedy cabbage

- “I like money”

- loan shark

- probably has teeth marks on his cabb-ose

- Dottore ate his ass is what I’m saying

Cotton-candy cabbage

- blue

- very soft and fluffy

- sharp teeth (likes to bite)

- secretly kidnaps baby cabbages in the dead of night to throw them into a pot and test new stew recipes. I can still hear their screams.

- very soft and fluffy

Cow cabbage

- bites H A R D

- quadratic scaling but actually he just multiplies indefinitely

- is a plague

- heeheeemoomoo

Robo-cabbage

- the actual cabbage is the ear

- uses the bodies of dead cabbages to form her taxi cabbage

- always annoyed

Angel cabbage 🥺����❤️🥰😍

- can manipulate minds (allegedly)

- the most beautiful singer and adorable being to ever bless cabbagekind

- likes scrapbooking

Arleccabbage

- the chef probably barks for her

- but what else is new

- hidden stash of baby cabbages she uses to feed the dottores

- does not like scapbooking

Daddage

- old man but cabbage

- slightly past expiration date

- found a gremlin cabbage outside the garden

- “orange is a normal cabbage color, he won’t taste any different. Dottore, stop trying to eat him.”

- orange is not a normal cabbage color

Dilf cabbage

- got nommed on

- was not very tasty

- might be rotting

Cabbatano

- is that actually a cabbage /gen /srs

- I think it’s looking at me

- help

Ghost cabbage

- like a ghost pepper

-except cabbage

- hot like pepper, dead like ghost

- remembers the cabbaclysm

- rest in peppers

Missingno.

- guys I swear there was another cabbage here

[compneuropapers]

Interesting Papers for Week 15, 2023

Cross-scale excitability in networks of quadratic integrate-and-fire neurons. Avitabile, D., Desroches, M., & Ermentrout, G. B. (2022). PLOS Computational Biology, 18(10), e1010569.

Modulation of working memory duration by synaptic and astrocytic mechanisms. Becker, S., Nold, A., & Tchumatchenko, T. (2022). PLOS Computational Biology, 18(10), e1010543.

Mathematical relationships between spinal motoneuron properties. Caillet, A. H., Phillips, A. T., Farina, D., & Modenese, L. (2022). eLife, 11, e76489.

The pupillometry of the possible: an investigation of infants’ representation of alternative possibilities. Cesana-Arlotti, N., Varga, B., & Téglás, E. (2022). Philosophical Transactions of the Royal Society B: Biological Sciences, 377(1866).

Olfactory responses of Drosophila are encoded in the organization of projection neurons. Choi, K., Kim, W. K., & Hyeon, C. (2022). eLife, 11, e77748.

Postsynaptic burst reactivation of hippocampal neurons enables associative plasticity of temporally discontiguous inputs. Fuchsberger, T., Clopath, C., Jarzebowski, P., Brzosko, Z., Wang, H., & Paulsen, O. (2022). eLife, 11, e81071.

Immature olfactory sensory neurons provide behaviourally relevant sensory input to the olfactory bulb. Huang, J. S., Kunkhyen, T., Rangel, A. N., Brechbill, T. R., Gregory, J. D., Winson-Bushby, E. D., … Cheetham, C. E. J. (2022). Nature Communications, 13(1), 6194.

Humans Can Track But Fail to Predict Accelerating Objects. Kreyenmeier, P., Kämmer, L., Fooken, J., & Spering, M. (2022). ENeuro, 9(5).

Ventrolateral Prefrontal Cortex Contributes to Human Motor Learning. Kumar, N., Sidarta, A., Smith, C., & Ostry, D. J. (2022). ENeuro, 9(5).

Magnitude-sensitive reaction times reveal non-linear time costs in multi-alternative decision-making. Marshall, J. A. R., Reina, A., Hay, C., Dussutour, A., & Pirrone, A. (2022). PLOS Computational Biology, 18(10), e1010523.

Differences in temporal processing speeds between the right and left auditory cortex reflect the strength of recurrent synaptic connectivity. Neophytou, D., Arribas, D. M., Arora, T., Levy, R. B., Park, I. M., & Oviedo, H. V. (2022). PLOS Biology, 20(10), e3001803.

Structured random receptive fields enable informative sensory encodings. Pandey, B., Pachitariu, M., Brunton, B. W., & Harris, K. D. (2022). PLOS Computational Biology, 18(10), e1010484.

Obsessive-compulsive disorder is characterized by decreased Pavlovian influence on instrumental behavior. Peng, Z., He, L., Wen, R., Verguts, T., Seger, C. A., & Chen, Q. (2022). PLOS Computational Biology, 18(10), e1009945.

The value of confidence: Confidence prediction errors drive value-based learning in the absence of external feedback. Ptasczynski, L. E., Steinecker, I., Sterzer, P., & Guggenmos, M. (2022). PLOS Computational Biology, 18(10), e1010580.

Psychedelics and schizophrenia: Distinct alterations to Bayesian inference. Rajpal, H., Mediano, P. A. M., Rosas, F. E., Timmermann, C. B., Brugger, S., Muthukumaraswamy, S., … Jensen, H. J. (2022). NeuroImage, 263, 119624.

Visual working memory recruits two functionally distinct alpha rhythms in posterior cortex. Rodriguez-Larios, J., ElShafei, A., Wiehe, M., & Haegens, S. (2022). ENeuro, 9(5).

Pitfalls in post hoc analyses of population receptive field data. Stoll, S., Infanti, E., de Haas, B., & Schwarzkopf, D. S. (2022). NeuroImage, 263, 119557.

Event-related microstate dynamics represents working memory performance. Tamano, R., Ogawa, T., Katagiri, A., Cai, C., Asai, T., & Kawanabe, M. (2022). NeuroImage, 263, 119669.

Rule-based and stimulus-based cues bias auditory decisions via different computational and physiological mechanisms. Tardiff, N., Suriya-Arunroj, L., Cohen, Y. E., & Gold, J. I. (2022). PLOS Computational Biology, 18(10), e1010601.

Correcting the hebbian mistake: Toward a fully error-driven hippocampus. Zheng, Y., Liu, X. L., Nishiyama, S., Ranganath, C., & O’Reilly, R. C. (2022). PLOS Computational Biology, 18(10), e1010589.

[octochick]

genshin spiderbit

Cellbit - dendro catalyst, 3 attack string, 2 modes, 'pen' mode and 'knife' mode, gets stacks when attacking with pen mode that are used up to buff when the skill is used. burst has the same 2 modes, pen mode gives buffs to the team and knife mode augments his own damage and does lots of damage. damage mainly comes from autos. when has 3 stacks of pen, if the character is changed and is a meele weapon, their attacks will be imbued with dendro. if 'knife' autos come in contact with pyro they will deal extra dendro damage depending on the number of stacks. his autos also have a slightly smaller icd (1 sec instead of 2.5, every other attack instead of every third)

Roier - pyro sword, 4 attack string, each button press is 2 damage instances, skill is a dash like yelan that marks and connects the enemies in a web-like thing. the web periodically pulses, dealing pyro damage and pulling the enemies towards the center of the web. without anything, the web pulses 3 time (7.5 secs), but if a burning or pyro swirl reaction is triggered the duration of the web is extended, up to 25 secs. burst is coordinated attacks like xingqiu or yelan for around 7 seconds. if the attacks hit an enemy marked by the web, the web will pulse, and when the burst ends, the web timer is reset. has a lot about quadratic scaling in his kit

[gayarograce]

🌻

(Oh man, the mortifying ordeal of actually having to pick something to talk about when I have so many ideas...)

Uh, OK, I'm talking about galactic algorithms, I've decided! Also, there are some links peppered throughout this post with some extra reading, if any of my simplifications are confusing or you want to learn more. Finally, all logarithms in this post are base-2.

So, just to start from the basics, an algorithm is simply a set of instructions to follow in order to perform a larger task. For example, if you wanted to sort an array of numbers, one potential way of doing this would be to run through the entire list in order to find the largest element, swap it with the last element, and then run though again searching for the second-largest element, and swapping that with the second-to-last element, and so on until you eventually search for and find the smallest element. This is a pretty simplified explanation of the selection sort algorithm, as an example.

A common metric for measuring how well an algorithm performs is to measure how the time it takes to run changes with respect to the size of the input. This is called runtime. Runtime is reported using asymptotic notation; basically, a program's runtime is reported as the "simplest" function which is asymptotically equivalent. This usually involves taking the highest-ordered term and dropping its coefficient, and then reporting that. Again, as a basic example, suppose we have an algorithm which, for an input of size n, performs 7n³ + 9n² operations. Its runtime would be reported as Θ(n³). (Don't worry too much about the theta, anyone who's never seen this before. It has a specific meaning, but it's not important here.)

One notable flaw with asymptotic notation is that two different functions which have the same asymptotic runtime can (and do) have two different actual runtimes. For an example of this, let's look at merge sort and quick sort. Merge sort sorts an array of numbers by splitting the array into two, recursively sorting each half, and then merging the two sub-halves together. Merge sort has a runtime of Θ(nlogn). Quick sort picks a random pivot and then partitions the array such that items to the left of the pivot are smaller than it, and items to the right are greater than or equal to it. It then recursively does this same set of operations on each of the two "halves" (the sub-arrays are seldom of equal size). Quick sort has an average runtime of O(nlogn). (It also has a quadratic worst-case runtime, but don't worry about that.) On average, the two are asymptotically equivalent, but in practice, quick sort tends to sort faster than merge sort because merge sort has a higher hidden coefficient.

Lastly (before finally talking about galactic algorithms), it's also possible to have an algorithm with an asymptotically larger runtime than a second algorithm which still has a quicker actual runtime that the asymptotically faster one. Again, this comes down to the hidden coefficients. In practice, this usually means that the asymptotically greater algorithms perform better on smaller input sizes, and vice versa.

Now, ready to see this at its most extreme?

A galactic algorithm is an algorithm with a better asymptotic runtime than the commonly used algorithm, but is in practice never used because it doesn't achieve a faster actual runtime until the input size is so galactic in scale that humans have no such use for them. Here are a few examples:

Matrix multiplication. A matrix multiplication algorithm simply multiplies two matrices together and returns the result. The naive algorithm, which just follows the standard matrix multiplication formula you'd encounter in a linear algebra class, has a runtime of O(n³). In the 1960s, German mathematician Volker Strassen did some algebra (that I don't entirely understand) and found an algorithm with a runtime of O(n^(log7)), or roughly O(n^2.7). Strassen's algorithm is now the standard matrix multiplication algorithm which is used nowadays. Since then, the best discovered runtime (access to paper requires university subscription) of matrix multiplication is now down to about O(n^2.3) (which is a larger improvement than it looks! -- note that the absolute lowest possible bound is O(n²), which is theorized in the current literature to be possible), but such algorithms have such large coefficients that they're not practical.

Integer multiplication. For processors without a built-in multiplication algorithm, integer multiplication has a quadratic runtime. The best runtime which has been achieved by an algorithm for integer multiplication is O(nlogn) (I think access to this article is free for anyone, regardless of academic affiliation or lack thereof?). However, as noted in the linked paper, this algorithm is slower than the classical multiplication algorithm for input sizes less than n^(1729^12). Yeah.

Despite their impracticality, galactic algorithms are still useful within theoretical computer science, and could potentially one day have some pretty massive implications. P=NP is perhaps the largest unsolved problem in computer science, and it's one of the seven millennium problems. For reasons I won't get into right now (because it's getting late and I'm getting tired), a polynomial-time algorithm to solve the satisfiability problem, even if its power is absurdly large, would still solve P=NP by proving that the sets P and NP are equivalent.

Alright, I think that's enough for now. It has probably taken me over an hour to write this post lol.

[quiet-art-kid]

The grades I think I would get if I was in the pirate training program

Bbad spelling: 10/10 I would fucking ace that class. My spelling is so horrible that in fourth grade I was put in the “special class” for spelling. I can’t spell their properly for fuck’s sake, I always misspell it as thier. 

Basic burglary: 6/10 I think I’d be pretty good at stealing and even better at gaslighting the victims like “Where is your bracelet? I haven’t seen it. No, you didn’t put it on today. Oh, this? This was a gift from my cousin. Why does it have your name? It's a hand me down duh, my cousin has the same name as you. If you’re that upset about losing your bracelet, I can help you look for it, because I’m just that nice.”

Baseball: 0/10 I will activate my quiet, nerd, introvert cloak of invisibility (standing so still that adults forget I exist)  during all sports. It’s worked before, it will work again. 

Beginner Burping: 2/10 I could do it, just not very well.

Boarding an enemy ship: 1.5/10 - 6/10 This one sounds like it requires a lot of stamina and depending on if I’m in a more manic or depressive episode my grade would fluctuate.  

Dragon training: 1/10 I would have a bunch of fun debating my dragon about the meaning of life and free will. I would not train the dragon, I would educate them. My dragon would be able to do quadratics and tell you why true democracy can never work on a large scale, but would not be able to catch fish.

Frightening foreigners: 5/10 I can be scary when I want to be, however my brand of frightening is…different from the vikings. It would also require quite a bit of effort, half the time I’d just end up being a resting bitch face nihilist. 

Hammer throwing: 1.5 /10 I have been working out regularly, so I could probably pick up the hammer and toss it like ½ a foot in front of me, but not much more. 

Herding reindeer on dragonback: 3.5/10 I would be so-so at it (a bit below because I don’t love getting near large animals) 

Herding sheep: 4/10 I feel like sheep would be easier to get closer to. 

Hunting-With-Bows-and-Arrows-on-Skis: 1/10 I’m a shit archer but not a terrible skier (which would be the only thing saving me from a 0/10) 

Pointless graffiti: 8/10 I would get 2 points taken off for accidentally creating moving and meaningful graffiti. 

Robbery: 4.5/10: I am good at stealth and plans, but I can be clumsy. 

Senseless Violence: 5/10 I am not normally a violent person, but if need be I won’t hesitate to throw hands. My older brother would fight with me when we were kids for fun. I also have anger issues. 

Shipbuilding: 4/10 I would not follow the instructions but through sheer luck, determination, and will power would get a working boat, that is simultaneously an artistic masterpiece. 

Shouting: 1.5/10 I literally can’t raise my voice nine times out of ten because of my trauma. I’d also probably start crying in the middle of class. 

Spitting: 0/10 I’m not even trying on this one. 

Sword Fighting: 4/10 I’d be average at it (just slightly below)

Sword Fighting at Sea: 3/10 I’d be even worse due to the rockiness of the boat. 

Advanced rudery: 7/10 I …watch way too much r/roast me. 

Weaponry: 2/10 I’m not sure what this class entails but I doubt I’d be good at it. 

[typhlonectes]

A new early diverging thalattosuchian (Crocodylomorpha) from the Early Jurassic (Pliensbachian) of Dorset, U.K. and implications for the origin and evolution of the group

Eric W. Wilberg,Pedro L. Godoy,Elizabeth F. Griffiths,Alan H. Turner &Roger B. J. Benson

ABSTRACT

Among archosaurs, thalattosuchian crocodylomorphs experienced the most extensive adaptations to the marine realm. Despite significant attention, the phylogenetic position of the group remains uncertain. Thalattosuchians are either the sister-group to Crocodyliformes, basal mesoeucrocodylians, or nest among longirostrine neosuchians. 

The earliest definite thalattosuchians are Toarcian, and already possess many synapomorphies of the group. All phylogenetic hypotheses imply a ghost lineage extending at least to the Sinemurian, and a lack of older or more plesiomorphic forms may contribute to the uncertain phylogenetic placement of the group. 

Here we describe a new species, Turnersuchus hingleyae, gen. et sp. nov., from the early Pliensbachian Belemnite Marl Member of the Charmouth Mudstone Formation (Dorset, U.K.). The specimen includes partially articulated cranial, mandibular, axial, and appendicular elements. It can be attributed to Thalattosuchia based on the following features: distinct fossa on the posterolateral corner of the squamosal; broad ventrolateral process of the otoccipital covering the dorsal surface of the quadrate; large supratemporal fenestrae lacking a flattened skull table; broadly exposed prootic; orbital process of quadrate lacking bony attachment with the braincase. This specimen represents the earliest thalattosuchian currently known from diagnostic material. 

Phylogenetic analyses of two published datasets recover Turnersuchus as the earliest diverging thalattosuchian, and sister to Teleosauroidea + Metriorhynchoidea. Bayesian tip-dating analyses suggest a Rhaetian or Sinemurian divergence of Thalattosuchia from other crocodylomorphs, depending on topology, with confidence intervals spanning from the Norian to the Pliensbachian. 

The new specimen extends the fossil record of Thalattosuchia, but the time-scaling analyses demonstrate that a significant ghost lineage remains.

Read the paper here:

https://www.tandfonline.com/doi/abs/10.1080/02724634.2022.2161909

[vox-ex]

of hearts and wavelengths

Supercorptober Day 16 “sparks”

thanks to kmsdraws for the prompts

Lena finds a way to bring Kara home from the phantom zone. 

////

It was all a matter of getting the science right. It seemed like it was always a matter of getting the science right.

Step one, figure out the propellant.

Step two, figure out how to start the propellant.

Rocket engines are reaction engines, producing thrust by ejecting mass rearward. She’s come to learn earth science might not be right about everything, but it was right about some things — or at least good enough at some things.

Step three, ignition.

She could have all the fuel and oxidizer sitting around, intermixing all they wanted to, but until she could figure out how to give it that little bump of energy it needed, well...in simple terms, no spark no boom, and no boom, no go.

Lena stared at the equations scribbled on the papers in front of her. Her notes and Brainy's blending together until it was hard to discern one thought from the next. It should be possible. It had to be possible.

But the ship they had was too small, too fragile. Meant for shorter trips that spanned at most a couple lightyears, but never intended to cross dimensions. They needed the fuel to burn hotter and have the engines last longer. It was one thing to get to Kara; it was another to make sure that if they did, there was a way to get back with her too.

She leaned back in her chair and rubbed her eyes, her mind both tired and restless all at once. She stood up and stretched, walking over to the window. She looked out at the city and the sky stretching far beyond it. It was hard not to think about Kara in all that darkness. It was hard not to miss her.

But she was coming back. Lena was going to make sure of it.

It was just a matter of getting the science right.

Lena sat back down at her desk and got to work.

Step three, ignition.

All they needed was a spark.

J'ohn and Brainy had explained the ship's current engines and the way they worked. They were good at getting going. But there had to be a way to keep them firing over and over, to turn the current single-stage engines into multi-stage engines that would keep them going far enough, fast enough.

She needed something big enough but not so big it would burn through everything around it. Something that would be easy to replicate on a larger scale.

How many different ways can you start a fire?

There was always the old-fashioned way. Some kind of heat. But that would take too long on the scale she was thinking of, and there would be no guarantee it would work every time.

She could shove a giant spark plug into the engine system. But she needed to figure out how to contain that much electricity and control it with the precision they needed?

Chemical reactions were messy and unpredictable and could definitely blow them up.

Lena looked back at her notes and started working on the equations again. Arcing lines and inelegant quadratics scribbled across graph paper, her hands trying to force the numbers and patterns into answers that her heart so desperately wanted.

She was so focused on her work that she didn't hear Brainy come in until he spoke.

"Lena?"

She looked up at him, a little startled. "Hey, Brainy, sorry I didn't hear you come in."

"It's quite alright. I didn't mean to disturb you." He paused for a moment before continuing. "I just wanted to check in on you."

She leaned back in her chair and rubbed her eyes again. "Just trying to figure all this out."

Brainy stood in the doorway, hands fidgeting in a way that reminded her a little of Kara.

He took a step into the room and then back out. Hesitating. Unsure.

"May I?"

He pointed to the papers on her desk, the ones she

"Of course; I mean, I'm not sure there's anything new or useful, but please."

Brainy leafed through her notes while she watched. He stopped at one page in particular and stared at it for a long time before looking up at her.

"This could work."

She frowned, not understanding what he was talking about. "What?"

"Your idea; it could work." He pointed to the paper again. "With some modification, of course, and ensuring we can find a way to contain it."

Lena's heart started beating faster.

"But this could work," he said again, looking up at her, "this could definitely work."

Lena reached out, taking her notes as Brainy handed them back to her. She looked at them as if she had never seen them before. As if she hadn't been the one to write them.

Brainy was right; with some modification, the idea could work. They might actually be able to make this engine. They might be able to bring Kara home.

Lena felt a small smile spread across her face, and she leaned back in her chair, the aching familiarity of hope tugging somewhere deep in her chest.

"And if I may...It seems quite fitting too."

Lena raised an eyebrow, not sure what he was getting at.

"Look at the wavelengths," he said simply, pointing to her scribbled notes.

She looked down at the numbers she had been going back and forth with for so long.

And suddenly she saw it. The one solution she had apparently been overlooking.

The last way to start a fire.

Light — photon energy.

Photon energy tuned to the wavelength needed to excite the propellant molecules directly.

The same wavelength as brightest of the Sun’s rays

The same wavelength of Kara's heat vision.

"I think she would approve," Brainy said softly.

Lena swallowed hard, "Yeah," she said, her voice rough, fingers tracing the numbers on the paper as if to make sure they wouldn't suddenly disappear.

Step four trajectory.

If figuring out how to go was hard, the coming back was harder.

Because it turns out the science was the easy part.

The detachment of knowing what to do and how to do it, replaced by the weight of what it means to actually go through with it.

The idea of seeing Kara again, the feeling of having her there in front of her once again.

Bleeding, aching, crying — breathing, whole.

Lines and numbers and graphs and formulas can't won't ever be able to tell you what that will feel like. Won't ever be able to predict the trajectory of your heart with the same precision as an object hurtling lightyears through space.

There is no calculus for that.

But the gravity pulling her into Kara's arms now is the same gravity that dragged her across space.

And that feels like proof of the kind of elegant universe she didn't always believe in but always had hoped for — one infinitely more complex and infinitely more simple all at once.

[eonars]

in wow im like hmm ok given unholy death knight's quadratic scaling i need to save army for a large pull and also save mirror for army and use unholy assault right at the start so i have enough wounds to instantly pop apoc and get the 4set bonus magi and make sure to put defile down before building my wounds back up to maximize cleave and then in bg3 ive just been like idfk man one of you go over there and handle it

[intimidating-fettuccine]

Me to Slender (just to fuck with him) be like:

On a scale of 1 to 10, what is the quadratic equation to Einstein's theory of gravity?

.

- 🐺

Me who is too tired to even begin to process any of those words myself

[msexcelfractal]

Hausdorff dimension in fractals

Doubling a line makes 2 lines. Doubling a square makes 4 squares. Doubling a cube makes 8 cubes.

Have you heard of the square-cube law? If you double an object's linear dimensions its volume is multiplied by 8, but its surface area is multiplied by 4. For an object of dimension D, doubling its linear dimensions multiplies the object by 2^D. D is the object's Hausdorff dimension. Unlike definitions of dimension that rely on orthogonality, D can take non-integer values. And it does, for some of your favorite fractals!

This Sierpiński gasket is made of 3 subunits, each of which has half its linear dimensions. That means when you double a gasket's linear dimensions, you triple its area. That's more than a line gets but less than a square gets. The gasket has dimension log_2(3) ≈ 1.585. This fractal is one-and-a-half dimensional!

One gets the sense that Sierpiński's carpet is more substantial than his gasket. It's composed of 8 subunits at 1:3 scale, so 3^D = 8 and D ≈ 1.893. It's almost 2-dimensional. Much closer than the gasket at least.

Oh god, act casual. It's elementary automaton #150. Want to take a crack at it? Answer under the cut.

This animal is made of 2 half-scale subunits and 4 quarter-scale subunits. If we let the little subunits have area 1, the big subunits have 2^D and the whole shebang is 4^D. We obtain 4+2*2^D=4^D. The quadratic formula gives 2^D=1+sqrt(5), so D≈1.694.

Now I intend to get #150 tattooed on my torso. If a 3 x 3 in tat is $100 then by my calculations the 6 x 6 in should be 100*2^D = $323.61 instead of $400. But hey, I don't make the rules!