## On Sci-Fi Horror (or The Attack of the Nexocytes)

When I was in high school, I really wanted to be a writer. I knew I’d have to write nonstop and suffer for my work if I wanted to be successful at it, which probably explains why I wound up becoming a scientist—it’s like being a writer in the sense that you have to write all the time and that your career prospects are realistically garbage, but different in the sense that you don’t get a cultural pass to wear scarves all the time and smoke cigarettes outside of snowy cafés without getting branded as a massive tool.  But even though I’ve read as much Flaubert and Forster on the subway as the next guy (and you better believe I was rocking scarves and a pensive look when I was doing that), I’ll always be a big fat sucker for outlandish sci-fi horror. Whether it be about a group of hyperintelligent seagulls and shark-punching frogmen handling a worldwide birdpocalypse caused by an Egyptian god or a concept-eating basking shark let loose in a hyperdimensional Portland, I always wanted to someday write about an utterly bonkers world-ending scenario or something of that ilk. And in this entry, I’ll try to pitch you my idea for it—the nexocyte.

Cue scary trumpets.

One of the longest-lasting sci-fi horror tropes consists of self-replicating nanobots; nanoscopic little machines that are able to create new copies of themselves without dying in the process. These little nanobots are usually referred to as “grey goo“, and one recent execution of this concept was in the (profoundly mediocre) 2008 remake of The Day The Earth Stood Still. Take a look at the scene involving them below and enjoy one of the few sci-fi action scenes that portrays lab safety protocols semi-accurately:

This flawless self-replicator concept has been around for ages, but for all its countless variants, it’s always stuck to the same simple formula; after they accumulate enough “food”, they’ll generate a copy of themselves while leaving the original “source” individual intact.

Although this seems harmless enough, you’ll find that the number of grey goo nanites in your tacky secret military bunker can get real big real fast. Take it from the guy who invented the concept:

Imagine such a replicator floating in a bottle of chemicals, making copies of itself…the first replicator assembles a copy in one thousand seconds, the two replicators then build two more in the next thousand seconds, the four build another four, and the eight build another eight. At the end of ten hours, there are not thirty-six new replicators, but over 68 billion. In less than a day, they would weigh a ton; in less than two days, they would outweigh the Earth; in another four hours, they would exceed the mass of the Sun and all the planets combined — if the bottle of chemicals hadn’t run dry long before.

But note that these replicators are, in some sense, “dumb”; their reproductive process is entirely independent of the state of the group, which you would expect if each of these little robobugs didn’t know anything about the group/collective/colony it belongs to. But what if it did?

Let’s consider some strange new type of replicator, which I’m going to call a nexocyte, that does know the state of the colony it belongs to thanks to some kind of intra-colony communication network. As a result, its reproductive process can be informed by the state of the colony, and in fact, it may try to clone the colony itself through its (presumably long and arduous) reproduction process.

Although the illustrations don’t make it seem like the nexocytes and the nanites are too different (only a factor of 2 off after the second reproductive cycles), the difference very quickly adds up when you do the math. In fact, I’ll do the math for you:

Look at thatthe nexocytes have already hit 65,000 while the nanites haven’t even clocked 50. And, just in case you’re curious, the total number of nexocytes after the sixth reproductive cycle is precisely 18,446,744,073,709,551,616. Let me repeat that; after only six reproduction cycles, the nexocyte colony numbers 18 quintillion, 446 quadrillion, 744 trillion, 73 billion, 709 million, 551 thousand, six hundred and sixteen individuals.

To show precisely how catastrophic the existence of such a replicator would be, let’s envision a scenario like the one quoted above, where each nexocyte has the same mass and volume as an HIV virus, and replicates freely without any concerns for food, chemicals, or the laws of physics. If the timing for the reproductive cycles is the same as above (1000 seconds between each), here’s what the result of each cycle looks like:

### Cycles 0-5 (0 seconds to 1 hour and 23 minutes)

Everything here is still well in the microscopic range; even though there are a huge number of nexocytes in our colony by Cycle 5, they’ll collectively be around the size of a dust mite and weigh accordingly tooalmost completely imperceptible.

### Cycle 6 (1 hour and 40 minutes)

Now we’re getting somewhere. After the sixth reproductive cycle, our little colony is not so little anymore, weighing in at about 40 pounds and measuring up to the size of a decent paint can (4.5 gallons). For some unlucky sci-fi horror protagonist, it will probably be very shocking to see a weird paint can-sized lump of goo pop out of (what appears) to be thin air, but all in all this isn’t so bad! If our protagonist doesn’t get laughed off by emergency services, the authorities might be able to close in on this thing just as it undergoes its next cycle 16 minutes(ish) later. That’s fine, though—I mean, how big could this thing get?

### Cycle 7 (1 hour and 57 minutes)

Goodbye protagonist. The nexocyte colony after Cycle 7, if it keeps a nice spherical shape, will be about 840 kilometers wide and weigh in at a whopping 3.4 *10^20 kilograms, which is enough to wipe almost all of the state of New York off the map. Our cute little “nexosphere” is now comparable to the dwarf planet Ceres, and our colony just about makes the cut to be called a dwarf planet (if it was zipping around in orbit instead of crushing our protagonist’s internal organs). This will definitely grab your omniscient secret government of choice’s attention, and if we’re very lucky, they’ll get their ducks in a row and let more than a couple of ballistic nukes rip on this thing by the sixteen minute mark. Because if not…

### Cycle 8 (2 hours and 13 minutes)

…goodbye solar system. Our former little dwarf planet is now a sphere bigger than most small galaxies and about a hundredth the width of the Milky Way, with a radius of 309.1 light-years. That’s right: it would take light 309.1 years to travel from one end of our nexocyte colony to the other. Luckily, it won’t splat over the entirety of the Milky Way, and will definitely (and finally!) grab the attention of the advanced alien civilization of your choice. And luckily, they won’t even have to do that much to get rid of the nexocytes either!

Because they’ll become a black hole.

See, it turns out that our galactic-sized ball of nexocyte is sufficiently massive (35,000 times the weight of the entire observable universe) to cause it to become a black hole orders of magnitude bigger than any black hole that could ever plausibly exist. Our alien friends will simply have to enjoy whatever of life’s simple pleasures they can get before they’re sucked into the black hole/thrown off their orbit/fried with ionizing radiation/etcetra as the Milky Way slowly but surely collapses from the sudden disruptive presence of the nexocyte singularity.

So there you go; our colony of nexocytes may not be as long-lived as the nanites’, but we did get to destroy the Solar System in 2 hours, 13 minutes and 20 seconds (with the Milky Way getting sucked in soon after).  Luckily for us, this kind of replicator is completely implausible, because it would take too much time and resources for it to replicate its colony! The only way this replicator concept would ever even potentially get off the ground is if it existed inside of a medium with nearly unlimited amounts of energy compared to each nexocyte’s energy consumption, and where information exchange between nexocytes could be extremely fast and efficientand even then it would just wind up wrecking the place at breakneck speed. Good thing a place like that doesn’t exist, huh?

## On Pointless Pastimes

Everybody has a strange pastime. Whether it’s something routine, like watching really old cartoons, or something more exoticlike intentionally calling every barista you interact with “Greg”—these quirky habits have the tendency of inexplicably making you a little bit happier than you were before (and potentially causing others to look at you like you escaped from an insane asylum). The different kinds of weird hobbies available to us has grown exponentially in the modern age, and you can basically be sure that for any kind of bizarre activity, you can find a poorly maintained hobbyist forum from the early 2000’s for it. (Haven’t had any luck finding a community of people who intentionally misname baristas, though.)

Some of the oddest pastimes take something easy and come up with a way to make it really hard for no apparent reason. For example, ironing your clothes is usually very boring, but you’ll probably get a bunch of YouTube views if you film yourself doing it while hanging off of a cliff. For the people who pursue these, a lot of the motivation comes from being the first or only person to succesfully do that strange specific action; you can rest assured that the guy who created the largest ball of paint by repeatedly coating a baseball won’t have his record taken from him anytime soon, even though I can make an even bigger paint ball by just pouring a bunch of paint into a spherical tank and letting it dry.

For this blog post however, I want to focus on two particular odd pastimes of this type that piqued my interest, the first of which I like to call constrained Super Mario 64. The gist of it is that a very dedicated group of gamers decided they would try and see whether or not they could beat specific levels in Super Mario 64 without ever performing important actions, like moving the joystick or pressing the A button. In fact, one particular guy has been trying for years to solve the “open problem” of figuring out how to beat the entire game without ever pressing the A button. It may be the case that doing so is impossible, but no one can say the guy hasn’t tried: just take a look at his channel in the hyperlink above and you’ll see all of his insanely complex attempts at beating levels without pressing the A button.

My personal favorite, and the video that made this guy Internet famous, is his successful attempt at beating this one specific level without pressing the A button. (Technically, he’s left it pressed since before entering the level, but we don’t need to be too pedantic.) Words can’t describe the amount of effort, dedication, and ingenuity he spent on doing this: you’ll have to see this work of art unfold for yourself. The video explaining his techniques below is about a half-hour long; you can find the much shorter uncommentated version here. If you do decide to watch it though, buckle up.

I was literally more excited watching the execution of this than I was when they found the Higgs boson (and I saw it live!). The fact this man was not immediately hired by NASA to coordinate rocket launches after the making of this video convinced me that there is no such thing as cosmic justice. If I could take any one person on an all-expenses-paid trip with me to the Bahamas, I would either take this guy or my favorite barista Greg.

In any case, I have nowhere near the skill or technical know-how to play Super Mario 64 like this, and every problem of this type in constrained SM64 that’s considered difficult has probably already been described; after all, there are only so many buttons you can’t press. As a result, if I want to get famous off of a weird pastime, I need to find some other strange activity which has undiscovered problems to solve, and that brings me to the second topic of this blog post: number theory.

Number theory is the study of numbers (great writing, Arnaldo), in particular the study of groups of numbers and facts about them. Some facts are easy to show, and some aren’tbut luckily for me, number theory has a massive amount of undiscovered problems! See, number theory is just like constrained Super Mario 64; it is extremely difficult, very interesting, mostly fun, and largely pointless (except for some key applications in cryptography*). The key difference, though, is that there’s only so much Super Mario 64; there are no limits to the amount of numbers and number groups.

*If you get all 120 stars without pressing the A button, you can find Yoshi on the castle roof and he’ll give you the private key to an offshore cryptocurrency wallet.

Perhaps the best thing about problems in number theory is that, as long as it’s not easy and it’s not impossible, I can basically claim some arbitrary unsolved problem is as important as any other famous problem because no problems are really “important” in any concrete objective sense. It’s like saying that beating a Super Mario 64 level without moving the joystick is more important than beating it without pressing the A button; one or the other might be easier, but they’re both pretty damn impressive, and doing either is ultimately pointless.

Easier said than done, you might think. Well, why don’t we actually take a crack at finding an “important” number theory problem? Let’s give it a shot by following these key steps:

1. Find topics that are “hot” in number theory.
2. Find an arbitrary specific problem involving these “hot” topics.
3. Show this problem isn’t easy or equivalent to another known “important” problem.

We first need to look at what’s “hot” in the field of number theory, and perhaps the hottest topic in number theory is the study of what are called prime numbers. (It’s so hot that Wikipedia has an entire section on unsolved prime number theory problems!) These are numbers that can’t be divided by any number other than 1 or themselves without creating a bunch of decimal gunk. An example of a big number that’s prime is 89: try dividing it by any number other than 1 or 89 and you’ll always get a number with stuff past the decimal point. For clarity, the first few prime numbers are:

$P_{i} = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, ...]$

Another hot topic is the Fibonacci numbers; these are a bunch of numbers on a list defined so that the next number on the list is equal to the sum of the two last numbers. By defining both the first and second Fibonacci numbers as 1, the list of Fibonacci numbers begins as:

$F_{i} = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...]$

Both prime numbers and Fibonacci numbers have been studied to death, and pop up very often in pop-math books and related media; I even vaguely recall seeing the Fibonacci numbers show up in The Da Vinci Code*. However, one thing that isn’t known (and is considered a well-known “important” problem to number theorists) is whether or not there are an infinite number of prime numbers that are also Fibonacci numbers. We can certainly spot a few prime numbers in the starting Fibonacci numbers I listed: 2, 3, 5, 13, 89, and 233.

*I always assume that science and math topics reach “peak pop-sci” when featured in a Dan Brown book. I send Mr. Brown emails every day about how cool low-Reynolds number fluid dynamics is, but he hasn’t taken the bait yet.

Anyways, figuring out the number of primes that are also Fibonacci numbers is a well-known problem; in order to come up with a new problem, we need to be a little bit more specific. Let’s think about the following list of related (and completely arbitrarily defined) numbers:

Start the list off by picking some prime number $a$. Pick the next number on the list by finding the $a$-th Fibonacci number. Then find the Fibonacci number corresponding to that number and put it on the list.

That’s it! This is just a list of specific Fibonacci numbers. To get a more intuitive sense of this list of numbers, I’ll call this list of numbers the “pointless sequence” $T$ and start rattling off the first couple of numbers on the list if I pick, say, $a = 7$:

$T_{1} = 7$

$T_{2} = F_{7} = 13$

$T_{3} = F_{13} = 233$

$T_{4} = F_{233} = 2211236406303914545699412969744873993387956988653$

Jeez, that got out of hand really quickly! It seems like our arbitrary list is pulling in big numbers even at the start. But that’s great for number theorists; the bigger the numbers involved, the more difficult it is to deal with them, and the more challenging and “important” a problem is. One thing you may notice is that, if I pick $2$ or $3$ as my starting value, this sequence of numbers will just eventually start spouting out $1$ forever. If I picked $5$, it would just keep spouting out $5$ forever, but if I pick any prime number bigger than that, I’ll start seeing the crazy blow-up we saw for $a = 7$.

Another thing you may notice is that those first three numbers on our list for $a = 7$ are prime! (The fourth one unfortunately isn’t.) We can then ask pointless questions about this list of numbers and hope we hit on a tough one, like if there exists an $a$ other than $5$ so that every number on this list would be prime. Because mathematicians don’t like problems in the forms of questions, we can guess that this isn’t true and reduce our problem to answering whether or not the following pointless conjecture is true; I’ll even put it into videogame format to make it pop a bit more.

Now that we have Step 1 and 2 out of the way, let’s proceed to Step 3 and check if this problem is easy or equivalent to another problem, particularly the “important” problem of whether or not there are an infinite number of Fibonacci primes. If there was a finite number of Fibonacci primes, then this sequence would have to eventually hit a non-prime and our problem would be solved. (Lucky for us it’s unsolved!) However, if the number of Fibonacci primes was infinite, it wouldn’t tell us anything about whether or not our list of specific numbers would eventually have a non-prime, which means our problem isn’t equivalent. Score!

So we know verifying the conjecture isn’t equivalent to this other problem, and we know that showing it’s true isn’t easy (because we’d solve this other nasty problem about infinite Fibonacci primes if we did). However, we need to figure out if showing that it’s false is easy, and that isn’t something we can check straightforwardly; so we’ll just have to drop it onto a math forum like stackexchange and see if anyone berates us for wasting their time on an easy problem.

Once we’ve completed all three steps, now we have to go through the hard process of actually trying to solve it; and for that, there’s no steps or rules other than staying dedicated, being creative, and enjoying it every day. In my case, I think it’ll probably be best if I just stick to calling baristas Greg.

## On the Benefits of Being a Dumb Tourist

I’ve stayed at a fair share of different places over the last few years, and using public transportation takes the cake for being the most stressful and annoying day-to-day experience in every place I’ve been to. From riding 5-and-a-half hours every week in a packed Chevy Astro through hot Puerto Rican highways to starting your workweek at Berkeley with the fresh sight and smell of body parts, I’ve never had a positive relationship with public transportation (and don’t expect that to change anytime soon). However, for someone who can’t afford to buy a car—and who is universally described as driving “like a grandmother politely trying to get to the hospital while having a heart attack”—it is a regrettably indispensable part of my life.

As a result, I’ve had to spend a considerable amount of time thinking about how to maneuver the crowded Roman trains and smelly New York buses, and have stumbled onto some weird tricks that might be of use for both tourists and daily commuters. For this post specifically, my intent is to show you the “paradox” that, when trying to get on a packed metro train, being a dumb tourist is better than being a smart one; and I’m going to do it by using something just as annoying, stress-inducing and indispensable as public transportation. Statistics.*

*Cue Inception horns and distant screams.

If mathematics were a family, probability & statistics would be the bizarre great-uncle that won’t stop talking about how taxidermy is a spiritually fulfilling hobby at the dinner table. It is a field of study that is simultaneously too trivial for “real” mathematicians (they’re too busy writing proofs no one understands) and so strange that one of the best mathematicians of all time didn’t believe a simple statistics result until someone showed him a computer simulation proving it. But rather than go into any detailed description of this curious field of math, I’ll just give you a small primer on the basics of this strange field before we delve into any commuting weirdness.

Perhaps the two most important pieces of information in the statistical sciences are the long-term average and the single likeliest outcome. The names are pretty straightforward, but just in case, I’ll explain them with a six-sided die.

1. The single likeliest outcome is just that. For one six-sided die, there isn’t any single likeliest outcome because you have an equal chance of getting any number between 1 and 6 (unless you’ve been loading your dice, you cheater). It’s easy to spot in an outcome graph, because it’s the outcome that happens the most.
2. The long-time average is a little more detailed, but not very: it’s the average of your results after you obtain a very large amount of them! For a single six-sided die, that number is 3.5. You can’t spot this one in an outcome graph, but you can deduce/guess it if the shape is simple.

Now that we’ve got our statistics bases covered, allow me to illustrate the promised “dumb tourist paradox” through my experience living in the Bay Area. Trying to get on a BART train (the Bay Area’s metro system) during the busy hours was mostly a game of chance; you had to hope you picked a waiting spot close to where the train door lands or you’re looking at a 15 minute wait for the next one to roll in.

However, let’s say you knew that the train door always pops up within the same 100-foot strip of train station, but you don’t know exactly where. Assuming there’s an equal chance of it showing up anywhere in the strip, the instinctively smart thing to do would be to always wait smack-dab in the center of it; that’s the position that puts you closest to the train door in the worst-case, and it certainly feels like it’s your best bet.

In this scenario, you might claim you’re making the smartest choice, so let’s call this the smart tourist scenario. Now, instead of using some fancy math theorems to tell you what the most likely distance and long-term average distance are in this case, I’m going to be 100% thorough and actually simulate it! Let’s take a look at what being a smart tourist comes out to when you simulate the train arriving a million times:

There are two things to take away from this graph. First, since the graph indicates that the train stopped everywhere about the same number of times, there’s no single likeliest outcome. It’s equally likely for the train door to land right in front of you than it is for it to wind up 50 feet away! Second, if you used the train over and over, your average distance from the train door would be 25 feet (which you could calculate by finding the average of all the distance outcomes). Nothing unexpected here.

Now we’re going to go into “paradox” territory. Let’s say you take a page from your weird great-uncle’s book and, instead of carefully planning things out, you just decide to randomly pick a spot inside of the 100-foot strip to wait in.

In this case, you’re not making any decision at all about what’s best or not; you’re just randomly waiting somewhere. Let’s call this the dumb tourist scenario, and here’s what that looks like when you pick random spots a million times:

The simulations don’t lie: the likeliest outcome now is that the train stops right in front of you, and the average distance between you and the train will be about 33 feet.

Comparing both scenarios, there’s nothing weird going on if you commute all the time; the long-time average distance is bigger when you randomly wander around the train station (33 ft) versus when you wait in the middle (25 ft), so doing the smart thing is still your best bet in that case. But, when you’re a tourist and only plan on riding the train once or twice, this somehow seems to imply that it’s better to randomly pick a spot to wait in than to pick the best logical spot!

This “dumb tourist paradox” is profoundly counter-intuitive on many levels; how can a “dumb” action turn out to be better than a “smart” one? How can my random action cause the train to usually arrive closer to me? How can I understand this result intuitively? Well, I could try to calm you down by pointing out that being a dumb tourist has two negatives, which is that 1) your long-time average distance is larger and that 2) you have a nontrivial chance of having the train show up more than 50 feet away from you, which is impossible for the smart tourist. If you’re like me, though, you are probably still very puzzled.

The answer, however, is pretty mundane—even though it’s certainly true that the single likeliest outcome is that the train door stops directly in front of a dumb tourist, whereas there isn’t a likeliest outcome for a smart tourist, the actual chances of the train arriving directly in front of a smart tourist and a dumb tourist are effectively the same. As a result, you can’t actually game the system—it just ultimately looks like you can. If that wasn’t obvious to you, you can take solace from the fact that the smartest man who ever lived once said that “in mathematics you don’t understand things, you just get used to them”, and my advice is the following: get used to it. This is by no means the only “paradox” in the statistical sciences, as great many others are known to exist, and they’ve puzzled everyone just as much as this little factoid does. The best thing you can do is to learn about them and why they happen so that you don’t get surprised by them (or more importantly, make wrong assumptions because of them). And who knows! With time you may find some new ones yourself, if you decide to formally study statistics—or if you commute enough.

## On Writing Nonsense and Getting Away With It

Roald Dahl was a master of the written word, and this was perhaps most exemplified by his ability to use nonsense words like “flushbunkled” and “frothbuggling” without making the reader question whether or not they are viewing the product of an elderly Welshman having a mild stroke on his typewriter. Regardless of how silly they sound to us, such nonsense words have a rich history in the English language; in fact, they form a large part of it! Back in the 16th century, Shakespeare is claimed to have invented over 1700 words that are sure to have made a few English eyes squint back in the day. Examples include fracted, propugnation, and fairly hilariously, elbow. (What the hell were they calling elbows before Shakespeare came along? Did the English just point to their elbows and go “I have some pain in my…um…well, you know what I mean”?)

In any case, this type of creativity is not limited to just dead English writers, as mathematicians have often attempted to transcend the boundaries of the established and the intuitive to make use of similar nonsensical concepts in their equations. In this entry, I’ll talk about the most commonly dreaded example of this; the often-frightsome complex/imaginary numbers.

Whenever imaginary numbers were brought up in high school math class, I’m sure mostly everyone wanted to leap up dramatically from their desk and shout “Why the hell are we studying imaginary numbers? What is the purpose of this? Why don’t we learn things like how to balance a checkbook/do taxes/apply for a job?”

If I were a high-school math teacher*, my response would be “You’re right! You shouldn’t be studying imaginary numbers.” There really is no reason for a general high-school math course to cover them, and the discussion of this kind of thing is best left to STEM-track math courses for everyone your class liked to hoist from the school flagpole. I would be happy to leave you to your Balance a Checkbook 101 class, where you can revel in the fact that you are taking a class for something that is both mind-numbingly tedious and so conceptually simple that you learned all the skills to do it when you were 8 (except for the skill to realize that you don’t need a state educator to remind you how to add and subtract).

*I tragically don’t qualify to be a high-school math teacher, as all high-school math teachers are mandated by the state to have bushy mustaches, square-rim glasses, and an unironically ugly wool sweater. (Wool sweaters are expensive.)

The point of this entry, however, is not to tell you whether or not you should know about imaginary numbers; in fact, I’m not even going to try and explain what they are. My goal here is to try and explain why they’re useful to Melvins like me. And, like your high-school math teacher trying to explain why model trains are a fun and interesting hobby, I probably won’t do a good job of it.

The gist of it is that, like nonsense words, the importance of imaginary numbers lies not in what they are but rather what they do; how they interact with the rest of the normal parts of the medium, be it literature or mathematics. It doesn’t matter what the Gizzardgulper meant when he squawked “I is slopgroggled” in The BFG, it matters what this implies about the giant’s ability to speak the English language and the richness of context such a simple statement can provide to a book. In the same sense, imaginary numbers would just be some daydream an Italian guy had in the 15th century if they didn’t let us use the very simplest tools in math (adding, multiplying, etc.) to perform some interesting and useful tricks.

Take, for example, what happens when you multiply $2$ by itself some number of times. (I’ve graphed the results below.) Nothing strange or unexpected here; feel free to check that $2*2*2 = 8$ and $2*2*2*2*2 = 32$.

Now let’s try to do the same thing for the imaginary/complex number $2^{i}$. What $2^{i}$ actually is doesn’t matter; what matters is what the values of the multiplications are once I get rid of all the gunk that has $i$‘s on it.

See that? The value is oscillating! We’d see the same thing if we tried multiplying something like $3^{i}$ or $4^{i}$, except the frequency of the oscillations would change.

As it turns out, the chief usefulness of imaginary numbers is that they make it very easy to describe things that oscillate*, and this is what makes them show up everywhere from electrical engineering (currents in wires tend to oscillate, hence why AC stands for alternating current) to quantum mechanics (the central object of QM has wave-y behavior). Imaginary numbers are not required to describe any of those phenomena, but trying to avoid them requires altering your math so much that you generate things almost as nonsensical as them anyways. Attempting not to use them because they don’t “feel” right is just as gauche and annoying as it would be for you to keep calling your elbow “the thing that connects your upper arm thingy to your lower arm thingy”.

*Clever mathematicians have found other surprising uses for complex numbers, such as finding the areas under curves that are mathematically difficult to deal with, but these applications are more esoteric than anything.

Imaginary numbers are certainly not the only nonsensical objects that mathematicians have come up with; they stand in company with a bounty of strange concepts that have been invented over the years, like numbers that represent the size of infinities or numbers that aren’t really numbers at all. And the fact you can’t conjure up an image of $2^{i}$ apples shouldn’t deter you from thinking these ideas are somehow different from the numbers you’re familiar with! After all, Shakespeare also invented words like moonbeam, submerge and obscene. These words would’ve sounded just as strange to 16th century Englishmen as fracted and propugnation; the only reason we find them normal is because we’ve always used them. If we did the same for complex numbers, it might be possible for us to easily imagine balancing $2^{i}$ apples on those jointed pointy things at the end of our hands.