Solutions Manual II: Separation and Combination
This is the second installment of Solutions Manual. The problem this week:
PROBLEM: Is it easier to put things together or split them apart? Why or why not?
This problem was inspired by the sequence of events after a vase shatters: shards fall, they hit the ground in a mess. This is natural. What if, instead, it was natural for the shards to rise, gathering into a vase? Or is the splitting vase also a fusion—of the specific arrangement of shards on the floor? Were the shards drawn apart, or were they simply drawn towards the ground? Forces of attraction, forces of repulsion. How, considering both, can we view the world’s array of destruction, creation, combinations, and separations?
SOLUTION 1 (Daniel Li, Princeton ’17, Mathematics):
Let’s use number theory to approach the problem. A previous post in this series alluded to the difficulty of factorizing integers as an example response to this question, but that’s an algorithmic take. We’ll use a theoretical viewpoint instead.
Computational difficulties aside, the fundamental theorem of arithmetic splits every positive integer n into a product of primes:
With this decomposition in hand, we’re naturally led to wonder whether we can answer number-theoretic questions by
- first answering them for each prime p individually,
- then putting all these answers together.
Part 1 of this strategy leads to the world of p-adic numbers, an alternative number system where surprising equations like:
hold true (for p = 2)! Given tricks like these, it may come as no surprise that mathematicians have been quite successful at carrying out part 1. However, part 2 is much harder. As an illustration, consider the Langlands conjectures, which predict a deep correspondence between number theory and harmonic analysis. In the p-adic world, the Langlands conjectures for all dimensions were proved in 2001, but in the world of the integers, they have only been fully proven in the one-dimensional case — and that was back in 1927!
SOLUTION 2 (Rebecca Lowy, Princeton ’16, Geosciences):
Let’s start with the difference between a physical and chemical change. A physical change is like a salad — even after mixing your ingredients, you can still take your tomatoes away from your lettuce. Equivalently, salt will dissolve completely in water; but pop it on the stove, and soon enough, you’ll be left with the salt, no worse for wear. It’s easy to toss together a salad (as it’s easy to dissolve salt in water), but more difficult to pick out your favorite ingredient (and even more difficult to get your salt back). But separation can be done, just with more time and energy.
A chemical change shows this more starkly; heat up some sugar in a pan (C12H22O11 + ΔH), and it turns into caramel — and I promise that you won’t be getting your plain old sugar back any time soon. Chemical changes are like pizza; it’s impossible to get fresh mozzarella from a melty slice. The point is simple; in chemistry, it is nearly always far more difficult to split things apart than to put things together. In some cases, it is literally impossible. That’s not a bad thing though; this tendency towards difficult separation is what keeps your salt from exploding in your water, and what provides us with amazing new metals, pigments, and medicines every day — the hope that throwing together a few ingredients and adding some heat might give us the best pizza ever.
SOLUTION 3 (Ryan Budnick, Princeton ’16, Linguistics):
Where do new words come from? Many are borrowed from other languages (“algebra,” “kangaroo”), some invented whole cloth (“zip,” “googol”), but most are built from existing parts of a language.
For example, we regularly combine two words, like in “blackboard” and “sleepwalk.” The parts don’t even have to be full words: “darkness” and “sadness” both contain “-ness,” which can productively form new words, like “wokeness.” Two adjacent words can fuse after repeated usage, giving words like “because” (“by+cause”), “already” (“all+ready”), and “gonna” (“going+to”).
We also split words where they were previously jointless. “Donation” was borrowed from French, but reanalyzed as “donate+ion” by analogy with words like “inflation” (“inflate+ion”), creating the new word “donate.” “Sass” is similarly derived from “sassy,” and “burger” from “hamburger.”
Which is easier? Fusing words is more common — so much that linguists have an aphorism: “Today’s morphology is yesterday’s syntax” (what used to be separate words are now parts of one word.) Some asymmetry can be explained by how we learn new words. Fusions arise through under-parsing (failing to see joints in a phrase), while fissions arise through over-parsing (finding imaginary joints in words). Efficiency and laziness cause conservative parsing and word fusion.
SOLUTION 4 (Siyi Reed, Rhode Island School of Design ’16, Painting):
I’ve taken pictures of myself wearing clothes I don’t wear anymore. Some I keep for sentimental reasons. I look at myself and think about what my image might translate to another person. How does clothing matter, in the presentation of ourselves? Is there something about those clothes that doesn’t match the image of what I want to be perceived as, versus the image I think they create? I was choosing clothes based on what I thought what was supposed to “fit” my body and my image, but of what? I’ve started to become comfortable wearing clothes where I was drawn to their presentation, more than if they fit “me”, since I couldn’t decide what I was. Seeing myself wearing the clothes I don’t find suitable, gave me a clear perspective of the image each one provided. Some of them I could easily let go of, and some still drew my attention. This prompt became a system of comparison of any subject I applied it to, to observe where my value is placed within the variables I have. Something is put together, and another thing is simultaneously taken apart. So I can see clearly, how important each thing I’ve made is.
SOLUTION 5 (Aditya Trivedi, Princeton ’16, Physics):
Physics tells us that it is always easier to split things apart than to bring them together. The Second Law of thermodynamics states that the amount of entropy in a closed system will never decrease. Entropy is a measure of disorder — given a group of objects in a system, the system will always favor the configuration that allows the largest possible number of object-states. Thus, if you take a deck of cards out of it’s pack and throw it on the ground, it’s highly unlikely for it to remain in the stacked deck position — the cards will likely spread all over the place. Nuclear physics gives us more insight into this question. Nuclear fission — or splitting an atom into two — is much more viable as an energy source than smashing two atoms to create another, heavier atom. Fission can easily create a chain reaction — the energy released by splitting one atom goes towards splitting several more, so on, so forth. In contrast, the energy required to speed up atoms in fusion is so high that a commercially viable chain reaction has not yet been developed, and likely won’t be for several decades.
SOLUTION 6 (Cissy Chen, Princeton ’16, Computer Science — using Fiction lens):
I’m watching through the window my favorite dance. There’s no music, no sound. Two humans approach each other, bodies bent like ferns, faces beaming from a small light source in their hands. They’ve been moving closer to each other while still centered on their own little suns. They meet each other on the sidewalk, one on the left, one on the right. They almost graze arms, and their backs straighten — the story of my two dancers begins.
A whiff of her dry shampoo mixed with winter sweat: two scents traveling like electricity straight to his brain, and his head instinctively lifts. She smiles politely as he does, their lips hung up by invisible strings called propriety. They pass each other, stepping in each other’s just-taken footsteps, and they sink and sway, ferns in the wind. They look back, slow down, negotiate words to each other. Each is afraid to say too much or too little, to walk towards or away. They flutter-dance there for a moment. I can’t pull my eyes away.
