The fastest human in the world, according to the Ancient Greek legend, was the heroine Atalanta. Although she was a famous huntress who joined Jason and the Argonauts in the search for the golden fleece, she was most renowned for the one avenue in which she surpassed all other humans: her speed. While many boasted of how swift or fleet-footed they were, Atalanta outdid them all. No one possessed the capabilities to defeat her in a fair footrace. According to legend, she refused to be wed unless a potential suitor could outrace her, and remained unwed for a very long time. Arguably, if not for the intervention of the Goddess Aphrodite, she would have avoided marriage for the entirety of her life.

Aside from her running exploits, Atalanta was also the inspiration for the first of many similar paradoxes put forth by the ancient philosopher Zeno of Elea about how motion, logically, should be impossible. The argument goes something like this:

• To go from her starting point to her destination, Atalanta must first travel half of the total distance.
• To travel the remaining distance, she must first travel half of what’s left over.
• No matter how small a distance is still left, she must travel half of it, and then half of what’s still remaining, and so on, ad infinitum.
• With an infinite number of steps required to get there, clearly she can never complete the journey.
• And hence, Zeno states, motion is impossible: Zeno’s paradox.

While there are many variants to this paradox, this is the closest version to the original that survives. Although it may be obvious what the solution is — that motion is indeed possible — it’s physics, not merely mathematics alone, that allows us to see how this paradox gets resolved.

A sculpture of Atalanta, the fastest person in the world, running in a race. If not for the trickery of Aphrodite and the allure of the three golden apples, which she stopped to pick up each time her opponent dropped one, nobody would have ever defeated Atalanta in a fair footrace.

The oldest “solution” to the paradox was put forth based on a purely mathematical perspective. The claim admits that, sure, there might be an infinite number of jumps that you’d need to take, but each new jump gets smaller and smaller than the previous one. Therefore, as long as you could demonstrate that the total sum of every jump you need to take adds up to a finite value, it doesn’t matter how many chunks you divide it into.

For example, if the total journey is defined to be 1 unit (whatever that unit is), then you could get there by adding half after half after half, etc. The series ½ + ¼ + ⅛ + … does indeed converge to 1, so that you eventually cover the entire needed distance if you add an infinite number of terms. You can prove this, cleverly, by subtracting the entire series from double the entire series as follows:

• (series) = ½ + ¼ + ⅛ + …
• 2 * (series) = 1 + ½ + ¼ + ⅛ + …
• therefore, [2 * (series) – (series)] = 1 + (½ + ¼ + ⅛ + …) – (½ + ¼ + ⅛ + …) = 1.

That seems like a simple, straightforward, and compelling, solution, doesn’t it?

By continuously halving a quantity, you can show that the sum of each successive half leads to a convergent series: one entire “thing” can be obtained by summing up one half plus one fourth plus one eighth, etc.

Unfortunately, this “solution” only works if you make certain assumptions about other aspects of the problem that are never explicitly stated. This mathematical line of reasoning is only good enough to robustly show that the total distance Atalanta must travel converges to a finite value, even if it takes her an infinite number of “steps” or “halves” to get there. It doesn’t tell you anything about how long it takes her to reach her destination, or to take that potentially infinite number of steps (or halves) to get there. Unless you make additional assumptions about time, you can’t solve Zeno’s paradox by appealing to the finite distance aspect of the problem.

How is it possible that time could come into play and ruin what appears to be a mathematically elegant and compelling “solution” to Zeno’s paradox?

Because there’s no guarantee that each of the infinite number of jumps you need to take — even to cover a finite distance — occurs in a finite amount of time. If each jump took the same amount of time, for example, regardless of the distance traveled, it would take an infinite amount of time to cover whatever tiny fraction-of-the-journey remains. Under this line of thinking, it might still be impossible for Atalanta to reach her destination.

One of the many representations (and formulations) of Zeno of Elea’s paradox relating to the impossibility of motion. It was only through a physical understanding of distance, time, and their relationship that this paradox was able to be robustly resolved.

Many thinkers, both ancient and contemporary, tried to resolve this paradox by invoking the idea of time. One attempt was made a few centuries after Zeno by the legendary mathematician Archimedes, who argued the following.

• It must take less time to complete a smaller distance jump than it does to complete a larger distance jump.
• And therefore, if you travel a finite distance, it must take you only a finite amount of time.
• If that’s true, then Atalanta must finally, eventually reach her destination, and thus, her journey will be complete.

Only, this line of thinking is not necessarily mathematically airtight either. It’s eminently possible that the time it takes to finish each step will still go down: half the original time for the first step, a third of the original time for the next step, a quarter of the original time for the subsequent step, then a fifth of the original time, and so on. However, if that’s the way your “step duration” decreases, then the total journey will actually wind up taking an infinite amount of time. You can check this for yourself by trying to find what the series [½ + ⅓ + ¼ + ⅕ + ⅙ + …] sums to. As it turns out, the limit does not exist: this is a diverging series, and the sum tends toward infinity.

The harmonic series, as shown here, is a classic example of a series where each and every term is smaller than the previous term, but the total series still diverges: i.e., has a sum that tends toward infinity. It is not enough to contend that time jumps get shorter as distance jumps get shorter; a quantitative relationship is necessary.

It might seem counterintuitive, but even though Zeno’s paradox was initially conceived as a purely mathematical problem, pure mathematics alone cannot solve it. Mathematics is the most useful tool we have for performing quantitative analysis of any type, but without an understanding of how travel works in our physical reality, it won’t provide a satisfactory solution to the paradox. The reason is simple: the paradox isn’t simply about dividing a finite thing up into an infinite number of parts, but rather about the inherently physical concept of a rate, and specifically of the rate of traversing a distance over a duration of time.

That’s why it’s a bit misleading to hear Zeno’s paradox: the paradox is usually posed in terms of distances alone, when it’s really about motion, which is about the amount of distance covered in a specific amount of time. The Greeks had a word for this concept — τάχος — which is where we get modern words like “tachometer” or even “tachyon” from, and it literally means the swiftness of something. To someone like Zeno, however, the concept of τάχος, which most closely equates to velocity, was only known in a qualitative sense. To connect the explicit relationship between distance and velocity, there must be a physical link at some level, and there indeed is: through time.

If anything moves at a constant velocity and you can figure out its velocity vector (magnitude and direction of its motion), you can easily come up with a relationship between distance and time: you will traverse a specific distance in a specific and finite amount of time, depending on what your velocity is. This can be calculated even for non-constant velocities by understanding and incorporating accelerations, as well, as determined by Newton.

Because we’re not bound by the qualitative thought patterns that would’ve come along with the mention of the word τάχος here in the 21st century, we can simply think about a variety of terms that we are familiar with.

• How fast does something move? That’s what its speed is.
• What happens if you don’t just ask how fast it’s moving, but if you add in which direction it’s moving in? Then that speed suddenly becomes a velocity: a speed plus a direction.
• And what’s the quantitative definition of velocity, as it relates to distance and time? It’s the overall change in distance divided by the overall change in time.

This makes velocity an example of a concept known as a rate: the amount that one quantity (distance) changes dependent on how another quantity (time) changes as well. You can have a constant velocity (without any acceleration) or you can have a velocity that evolves with time (with either positive or negative acceleration). You can have an instantaneous velocity (i.e., where you measure your velocity at one specific moment in time) or an average velocity (i.e., your velocity over a certain interval, whether a part or the entirety of a journey).

However, if something is in constant motion, then the relationship between distance, velocity, and time becomes very simple: the distance you traverse is simply your velocity multiplied by the time you spend in motion. (Distance = velocity * time.)

When a person moves from one location to another, they are traveling a total amount of distance in a total amount of time. Figuring out the relationship between distance and time quantitatively did not happen until the time of Galileo and Newton, at which point Zeno’s famous paradox was resolved not by mathematics or logic or philosophy, but by a physical understanding of the Universe.

This is how we arrive at the first correct resolution of the classical “Zeno’s paradox” as commonly stated. The reason that objects can move from one location to another (i.e., travel a finite distance) in a finite amount of time is not because their velocities are not only always finite, but because they do not change in time unless acted upon by an outside force: a statement equivalent to Newton’s first law of motion. If you consider a person like Atalanta, who moves at a constant speed, you’ll find that she can cover any distance you choose in a specific amount of time: a time prescribed by the equation that relates distance to velocity, time = distance/velocity.

While this isn’t necessarily the most common form that we encounter Newton’s first law in (e.g., objects at rest remain at rest and objects in motion remain in constant motion unless acted on by an outside force), it very much arises from Newton’s principles when applied to the special case of constant motion. If you consider the total distance you need to travel and then halve the distance that you’re traveling, it will take you only half the time to traverse it as it would to traverse the full distance. To travel (½ + ¼ + ⅛ + …) the total distance you’re trying to cover, it takes you (½ + ¼ + ⅛ + …) the total amount of time to do so. And this works for any distance, no matter how arbitrarily tiny, you seek to cover.

Whether it’s a massive particle or a massless quantum of energy (like light) that’s moving, there’s a straightforward relationship between distance, velocity, and time. If you know how fast your object is going, and if it’s in constant motion, distance and time are directly proportional.

For anyone interested in the physical world, this should be enough to resolve Zeno’s paradox. It works whether space (and time) is continuous or discrete; it works at both a classical level and a quantum level; it doesn’t rely on philosophical or logical assumptions. For objects that move in this Universe, simple Newtonian physics is completely sufficient to solve Zeno’s paradox.

But, as with most things in our classical Universe, if you go down to the quantum level, an entirely new paradox can emerge, known as the quantum Zeno effect. Certain physical phenomena only happen due to the quantum properties of matter and energy, like quantum tunneling through a thin and solid barrier, or the radioactive decay of an unstable atomic nucleus. In order to go from one quantum state to another, your quantum system needs to behave like a wave: with a wavefunction that spreads out over time.

Eventually, the wavefunction will have spread out sufficiently so that there will be a non-zero probability of winding up in a lower-energy quantum state. This is how you can wind up occupying a more energetically favorable state even when there isn’t a classical path that allows you to get there: through the process of quantum tunneling.

By firing a pulse of light at a semi-transparent/semi-reflective thin medium, researchers can measure the time it must take for these photons to tunnel through the barrier to the other side. Although the step of tunneling itself may be instantaneous, the traveling particles are still limited by the speed of light.

Remarkably, there’s a way to inhibit quantum tunneling, and in the most extreme scenario, to prevent it from occurring at all. All you need to do is this: just observe and/or measure the system you’re monitoring before the wavefunction can sufficiently spread out so that it overlaps with a lower-energy state it can occupy. Most physicists are keen to refer to this type of interaction as “collapsing the wavefunction,” as you’re basically causing whatever quantum system you’re measuring to act “particle-like” instead of “wave-like.” But a wavefunction collapse is just one interpretation of what’s happening, and the phenomenon of quantum tunneling is real regardless of your chosen interpretation of quantum physics.

Another — perhaps more general — way of looking at the quantum version of Zeno’s paradox is that you’re restricting the possible quantum states your system can be in through the act of observation and/or measurement. If you make this measurement too close in time to your prior measurement, there will be an infinitesimal (or even a zero) probability of tunneling into your desired state. If you keep your quantum system interacting with the environment, you can suppress the inherently quantum effects, leaving you with only the classical outcomes as possibilities: effectively forbidding quantum tunneling from occurring.

When a quantum particle approaches a barrier, it will most frequently interact with it. But there is a finite probability of not only reflecting off of the barrier, but tunneling through it. If you were to measure the position of the particle continuously, however, including upon its interaction with the barrier, this tunneling effect could be entirely suppressed via the quantum Zeno effect.

The key takeaway is this: motion from one place to another is possible, and the reason it’s possible is because of the explicit physical relationship between distance, velocity, and time. With those relationships in hand — i.e., that the distance you traverse is your velocity multiplied by the duration of time you’re in motion — we can learn exactly how motion occurs in a quantitative sense. Yes, in order to cover the full distance from one location to another, you have to first cover half that distance, then half the remaining distance, then half of what’s left, etc., which would take an infinite number of “halving” steps before you actually reach your destination.

But the time it takes to take each “next step” also halves with respect to each prior step, so motion over a finite distance always takes a finite amount of time for any object in motion. Even today, the Zeno’s paradox puzzle still remains an interesting exercise for mathematicians and philosophers. Not only is the solution reliant on physics, but physicists have even extended it to quantum phenomena, where a new quantum Zeno effect — not a paradox, but a suppression of purely quantum effects — emerges.

For motion, its possibility, and how it occurs in our physical reality, mathematics alone is not enough to arrive at a satisfactory resolution. As in all scientific fields, only the Universe itself can actually be the final arbiter for how reality behaves. Thanks to physics, Zeno’s original paradox gets resolved, and now we can all finally understand exactly how.

A version of this article was first published in February of 2022. It was updated in January of 2026.

This article Zeno’s Paradox resolved by physics, not by math alone is featured on Big Think.

Power on your PCs, my gentle users, because I just found a fresh Excel file to overcomplicate. Hoo boy, I can’t wait to rework every cell of “Company Staffing.xlsx.”

Most peons at this company think a spreadsheet is just a tool to create a budget. Not me. Not us. You see, there’s one of us in every organization. Though it’s nowhere in our job descriptions, we spend hours crafting Gordian knots of obscure Excel features so that even the simplest files become unrecognizable monstrosities.

Before we do anything with these measly kilobytes, we need to duplicate this file. Several times. Then we add an underscore, “NEW,” and a different numbering convention. The filename should evoke the image of an overbaked Feast of Assumption turducken.

There. We’re ready to open “Company Staffing_NEW_FINAL_003.xlsx.”

Sweet mother of Steve Ballmer, we have only three columns here: “Name,” “Hire date,” and “Salary.” Time to really balloon this “dataset.” With one well-spent afternoon, I can 5X this amateur foray into spreadsheet-making and split that puny “Name” column into “First name,” “Nickname,” “Mother’s maiden name,” “Middle name,” and “Surname.” Have to make some educated guesses for most of these values, of course. God, I obfuscate so much for this company.

These greenhorns are so lucky to have a power-user like me. No one asked, but I’m going to add a Pivot Table. Don’t know what that is? It’s just an advanced feature I learned from one of my many yellowed manuals that will make looking at this list feel like lifting Russian nesting dolls. Only under each doll lie increasingly larger horned dolls, with monospaced-font tattoos of VLOOKUP function incantations.

When I’m done, the whole thing will be the spreadsheet-equivalent of a Picasso. Except instead of having one crooked nose, you get twenty misshapen noses along with a bunch of unnecessary ways of sorting and filtering the noses.

Speaking of fine art, the newbie who gave me this canvas didn’t pick a theme. Holy guacamole, I am so excited to click on that “Layout” tab. I’m thinking fuchsia and teal zebra stripes for the row backgrounds. Ah, that’s better.

My coworkers are really going to be late when I fire this baby off in an email two minutes before our next all-hands meeting. They always need a lot of time to process my changes.

Ugh, the boss keeps asking to meet with me and HR. I wish she realized I was deep in the weeds, making this dim doc into Frankenstein’s monster of Excel. She doesn’t even realize that once I’m done, I’ll be the only one who can maintain this thing.

Geez, I almost forgot to freeze one of the columns for no reason. Let’s go with “Hire date.”

Almost done revamping the look of this number dump. But we need more columns. I yearn to see the triple alphanumeric cell name AAB:012 in all its glory.

If I play my cards right, people are going to have to scroll so far horizontally that their wrists cramp from dragging their way across the screen.

Whoa, I found another tab. “Planned Layoffs – Sheet 2.” Hey, why’s my severance pay “#VALUE!”?

Elon Musk promised that his social media company X would be “the everything app,” but these days “everything” seems to only include slop, fascist propaganda, and abuse. Increasingly, the social media site has been awash in vulgar and non-consensual sexual images that users are creating with X’s built-in AI tool, Grok. As The Guardian’s Nick Robins-Early wrote:

Many users on X have prompted Grok to generate sexualized, nonconsensual AI-altered versions of images in recent days, in some cases removing people’s clothing without their consent. Musk on Thursday reposted an AI photo of himself in a bikini, captioned with cry-laughing emojis, in a nod to the trend.

And as 404 Media uncovered, the abuse this software is enabling is likely far worse than it appears and is in many ways merely the latest escalation of an online creep problem that’s as old as the internet.

It’s horrendous, from top to bottom, especially for women who are being aggressively targeted by X users just for existing online.

The writer Ketan Joshi picked up on a strange pattern of language and usage in the media coverage of this scandal. Joshi posted a thread on Bluesky gathering examples “of major media outlets falsely anthropomorphising the “Grok” chatbot program and in doing so, actively and directly removing responsibility and accountability from individual people working at X who created a child pornography generator.” The example headlines and articles Joshi found include phrases like “Grok apologizes,” “Grok says,” or “Elon Musk’s AI chatbot blames.” The articles go further in some cases, giving the software agency by quoting it as “writing,” “saying,” and “posting.”

The problem here, as Joshi wrote, is that this framing shifts responsibility away from the people who are using and platforming this software. Implying that the chatbot and image generation program itself is accountable allows people to hide from their own culpability in the bot’s shadow.

This has been a trend in how AI is discussed for a while. The media’s language and framing are often overly deferential to the tech industry’s own marketing hype—imagine blaming a toaster for a burned slice of multigrain just because a salesman assured you about the Bread Safe Smart Sensor technology. This tendency to assume that these programs are as capable as we’re being told isn’t unique to AI—think of “smart bombs”—but the trend in usage doesn’t seem to be getting any better.

The word “artificial” in AI is accurate, though. These programs are not natural, they’re human-made artifices conceived, created, and maintained by people. Allowing creators, engineers, and executives to evade accountability for their decisions, just because we imagine that the toasters they made are awake, will only degrade the internet further.

I think 2026 will be the nadir of social media. Without changes, these online platforms will be squeezed into more horrible and unpleasant forms by the pressures of AI maximimalists, extractive data miners, and fascistic supporters of a “clicktatorship” who care above all else about creating and curating displays of made-for-TV violence. A better internet is not impossible, though. We can name the people behind these problems, and we can do something about it.

The viral warning that “a computer can never be held accountable” from a 1979 IBM training document has never been more resonant. The problem with Grok and other programs isn’t that it’s escaped containment like Skynet, the problem is more akin to an owner who has let their aggressive dog off its leash.

People who live in a society with you and me are putting these tools to malicious uses. They are people who take time in their day to craft and share abusive images of kids and strangers, and who delight in the pain those images cause. They are people who post slop of themselves next to cry-laughing emojis, desperate to be the funny one for once. They are people who blew off meeting up with friends so they could stay up late into the night to program these tools, who got bored and zoned out in long meetings to discuss implementing this software, and who are right now ignoring texts about why they’re letting the platforms they’re responsible for flood with filth.

None of this is the toaster’s doing. We shouldn’t allow the marketers and their apologists let those who are really responsible avoid their time in the spotlight.

Reducing classroom distractions during a lesson is essential to any definition of effective teaching, much less student learning. With cell phones ubiquitous among students, distractions multiply. What, for example, do some teachers do before or during a lesson to manage cell phone use?

Veteran math teacher Tony Riehl wrote a post on this subject. It appeared May 22, 2017 . He has taught high school math courses in Montana for 35 years. I added math teaching blogger Dan Meyer’s comments on Riehl’s post.

I learned early on with cell phones, that when you ask a student to hand you their phone, it very often becomes confrontational. A cell phone is a very personal item for some people.

To avoid the confrontation I created a “distraction box” and lumped cell phones in with the many other distraction that students bring to class. These items have changed over time, but include “fast food” toys, bouncy balls, Rubics cubes, bobble heads, magic cards, and the hot item now are the fidget cubes and fidget spinners.

A distraction could be a distraction to the individual student, the other students or even a distraction to me. On the first day of the year I explain to my students that if I make eye contact with them and point to the distraction box, they have a choice to make. If they smile and put the item in the box, they can take the item out of the box on the way out of the room. If they throw a fit and put the distraction in the box, they can have it back at the end of the day. If they refuse to put the distraction in the box, they go to the office with the distraction.

On the first day of the year we even practice smiling while we put an item in the box. The interaction is always kept very light and the students really are cooperative. It has been a few years since an interaction actually became confrontational, because I am not asking them to put the item in my hand. I even have students sometimes put their cell phone in the box on the way in the door because they know they are going to have trouble staying focused.

This distraction box concept really has changed the atmosphere of my room. Students understand what a distraction is and why we need to limit distractions….

This Is My Favorite Cell Phone Policy

By Dan Meyer • May 24, 2017

Schools around the world are struggling to integrate modern technology like cell phones into existing instructional routines. Their stances towards that technology range from total proscription – no cell phones allowed from first bell to last – to unlimited usage. Both of those policies seem misguided to me for the same reason: they don’t offer students help, coaching, or feedback in the complex skills of focus and self-regulation.

Enter Tony Riehl’s cell phone policy, which I love for many reasons, not least of which because it isn’t exclusively a cell phone policy. It’s a distractions policy.

What Tony’s “distraction box” does very well:

• It makes the positive statement that “we’re in class to work with as few distractions as possible.” It isn’t a negative statement about any particular distraction. Great mission statement.
• Specifically, it doesn’t single out cell phones. The reality is that cell phones are only one kind of technology students will bring to school, and digital technology is only one distractor out of many. Tony notes that “these items have changed over time, but include fast food toys, bouncy balls, Rubik’s cubes, bobble heads, magic cards, and the hot items now are the fidget cubes and fidget spinners.”
• It acknowledges differences between students. What distracts you might not distract me. My cell phone distracts my learning so it goes in the box. Your cell phone helps you learn so it stays on your desk.
• It builds rather than erodes the relationship between teachers and students. Cell phone policies often encourage teachers to become detectives and students to learn to evade them. None of this does any good for the working relationship between teachers and students. Meanwhile, Tony describes a policy that has “changed the atmosphere of my room,” a policy in which students and teachers are mutually respected and mutually invested.

Programmers mostly know about Regex and dislike it. Normal people are mostly blissfully ignorant of the existence of RegEx.

To me, RegEx is like superpower that’s available to everyone, but most don’t bother to develop, because you have to eat lots of spinach.

What is RegEx (Regular Expressions)? It’s a language for describing patterns in text. If a human can describe a pattern (ie, the phone book has numbers with separators) then regex can usually describe that pattern!

You don’t need to be a programmer to use regex because it’s built into most text editors. Even LibreOffice and Word!

Here’s a gif that shows the power of regex:

Regex’s power is even greater when combined with other tools (like multiple selections).

Regex is horrible because the syntax is hard to learn. But once you get it, the syntax is easy. Then, regex is horrible because the syntax isn’t the same for different regex “engines”.

There’s POSIX regex with basic (BRE) and extended (ERE) versions, and Perl/PCRE regex, and Vim regex (which I still haven’t figured out) with its \verymagic and \nomagic modes, and probably countless more. (I haven’t found any tool that converts between Regex variants. This seems silly.) Then there’s the fact that the regex engine used by various hip software like Helix and Ripgrep is a Rust crate that doesn’t support regex features that I consider table stakes: lookahead and lookbehind.

But if regex is worth using despite all that, it must be good! And it is.

Here’s my suggestion.

If you’re a programmer, write a tool that can translate between regex flavors.

If you’re a programmer who isn’t super familiar with regex, regexr.com is the best tool I’ve found for testing and debugging regex.

If you’re a normal person, next time you find yourself looking for a pattern – ie, you know some digits of a phone number, or want to find all jpegs without a date in the filename – that you can describe with words, or performing the same operation a bunch of times in a text file, ask a nearby programmer to help you write your first regex. Or email me.

I sometimes resort to a rather common measuring technique that is neither fast, nor accurate, nor recommended by any standards body and yet it hasn't failed me whenever I have had to use it. I will describe it here, though calling it a technique might be overselling it. Please do not use it for installing kitchen cabinets or anything that will stare back at you every day for the next ten years. It involves one tool: a sheet of A4 paper.

Like most sensible people with a reasonable sense of priorities, I do not carry a ruler with me wherever I go. Nevertheless, I often find myself needing to measure something at short notice, usually in situations where a certain amount of inaccuracy is entirely forgivable. When I cannot easily fetch a ruler, I end up doing what many people do and reach for the next best thing, which for me is a sheet of A4 paper, available in abundant supply where I live.

From photocopying night-sky charts to serving as a scratch pad for working through mathematical proofs, A4 paper has been a trusted companion since my childhood days. I use it often. If I am carrying a bag, there is almost always some A4 paper inside: perhaps a printed research paper or a mathematical problem I have worked on recently and need to chew on a bit more during my next train ride.

Dimensions

The dimensions of A4 paper are the solution to a simple, elegant problem. Imagine designing a sheet of paper such that, when you cut it in half parallel to its shorter side, both halves have exactly the same aspect ratio as the original. In other words, if the shorter side has length \( x \) and the longer side has length \( y , \) then \[ \frac{y}{x} = \frac{x}{y / 2} \] which gives us \[ \frac{y}{x} = \sqrt{2}. \] Test it out. Suppose we have \( y/x = \sqrt{2}. \) We cut the paper in half parallel to the shorter side to get two halves, each with shorter side \( x' = y / 2 = x \sqrt{2} / 2 = x / \sqrt{2} \) and longer side \( y' = x. \) Then indeed \[ \frac{y'}{x'} = \frac{x}{x / \sqrt{2}} = \sqrt{2}. \] In fact, we can keep cutting the halves like this and we'll keep getting even smaller sheets with the aspect ratio \( \sqrt{2} \) intact. To summarise, when a sheet of paper has the aspect ratio \( \sqrt{2}, \) bisecting it parallel to the shorter side leaves us with two halves that preserve the aspect ratio. A4 paper has this property.

But what are the exact dimensions of A4 and why is it called A4? What does 4 mean here? Like most good answers, this one too begins by considering the numbers \( 0 \) and \( 1. \) Let me elaborate.

Let us say we want to make a sheet of paper that is \( 1 \, \mathrm{m}^2 \) in area and has the aspect-ratio-preserving property that we just discussed. What should its dimensions be? We want \[ xy = 1 \, \mathrm{m}^2 \] subject to the condition \[ \frac{y}{x} = \sqrt{2}. \] Solving these two equations gives us \[ x^2 = \frac{1}{\sqrt{2}} \, \mathrm{m}^2 \] from which we obtain \[ x = \frac{1}{\sqrt[4]{2}} \, \mathrm{m}, \quad y = \sqrt[4]{2} \, \mathrm{m}. \] Up to three decimal places, this amounts to \[ x = 0.841 \, \mathrm{m}, \quad y = 1.189 \, \mathrm{m}. \] These are the dimensions of A0 paper. They are precisely the dimensions specified by the ISO standard for it. It is quite large to scribble mathematical solutions on, unless your goal is to make a spectacle of yourself and cause your friends and family to reassess your sanity. So we need something smaller that allows us to work in peace, without inviting commentary or concerns from passersby. We take the A0 paper of size \[ 84.1 \, \mathrm{cm} \times 118.9 \, \mathrm{cm} \] and bisect it to get A1 paper of size \[ 59.4 \, \mathrm{cm} \times 84.1 \, \mathrm{cm}. \] Then we bisect it again to get A2 paper with dimensions \[ 42.0 \, \mathrm{cm} \times 59.4 \, \mathrm{cm}. \] And once again to get A3 paper with dimensions \[ 29.7 \, \mathrm{cm} \times 42.0 \, \mathrm{cm}. \] And then once again to get A4 paper with dimensions \[ 21.0 \, \mathrm{cm} \times 29.7 \, \mathrm{cm}. \] There we have it. The dimensions of A4 paper. These numbers are etched in my memory like the multiplication table of \( 1. \) We can keep going further to get A5, A6, etc. We could, in theory, go all the way up to A\( \infty. \) Hold on, I think I hear someone heckle. What's that? Oh, we can't go all the way to A\( \infty? \) Something about atoms, was it? Hmm. Security! Where's security? Ah yes, thank you, sir. Please show this gentleman out, would you?

Sorry for the interruption, ladies and gentlemen. Phew! That fellow! Atoms? Honestly. We, the mathematically inclined, are not particularly concerned with such trivial limitations. We drink our tea from doughnuts. We are not going to let the size of atoms dictate matters, now are we?

So I was saying that we can bisect our paper like this and go all the way to A\( \infty. \) That reminds me. Last night I was at a bar in Hoxton and I saw an infinite number of mathematicians walk in. The first one asked, "Sorry to bother you, but would it be possible to have a sheet of A0 paper? I just need something to scribble a few equations on." The second one asked, "If you happen to have one spare, could I please have an A1 sheet?" The third one said, "An A2 would be perfectly fine for me, thank you." Before the fourth one could ask, the bartender disappeared into the back for a moment and emerged with two sheets of A0 paper and said, "Right. That should do it. Do know your limits and split these between yourselves."

In general, a sheet of A\( n \) paper has the dimensions \[ 2^{-(2n + 1)/4} \, \mathrm{m} \times 2^{-(2n - 1)/4} \, \mathrm{m}. \] If we plug in \( n = 4, \) we indeed get the dimensions of A4 paper: \[ 0.210 \, \mathrm{m} \times 0.297 \, \mathrm{m}. \]

Measuring Stuff

Let us now return to the business of measuring things. As I mentioned earlier, the dimensions of A4 are lodged firmly into my memory. Getting hold of a sheet of A4 paper is rarely a challenge where I live. I have accumulated a number of A4 paper stories over the years. Let me share a recent one. I was hanging out with a few folks of the nerd variety one afternoon when the conversation drifted, as it sometimes does, to a nearby computer monitor that happened to be turned off. At some point, someone confidently declared that the screen in front of us was 27 inches. That sounded plausible but we wanted to confirm it. So I reached for my trusted measuring instrument: an A4 sheet of paper. What followed was neither fast, nor especially precise, but it was more than adequate for settling the matter at hand.

I lined up the longer edge of the A4 sheet with the width of the monitor. One length. Then I repositioned it and measured a second length. The screen was still sticking out slightly at the end. By eye, drawing on an entirely unjustified confidence built from years of measuring things that never needed measuring, I estimated the remaining bit at about \( 1 \, \mathrm{cm}. \) That gives us a width of \[ 29.7 \, \mathrm{cm} + 29.7 \, \mathrm{cm} + 1.0 \, \mathrm{cm} = 60.4 \, \mathrm{cm}. \] Let us round that down to \( 60 \, \mathrm{cm}. \) For the height, I switched to the shorter edge. One full \( 21 \, \mathrm{cm} \) fit easily. For the remainder, I folded the paper parallel to the shorter side, producing an A5-sized rectangle with dimensions \( 14.8 \, \mathrm{cm} \times 21.0 \, \mathrm{cm}. \) Using the \( 14.8 \, \mathrm{cm} \) edge, I discovered that it overshot the top of the screen slightly. Again, by eye, I estimated the excess at around \( 2 \, \mathrm{cm}. \) That gives us \[ 21.0 \, \mathrm{cm} + 14.8 \, \mathrm{cm} -2.0 \, \mathrm{cm} = 33.8 \, \mathrm{cm}. \] Let us round this up to \( 34 \, \mathrm{cm}. \) The ratio \( 60 / 34 \approx 1.76 \) is quite close to \( 16/9, \) a popular aspect ratio of modern displays. At this point the measurements were looking good. So far, the paper had not embarrassed itself. Invoking the wisdom of the Pythagoreans, we can now estimate the diagonal as \[ \sqrt{(60 \, \mathrm{cm})^2 + (34 \, \mathrm{cm})^2} \approx 68.9 \,\mathrm{cm}. \] Finally, there is the small matter of units. One inch is \( 2.54 \, \mathrm{cm}, \) another figure that has embedded itself in my head. Dividing \( 68.9 \) by \( 2.54 \) gives us roughly \( 27.2 \, \mathrm{in}. \) So yes. It was indeed a \( 27 \)-inch display. My elaborate exercise in showing off my A4 paper skills was now complete. Nobody said anything. A few people looked away in silence. I assumed they were reflecting. I am sure they were impressed deep down. Or perhaps... no, no. They were definitely impressed. I am sure.

Hold on. I think I hear another heckle. What is that? There are mobile phone apps that can measure things now? Really? Right. Security. Where's security?

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