How Metric Paper Works & The Whole of the Universe

This is not just a sheet of paper; it is an invitation to everything that exists. For this paper is metric, which has a special property other pages don't—dividing into each twain is half the whole. Uh, obviously, I guess, but each half is also the same shape as the whole. The ratio of its sides is identical. This only works for metric, though. For origami and letter paper, their halves are half as big but different shapes, which is far less satisfying.

A full sheet of metric is named A4, and each half A5, and each half of that A6—all the same shape again. And so it goes in the opposite direction: double A4 to get same-shaped A3, again to A2, to A1, to zero, which is one square meter of paper exactly and exactly the same shape as A4. With metric paper, everything from posters to postage can be designed on the A4 before you and scaled up or down perfectly. No fuzzing with the margins, and everything can fit on a roll of A0 exactly for printing. It's so satisfying and practical and mathematical. Each sheet of A4 is so hypnotizing.

This A4 is a door to the exponential spiral of everything. Begin with one sheet of A4 metric paper. Divide in half, then half the half again and again, always creating the same shape—dividing down to the size of a B, and then smaller to the size of the organs of the bee and the detailed limit of human vision. But no need to stop here. At 24 divisions are the cells of the bee, each just as alive as the animal they form and each with their own organs carrying out the business of animating inert matter, defending themselves from viruses.

Things so small as to hover on the edge of what can be alive. 32 divisions down and we enter the mechanical brain of this animal cell, containing chromosomes of DNA—the operating system of life, tightly coiled. But it's here the lights go out. Just 39 divisions is already smaller than visible light, and seeing in a meaningfully human way with limited human eyes is meaningless.

So let us turn on atomic vision to explore what's here. 56 divisions down is the width of a strand of DNA, where hexagonal arrangements of atoms interact mechanically to build living things—to build more DNA. 64 divisions down is the size of a single hydrogen atom, and we fall through its electron orbital to enter the interior and discover the truth of matter: it is made of mostly nothing. Everything you touch, everything physical you care about, is an electron cloud creating the illusion of something over nothing. At the center of the atom, there is but the tiniest speck of matter—the hydrogen nucleus—a single proton, 1/100 the size of the atom that contains it.

Before yet even a hundred half divisions down from the A4 where we started begins to stretch human comprehensibility, the proton isn't a sphere but a sea of quarks appearing and disappearing, averaging out to three bound together with the strongest force. Here we hover above the edge of the abyss of quantum deep before plunging down into the nothing. Falling for twice as many divisions down as we have already traveled until we reach quantum madness at the very floor of the universe and the limit of our current theories—the Planck length.

There is nothing here, yet this is the smallest size, the least distance that can be moved or measured—a sort of reality pixel, which is best not to think about. So let us fly back up the quantum well, doubling and doubling to return to the world of quarks and nuclei and atoms and light and the illusory cloud of the physical. Back to life, back to our sheet of A4 paper, which now unfolds itself to a sheet twice the size, doubling to the size of a desk, then a room, then a home—the building the home is in, the street the building is on—continuing this same shape spiral to double to the size of the neighborhood around the street, then the size of a town, a small city—a thing which is not alive but a meta-organism which lives through the actions of its human cells.

Just 36 doublings of a sheet of A4 is the size of the largest cities, and it is at this scale the accomplishments of the human species are most visible and impressive. But just a few doublings later, every structure that humans have wrought fades to invisibility, and only the light we create can be seen. At twice the distance the Earth is wide are the farthest satellites, passing light around Earth for instant Terran communication.

Sixty doublings brings the moon's orbit into view, and at this scale the speed of light, instant just a moment ago, is now slow enough to watch move a beam from Earth to the moon, taking one second. As far as we know, this is the speed limit of the universe—nothing can travel faster. Communication between Earth and the moon will always have a delay, and as we resume doublings, it will not be long before the speed of light proves heartbreakingly slow.

Past 68 doublings, Earth becomes too small to see, so we must draw her orbit instead. And at 76 doublings, we can see the orbits of Venus and Mars. Light now takes two minutes to reach Venus and 14 to reach Mars. 80 doublings is the distance to the sun, our home star, and doublings beyond this reveal the orbits of the outer planets and eventually the whole of the solar system, which light takes eight hours to cross. Beyond the orbit of Pluto is the Voyager 1 space probe—the farthest any object produced by human civilization has traveled. Yet at the scale of all that exists, our species has explored nothing.

There are yet still 20 doublings to go before we've left our home star. After what would take a month traveling at the speed of light, we entered the Oort cloud. I see planetesimals occasionally knocked from orbits to fall inward, becoming comets. It would take another year at light speed to leave the Oort cloud and the realm of the sun's dominance.

124 doublings is wide enough to reach our closest neighboring stars, but at this scale, they are only points of light. And thus, the truth of space: it is made of mostly nothing. Stars with their own planets and potentially their own life are light-years apart, with not but stray atoms of hydrogen in between. Though as we continue doubling, the illusion of something solid appears—our galaxy, this collection of 100 billion stars shining into the dark to create the appearance of a cloud, at the speed of light, 100 millennia across.

But even our galaxy shrinks to a point, and now the other points of light are galaxies themselves, each with billions of their own stars. As the doublings continue, the galaxies form galactic filaments—the largest known structures, the last threads of light in the void. At 184 doublings of a sheet of A4 paper is the end of everything that can be for us. We have reached the edge of the observable universe, a sphere containing two trillion galaxies, yet even this may not be the end of everything that there is.

Beyond the observable universe is the unobservable universe, in which exist objects so distant that light traveling from them, beyond this border, will never reach Earth before the death of every living thing everywhere, caused by the extinction of every star and the end of the universe itself. Just how many doublings exist beyond is unknowable, but potentially enough to shrink everything that could ever be for us into nothing—in infinite nothing.

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