Einstein once wrote an introduction to a book on astronomy. He mentioned this idea, he called it an idea of pure genius. This is Terence Tao, one of the world’s most renowned mathematicians. What he’s referencing right now is how Kepler deduced the shape of Earth’s orbit, which was astoundingly more clever than I had realized. Tao’s career has been remarkably collaborative, spanning an unusually wide breadth of topics. When asked what topics would benefit from a strongly visual presentation, he proposed cosmic distances — the story of how humanity first figured out the sizes of objects from the Earth to the universe, and how each measurement unlocks the next. Tao: “I always liked astronomy as a child. Found red dwarfs and neutron stars and so forth fascinating.”
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As a kid he just read books and accepted “the distance to Mars is 18 million miles.” Science communication at its best is less about presenting facts memorably, more about showing how we know what we know. With cosmic distances, the sheer scales involved are awesome — but what deserves equal awe and is much less commonly highlighted is how clever the reasoning is at each step. Tao: “It was always some mathematical trick. And it was always a cool trick. If you want to measure the distance to x, you can never just look at x. You have to look at y and how x impacts y.” The technique needs data (technology), but then it’s mathematics. This video covers steps up to the planets and Kepler’s genius. Next part continues to the most distant galaxies.
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The first rung of the ladder is the radius of the Earth — a classic tale. To even ask presumes the Earth is a sphere. The first really convincing proof came from the Moon, and this is a constant theme in the distance ladder: to measure one object you have to use a reference object some distance away. We’re stuck on Earth — if we could step away and look from many angles, we could see it’s round. But from one angle a sphere could be a flat disc; from a different angle the disc would be an ellipse. Mathematician’s note: there’s a nice geometric argument that if every projection of a convex body is a circle, then it must be a sphere. In 2D this is not true (counterexamples exist), but in 3D there’s enough perspective. Aristotle could do this using the Moon, because lunar eclipses show Earth’s shadow as always a circular arc. A composite photo of the Moon entering and exiting a lunar eclipse makes the round shadow visible — no telescopes, sextants, or spacecraft needed.
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These compositions also reveal the relative size of Earth and Moon. But first: how big is the Earth? Eratosthenes is the first known. Story: he had read of a well in Syene where on the summer solstice you could look down and see the Sun reflected in the water. He waited for the summer solstice in Alexandria, looked down his own well — and there was no reflection. He knew from Aristotle the Earth was round. Treating the Sun as far enough away that all rays are effectively parallel: at any moment a line from Earth’s center to the Sun passes through one surface point experiencing the Sun directly overhead. The summer solstice is special because Earth’s axis is tilted directly toward the Sun, so the Tropic of Cancer line predictably experiences this phenomenon. Syene sits on this line; Alexandria is north of it.
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So at noon in Alexandria, Eratosthenes’ “straight up” was at some angle to the Sun’s rays. Using a gnomon (portable sundial-protractor) he measured ~7 degrees off vertical. Knowing Syene was simultaneously experiencing Sun directly overhead, the arc length between Alexandria and Syene corresponds to 7 degrees. So 7°/360° equals (distance Alexandria-Syene)/(circumference). Distance was reported as 5000 stadia (~500 miles, depending on conversion). With conventionally accepted conversions, accuracy is about 10%. Tao: “You can find sources online that claim his estimate was more accurate than this, but if you selectively choose which conversions to use, you can kind of p-hack your way into a better number.” 10% is pretty good with no technology. The remaining question: how did he know the distance? Possible: Nile merchants knew typical sail-day distances. Joke: he hired what we’d call a graduate student to pace it. From that one graduate student you can get all the way to the diameter of the universe in principle.
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Now to the Moon. Use eclipses again. During a lunar eclipse, Earth’s shadow is roughly twice the radius of Earth (penumbra/umbra refinement aside). Lunar eclipses last no longer than ~4 hours; Moon takes one lunar month (~28 days) to traverse its orbit. Ratio of 28 days to 4 hours, with the necessary geometry corrections for Moon’s diameter and the fact that not all eclipses pass through the shadow’s center, gives the relation between distance to the Moon and Earth’s radius. Slightly more accurate eclipse-traverse time is ~3.5 hours. Aristarchus measured the distance to the Moon at ~60 Earth radii. The actual orbit varies between 58 and 62 — as good as one could hope.
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For the Moon’s actual size: with no photography you can’t easily get it from the eclipse photos. The Greeks had a different method. Watch a full Moon rising and time how long it takes — about 2 minutes. You’re not really watching the Moon move through its orbit; rather, the Earth’s rotation scans your line of sight over the Moon (~24 hours total, slightly less because Moon moves). Ratio of 2 minutes to 24 hours gives the ratio between Moon’s radius and Moon’s distance — and since they had the distance, they had the size. With almost no technology, decent estimates of both.
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The Sun was trickier. Doesn’t matter for the argument whether Sun goes around Earth or vice versa. Same two unknowns: size and distance of Sun. Eclipses help via a striking coincidence: during a solar eclipse, Moon and Sun appear almost exactly the same size. So the ratio (Moon radius / Moon distance) equals (Sun radius / Sun distance). The Greeks knew the Moon ratio. So if they could pin down either Sun size or Sun distance, they’d have the other.
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Method to get distance to the Sun: phases of the Moon. The Sun illuminates half the Moon. From Earth we see half the Moon, but a different half — that’s why phases happen. (This also tells us the Moon is round; if it were flat we’d see only “lit” or “dim”, no phases.) Half-Moon does NOT occur halfway between New Moon and Full Moon. It occurs when Earth-Sun-Moon form a right angle AT THE MOON, not at the Earth. The angle separating the true halfway-point timing from the actual half-Moon timing depends on Sun’s distance: smaller angle means farther Sun. Trigonometry: distance to Sun = distance to Moon / sin(that angle).
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This is where the Greeks hit a technological wall. Aristarchus thought half-Moons occurred 6 hours before the midpoint between New and Full. Actual discrepancy is about half an hour. He’s off by an order of magnitude — they had no clocks (sundials don’t work in the dark) and no telescopes. The math was sound; the technology wasn’t. He concluded Sun was 20× the Moon’s distance and 7× the Earth’s size. True values: ~370× and 109×. But qualitatively his conclusion was correct: the Sun is much bigger than Earth, so why would the Sun go around the Earth? He proposed the heliocentric model — first to do so. Copernicus’ famous book credits him.
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The Greeks dismissed Aristarchus for good mathematical reasons (limited by technology). Their objection: if Earth went around the Sun, the apparent positions of stars should shift seasonally (parallax) — and they didn’t see any shift. Same parallax phenomenon as nearby trees moving faster than background mountains when you drive. The Greeks reasoned: if Aristarchus is right, the stars must be thousands of times farther away than we currently think — and that would be absurd. Tao: “Even when you have the math right, you don’t necessarily get to the truth, because of course the universe is in fact not just thousands of times larger than the theory, but actually billions and trillions of times larger.”
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Jump ahead to Kepler — the most genius step of the ladder. Kepler built on Copernicus, who had worked out the planets move around the Sun in circular orbits, and figured out the periods (Earth: 1 year; Mars: 687 days; etc.). These numbers were arguably Copernicus’ most important contribution. He computed them from Babylonian observations — centuries of data on where planets appeared among stars, finding when patterns repeat, factoring in Earth’s movement.
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Kepler wanted relative sizes of all orbits. He had a pet theory linking orbital ratios to nested platonic solids (sphere inscribed in octahedron, sphere around it, icosahedron around that, sphere, dodecahedron, sphere, tetrahedron, sphere, cube, sphere — six spheres for six planets). Beautiful if true. He needed data: Tycho Brahe, an eccentric wealthy aristocrat, had been given an island (with peasants) by Denmark to build the observatory Uranenburg. Brahe had decades of observations but wouldn’t share. Kepler stole the data. The platonic-solid theory was off by a couple percent. He couldn’t make his theory fit. Couldn’t even make Copernicus’ circular orbits fit.
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Critical context: throw out the assumption that orbits are circular and the data becomes very hard to interpret. Brahe didn’t record exact 3D positions — he recorded where each planet appeared in the sky (which constellation, what date). From that — just sequences of angles — Kepler deduced the shapes of all orbits, including Earth’s. The setup: Sun fixed, Earth in some orbit, Mars in some orbit. From Earth on a given date you can compute the direction to the Sun (knowing the date) and the direction to Mars (from observation). But you don’t know either distance. Two angles, two unknown orbits — looks unsolvable. Even granting everything’s planar (planets on a zodiac is roughly true).
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Kepler’s trick: solve a simpler problem first. If Mars were nailed to a fixed point in space, the Earth-Sun and Earth-Mars angles on a given night would be enough to triangulate Earth’s position relative to those two fixed points. Many nights of data would trace out Earth’s orbit relative to fixed Mars and fixed Sun. But Mars does move — so how do you triangulate when something moves?
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The genius: Kepler knew from Copernicus that every 687 days (Martian year), Mars returns to where it was. So if you sample Brahe’s data at 687-day spacings, Mars IS effectively a reference point in that subsampled time series. Brahe observed for 10 years — just enough data. Five samples spaced 687 days apart give five Earth positions relative to that one Mars location. Then take a different starting day a couple days later — Mars only moved a little — and get five more Earth positions relative to that nearby (but slightly shifted) Mars location. Each “five-point set” is conditional on one Mars location; adjacent sets must shift only slightly.
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Kepler had a massive jigsaw puzzle: each piece is five Earth positions conditioned on a mystery Mars location, and adjacent pieces (one day apart) must fit together coherently. He pieced them into a coherent Earth orbit alongside a coherent Mars orbit. This gives only shapes — not absolute distances — but he saw what no one had: Earth’s orbit is an ellipse, not a circle. He also derived equal-areas-in-equal-times. With Earth’s orbit shape known, deducing Mars (or any planet’s) orbit becomes easier: take Mars angle measurements on five nights spaced 687 days apart (Mars is at the same spot), and triangulate Mars relative to Earth’s now-known orbit. Repeat across adjacent time series for the full Mars orbit. Einstein wrote an introduction to a book on astronomy and called this idea pure genius. It required Tycho’s data, plus Copernicus’ data going all the way back to the Babylonians.
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Still no absolute distances — only shapes of all orbits relative to Earth’s. Kepler and contemporaries: “It’s like they could draw the exact picture, but they didn’t know the size of the paper.” From this point onward, astronomers hunted any distance in the solar system they could measure precisely — one distance would be enough to lock everything else into place. Next part: how that was first done, plus how the AU lets you deduce the speed of light, distance to nearest stars,
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and ultimately the distance to the farthest observable galaxies. If you want to stay updated for the next part, make sure to follow 3Blue1Brown on whatever platform it is that you most like to follow Steph on. There’s an email list, just going to throw that out there.