34:Tda7,md`# We need more tiny knowledge projects—here’s one *Crowdsourcing dynamic illustrations for a new astronomy book by Terry Tao and Tanya Klowden.* The other day I was reading the [sample chapter](https://terrytao.files.wordpress.com/2020/10/sample_chapter.pdf) of a new book by [Terence Tao](https://www.nytimes.com/2015/07/26/magazine/the-singular-mind-of-terry-tao.html) and Tanya Klowden, on the so-called [cosmic distance ladder](https://en.wikipedia.org/wiki/Cosmic_distance_ladder). It’s a lovely piece of writing. The chapter tells the story of how Johannes Kepler, using data collected by Tycho Brahe, solved the puzzle of planetary motion. It puts us in the discoverers’ shoes: instead of working backwards from Kepler’s equations, we work forwards from the inexplicable behavior of Mars’s orbit, which became his obsession. It’s a human story—we learn, for instance, that Kepler supported himself with a day job as an astrologer, and that Brahe lost his nose in a duel—but it’s told with rigor and clarity. I think it is maybe the best introduction to astronomy I’ve read. If it has a weakness, it’s that there’s only one illustration, appearing in an endnote. Klowden has told me that they’re scouting spots for more in the final version, but that space is limited. And, of course, there can’t be any animations in print. They’ve talked about an online supplement, but aren’t sure what form it might take. This is a situation that many science writers probably find themselves in. Their material is ripe for a kind of illustration that’s both hard to make and hard to find a place for. But as I read the sample chapter I got to thinking... It begins this way: >

If you have ever been told one piece of astronomy trivia, it is probably that the word “planet” comes from the Greek πλάνητες ἄστέρες (*planētes asteres*, wandering star) or sometimes just πλάνῆται (*planētai*, wanderers). As the ancient Greeks studied the heavens, they observed that most of the stars seemed fixed into constant patterns, or constellations, in a great celestial sphere gently rotating around a single fixed axis (oriented along celestial north, which is very nearly where the north star Polaris is currently located). A handful of stars, however, seemed to follow their own unique path across the sky at their own speed. These *planētes asteres* did not wander into every quadrant of the sky, but traveled within a consistent band along with the Sun and Moon. This band was called the ecliptic (because it was in this band that eclipses happened) and in ancient Greece it happened to cross twelve constellations in the fixed heavenly sphere.

It puts you in the mind of a natural philosopher staring into the night sky. With no light pollution, it would have been crowded with thousands of luminescent pinpricks; it might have seemed like a living surface. The stars swept in circles across the sky so predictable you could steer your ship by them. But defying this order were the *planētes*, which seemed to run around like lost dogs. As I was reading, I wanted to *see* these wandering stars. I don’t know what Mars’s path through the sky looks like, and I wanted to have it animated. I wanted the so-called “ecliptic” pointed out to me. I wanted the illustration to be accurate, and, if possible, based on what Brahe would have actually seen in the sixteenth century. So I made one myself: `35:T530,html`

This GIF was built by scripting “dynamic image requests” to Fourmilab’s Your Sky tool. I set it to the coordinates of the Herrevad Abbey, where Tycho Brahe’s observations were made. The sky map starts a little after November 1st, 1576, and proceeds one day at a time every 150ms. The planets are represented by the little icons with symbols. Mars is the ♂ symbol. The moon is also shown.

I would have liked to use Observable itself to create this image, but alas, I couldn’t prevent it from flickering as I dynamically updated the <img> tag’s src attribute. Plus, I wanted to avoid clobbering the Your Sky server with a zillion image requests as clients loaded this page. The code for creating the animation is at this Github gist.

A few possible improvements: make the animation smoother; hide all the planets but Mars; draw Mars’s path over time; make it interactive (so readers can slide forward and backward through time, or show & hide planets); use Observable to generate the image so that it’s more easily hackable.

`36:T42f,md`Per [Cunningham’s Law](https://meta.wikimedia.org/wiki/Cunningham%27s_Law#:~:text=Cunningham's%20Law%20states%20%22the%20best,Cunningham%2C%20father%20of%20the%20wiki.), I’d like someone to improve on my picture, and to try their hand at illustrating other moments in the text, laid out below, where other pictures could help. I’d like to do this here in Observable—because anyone can edit this page and create a fork for me to merge; and because the platform is [especially suited](https://observablehq.com/@tmcw/enigma-machine) to dynamic interactive visualizations. **Update**: A flurry of activity has produced some outstanding improvements over my initial stab. Here's one courtesy of [@visnup](https://observablehq.com/@visnup), defined in [this notebook](https://observablehq.com/@visnup/re-we-need-more-tiny-knowledge-projects-heres-one). What's great about it—besides looking really good—is that all of its code is available / hackable. Follow the comment threads (click the little comment icons in the left sidebar) to see the work in progress.`37:Tdbf,md`I used to think of “knowledge projects” as involving a big mission, a big community, and a complex piece of software: the stuff of Wikipedia, Stack Overflow, or Genius (where I worked for several years). But the web is just as good for collaborations in the small. Not so long ago, I took a poem I wanted to understand better and sent it to a friend, and the two of us annotated it together online. Why don’t I do this more often? I had the same thought when I ran across [a guide to the aerosol transmission of COVID-19](https://docs.google.com/document/u/0/d/1fB5pysccOHvxphpTmCG_TGdytavMmc1cUumn8m0pwzo/mobilebasic?urp=gmail_link&gxids=7628#h.ml0fjyc1g62p). It was by far the best thing I’d read on the subject. It was prepared by a small group of scientists using a Google Doc.

I used to think of “knowledge projects” as involving a big mission, a big community, and a complex piece of software. But the web is just as good for collaborations in the small.

There ought to be more of these tiny knowledge projects. Often, in exploring complex subjects—immunology, say, or astronomy—I [find myself wishing](https://jsomers.net/i-should-have-loved-biology/) for more diagrams, animations, “interactives.” Why should these only ever be created by a small class of professional artists? And why should collaborative editing be restricted to wikis? Surely other forms—even books published by fancy presses—could benefit from reader improvements. Here’s to giving it a try. I’ll be watching this “living document” and accepting merge requests that improve the illustration above and the ones below. (The excerpts below are reproduced with permission. Klowden has given this project her blessing, though she makes no promises about using the illustrations that result. She added that taking someone else’s work and running with it was a time-honored tradition in science and literature. Certainly that’s the spirit of Tao’s own [blog](https://terrytao.wordpress.com/): a gem of the Internet where professional and amateur mathematicians collaborate in earnest in the comment threads, occasionally producing significant new results.)

The crowdsourced illustrations (a work in progress)

I thought the following excerpts seemed especially illustrate-able. But there may be others—feel free to add them. ### 1. The epicycle model >

Apollonius of Perga, another brilliant Greek mathematician, provided the next refinement. Known today primarily for some of the earliest work on conic sections, Apollonius was able to simplify the Eudoxan model by introducing the concept of the *deferent* and *epicycle*. Epicycles can be hard to explain, but we can fortunately draw upon our modern heliocentric understanding of the solar system. to more easily picture what had to be very difficult to describe in Hellenistic Greece. As we are taught in school, the Earth orbits around the Sun, and the Moon in turn orbits around the Earth. The orbit of the Earth corresponds to one of Apollonius’ deferents, and the smaller orbit of the Moon around the earth is an epicycle. If one then removes the physical Earth from this picture, so that the Moon now orbits an epicycle centered at some imaginary point that in turn traverses the deferent, then you have a pretty good idea of what an epicycle-based orbit would look like.

`38:T8a9,md` ### 5. Kepler’s puzzle >

As we have seen, since the observations of the Babylonians and Greeks, it was known that the planets lie (basically) in a single two-dimensional plane (the ecliptic). The Earth lies in this plane as well, which means that when looking at the planets in the night sky, every visible planet will appear to fall along a straight line. While a formal system of polar coordinates had not been defined yet, the ancient Greeks understood that to explain where in space something is, in two dimensions you need only an angle (*ecliptic longitude* in astronomical terms) and a distance. The records on Mars’ position in the sky from Copernicus and earlier astronomers gave ecliptic longitudes but were very sporadic, sometimes here, sometimes there, with large gaps between the observations. Brahe did what no one in the world had done before, recording the declinations and right ascensions of all the planets every single night (weather permitting) for fifteen consecutive years. This gave Kepler the ecliptic longitudes of Mars whenever it was visible, which he needed, but gave no distances, which was the other thing Kepler needed to define where Mars was in space. Kepler had another problem with this data. While Brahe firmly believed that the Earth was fixed as the center of the visible universe, Kepler’s heliocentric model meant that all the observations of Mars from the earth were from a moving vantage point. It didn’t really help that Kepler didn’t know what the orbit of the Earth was either.

> >

Kepler was stuck with two unknowns, the Earth’s orbit and Mars’ orbit and, at best, half a piece of data, the ecliptic longitude information, which gave the relative positions between the Earth and Mars for fifteen years. With too many unknowns and not enough equations it seemed like an impossible problem to solve. Kepler also realized that he would not be able to work out the orbit of Mars until he had figured out the orbit of the Earth. The earth’s motion kept changing our vantage point for observations of Mars so without knowing where in space we were, we couldn’t figure out where another moving object was relative to us.

`39:Teb8,md` ### 6. Kepler’s solution to the puzzle >

In between trundling off to give Emperor Rudolph his daily horoscope, Kepler came up with the ingenious idea that the way to get himself unstuck in this problem was, ironically, to stick something else in place. The sun was already a fixed point in his system and with a second fixed point, he would be able to figure out the position of the earth. In mathematics, the *angle-angle-side theorem* states that if you know two angles of a triangle and the side next to them, you know the entire triangle, and can therefore work out the location of one vertex relative to the other two. This technique, known as *triangulation*, was known since Phoenician times and is used today in our GPS systems. In Kepler’s system, with the sun fixed and the earth moving, the only remaining object in this system that he could easily fix in place was Mars itself.

> >

To fix Mars in place, Kepler needed some information he knew had already been worked out by Copernicus. The Babylonians had found the amount of time that it took Mars to return to the same place in the sky, called the *synodic* or *apparent period* (it appears in the same place relative to the earth but the earth has also moved in that time). This is 780 days. Copernicus knew that the earth went around the sun once every 365 days, which meant that the angular velocity of the earth was 1/365 rotations/day and the difference in angular velocity between the earth and Mars was 1/780 rotations/day. By subtracting the difference in angle of velocity of the two planets from the earth’s angle of velocity, Copernicus arrived at 687 days as the sidereal period of Mars, or the actual time it took Mars to orbit the sun and return to the exact same point in space. Kepler likely did not say “Bingo!” because that word hadn’t been coined yet, but this was precisely what Kepler needed. Bingo!

> >

Kepler now only needed to look at Mars’ position relative to the earth in intervals of 687 days, something Copernicus was never able to do because of the sporadic nature of the observations of Mars in his day. Kepler, however, had Tycho Brahe’s fifteen years of consistent planetary observations. From his other calculations related to what was to become his second law, Kepler already suspected that the orbits of the planets were not circular and that the Sun was not in the center of the orbit. He had put quite a lot of effort into both an elliptical shape, which he then set aside as being too computationally complex, and an egg shape, which just did not work at all. To define the shortest path as an ellipse rather than a circle, Kepler would need a minimum of four defined points. Brahe’s years of data would have ideally yielded approximately eight data points, but between Mars not being visible at night some portion of the year and the sky above the island of Hven not being perfectly cloudless at all times, Kepler ended up with five fixed points to give a good approximation the orbit of the Earth and produced an orbit that was just very barely elliptical. By picking another fixed position for Mars, he could produce a new set of points and further refine Earth’s orbit, confirming that these points fell upon the ellipse he had found. He continued in this manner (complaining bitterly about having to repeat his calculations some seventy times) until he had reasonably defined the Earth’s orbit. Kepler could then use the same method, fixing the Earth at one point along its orbit around the Sun, to define Mars’ orbit also. The orbit of Mars, very helpfully, is the most eccentric (which is to say, the least circular) elliptical shape which made Kepler’s computational results very clear.

`3a:[] 2f:{"title":"We need more tiny knowledge projects—here’s one","cells":[{"id":0,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":697,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":366,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":709,"value":"$36","pinned":false,"hidden":false,"mode":"ojs"},{"id":731,"value":"viewof t","pinned":false,"hidden":false,"mode":"ojs"},{"id":688,"value":"visnupPlanets","pinned":false,"hidden":false,"mode":"ojs"},{"id":99,"value":"$37","pinned":false,"hidden":false,"mode":"ojs"},{"id":449,"value":"md`<Why don’t we actually show what’s being described in these last few sentences? What do the resulting orbits actually look like? How did they deviate from the orbits we now know to be correct?>`","pinned":false,"hidden":false,"mode":"ojs"},{"id":456,"value":"md`\n### 2. Kepler’s Platonic solids\n\n>

Kepler already knew that the planets moved in some regular fashion, and wanted to fit this movement into an orderly progress about the Sun. He first tried to create a system in two dimensions, with the circular orbit of each planet layered between regular polygons. When he could not make this system align with observation, he expanded it into a third dimension, with the orbit of each of the six (known) planets arranged on the surface of a sphere that was itself nested inside one of the five *Platonic solids*.

\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":459,"value":"md`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":576,"value":"md`<Can we do better than this lovely image—which is beautiful, but actually kind of hard to parse. Perhaps we could show the orbits themselves more legibly on top of the Platonic solids model? Maybe the two-dimensional scheme that Kepler started with is easier to draw?>`","pinned":false,"hidden":false,"mode":"ojs"},{"id":475,"value":"md`\n### 3. The Rudolphine Tables\n\n> These *Rudolphine Tables* would chart the planetary motion through the constellations in great detail, allowing anyone who used them to draw up much more accurate horoscopes than they were able to using the current error-ridden charts that drew on Copernicus’ scant data. With ongoing rumblings of war across Europe, Emperor Rudolph saw great value in better astrological predictions just as Frederick II had before him and was happy to fund compilation and publication of these tables by Brahe and his staff.\n> \n>

After some months of this routine of Kepler picking the Mars observations from the greater body of data page by page, [. . .]

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":482,"value":"md`<What would Kepler have actually seen while poring over these tables? Are there archival images available somewhere? It would be especially great to see the Mars data highlighted somehow.>`","pinned":false,"hidden":false,"mode":"ojs"},{"id":467,"value":"md`\n### 4. The problem of Mars\n\n>

Finally, Kepler felt he had everything he needed to sort out Mars’ orbit and show that it agreed with the tidy circular orbits of the Copernican model. As he really dug into the data this time, he quickly ran into two significant problems. The first was that now that he had Brahe’s observations of Mars laid out neatly before him, it was very apparent that the path that Mars took through the night sky did not fit the circular orbits he expected.

\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":469,"value":"md`<If Kepler were to sketch Mars’s path at this point, given the data he had on hand, what would it have looked like? Is there a real sketch in his notebooks somewhere, the dots as yet unconnected?>`","pinned":false,"hidden":false,"mode":"ojs"},{"id":488,"value":"$38","pinned":false,"hidden":false,"mode":"ojs"},{"id":492,"value":"md`<Can we illustrate the ecliptic plane in two dimensions—showing the planets running in straight lines—and compare that model to the curve we see in the Your Sky map above?>`","pinned":false,"hidden":false,"mode":"ojs"},{"id":584,"value":"md`<What did Brahe’s ecliptic longitude data look like? It left Kepler with an incomplete picture. Can we sketch his mental model—the little animation that might have been playing in his head? In other words, can we visually represent the uncertain picture he had at this time?>`","pinned":false,"hidden":false,"mode":"ojs"},{"id":495,"value":"$39","pinned":false,"hidden":false,"mode":"ojs"},{"id":500,"value":"md`<This is the crux of it, and it would be wonderful to have a dynamic illustration, perhaps an interactive one, of Kepler’s iterative triangulation method—by which he took a handful of Brahe’s fixed points for Mars and used it to approximate Earth’s orbit, then Mars’s in turn.>`","pinned":false,"hidden":false,"mode":"ojs"},{"id":717,"value":"md`## Code, imports, etc.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":729,"value":"maxWidth = 600","pinned":false,"hidden":false,"mode":"ojs"},{"id":320,"value":"import { chart as visnupPlanets, update, viewof t } with { maxWidth as width } from \"@visnup/re-we-need-more-tiny-knowledge-projects-heres-one\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":727,"value":"update","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W3a","stdlib":"1","titling":"auto"}