from Which Differential Equations are Easy to Solve? by John Starrett
Posts tagged with projection
Vertical cross-section of the Sierra Madres, near Oaxaca City.
You can think about the landscape as a scalar field in 2-D (puncture the plane if you need to do this to the whole Earth) with the height being a numerical quantity assigned to any point. In that case the image above records the values the scalar field takes along the 1-dimensional γ. (If the scalar field were discontinuous, that would mean you either just biked off a cliff, or biked into a wall.) Or you can think about it in 3-D — looking around from your bicycle as you ascend the mountain ridge.
Since people liked my last opinion piece on
#big data, here’s another one.
Imagine there was a technology that allowed me to record the position of every atom in a small room, thereby generating some ridiculous amount of data (Avogadro’s number is 𝒪(10²³) so some prefix around that order of magnitude — eg yoctobytes). And also imagine that there was a way for other scientists to decode and view all of that. (Maybe the latency and bandwidth can still be restricted even though neither capacity nor resolution nor fidelity nor coverage of the measurement are restricted — although that won’t be relevant to my thought experiment, it would seem “like today” where MapReduce is required.)
Let’s say I am running some behavioural economics experiment, because I like those. What fraction of the data am I going to make use of in building my model? I submit that the psychometric model might be exactly the same size as it is today. If I’m interested in decision theory then I’m going to be looking to verify/falsify some high-level hypothesis like “Expected utility" or "Hebbian learning". The evidence for/against that idea is going to be so far above the atomic level, so far above the neuron level, I will basically still be looking at what I look at now:
If I’ve recorded every atom in the room, then with some work I can get up to a coarser resolution and make myself an MRI. (Imagine working with tick-level stock data when you really are only interested in monthly price movements—but in 3-D.) (I guess I wrote myself into even more of a corner here, if we have atomic level data then it’s quantum, meaning you really have to do some work to get it to the fMRI scale!) But say I’ve gotten to fMRI level data, then what am I going to do with them? I don’t know how brains work. I could look up some theories of what lighting-up in different areas of the brain means (and what about 16-way dynamical correlations of messages passing between brain areas? I don’t think anatomy books have gotten there yet). So I would have all this fMRI data and basically not know what to do with it. I could start my next research project to look at numerically / mathematically obvious properties of this dataset, but that doesn’t seem like it would yield up a Master Answer of the Experiment because there’s no interplay beween theories of the brain and trying different experiments to test it out — I’m just looking at “one single cross section” which is my one behavioural econ experiment. Might squeeze some juice but who knows.
Then let’s talk about people critiquing my research paper. I would post all the atomic-level data online of course, because that’s what Jesus would do. But would the people arguing against my paper be able to use that granular data effectively?
I don’t really think so. I think they would look at the very high level of 𝒪(100) or 𝒪(1) data that I mentioned before, where I would be looking.
Now think about either the scientists 100 years after that or if we had such perfect-fidelity recordings of some famous historical experiment. Let’s say it’s Michelson & Morley. Then it would be interesting to just watch the video from all angles (full resolution still not necessary) and learn a bit about the characters we’ve talked so much about.
But even here I don’t think what you would do is run an exploratory algorithm on the atomic level and see what it finds — even if you had a bajillion processing power so it didn’t take so long. There’s just way too much to throw away. If you had a perfect-fidelity-10²⁵-zoom-full-capacity replica of something worth observing, that resolution and fidelity would be useful to make sure you have the one key thing worth observing, not because you want to look at everything and “do an algo” to find what’s going on. Imagine you have a videotape of a murder scene, the benefit is that you’ve recorded every angle and every second, and then you zoom in on the murder weapon or the grisly act being committed or the face of the person or the tiny piece of hair they left and that one little sliver of the data space is what counts.
What would you do with infinite data? I submit that, for analysis, you’d throw most of the 10²⁵ bytes away.
You know what’s surprising?
I guess lo conocí but no entendí. Like, I could write you the matrix formula for a rotation by θ degrees:
I guess 2-D linear mappings ℝ²→ℝ² surprise our natural 1-D way of thinking about “straightness”.
quasiperiodic tilings from a 15th-century Uzbekistani madrasa
Yeah, the same quasiperiodic tilings that theoretical physicist Roger Penrose wrote about in the 20th century. The same quasiperiodic tilings that Tony Robbin says are the projections of a high-dimensional cubic lattice onto 2-D.
What now, Christian culture?
ARTICLE IN: Saudi Aramco World, via Artemy Kolchinsky
PS Saudi Aramco does $233 billion in sales each year. For reference the total value of Facebook is $25b. So Saudi Aramco transacts 9 Facebooks each year. What now, The Social Network?
This is for my homies in maths class. I know many of you fear the
They’re just a way to convert from circular coordinates to square coordinates.
You want to get to the red dot. If you’re at sea you set your bearing
θ and sail straight there. If you’re in the city you walk north and then east.
Cosine tells you how far you have to walk crosstown; sine is how far uptown.
That’s basically it. If you think uptown and crosstown look like legs of a right triangle, you’re thinking along the right track to understand where the many sin & cos formulæ come from.
Here’s a GPL’ed picture of why we sometimes think of cosine as a wave: