Author: FranklyWrong

  • [Little Demos] 04 – What Curve Do Shadows Trace (On a Spherical Earth)?

    This was a cancelled “Work-At-Home” experiment for physics undergraduates, due to local weather conditions, and a second thought that this project is better suited for high school students.

    Conic Sections

    You may skip to the next section if you are comfortable with analytic geometry or related mathematical fields.

    If you use a plane to cut the surface of an infinitely extending cone, the resulting curve of intersection is called a conic section. Depending on the relative positions of the plane and the cone, you might get an ellipse, a parabola, or a hyperbola.

    On the Cartesian plane, any quadratic equation of x and y (as long as it’s not degenerate) corresponds to a conic section. That means that all conic sections follow the form:

    Ax2 +Bxy+Cy2 +Dx+Ey+F =0

    Merits of Geocentrism

    The earth orbits the sun with a persistent axial tilt of roughly 23.5 degrees. As a result, the latitude at which the sun directly shines overhead at solar noon (the subsolar point), oscillates between -23.5 and 23.5 degrees during the course of a year. You might have seen these photos from tropical regions, such as Hawaii, where the shadows of things disappear on certain days at noon, giving you a surreal feeling.

    On this note, the long-superseded Geocentric view of the universe did leave one practical legacy: for the slow-moving creatures on the surface of the planet such as ourselves, thinking like Ptolemy of Alexandria gives us a useful framework to describe the paths that celestial bodies take during the day.

    A sphere is locally flat, and so we picture the ground on which we stand as the x − y plane, with positive x- direction being east and positive y-direction being north. Readers in the Southern Hemisphere can simply flip the signs.

    We then regard the rest of the universe a large sphere that surrounds us. It has two poles that are directly above the earth’s north and south poles, and the ground plane cuts into it, forming a hemispherical “sky dome”. All celestial objects, the sun included, trace almost circular paths on the dome as a result of the earth’s rotation. 

    This coordinate system is illustrated below.

    The northern celestial pole conveniently has a bright star system nearby – Polaris – and deducing one’s latitude would be as simple as measuring the elevation angle of that star. That’s not relevant to today’s little demo, of course.

    The Shadow

    In the coordinate system defined above, we can consider a rod of unit length standing upright relative to the ground.

    No demos around here are complete without a bad Blender animation …

    Assuming the earth is perfectly spherical, it is not hard to show that the tip of the rod’s shadow will move on the ground according to the following equation,

    (􏰀x2 +y2 +1) 􏰁sin2a=(ycosb−sinb)2

    This is a family of conics (one or a pair) parametrised by two angles:

    • a, the latitude of the sub-solar point,
    • b, the latitude of the rod.

    Henceforth, if we can get the curve traced by the shadow on the East-North coordinate system, we may employ nonlinear curve fit to find a and b. In other words, we can locate ourselves in space and in time (restricted to somewhere on contemporary earth …) by staring at shadows.

    Of course, to see shadows change with your own eyes, you must be very bored and wait for hours. So we stare at the model instead, like happy little theorists that we are.

    For most of us, notice that the a and b combinations usually describe a hyperbola, going readily to infinity to both sides, which corresponds to sunrises and sunsets. At the same time. it never hurts to remind ourselves that, when the length of the shadow is comparable to the curvature scale of the earth, our model breaks down.

    For high latitudes, it’s actually possible for the tip of the shadow to trace an ellipse, which corresponds to the Midnight sun, where the sun never sets during summer. Of course, the transition case of a parabola is also possible. Question for you: Where and when does that occur?

    I’ve prepared an interactive notebook (with one extra simplification model where we work out a values according to day of the year):

    Demo on Desmos Graphing Calculator.

    I skipped a lot of historical contexts in the interest of time, but as you may have guessed, what I described here is exactly how sundials work…

    Lastly…

    Take this, flatearthers!
    (I was one of them until I realised some F. E. believers were actually serious… Is this a hint of the next Aperiodical blog post?)

  • [Lux] Construction Conjunction

    In Series …
    Photo stories
    Artistic, scientific, social, or otherwise nontrivial moments that I freeze, served with related thoughts.

    The order of nature that we find ourselves developing from, is sometimes contrasted with the order that we strive to construct for ourselves after (and despite) realising the frequent lack thereof if one is ever not careful.

    I thank the language of physics for unifying my appreciations of both.

    …we give little thought to the machinery that generates the sunlight that makes life possible, to the gravity that glues us to an Earth that would otherwise send us spinning off into space, or to the atoms of which we are made and on whose stability we fundamentally depend…

    Carl Sagan, Introduction to A Brief History of Time, 1st Ed.

    In another recent note, 366 days ago, I turned in my honours / senior thesis, and wrote down the first paragraphs of my statement of purpose in hep-th. So much for that thought, but I am glad I kept going in physics.
    — And I got to sneak into the same mathematics office again in order to recreate a one-year-old photograph. Try the new slider widget below!

    What’s so bright in my eyes?

    The ceiling lamp.

  • [CS] A Story of Blank, Part I

    … What else can you do while waiting for cosmological simulations to finish other than drawing or reading? Asking for a friend.

    (more…)
  • One Try

    The idea for this essay came up during a discussion with one of my grad school referees after the 2019 season. Namely, if I so eagerly want to “save the world”, why study theoretical physics?

    The following essay was the first part of my attempt to answer that question… The rest, of course, needs to be in action.


    How are we sure, that Homo, our genus, is the maker of the first technical civilisation on this planet? This question persisted in my head, probably since my primary school readings of fiction and dubious mystery books.

    I thought I would (as a noble form of procrastination) outline a few facts that I believe supports the case that we are the first, and, quite probably, the last, to emerge from this planet to our current level of technical (and scientific) proficiency.

    Looking at our own past, some preliminary evolutionary considerations can be made. The environmental and thermodynamic conditions required for “intelligence” to be in favour was indeed rare. We owe our bipedalism, our properly placed thumbs, our sugar-craving brains, and the worlds we managed to shape with those gifts, to the K-T comet that cleared the stage for us long before our story, to the receding forests on the African highlands, to the frozen northern Pacific ocean, and to the tigers and lions that didn’t like the taste of monkey flesh too avidly … to name a few.

    I have to accredit Harari’s Sapiens, for inspiring most of those above personal notions — which might be a sign to you that my thoughts on this subject have been shallow and stagnant. Take my rumblings with a grain of salt.

    Still, the history of earth’s biosphere is way longer than the measly millions of years that the primates have arrived on stage. That much I do know. Time erodes a lot of things, including our trust in time itself. So, you might ask, longingly, maybe, could technology and science have happened to some entirely different branches of life than us?

    Well, we know of ants that make tools and farm aphids. But probably not.

    Other than the so-far general lack of archaeological footprints left by any long-gone technocratic societies, the most convincing observation, to me, is the (historic) prevalence of easily accessible natural resources.

    We found, happily accepted, and sometimes wasted, the surplus of earth’s carbon cycle over tens of millions of years (…oil and coal…story for another day). But our luck did not end there. For most of our early history, we had inorganic minerals in shallow caves, if not right on the surface. And that, to me, is a hint that nobody prior has seen much use in them.

    The laws of nature dictated that our societies progressed from[1] bronze to iron, i.e. against the reducing agent strength ladder. Often, after metal tools became prevalent, agriculture evolved on the massively expanding farmlands, and industry emerged with the express aim to produce increasingly complex or powerful tools. The logic might have held up to an practically exclusionary extent in our history — civilisation did not thrive at places that generally lacked minerals, even though many of those states did have excellently arable land and massive settlements.

    Today, far more iron-rich and copper-rich ores come from inhospitable or technically challenging environments than when we started mining millennia ago. Among other reasons, we’ve largely dug up the ones more easily within our reach. It perhaps stands to reason, then, that if humanity is to be wiped off the surface of earth in our current era, whichever species[2] that emerges in another million years might not have any minerals to use.

    They can siphon and recycle our rubbles. You think to yourself. Indeed, it might be sufficient, over geological timescales, for pulverised metropolis and dilapidated recycling plants to return some of their constituent metals to their natural state. But I doubt if this is good enough.

    To begin with, without obvious hints of the way forward, in other words, without a easily accessible experience through which the future foragers can understand why these shiny bits from the ground promises a higher level of productivity, it’s possible our tech relics become more of an ornament than a industrial resource, like humans had done with various cave minerals for thousands of years ourselves. Secondly, the metals we discussed so far are far from the whole story. I will present one example of the missing pieces to try to bring my arguments together.

    By this point[3], many of you might have heard of how Napoléon treated his favorite visitors with alumin(i)um cutlery, and only less important guests of his got gold plates. But today, less than 200 years later, I am typing this essay up on an Aluminum laptop keyboard, protected from some 6th-floor high wind by a presumably aluminum alloy window frame. What made this dramatic cost drop possible?

    Credit human ingenuity as you please, (and I probably tend to agree), but the fundamental drive might be this naturally occurring chemical, Cryolite (Na3AlF6). It readily mixes with aluminium oxide, and it significantly lowers the melting point of the latter. This mineral was mined completely dry from the face of the earth in 1987, by the way. Only industrially prepared alternatives are available for use since[4].

    The ability to synthesise cryolites is only a tiny node on human’s industrial expertise today, but once upon a time, discovering cryolites in nature, was the key for us to access the entire branch of aluminium-based technology. I cannot fathom how many little things like these played similar roles in our past, and how many achievements would have been impossible if those little factors weren’t here.

    I suppose that trivia like these make me appreciate the preciousness and uniqueness of our one try at achieving cosmic greatness. The chances we took to entangle our legends with the threads of our planet, the rare finds that we stumbled upon before we knew better, and the rivers we crossed[5]. But, at the same time, the likely outcome for our story to be followed by endless stagnation and sorrow if we fail[6].

    I remain optimistic, that progress in science and technology solves our existential threats, though slowly, sometimes backtracking, and not exempt from strifes and struggles. Depleted resources? We most likely can find alternatives. Dead end of knowledge, and, by extension, a lack of vision for the future? That curse is eternal.

    FW, 100 seconds before midnight.

    Footnotes and references

    [1] Other than the obvious omissions (tin, lead), there are also gold and silver. However, as they require little to no chemical changes to be useful, I would, subjectively, count them as something no more challenging than rocks.

    [2] Ant people and squid people look equally appealing to me. But again, the selection pressure on brains is a cosmically delicate thing. Boltzmann knows best.

    [3] My sense of how well people around me are in knowing and using science is heavily biased as I progressed through my own education and travelled to various institutions. This is just a general comment and I will probably discuss it in full next week, when my #DailyChemistry stored on Tencent Weibo gets permanently deleted as the service shuts down. Still, I need to point out that if you’ve come for the chemical engineering… You are reading the wrong blog.

    [4] Cryolite lowered the temperature required to get Aluminium via electrolysis from 2000 degrees Celsius to about 1000. From as much I can gather from my high school memory, the reaction is done in three scalable steps. You get HF by boiling Fluorites (CaF2, that glowing ore in Minecraft) in sulfuric acid. Then the HF is taken to react with an Al(OH)3 solution, after which the whole thing is heated in the presence of either NaCl or Na2CO3 to make the crystal.

    [5] Hi Carl.

    [6] I do not worry too much. Our ancestors have definitely taken risks like this before. On the wild grass plains, some humble animals, not the strongest, not the fastest, and without almost all ferocity, made it to become us, against a fate of extinction. Of course, my Bayesian philosophy suggest that luck might be independent between risk taking… Take good care of your nuclear button, if you have one.

  • A Walk In The Garden – IrPt

    Not all songs deserve a blog mention, but this one I specifically have a dream-inspired short story to follow up in late 2020.

    And indeed, in that dream, I read a novella named “Little People” at a mishmash international airport, waiting to venture somewhere afar (and COVID-19-free). At least part of the story plays around my fascination with passenger aviation and people (person?) I love being “one hug away”, whatever the last phrase meant. This was how much I could jot down after waking up from that dream, anyway.

    What if the purpose of life is in dreams, and we wander around in this world just to seek resources to fuel those dreams?
    Meh. Unproductive idea.

    Also some generated tablature for the musically gifted to play with:

    Tablature Generated by Logic Pro X.