Riddle: A Bignomial Taboo

Okay. So, I have another riddle for you. This one is not really physics based, but still it can be solved with logic alone (no math). I’ll give it to you in storybook form just for fun.

Once upon a time there was a forest, and in this forest there lived a sufficiently large number of gnomes (the exact number of gnomes doesn’t matter). They were extremely logical beings and valued the needs of the group far above their own individual needs, so much so that no mirrors existed in their village, so as to keep the focus of their attention away from themselves. They valued homogeneity, which was just as well since all of them were identical… well, except for one thing: their hats. For some inexplicable reason, while most of the gnomes had red hats, there were a certain number of them (say “N”) who had blue hats. These blue hatted gnomes didn’t themselves know that they had blue hats (for lack of mirrors, you see) and it was a taboo subject of the highest degree. No gnome would EVER give any indication — verbal or otherwise — as to the color of another gnome’s hat.

One day, at one of the gnome village meetings, where all the gnomes gathered to discuss serious matters, they decided as a group that because they valued homogeneity so much, it would be better for the village if all of the blue hatted gnomes left and lived elsewhere. Nothing more was discussed. No gnomes were singled out as having blue hats. The blue hatted gnomes were simply expected to leave as soon as they knew they had blue hats.

How many village meetings passed before all of the blue hatted gnomes left?

This is a really tricky riddle. Just remember, the solution has nothing to do with the gnomes using sign language to tell other gnomes about their hats, or using spoons as mirrors to see themselves. It’s a much more elegant and logical solution. If you haven’t heard it before, feel free to bounce ideas back and forth in the comments.

Adventures of the Learning Assistant (Part 4)

(Here are Part 1, Part 2 and Part 3 in case you missed them).

Well… I don’t think my students have this much conceptual difficulty, but I thought I’d start this post off with a bit of comedic relief.

Anyways, a week has passed since the last practical but I’m not behind in posting because this is the undergraduate reading week. My mind has been bubbling with ideas since then; most of which are the result of a TA brainstorming session we had on Friday. A group of enthusiastic learning assistants (lured partly by free pizza) gathered to share their ideas, concerns and advice with a representative from the undergraduate education department (or something of that sort). It looks like there are a lot of problems with this new physics curriculum (IE: the “practical sessions”). We conveyed a lot of worries and put forth a lot of suggestions which I will try to summarize here.

What I should first mention is that I am not alone in my difficulties. Almost all of the other learning assistants are, like me, having difficulties. Here are, in my opinion, the top three problems that we and the students are having with this course:

1. The TAs and the students have a severe lack of feedback from each other.
2. Students won’t ask questions about anything they’ve been having trouble understanding in class, on an assignment, or anything outside the lab activity.
3. The students have difficulty finishing the activities before the end of the practical. This leaves almost no time for theoretical (tutorial-like) questions.

So. Problem 1 (The biggest):

Students need feedback on their work so that they can narrow down what it is they don’t understand. I think one of the hardest things about learning (in a student’s reference frame) is figuring out what it is you don’t know. But students are not given any feedback on their lab book (aside from an initial trial grading of the first activity). The reason for this is that not all activities in their lab books will get graded, and the choice of which ones will be graded is kept secret until midterm. The problem is that the TAs haven’t been told either… so we can’t go through on a weekly basis and put comments in the lab books because we haven’t been assigned enough hours to do that much “correcting”. Hopefully this will be easily taken care of by simply telling the TAs to grade a subset of the week’s activities on a regular basis.

Students need meaningful feedback when feedback is given to them. It’s quite deceiving when a computer tells a student that they’ve gotten the question 100% right when the truth of the matter is there are many things they still don’t understand. But this is what is happening. Each week the students are expected to complete an online assignment hosted by the Mastering Physics website. The problem with these assignments, I think, is best conveyed using the analogy of — and I apologize to the students for this analogy — trying to teach a donkey the way into town by leading it with a carrot on a stick, then expecting it to be able to make the journey on its own. The questions on the Mastering Physics assignments are good questions BUT they are asked in such a way that holds the students hands and practically gives them the answers to each step. This severely reduces the effectiveness of the questions. When I asked one of the students if she had trouble with the Mastering Physics questions she replied, “Well, I got the question right … but I still don’t understand what I did“. Other TAs and even past students have told me similar stories about these assignments.

… but perhaps these assignments are intended to be more useful as feedback for the TAs, you say?

If this were the case, then at least their existence would have some merit. The fact that the vast majority of the students in my section get above 90% on every question should illustrate that this is not very helpful as feedback for us. Apparently this Mastering Physics site has been used for years, much before the recent curriculum change. I think it was part of an effort to recycle old bits of curriculum that is falling short.

It also looks like a good example of an over-reliance on technology to improve education. Computers don’t teach people; people teach people. Fancy gadgets, clickers and advanced quizzing systems are a great idea, but they themselves are not enough. They need to be used effectively. I think this curriculum is still in its early stages of metamorphosis and everyone is still trying to figure out more effective ways of using the new technology and new teaching methods.

Possible solution to all three problems:

One fantastic solution to this problem came together as a melange of a few suggestions in the TA brainstorming session. The organizers of this course got rid of the formal tutorial sessions because they deemed them ineffective and thought it would be more effective to work that kind of material into the practicals. The way we are currently doing this is not working. Instead, what would be more helpful is to have “theoretical” questions as part of the lab activities. There are several benefits to this if it is conducted well.

Firstly, it would encourage the students to work out questions as groups inside the practical sessions. They could get immediate feedback from the TA, and if they worked out the question on their fancy new whiteboards (which I found to be an effective method when I tried it last week) the TA could immediately gather feedback from them in terms of conceptual difficulties and so forth. The questions could be made more difficult without the “hand holding” formulation because if they truly got stuck, they could ask the TA who would be able to gauge what hints were just enough to get the group back on track.

Secondly, the questions could be directly related to the lab activities they would do immediately after. This could solidify their understanding and also make it more interesting. They would be able to see the physics happen on paper, and then in real life. From a purely personal perspective, I frequently found the classroom material to be detached from “real life” physics when I was an undergraduate. It would be nice for the students to see a strong connection between the two through the curriculum.

Thirdly, and most importantly, it would give the students a taste of real science. IE: using a model to derive a prediction (hypothesis) and test it out in the lab. This would also mean that students could be expected to come up with their own experiment (perhaps with the TA’s help) in order to test their prediction. This would eliminate the mundanity of following lab activity instructions step-by-step with no real thought behind it (as was very common for me in my undergraduate days).

In all, I think it’s been a productive week for me as a learning assistant. I’ve pretty much given up on addressing them as a class in an attempt to gather conceptual difficulties. Instead what I found more useful was to visit each workstation  individually. They are much less shy when I do that. That, in conjunction with having them work out a tricky problem on their whiteboards, will hopefully generate a better feedback loop between us.

I’d love to see more of these TA brainstorming sessions for other courses. I think much can be gained from a diverse group of minds and some free pizza.

Let’s get philosophical: what’s your existential preference?

Quantum mechanics is weird. It gets even weirder when you try to interpret what the theory is telling you about “reality”. In fact, I’m taking a course at the moment called: Interpretations of Quantum Mechanics. I’m hoping eventually I’ll get some blogable material out of it.

For now, I have a question for you all. It will be a purely subjective question (not like last time). I’ve blogged about the random nature of the quantum world before, and I’ve also given an account of an experiment that demonstrated the requirement (under certain assumptions) for an inherently random world, but the nature of reality is still a hotly debated topic in the world of physics. Some physicists reject the notion of a world that is fundamentally random and instead consider the possibility that we’re not seeing the whole picture. They come up with, so called, hidden variable theories that attempt to explain away the randomness by postulating some hidden property in the small world that we can’t directly measure. I’ve also recently come across a paper that hypothesises that the (random) quantum mechanical nature of the very very small could be an emergent phenomenon; that is to say (in pedestrian terms) we aren’t squinting hard enough to see all of the information about a quantum system and this lack of information results in seemingly random behaviour.

I wish I understood these things well enough to explain them here… but I don’t. Instead I’d like to know what your personal preference is for reality and why.

Which description of reality are you (secretly?) cheering for? Are you more comfortable with the completely non-random deterministic view of the world, or are you instead enjoying the idea of a world built on random behaviour? AND WHY?

Adventures of the Learning Assistant (Part 3)

(Here is Part 1 and Part 2, in case you missed them).

Sorry for the silence this week… you know how it is.

Before I begin, looks like the MIT physics department is having a few troubles of its own with the new physics curriculum. And, don’t forget to check out the First Excited State for Week 2 of the teaching journal.

I need your help. The practicals are becoming slightly tricky in terms of grabbing students’ attention for tutorial-like situation. As I mentioned last week, the students are vastly more motivated to do the activities than ask tutorial questions because they will be getting graded on the activities. We have the ability to create quiz questions on their workstation computers and we’ve tried creating a quiz question to try to draw out questions from the students. What actually happened was they spent a little while on the question, guessed if necessary (since it wasn’t worth any grades) and then didn’t ask any questions (probably for fear of not having enough time to do the lab activity).

I’ve been toying with a few ideas to try to get their participation in asking questions. The first is instead of asking the whole class a question and going over the solution, to instead go around and ask each workstation one at a time. It would take up the same amount of time for the students. The upside is that they will be much less shy and almost certainly reveal any gaps they have in their understanding. The downside is that any enlightening bit of information will be confined to that table.

In order for the whole class to benefit, I’d have to somehow engage the whole class in problem solving. One general idea I’ve had in that respect is to do away with a multiple choice quiz type question (and eliminate half hazard guesses) and instead ask an involved/conceptual problem. They could then write their answers/ideas on their whiteboards and share their ideas with the rest of the class. Alternately, if they are too shy to speak up, I could go around the class while they are working on the activities and look at the ideas on their whiteboards and discuss with them.

I think the problem here is shyness and time constraints. I’m wondering if any of you have ideas to get students to participate in sharing their conceptual difficulties with the class. Also, I’m wondering if any of you have any ideas for relatively short, interesting questions on the subject of static electricity.

Adventures of the Learning Assistant (Part 2)

(Here is Part 1, in case you missed it).

Looks like there’s also a duality in the blogosphere. Over at The First Excited State, our favorite semi-anonymous author is joining me in this teaching assistant blogothon with his weekly Teaching Journal.

Anyways, another week, another practical session. As I mentioned in Part 1, this week the students measured the speed of sound. So far, the activities seem to be on the right track. They encourage a bit of playfulness and try to help students get some physical intuition about the concepts they learn in class. This week, for example, on of the questions asked the students to play around with the microphone; whistle into it, speak into it, etc, and look at the resulting waveform on the computer screen. It’s interesting to see how the students react to this type of question. One of the students apparently sang into the microphone in an enthusiastic operatic manner and when he noticed that he was being watched by a TA, he expressed very apologetic sentiments. I think it was a small illustration of a student conditioned to believe in the myth that you can’t be learning if you’re having fun. I try to encourage such playfulness. I went around the room telling students to try getting two people to whistle into the microphone at slightly different pitches. I demonstrated this to one of the workspace groups and they were impressed that they could actually see the beats show up on the computer screen.

That being said, there are some problems creeping up. The most prevalent is time constraint. These practicals are supposed to replace the labs AND the tutorials. Each week we have two hours to try to fit in these activities and a little problem session. So far, the activities have taken the students the full two hours. Since students are being graded on the activities and students tend to take a very grade-oriented view of education, the TAs and the students both feel pressured to just ignore the problem sessions and do the activities.

Fortunately, we’ve been given the freedom to grade the students’ workbooks as we see fit. If the majority of students don’t have time to finish all of the “required” activities, then we have the authority to issue grades which compensate for this. The wonderful fact about the grading scheme is that it is on a scale of: 0-4. This means that the majority of the time, the majority of the groups will get a 3. This not only takes pressure off of the TAs that grade them but also it removes much of the competitive pressure on the students. We’re, after all, trying to remove the grade-hungry attitude some of these students have to education. I am going one step further and not showing the students their grade unless they explicitly ask me for it. I’m hoping this will force them to pay attention to the detailed feedback I give them in their workbooks, which, unlike an obscure number, is what will really help them know how they’re doing in the course.

Finally, I’d like to point you to a post on the School of Everything blog to do with something I’ve been thinking about for a little while: adapting teaching methods to reflect the diversity in ways people learn. It goes over the great uncertainty in classification of learning styles and the difficulty this causes in trying to generate a teaching style that accounts for the diversity of people’s minds. I found it very interesting, and thought you might too.

Physics Riddle — Of Rope and Wood

I thought it was about time I gave you (yes you!) a physics riddle to go with your morning coffee. I’ll state the problem first and then give a few hints below the fold. Readers with a familiarity with ideas in physics can probably solve this without any hints. If you are unfamiliar with physics, don’t be ashamed to check out the hints. Most importantly: this riddle can be solved without any equations. Feel free to post your solutions in the comments. I’ll give the solution as a comment later on.

Okay. Here it is:

Not to scale...

You have two objects:

• A rope of length a given length.
• Two pieces of (let’s say…) wood joined by a bolt. Together these pieces of wood stretch out to be the same length as the rope.

These objects are the same length, same mass and the same mass per unit length. You now hang them by their endpoints so that they hang side by side (as shown in the picture). The horizontal distance between the endpoints on which they hang is the same for the wood pieces as it is for the rope.

Which object has the lower center of mass?

Note: If you are unfamiliar with the concept of center of mass, check out Rhett’s post on DotPhysics here. You don’t need to understand it all for the purposes of this problem, just know what center of mass is.

Adventures of the Learning Assistant (Part 1)

Well, the first of my first “physics practicals” were this week. By this I’m referring to the TA job I’ve been raving about. I promised pictures of the shiny new rooms we get to use, so without further ado:

Behold!

So hopefully these pictures will help you understand why I say the new rooms feel like a sportscar.

… okay, so if they feel like a sportscar, how’s the mileage, you ask?

A fair question. The way these rooms are constructed make them ideal for group interaction. They take focus off of the LA, which is as it should be. The LA is not a lecturer. But for that same reason it is very difficult for the LA to hold students’ attentions if they are telling them something important. To compensate for this they have a wireless microphone and speakers installed to give LAs voices a sort of omnipresence in the room. In addition to that the LAs have the ability to control the students’ computers (individually or in bulk) from the main computer at the front of the room; projecting information onto them, creating mini quizzes, taking full control, writing on them, etc. Overuse of these tools could result in the students going through a whole practical without interacting directly with the LA. I see this as a potentially bad thing. So what I’ve tried to do is avoid using the microphone altogether. Nothing says I need to address the class from the front of the room. I just walk to the middle where everyone can hear me better.

This is what I did the first day, and before I opened my mouth I suddenly felt that sensation I had been warned about by my TA friends: the moment of dread. All of those eyes of students in a required course, some of whom hate physics and don’t want to be there, staring at me, expecting me to do something… after about five seconds it passed and I broke the silence with an overly enthusiastic “HI!”. (I might have scared a few). After that I radiated as much enthusiasm and personality as I could muster. One of the first questions I asked them was: “who here absolutely hates physics?”. Out of a class of about thirty students, seven hands shot up. I’m focusing on those seven. If I can make them curious about physics, the rest will be a piece of cake.

The first practical’s activities were a bit of a drag. They mainly involved analyzing flash simulations of waves. Next practical, however, will be fun. I’ve got it all planned out. The scheduled activity for that practical will be measuring the speed of sound using a standing sound wave in a closed tube. The physics and process behind that experiment is completely analogous to my post about measuring the speed of light with chocolate and a microwave. They will use a microphone to find the pressure nodes (quiet bits: reverse analog of the soft bits of the chocolate), and use this to measure the wavelength for a given frequency (pitch).

My plan is to begin the practical by showing this youtube video. It’s a video of a Ruben’s Tube (if you haven’t seen a Ruben’s Tube you must watch that video). The physics behind the shape of the flame in a Ruben’s Tube is the same physics they will be using in their activity. With a Ruben’s Tube you could just take a ruler and measure the wavelength directly since you can see the shape of the wave in the fire. Unfortunately for the students, they won’t have that spectacular representation of the wave and will have to resort to using a microphone to find the quiet bits.

I’ve been keeping notes of ideas I have to make the practicals better. My plan is to get as much feedback from the students as possible. Hopefully some fine tuning will get everyone’s enthusiasm resonating throughout the practicals.

Eternal life – Dyson vs. Krauss

I’ve been meaning to post this for a while, but kept putting it off because I anticipated it being a rather long post. Several months ago I attended a lecture given by Lawrence Krauss at the CUPC. He gave us an overview of a “debate” he had with Freeman Dyson about whether or not life could exist forever. Keep in mind, this is not an argument for the likeliness of eternal life, it’s just simply addressing the possibility of it. In physics, the questions about whether or not something is even remotely physically possible are, many times, the most fun! And the ideas Krauss shared with us that originated from his back-and-forth with Dyson were so fun and interesting that I thought I’d take a stab at reproducing an overview of it all here. Keep in mind, I will be glazing over all of the mathematics and so if you want a more in depth look at the derivations of these results you should probably check out the original papers (here is Dyson’s; here is Krauss’s). They are enjoyable to read if you have a physics background (and maybe even if you don’t). So here it goes. Dyson vs. Krauss. But before we begin this faceoff, we need to buckle down and tend to a question that is begging to be answered:

…what do we mean by “life”?

Firstly, I must mention that we are not talking about eternal life for a single being. This debate was focused on eternal life for, say, a civilization albeit one that may evolve. Secondly, living things come in many shapes and forms, some of which we may not yet be aware of. It seems unreasonable to make the assumption that all forms of life are like those on earth; carbon based, dependent on water to survive, etc. In any case, Dyson and Krauss are both physicists and so for the purposes of their debate they were more concerned with the physics of “life” than its biology. Let me put it like this: we are not really concerned with the biological processes that lead to the thought “I think therefore I am”, we are simply concerned with the existence of the thought itself to define “life”. In other words, by “life” we really mean consciousness, or more simply, computation. Consciousness seems to have a lot to do with the firing of neurons which go about processing information much like a computer (or perhaps a quantum computer). Whether or not consciousness is really akin to some kind of computer program is a whole new debate in itself (perhaps some neuroscientist readers can comment on this). Despite this, computation must at least have a lot to do with consciousness and so surely by investigating the eternal existence of computation we won’t be doing too badly.

So, what restricts us from running a computer program for all time? Well, the first barrier is: energy. Hopefully you are familiar with the fact that the universe is expanding. Not only is it expanding, it is expanding at an accelerated rate. It turns out that this puts a constraint on the amount of energy any civilization can harvest to keep them alive (computing). With a finite amount of energy available one might give up at this point and declare that life, which requires energy to sustain itself, can’t exist for an infinite amount of time. Dyson, however, was still optimistic. He realized that living things are less concerned with physical time and are more concerned with, what he calls, subjective time. Living things measure time by the number of thoughts they have, so if a civilization can have an infinite number of thoughts using only a finite amount of energy, one could say that they have achieved eternal life. This subjective time depends on the temperature at which the entity operates. So if we assume that the civilization has the ability to change its temperature at whim, at first glance it seems like the civilization can have an infinite number of thoughts (live for an infinite subjective time) if it keeps decreasing its temperature for all time (getting closer and closer to absolute zero, but never exactly zero). That strategy (again, at first glance) will allow an infinite number of thoughts using only a finite amount of energy.

So, is this strategy really possible? Well, in answering this question we come to the next roadblock: heat dissipation. Computation generates heat (there’s a reason your computer gets warm when you turn it on). Living things will also generate heat. Even if we ignore all of the heat generated from familiar biological functions and only focus on the heat generated from thinking, we still have a minimum rate for heat production of a living entity. This heat has to be radiated away at a rate greater or equal to the rate at which the heat is produced, or the entity will “die” (there’s a reason your computer’s CPU needs a fan). Dyson considered this and deduced that the best way to get rid of waste heat would be through electromagnetic radiation. However, going through the math he deduced that the rate of radiation of waste heat this way would depend on the temperature and the number of electrons of which the entity was made. And if the life form kept reducing its temperature in this way, there would eventually be a time when it could not radiate its heat fast enough with only a finite number of electrons. So, this couldn’t work. Did Dyson give up?

Nope.

Think about this: what if you really really wanted to go about running a computation on your laptop but your fan couldn’t cool it off quickly enough. What would you do? What Dyson would probably do, is run the computation for a while, put the computer into sleep mode, let it cool off, wake it up, continue the computation and then repeat this until the computation was done! That’s exactly what he suggested a civilization might try to do to live forever; namely periodically hibernate in order to get rid of the excess waste heat! The civilization could continually lower its temperature (decrease its metabolism) and periodically hibernate for longer and longer in order to have an infinite number of thoughts using a finite amount of energy.

A nice strategy… but this is where Krauss stepped in and poked a lot of holes in this argument. The first caveat comes from the necessity for some kind of alarm clock to wake up the civilization from its hibernation. Any alarm clock is inevitably going to be performing some kind of computation in order to calculate when it should “ring” and tell the life forms to wake up and smell the coffee. This alarm clock is subject to the same laws of physics as the life forms themselves and, as such, will eventually use up all energy reserves by the same arguments as above (since a hibernating alarm clock would defeat the purpose).

The second caveat comes from the fact that we are living in a universe which is expanding at an accelerated rate. It turns out that a universe with that property will be permeated by background thermal radiation (analogous to Hawking radiation) which means a lower cutoff for temperature. In short, in a universe undergoing accelerated expansion there is a minimum temperature, which means that Dyson’s strategy of continually reducing a civilization’s temperature won’t work.

Now, you may have heard a bit about quantum computers and be thinking: “… but quantum computation doesn’t necessarily require any energy. You can, in principal, do as many computations as you like without generating heat as long as you don’t measure the result”. If you did think of that, great! However, as Krauss pointed out, you’ll necessarily have to radiate heat if you want to do any erasing in order to prepare for a new computation. If you had an infinite amount of memory storage available you could ignore that point, but any civilization’s memory storage is limited by the number of particles it has access to, which is (as with the case of energy) limited in supply. Krauss sums up this point well.

Thus any civilization can have only a finite total memory available, and resetting registers is therefore essential for any organism interacting with its environment, or initiating new calculations. While an existence, even nirvana, might be possible without this, we do not believe it is sensible to define this as life.

So right now it looks as though life (as some form of computation), by its very nature, must end. Mortality is a necessity of life. I am actually fond of this wistful result. I find it gives life more meaning and makes it more precious… but that’s just me. What do you think?

Have a causally consistent new year

… so what? Another year, another orbit of the earth. What’s the big deal, you say?

Well! I have some New Year’s news for you. Firstly, did you know that this year we get an extra second? To deal with this I suggest counting like a computer scientist; …3, 2, 1, 0! Happy New Year!

This year will, apparently, be the International Year of Astronomy! Georgia at Earth & Sky Science has found a great way to celebrate: 365 days of Astronomy Podcasts.

If that’s not enough, why not follow your lightcone this year? I found a great site that provides an RSS feed of astronomical bodies you (yes you!) could possibly have influenced since your birth.

… what’s a lightcone, you ask?

A lightcone is the 4-D surface in space and time that a flash of light forms as it travels away from its origin. The name is actually a bit misleading. It’s less of a cone, and more of a hyper-cone; that is a cone in four dimensions. Imagine a flash of light. As it moves forward in time the light will move outwards in all directions. At any point in time the light will be confined to the area of a sphere. As time progresses, the sphere will grow in size at a constant rate. If we think of time as another dimension, and tried to draw a 4-D graph of the flash, it would be a hyper-cone.

If you are still confused, a good way to begin thinking about this is to imagine a very small circle lying flat on the ground. Now, imagine it growing and as it grows you move it upwards. Every time the circle grows one centimeter in radius, it moves one centimeter higher up. This traces out the 3-D cone you know and love. The upward direction represents time and the other directions represent a 2-D space. To generalize to a 3-D space with time, you change the idea of a growing circle to a growing sphere. Et voila: a lightcone.

… ok, so what’s so special about a lightcone, you ask?

Since a lightcone is the boundary on which a flash of light travels, and nothing can travel faster than light, the lightcone also marks the boundary of influence of a certain action. Let’s say, for example, you sneezed. Achoo. At some other time, in some other place, let’s say a book fell over. Could your sneeze have possibly caused the book to fall over? What you could do is mark two points on a graph; one representing the time and place of your sneeze and the other representing the time and place of the falling book. You could draw a lightcone originating at the sneeze point. If the other point is outside this lightcone then it is physically impossible for your sneeze to have caused the book to fall over.

You could also do the same for your birth. Draw a cone originating from earth at your birthday. Now draw points for all the stars in the sky at time: today. Any points inside your lightcone could have been influenced by your birth. The word “could” is in italics because it’s really saying: “sure, the laws of special relativity don’t disagree with you… but… there’s more to cause and effect than lightcones”. Still, it’s a fun way to learn about astronomy!

So this year be aware of your lightcone and keep track of the people and events inside it. The range of influence of your actions is probably a lot more vast than you originally thought…

I’m dreaming of a white and sparkling christmas

Canadians know snow. Recently, I’ve been around Toronto and Montreal and I’ve been exposed to a great deal of snow. Delayed street cars; slushy, slippery sidewalks; frosty faces; hail and hazard lights on highways. These are just some of the things I’ve dealt with this past week…

But there’s a warmer side to the white stuff which one of my dearest friends reminded me of two nights ago.  She drew my attention to the fluffy, freshly fallen snowflakes which sparkled under the illumination of the streetlights. She had heard that there was some interesting physics behind the glisten of snow (of which I had no knowledge). She asked me to, one day, explain it to her. I love questions like this, so I decided to do a bit of googling.

…what makes snow sparkle, you ask?

First, let’s take a look at snowflakes in general. They can come in simple hexagonal shapes or complex tree-like crystals. So much could be said about the anatomy of snowflakes, most of which I couldn’t tell you because I don’t know (but here’s a link that might help). The key point I would like to get across is best illustrated by the graph to the left. The graph shows temperature decreasing to the right and humidity increasing upwards. While the more complex, pretty snowflakes tend to form at high humidity, the type of crystals that sparkle are actually the simple ones near the bottom of the graph. These plates and prisms at low humidity have large flat surfaces which act like mirrors reflecting almost all light that hits them. These snowflakes are randomly scattered on the ground and will reflect the dimmed light reflected from the trees, cars, and whatever else happens to be around into your eyes. The majority of the snowflakes will appear to have some average brightness, however, some snowflakes will happen to be at the correct angle to reflect light emitted directly from a light source (like the sun, or streetlights). It is these snowflakes which will be almost as bright as the light source itself; they will stand out and glisten. As you move your field of view, these snowflakes will no longer be properly oriented to reflect the light source and, instead, other snowflakes will stand out. The sparkling will appear to move as you do, in a sort of spectacular specular reflection.

But wait! There’s more!

These simple snowflakes don’t only need to reflect, they can also refract light. Refraction, in case you’ve forgotten, is the bending of light as it enters or exits a different material. The angle of refraction depends primarily on the materials it is entering and exiting, but it also depends on the frequency (color) of the light. Different colors of light will bend different amounts as they enter and exit a snow crystal. White light (coming from the sun, or a streetlight) consists of many different colors of light, these colors will be separated by the snow crystal as it passes through a prismic snow crystal. This is called dispersion. So if you are far enough away from the sparkling snow that the colors have separated significantly, you may see certain snow crystals as being certain colors! The result is something like this:

So on the next dry winter night after a fresh snowfall, take a look outside and see if you can spot some spectacular specular reflection, or even the colorful dispersion of randomly assorted snowy prisms.