SciBlogs

Seeing spots before my eyes Marcus Wilson Jan 23

“Doctor, Doctor, I keep seeing spots before my eyes”

“Have you ever seen an optician?”

“No, just spots”.

The concept of seeing an optician floating across my field of view is a scary one indeed. However, the concept of seeing spots doing the same is one I’m coming to terms with. 

I had a talk to an opthamologist about this last week, as part of an eye check-up. He was very good, I have to say, and we discussed in detail some optical physics, particularly with regard to the astigmatism in my right eye (and why no pair of glasses ever seems quite right).  He also reassured me that seeing floaters is nothing, in itself, to be worried about. It’s basically a sign of getting old. How nice. He did though talk about signs of a detached retina to look out for (pun intented) – and did some more extensive than usual examination. 

So what are those floaty things I see? To use a technical biological phrase, they are small lumps of rubbish that are floating around in the vitreous humour of the eye. They are real things – not an illusion – although I don’t ‘see’ them in the conventional manner that I would see other objects. 

The eye is there to look at things outside it. Its lens focuses light from objects onto the retina, where light sensitive cells convert the image to electrical signals that are interpreted by the brain. But given that the floaters are actually between the lens and the eye, how am I seeing them?

There are a couple of phenomena going on. First of all, a floater can cast a small shadow onto the retina. You can see this effect by using a lens to put an image of something (e.g. the scene outside) onto a piece of card, and then put something between the lens and the image. Some of the light can’t get to the card, and so part of the image is shadowed. The appearence of the shadow depends on how close the object is to the card – if its right by the lens there will be very little effect – but if close to the card there’ll be a tight, well-defiined shadow. My experience is that these spots are definitely most noticable in bright conditions – presumably because the shadows on the retina then appear in much greater contrast than under dull conditions. 

Secondly, however, they can bend the light. Their refractive index will be different from that of the vitreous humour, and therefore when a light ray hits a floater it will bend, a little. The consequence is a defocusing of a little bit of the image, which wil be visible. If the floater stayed still, it would probably barely be noticable, but when it moves, the little bit of bluriness moves with it, and the brain picks up the movement rather effectively. 

The most interesting thing to me is that it just isn’t possible to look at these things. When I try, my eyes move, and consequently these bits of rubbish flit out of view. Rather like quantum phenomena, you can’t observe them without changing where they are and where they are moving to.  

 

 

 

Modes of a square plate Marcus Wilson Jan 15

Alison has drawn my attention to this video. It demonstrates vibrational modes of a square plate by using sand. At certain frequencies, there are well defined modes of oscillation, in which parts of the plate ‘nodal lines’ are stationary. The sand will find its way to these parts and trace out some lovely pictures. 

Vibrational modes are often illustrated through waves on a guitar string. Here, the string is held stationary at both ends, but is free to vibrate elsewhere. There is a fundamental frequency of oscillation, where the distance between the ends of the string is half of a wavelength (this ensures the displacement of both ends of the string is zero since they are clamped).  Since wavelength is related to frequency (frequency = speed/wavelength) that means if the wavelength is 2 L where L is the distance between the ends of the string, we have frequency = speed/2L.  

But that’s not the only possible mode. Another one would have L equal to a whole wavelength (equals two half wavelengths). Or one-and-a-half wavelengths (equals three half-wavelengths.) This gives us, rather neatly, frequency = n speed/2L, where n is an integer. We see that our ‘harmonics’ are just integer multiples of the fundamental frequency. Rather neat.

However, if you look at the frequencies given in the video, they appear to be all over the place. I challenge you to pull out the relationships between these (I’ve tried). There are a few reasons why the case shown on the video is considerably more complicated than the waves on the string. 

1. The boundary conditions. The edges of the plate aren’t clamped in place. This makes it less straightforward to define the modes geometrically. 

2. The plate is square, giving rise to ‘degeneracy’ in the modes. This term refers to two or more distinct modes having the same frequency. You can see it rather well with the 4129 Hz mode. Basically, there are horizontal stripes shown. But equally, with the same frequency, you could get vertical stripes. Why don’t the two occur together? They do. You can see the effect of having a little bit of vertical stripe most clearly at the far end of the plate, where the pattern becomes more square-like. Also, with a square, you can get two completely different types of mode with the same frequency. This occurs because what matters are the sums squares of pairs of integers. Broadly speaking (at least for a square clamped on the edges, which I must point out this ISN”T), our modes follow the relationship:

f = C sqrt(n^2 + m^2)

where C is a constant, ‘sqrt’ means square-root, and n^2 is n-squared. So, for example, not only is 50 equal to 5-squared plus 5-squared, it is also equal to 1-squared plus 7-squared (or 7-squared plus 1-squared). This gives us three  modes all competing to appear at exactly the same time. What happens then isn’t easy to tell. 

3. Non-linear effects. This a physicist’s code-word for ‘it’s all too difficult’. That’s not quite true – arguably most of the interesting physics research happening in the world is looking at non-linear effects. What this really means is that, if A and B are both solutions of a problem, then some combination of A and B is NOT a solution. A lot of physics IS linear – Maxwell’s equations in a vacuum is a good example – but a whole lot isn’t. With waves, the speed of the wave usually depends on frequency (i.e. is not constant) which means we lose the nice, integer-multiple relationship of our waves-on-a-string mode.

So, enjoy the video for what it is, and don’t try to analyze it TOO closely. 

 

 

The changing face of professional institutions Marcus Wilson Jan 12

I’m sure many readers will know that one of the hats I wear is the treasurer of the New Zealand Institute of Physics. NZIP is the professional organization for physicists within New Zealand. Its aim is to promote the interests of physics and physicists, at all levels, within the country. In addition to counting the beans, the role comes with a position on the council, and therefore I have a significant responsibility for looking after the institution.

With that in mind, I had the dubious pleasure of travelling to Wellington shortly before Christmas to represent NZIP at a meeting of council members of New Zealands ‘science’ societies at the Royal Society of New Zealand. I use ‘science’ in a very broad context here.  It was one of those ten-degrees-with-horizontal-rain summer days in the captial*, though that didn’t matter too much as the day was spent inside RSNZ’s rather nice new building. The day was actually very useful, as we talked through some of the common issues facing our science societies. 

One clear issue that many societies are facing (NZIP included) is dwindling membership. With dwindling membership comes dwindling income, making it harder for the society to do useful things. A great many socieities can’t afford paid staff and so run on volunteers who necessarily need to prioritize their time elsewere.  Dwindling income basically means loss of services that can be offered to members, such as travel grants, teaching materials, careers advice, prizes and so forth.  It’s a vicious cycle. 

But it’s not all bleak, so long as we are prepared to accept the message that is coming from the research in this area. A recent report from the Australasian Society of Association Executives [which I'm afraid I can't find openly on their website, so no link I'm afraid - you might have to pay membership fees for it ;-)  ] talks about the changing face of membership. The report is rather pessimistically and not entirely accurately  titled ‘Membership is Dead‘. It talks about the difference in expectations that a ‘Generation Y‘ person has from the Baby-boomer (I so hate those stereotyping terms – but the report uses them). What particularly got my attention is that the very things that Generation Y values [clearly defined value to them, responsiveness (which means hours or minutes or instant, not days or weeks), innovations, accessibility] are things that baby-boomer-dominated councils see as low priority, because they themselves don’t value them. In other words, the expectiations of Generation Y and Council members when it comes to what a science society is and does are vastly different. A Baby-boomer may happily pay their membership fees year-on-year because they feel it’s part of their duty as a professional to belong – a Y-er is less likely to take that generous line. If they can’t see the value to them, they don’t cough up. (For the record, I’m an X-er). It encourages institutions to work hard and getting younger people onto councils, by actively targeting undergraduate students, for example by giving them opportunities to assist with conference organization, website development and maintenance, tweeting on behalf of the society, etc. Then step back and let the younger people run it in a way that the younger people (=future members) want it run. 

So membership isn’t actually dead. Instead, we just need to accept that ‘membership’ is going to mean something different to our younger people and adapt to account for it. Because, if we don’t, our societies will dwindle away, to be replaced by something rather different. 

*Maybe I’m a bit harsh here. On our return from the South Island after our Christmas Holiday, the ferry Aratere (with full a complement of screws (propellors) and a fully-functional electrical system) took us across a flat Cook Strait and into a beautifully calm and sun-kissed Wellington harbour. As they say, you can’t beat Wellington on a fine day…

 

 

How do you teach creativity in physics? Marcus Wilson Dec 12

In the last couple of weeks, I’ve been using Hermite Polynomials in my work. I won’t go into what they are (look them up here if you like) suffice to say that they are one of many contributions to mathematics from Charles Hermite (1822-1901), who was himself one of  many french mathematicians whose work has laid a foundation for much of modern theoretical physics. A physicist would generally know these polynomials (when modulated by a gaussian function ) as the solutions to the 1-dimensional quantum harmonic oscillator, although that’s not why I’m using them. 

The 1-dimensional quantum harmonic oscillator problem is a textbook problem that gets inflicted on generations of students. I remembering suffering the algebra that went with it. At the University of Waikato, we save our second year students the algebra by just talking about the solutions, but then spring it on them in third year. For those who like that kind of thing, it’s an interesting analysis, but for those that don’t, it really is quite horrible. 

Perhaps that is what motivated Paul Dirac to come up with (in my opinion) a really elegant complementary approach to solving the 1-dimensional quantum harmonic oscillator problem. While his approach is easily found in text-books, what I haven’t been able to track down is a description of how he came up with it. The same seems to be true of many of the analyses that get wheeled out to students. While they look clean and tidy when presented now, I’m left with the question “How did they come up with this?”. That tends to be overlooked in favour of the end product. Did Dirac spend weeks pondering over this, thinking “there must be a better approach – the symmetry between p and x in this equation should surely be exploitable somehow…”, was it a sudden revelation, did he try twenty different approaches till something worked, or what? My text books don’t say. 

What Dirac did was to reformulate the problem in terms of ‘raising’ and ‘lowering’ operators. He realized the problem as a ladder of energy-levels, and showed rather elegantly that these energy levels were equally spaced. Moreover, some rather neat operators, that he defined, could move a quantum state ‘up’ or ‘down’ the ladder. That’s a very creative way of looking at the problem, and has been taken much further since then. For example, when analyzing problems with many electrons (which generally means just about anything electronic) we can now formulate the problem in terms of operators that create and destroy electrons. Whether electrons really are being created and destroyed is a moot point, but the formulation is a neat one that helps us to analyze what is going on. Theoretical physicists consider it a really useful ‘tool’ of the trade, even though the history behind its construction tends to be overlooked when we teach it. 

So what is the point of me telling you this? Well, it’s about teaching. Just how do you teach creativity, especially in something that is, on the face of it, as tedious as physics. Physics isn’t actually tedious (if it were I wouldn’t be sitting here writing this) but we do tend to make it unnecessarily so at times. I wonder whether that’s because that’s the easiest path to take for undergraduate teaching. At PhD level and beyond, there’s some really creative research going on, but do our undergraduates really see this? Likewise, from what I’ve seen at school science fairs, there’s some great creativity at primary and intermediate school level, but that then vanishes late in secondary school in favour of ‘content’. Somehow, we tend to smother out creativity and elegance in favour of ‘something-that-gets-the-job-done.’  But truly great physicists, Dirac included, have never ‘just-got-the-job-done’. 

Open-ended projects are a way to go (and we manage to some extent to do this with our engineering students), but, as many readers know, we run into trouble with time, the need to prepare students for exams, fitting in with timetabling requirements, and so forth. The problem may go much deeper than we think – indeed, does the whole secondary and tertiary education structure smother-out creativity from students (at least in physics)? 

And with that, have a creative Christmas, and Happy New Year to you all!  I’ll be heading southward next week to the Canterbury hills – a part of the country I haven’t been to before. 

 

 

Virtual labs: Are they virtually as good as real ones? Marcus Wilson Dec 03

I’ve been reading a paper by Majorie Darrah and others (full reference below) on the use of ‘virtual labs’ in Undergraduate Physics. At Waikato (along with lots of other universities) our first year physics students carry out laboratory sessions to help them learn physics concepts and practical skills. If you are someone who has run a first-year laboratory class, you’ll be well aware that these things are costly and time-consuming. If they’re not done well, they become an expensive way of wasting everyone’s time. 

Recently, there’s been a lot of work on ‘virtual’ laboratories. These are laboratory sessions that aren’t ‘hands-on’, but simulated on a computer. There are some pretty sophisticated ones out there. At our last NZ Institute of Physics conference, David Sokoloff, one of our keynote speakers, talked about some of these. The computer software allows a student to do pretty-much whatever would be done in a laboratory, but without the university having to purchase, set-up and maintain the expensive equipment. (And, from the student’s perspective, they are not constrained as to when they carry out the ‘lab’). 

So, do they work? I don’t mean does the software work, but does the virtual lab give the same benefit to the student as undertaking a real lab. In other words, do the students achieve the same learning outcomes? To test this, Darrah and her colleagues worked with 224 students at two universities. They were put into three groups – one group did the traditional hands-on labs in a laboratory, one group did the hands-on labs AND the virtual labs, and the third group did just the virtual labs. Their learning was tested with a quiz after the lab , an assessment of the student’s written lab report, and  tests. 

So what was the result? They found no difference between the groups. One of the universities conveniently carried out a test assessment both before and after the lab sessions and found that all groups improved as a result of the labs, be they real or virtual. That is certainly an encouraging result for the likes of Sokoloff, and those budget-pressed universities with lots of students to push through first year university physics. The problem of doing laboratory work has been one of the reasons why MOOCs for science and engineering have been fairly slow to get going, However, it may be reasonable to do away with this, if good virtual labs can be prodcued. 

But, there is a but. It’s a big but in my opinion, and one that, surprisingly, the authors fail to comment on. Their post-lab assessment of learning was based on a written test of the physics theory concepts that were covered in the lab. In other words, they were testing how well the laboratory (real or virtual) supplemented the teaching of physics theory done in lectures and elsewhere. What they weren’t testing were practical laboratory skills (e.g. how to wire up a circuit, track down problems with the apparatus, carry out experiments in a controlled manner, etc.) These are all important skills for a physicist. If universities as a whole shifted towards virutal labs in first year, where does that leave students in learning these other skills? The paper doesn’t comment. What I’d like to see is the same study done, but the students afterwards given a laboratory test – put them in a real lab and get them to do a real experiment, assessing some practical learning outcomes. Then what happens? It would be nice to try it out – but it will take a bit of organizing (not least acquiring some virtual labs and convincing my colleagues that it is a good idea.) So don’t expect a response from me soon.

Darrah, M., Humbert, R., Finstein, J. Simon, M. & Hopkins, J. (2014). Are virtual labs as effective as hands-on labs for undergraduate physics? A comparitive study at two major universities. Journal of Science Education Technology 23:803-814. doi 10.1007/s10956-014-9513-9 

 

Archimedes principle: think carefully Marcus Wilson Nov 14

Benjamin has recently acquired a ‘new’ book from Grandma and Grandad: Mr Archimedes’ Bath (by Pamela Allen – here’s the amazon link – the reviews are as interesting as the content). The story-line is reasonable guessable from the title. Mr Achimedes puts water into his bath, gets in, and the water overflows. What’s going on? So we’ve been doing some copycat experiments – not by filling the bath right up and having it slosh all over the bathroom floor (Waipa District Council – you can rest easy about water usage)  but filling up rather more sensible-sized containers and dropping objects in.

Archimedes principle is actually a little more involved than simply saying that putting an object in the water will raise the water level. It says that the weight of water displaced is equal to the force of buoyancy acting on the object.  This picture summarizes it. That is, if an object of 2 kg floats, then 2 kg of water will be displaced. If an object is unable to displace enough water for this to be the case, it will sink. That still should be pretty easy to get, especially if you’ve done some experimenting. However, it can still be the basis of some really hard questions. I had one in my third year  physics exams at Cambridge. In our ‘paper 3′, as it was called then, the examiners had free reign to ask about ANYTHING that was on the core curriculum from any of our years of study – plus ANYTHING that was considered core knowledge for entry into the degree (which meant basically anything at all you were taught in physics or general science from primary school upwards). This paper was feared like anything – it was basically impossible to revise for*. 

Here is a question then, as I recall it from the exam.

An ice cube contains a coin. The ice completely surrounds the coin. The cube is floating in a container of water.  The cube melts. Does the water level rise, fall, or stay the same? 

Think carefully before answering. 

Now, the icecube melting question is one that is often banded about. A floating icecube will displace its own mass of water (so says Mr Archimedes). When it melts, this water will occupy the ‘space’ that is displaced by the cube. Consequently, the water level will stay the same. A practical example of this is in the estimation of sea-level rises due to global climate change. When the ice floating on the Arctic Ocean melts, it does not cause a sea-level rise, since it is already displacing its own weight. However, the icecap on Greenland will cause a sea-level rise as it melts, since it is currently not displacing any of the sea (since it is sitting on land.) 

However, that is not the question that is asked. Our icecube has a coin inside it. What difference does it make? Well, the icecube-and-the-coin will still displace its weight of water since it floats. However, when the icecube melts, the coin sinks and no longer displaces the same amount of water as it did when it was frozen into the cube. Therefore the water level falls. That’s quite a subtle application of Archimedes principle. After the exam, a group of us sat arguing about it, till we collectively worked out what the right answer was (see – exams can be good learning experiences!). Unfortunately, at this point I realized my answer was wrong. Even still, I managed to get out of the degree with a first-class honours, so I couldn’t have done too badly on this exam overall.

*The other question I remember from this paper is ‘What is Cherenkov Radiation?’ I didn’t have a clue what Cherenkov radiation was when I sat the paper – I made up some waffly words and wrote them down and almost certainly received zero for the question.’  Later, one of my friends found a single, incidental sentence in a handout that was given out by our nuclear physics lecturer that identified what it was. That’s how nasty this exam was. 

Help! There’s no equation to use Marcus Wilson Nov 07

Today the University of Waikato is hosting a group of local secondary physics teachers. We’ve had an entertaining morning, with some sharing of ideas. As part of this, Rob Torrens, who teaches our large first-year engineering papers, talked a bit about life as a first-year engineering student. How does the school to university transition work? (or not.)   On a non-technical front, he talked about the need for students to begin to take some responsibility for their own learning. If you fail to submit an assignment, it might be a little while before it’s noticed and acted on.  At university there’s no ‘bell’ to tell you that you need to be at your next lecture – there may indeed be no-one even telling you to get out of bed in the morning. It’s easy fall off the radar if you’re not motivated. 

On a technical front, Rob talked about some of the skills developed at university that are new to many students. Mathematical modelling is one. He used the example of ‘mass balance’ in an industrial process. If you are drying grain, you put damp grain into your drier and extract dry grain from the end; this is achieved by drawing in dry air from outside, heating it up, passing it over the grain, and expelling the damp air. Mass balance says that mass isn’t created or destroyed in the process. But how is that represented for this particular process. There isn’t a ‘grain-drying mass-balance’  formula in most engineering text books. Students need to work it out for themselves. The mass of what goes in must equal the mass of what goes out, so:

M_grain_in + M_air_in = M_grain_out + M_air_out

We have an equation that we’ve constructed, just by thinking about the physical principles involved. Throw in some more consideration about the amount of water air can hold (and therefore what M_air_out – M_air_in can be, and we can find out useful things, like how much air we need to draw into the machine for each tonne of grain that goes through it. We’ve started the process of mathematical modelling. 

This is a skill aligned well with what the Physics Scholarship exam is about – where students need to think carefully through physics concepts before drawing from mathematical equations. 

Toddler does physics-art Marcus Wilson Oct 29

As we all know, a scientifically-minded toddler plus a piece of technology can lead to unexpected results. This is the result of Benjamin playing with a retractable steel tape measure at the weekend. How we came to break the case apart I don’t know, but the results are pretty (the cellphone shot in poor light doesn’t do justice to the artwork): 

20141027_145124.jpg

 

 

20141027_145132.jpg

I like the koru-shape made by the end. The measure has curled itself into a complicated form rather reminiscent of a protein structure, with sections of helices and straighter lengths. Although the mechanisms are different (protein structure has a lot to do with the intricaces of chemical bonding) the physical process is similar –  the structure works itself to a local minimum of energy. Just how this happens  is all rather complicated from a physics perspective. Perhaps the most obvious example of twists of this form is in telephone cords. The phenomenon has even lent its name to a type of structure seen in thin films – the ‘telephone cord buckle’. Unfortunately Benjamin didn’t give me any warning about what was going to happen – otherwise I’d have filmed it (and he would probably have retreated to a safe distance – the whole unravelling was pretty energetic). 

BUT…since Karen is an occupational therapist and has accumulated large numbers of free tape measures as corporate freebies in her career, we could maybe spare a few for high-speed filming.

Robot racing Marcus Wilson Oct 22

The Engineering Design Show is currently in full swing here, with the competitions for the various design projects. The white-line followers kicked off proceedings. They were pretty impressive, with all but one team successfully being able to follow the (very squiggly) line without mistakes. There were traps to confuse the robots – the line got thinner and thicker, crossed over itself, had abrupt corners and so on, but the robots were well programmed and coped with this easily. The winning group was impressive indeed. They had some very carefully optimized control parameters, meaning that the robot was (a) really straight and fast on a straight-line section but also (b) precise round the turns, slowing down just enough to take each turn at about the right speed. I think anyone would struggle to get something going quicker than this one. 

On show at the moment are the third year mechanical engineering students who have designed a pin-collecting machine. The idea is that the vehicle pulls still pins (about 5 cm in length, maybe 5 mm in diameter) out of a board – the one that collects all the pins in the quickest possible time and drops them back in the collecting bin is the winner. The most striking conclusion from this exercise is the emphasis on the old adage “To finish first, first you must finish”. A good proportion of the entries have died part way through the process – pins have jammed the mechanisms, the motors have failed, or, in one disappointing case, the machine collected the pins in lightning quick time and then failed to go back to deposit them in the collecting bin. Also, we’ve seen one machine disqualified for being downright dangerous – its first run saw it pulling pins out of the board and firing them across the room causing spectators to beat a hasty retreat. 

But the winner (or so it looks) has pushed their luck to the limit.  The “…first you must finish” line is actually not quite correct. More accurate would be to say “…second you must finish. First, you must start“. They’ve admitted to putting 5 volts over a motor rated at 3 volts in practice just before the event, and frying the motor. They then had to hurridly locate a replacement and install it while the competition was in progress. Missing their first two rounds, they appeared looking hot and sweaty just in time for their run in round 3 out of 4 and simply destroyed the rest of the competition. (Presumably it won’t be long before they destroy their new motor too, but it’s survived long enough to win, according to the rules, and that’s what counts.)

Overall the design show has been great fun to be a part of and has really demonstrated the skills that the students have acquired. Well one everyone involved!

Postscript 29 October 2014: We’re a hit with the Waikato Times!

Getting the terminology correct Marcus Wilson Oct 09

Yesterday I read a neat little report by one of our final year engineering students. As part of her final year project, she’d been looking at misconceptions in first-year students’ thinking about electromagnetism. Learning about electric and magnetic fields isn’t easy. For one thing, you can’t actually see them. Therefore it’s not at all obvious how something influences them. It’s not like learning mechanics  - where you can swing pendulums of different lengths and see for yourself the effect it has on the period of oscillation – these fields are invisible and therefore some indirect way of probing them is required. That adds its own problems. 

Most of the problems identified by the student weren’t terribly surprising. The theory of electromagnetism is full of horrible cross-products, which are a mathematical oddity in themselves* (try to read the Wikipedia article on them – I bet you won’t get very far). It’s hard relating experiment to theory when the theory is a struggle to grasp. Many misconceptions relate to whether fields and currents lie parallel or perpendicular to each other, and which generates a force and which doesn’t. 

But one problem that was identified by the research (based on formative tests) was the slap-dash approach to terminology. Many students used terms such as ‘magnetic field’, ‘B-field’, ‘flux’, ‘force’, ‘current’, extremely loosely. They have very specific, and different meanings, and they are not interchangable.  I heard a case of this in the lab today – a student talked to me about the force of the wire, when he meant the current in the wire. I think there are two questions here: 1. Using terminology loosely may simply be a consequence of not understanding what the terminology is trying to describe, and therefore is a symptom of  deeper problems with grasping the concepts. Alternatively, 2. The slopiness in using terminology may actually be the root cause of some of the students problems. How can you explain something if you’re not using words correctly? – you end up confusing yourself. I’m writing a journal article at the moment – and it’s obvious that the process of putting down my thinking on paper, in a precise manner that someone else can follow, does wonders for cementing my own understanding of it (or, sometimes, exposing my own lack of understanding of it when I thought I had grasped it.) 

It wouldn’t surprise me if both cases formed a feedback loop (vicious circle) where lack of understanding leads to poor use of terminology, which in turn prevents students acquiring the right understanding. I feel like a little research project is brewing here for next year…

*Cross-products would cease to be an oddity if we put them where they belong – in the dustbin. They are a consequence of a desparate attempt to represent areas as vectors. If we recognized areas for what they actually were – areas (or bivectors) - and worked with geometric algebra, physics theory would become so much easier. But, alas, we are stuck with historical conventions that are probably too far ingrained to break. 

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