Imaginary musings

Regardless of the school you went to, if you did final year maths then you’ll relate to the following anecdote. Your stories might differ in specifics, but this is pretty much the archetype of what goes on in the classroom. (No doubt it also rings a bell if you’re a teacher.)

• In a maths class, I asserted that the intelligence quotient of the classmate I was talking to was zero.
• Said classmate replied that my IQ was zero POINT zero.
• Me: “Yours is zero point zero times ten to the minus INFINITY.”
• Him: “Yours is MINUS zero point zero times ten to the minus infinity.”
• Me: “Yours is minus zero point zero times ten to the minus infinity *i*.”
• Finally he concluded that my IQ was just i, a wholly imaginary quantity, and that at least his was real.

I tell this story as a gentle way to raise the topic of imaginary (and complex) numbers, which I’ve been musing on lately. I think my latest bout of musings were triggered by reading this, which I read via a chain of links that I can’t remember now, but would have included this.

The subject of my muse was why most attempts to explain the applications of complex numbers strike me as unilluminating.

Classroom banter doesn’t count as an application. Neither do pretty fractal patterns — they’re lovely, but like a game, the rules don’t have to mean anything. Neither is it remarkable that one abstraction can be used in the service of another abstraction, as in esoteric results from advanced mathematics. What counts are situations where it’s fruitful to take a quantity derived from real world measurements, and relate it to the square root of a negative number.

I never studied any applications of complex numbers myself, having not done any physics beyond first year university. But I’ve occasionally listened to, or read, people trying to explain how imaginary numbers are useful in their fields, and I almost invariably have no idea what they’re talking about. They might assert that imaginary numbers are useful for such-and-such, but in a way that leaves me no wiser about what said context has to do with squaring something and getting a negative result.

So it occurred to me that perhaps most explanations fail because they show the audience an imaginary-shaped peg without first cutting an imaginary-shaped hole. Let me illustrate the point with reference to negative numbers, and then relate it to imaginary/complex ones later on.

Once upon a time, negative numbers were widely regarded as an affront to common sense. Quantities less than nothing? Crazy. But applications soon became apparent, and those applications tended to fit very particular patterns. One can sensibly answer questions like: if you’re looking for a scenario that lends itself to being modelled with negative numbers, then what sort of characteristics should you look for?

In my metaphor, the answer to that question is the negative-shaped hole, and a particular scenario which has those characteristics is the negative-shaped peg. And here’s the answer: you look for a scenario where every quantity has a corresponding opposite. The opposite of having a particular speed is having the same speed in the opposite direction. The opposite of being so high off the ground is being so far under it (and, preferably, upside down). The opposite of having so much money is being so far in debt. And so on.

If you have a scenario in which various quantities have corresponding opposites, then you have a scenario that lends itself to being modelled with the aid of negative numbers. There’s your hole, and you’ll find any number of pegs to fit. Note also that negative numbers don’t measure anything mysterious; simply flip the coordinate system around and they become positive.

If we carry this pedagogical approach over to the imaginary case, the question becomes as follows. Suppose you’re looking for a scenario that lends itself to being modelled with imaginary numbers. What characteristics would you look for?

And for a pretty good answer, consider that in the world of imaginary numbers, there exists a function such that f(f(x)) = -x. In fact there are two such functions (multiplying by i and multiplying by -i), but the point is made by translating that formula into everyday language. It means there is something you can do to a quantity twice and always get back the opposite of what you started with.

That’s probably the simplest imaginary-shaped hole you can find. If you have a scenario in which there’s something you can do to a quantity twice and always get back the opposite of the quantity you started with, then you have a scenario that lends itself to being modelled with the aid of imaginary numbers. As for pegs, no problem. Try anything involving rotations and waves.

In the case of rotations, the thing you can do twice and end up with the opposite of what you started with is to rotate a quarter of the way around. In the case of waves, it’s to wait for a quarter of a wavelength to go by. Given such tools, if a physicist’s eyes roll at an elementary blunder in this blog post, you can calculate what that physicist will be looking at by the end of it. You can almost as simply calculate the same thing using old-fashioned real numbers, of course, but with a good i you can fit into a single formula what otherwise requires an algorithm. And you can show that the principle neatly carries over to finer intervals — less than a quarter of the whole.

I haven’t mentioned Euler’s equation — an oft-celebrated result of a very clever procedure that makes it meaningful to raise numbers to imaginary powers, and is typically taught in first year mathematics at university. The relevant consequence is that i can be re-written as e to the power of πi/2, and hence that i to the power of some number n is the same as e to the power of nπi/2 [note: π is pi — but my blog happens to use a font that doesn’t draw it very well]. Re-writing everything as exponents of e may make things more complicated for algebra, but simplifies things for calculus. But because I haven’t done any calculus since the 1990s (and wasn’t much good at it then), I won’t go there.

(Exponents of e are also neater if you measure angles in radians rather than quarter-revolutions, but I’m pretty sure the reason mathematicians measure angles in radians is because it leads to trigonometric functions having neat power series and hence Euler’s equation etc, so the causality is kind of inverted there.)

I’m definitely not going to mention quantum physics. If you’re interested, try a blog by someone who understands it. You didn’t come here for that.

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If we are designing flags…

Every so often, the prospect of changing the Australian flag crops up in the media. Politically, I don’t much care: there are more important things than some piece of cloth on a pole. But creatively, if people are designing flags, then I want in on the action.

The latest revival of the conversation was triggered by a rather hideous design that’s not worth linking to (it’s only a catalyst, anyway). A better design along a similar line was Brendan Jones’s 1995 Reconciliation Flag, of which my only criticism is that the white line looks rather too divisive. I think it was the first I ever saw that involved the use of a boomerang.

Here’s a link to a list of 100 flags proposed by the public in the late 90s.

It’s been a while since we last had a flag debate in Australia, and likewise, a while since I last had a go at designing one of my own. Now, I happen to work as part of a graphic design team these days, but at home I don’t have access to the software we use at work, so I had to try and sketch a design using Windows Paintbrush.

So bearing in mind that this is better thought of as a rough sketch than a finished design, here is the best idea I came up with. [Update: I’ve fiddled a bit since the original upload, but still using Paintbrush.]

Every flag tells a story, and I took as my inspiration the rarely-sung second verse of Australia’s national anthem.

• “Beneath our radiant Southern Cross, we’ll toil with hearts and hands”
  So I’ve depicted the Southern Cross, which is also depicted on our current flag. The number of points on the stars is irrelevant, but five points is easiest in Paintbrush, so I went with that.
• “To make this Commonwealth of ours renowned through all the lands”
  The red boomerang (an idea borrowed from the Reconciliation Flag and others like it) is firstly a distinctly Australian symbol but also doubles as an upward-pointing arrow for ambition. More evocatively, picture a boomerang being tossed into the night sky, to fly amidst the stars of the Southern Cross.
• “For those who come across the seas, we’ve boundless plains to share”
  I’ve included blue for the ocean on both sides of the design. As for sharing boundless plains, it has to be said that Australian politics right now is anything but welcoming to outsiders. But personally, I like the idea of national symbols that can be used to shame us, by expressing values we’re supposed to have but seem to have forgotten.

The one thing obviously missing is any representation of Australia’s individual states and territories. But flags are supposed to be simple, and one must make decisions about what to put in and what to leave out. There is no rule that requires a country’s flag to indicate the number of states and territories it has, so I decided that this design would not. (An advantage is that if ever Australians decide to alter the number of states, it can be done independently of the flag.)

To defend the design from another perspective — without the boomerang, you’d have a simple flag with the Southern Cross in the middle, and colours representing the water and land of our island continent. Surveys show that many people would like to retain the Southern Cross on a new flag, although here it appears in a simplified, more abstract form. On the other hand, the main argument against using the Southern Cross is that it is not distinctly Australian, as the flags of New Zealand and Papua New Guinea also share the feature. The boomerang adds precisely that Australian element, and is integrated with everything else.

I sketched this for fun, and won’t try to convince anyone that we should adopt it as a nation. This blog post is simply my humble contribution to the national discussion. If you have some flag-related ideas you’d like to share with me — whether you’re Australian or not, whether you think we need a new flag or not — then you are welcome to continue that discussion in the comments.

Update: I found a site that lets you convert a flag image into an animation. Here’s what mine looks like in motion:

Update: I got thinking about how states’ flags might be constructed to match my design for the national flag, and the idea that worked best was to turn the design sideways, put it against the mast, extend the stripes, and add the state’s emblem on the other side. Here’s how that turns out for South Australia.

Update and Postscript: My supervisor at the graphic design place I work at mentioned that he likes Brendan Jones’s other design. It is certainly one of the better conservative proposals (i.e. ones that maintain continuity with the current Australian flag), and I would consider it serviceable — if a little stark for my taste and overreliant on stars. I decided to fiddle with it, creating a modified version with a 2:3 aspect ratio, a brighter shade of blue to contrast better with the black, and stars reduced in size to better fit the new width. (Here it is animated.)

General updates 2011: July (with Renaissance coffee mug for sale)

Everything’s going well in preparation for my upcoming holiday and computer upgrade.

I’m also planning to buy a laptop to with me. Mum will be staying home while Dad, my sister, her husband and myself go overseas, and the plan is to communicate via Skype — except that none of us have actually got Skype yet, so that’s the next thing to look at after I get the upgrade done. The plan has always been to upgrade sufficiently in advance of the holiday to make time to practice in between.

Renaissance coffee mug

I recently published a new product on Zazzle: a renaissance mug! The story behind this is that while I was browsing the web for a project at work I came across this book of Renaissance collages. My production supervisor thought it would be worth me creating a similar collage of my own, if not for the project then at least as an exercise, and this is what I came up with:

Reflections on prayer

I haven’t believed in God for a number of years now, but acknowledging that we are all shaped by our past, I occasionally like to talk about my former faith on this blog. Today I’d like to focus specifically on the subject of prayer.

I hope we can agree that the description of prayer found in The Last Hero by Terry Pratchett (“frightened people trying to make friends with the bully“) is a long way from the Christian understanding. But while it’s easy to talk about what prayer is not, discussing what it is requires grappling with paradoxes like why an omniscient God would need suggestions from humans. Back in the day I devoted considerable thought to theological questions like that, and I wrote some of those thoughts down in a document that I’m using as the main source for this post.

Prayer, according to what I consider the highest Christian understanding, is not about giving God suggestions. It’s more like tuning an aerial — that is, maintaining and refining the telepathic link between one’s self and God. Because God is understood in terms of moral perfection, and because there is no greater pursuit than to better ourselves morally, the Christian’s greatest aspiration is to think the thoughts of God. The point of prayer is to try to fill the mind with God’s thoughts, in part by putting into words the thoughts one believes to be “godlike” (such as compassionate wishes for other people). Deliberately focusing on godlike thoughts, it is thought, makes the mind more receptive to thoughts that come directly from God: the aerial-tuning analogy works pretty well here.

Interesting Stuff: Early July 2010

Here is my pick for this half of the month:

• The story of a writer who forgot how to read.
• A popular geology book review that tempts me. (Actually, it’s the front cover image that does most of the tempting.)
• Comedy video of the month.
• Serious video: Benoit Mandelbrot’s TED talk. The content isn’t particularly new for fractal lovers, but it’s a treat to hear it from the horse’s mouth.
• A really good article about masturbation.

That last item gives me an excuse to say that in an ideal world, there would be more sexual content on this blog. Not very much, mind you, but ideally I believe sexuality should be treated the same way as any other aspect of life: as a valid topic to talk about and share personal perspectives on. This would not be from an ego-boosting or thrill-seeking perspective, but from a conviction that sharing the diversity of human experience enriches us all.

Unfortunately, culture — and the memory of unpleasant encounters with people who are content to feel threatened by the unfamiliar — keeps me from being as open as I would like.

I would support the establishment of an International Blog About Sex Week, to give those of us who are naturally shy an incentive to be more forthcoming at least once a year…

[UPDATE: I’ve deleted comments, as extensive revisions to this post have reduced their pertinence. In addition, this post originally included my eclipse photograph, now posted separately.]

Internet essay from 1999

The following is an essay I wrote for a university course called “Professional English” in 1999. In 2001, it was published in an edition of The Autiser (the magazine of the Autism Association of South Australia).

The Internet has changed a lot since I wrote this, but despite its obvious datedness I think the essay is worth preserving. I’ve made some small (cringe-proofing) editorial alterations from the original, but not made any substantial changes. Comments are welcome if you’d like to discuss the Internet of today with reference to the topic of the essay.

Is the Internet Destructive to Human Relationships?

My original image of the Net was one of all these sad, lonely souls, sitting isolated at their computers … I now see it as the opposite, a means for people to reach out to each other.

— An Internet user

Misunderstood childhood questions

Topic for the day: What are some questions that you remember asking as a child, but which the adults failed to understand?

I’d like to share two.

— 1 —

The first was about the future of human society. I’d read about concepts such as “civilisation” and “barbarism” in books, and wondered if human society might someday advance to a point so different from what we usually refer to as “civilisation” that it merits a new name. This would not be a degenerate state, but a transcendent one that makes civilisation look primitive. If so, what would be the achievement that defines the beginning of that new stage?

I don’t remember how I worded the question, but I do remember Dad missed the point, insisting that the only way civilisation could be replaced by something else is if society regressed to a more primitive state.

Some tenets of faith

We all change our beliefs over the course of our lives, but the beliefs we once held remain part of us in the sense that they are part of our story, and helped shape the people we have become. They are, therefore, worthy of an occasional blog post.

I was raised in a Christian family (liberal Protestant), and during the period corresponding roughly with my teens, held a strong Christian faith of my own. I was also interested in theology, especially in how the more challenging parts of the Bible could be harmonised with what I knew to be true on other grounds (be that intellectual or moral).

Listed below are ten beliefs I held, or conclusions I came to, on the subject of the afterlife: Heaven and Hell, etc. They are not presented as a comprehensive overview (for example, they don’t specifically mention Christ), but they do summarise a lot of my thinking on these matters.

Interesting Stuff: Late July 2009

It’s now late July, and therefore a suitable time to publish my Late July collection of Interesting Stuff. However, I’ve been short of time lately, and I think I need to adjust my priorities. That doesn’t mean I’ll stop publishing the series, but I think I should devote less time each day to catching up on the latest news in science, and more time to other activities, both offline and on. In the meantime, installments of this series may be briefer than usual.

Over in the Real World, I am currently reading several books. As I’ve mentioned, I recently won a copy of a pre-release version of Chad Orzel’s How To Teach Physics To Your Dog, so I’m reading that, and I’m also reading a couple of books by Simon Singh that I bought when I attended his talk at the Adelaide Festival of Ideas on July 12, including Trick Or Treatment: Alternative Medicine On Trial.

Anyway, here are some online articles that have caught my attention recently.

Belated report on Skeptics Convention

About a fortnight ago I went to the Australian Skeptics National Convention. I’ve been putting off writing about it, largely because I expected to see a lot more material I could link to (such as other people’s reports, videos, etc). That hasn’t been the case, but there is a report in episode four of The Skeptic Zone.

The best part of the convention was the opportunity to meet people. I started a conversation with Martin Bridgstock almost immediately, and sat next to him during the first couple of talks. Martin was not a speaker at this convention, but I have a lot of respect for him based on videos of his talks at other events.