Homeopathy in the British Dental Journal

A while ago, the British Dental Journal published a quite good opinion piece entitled Unethical Aspects of Homeopathic Dentistry. The following issue was full of angry homeopaths, livid that someone would have the temerity to point out that homeopathy is perhaps a little bit silly, of which the most irritating was a lengthy opinion piece entitled Homeopathy and its Ethical Use in Dentistry. I wrote a letter in reply to this, and apparently so did a couple of other readers, because the current issue is full of letters from actual scientists, laced with varying degrees of sarcasm. I think my favourite line is from R Levy’s letter:

On the positive side, however, I am always happy to be reminded of the tale of the homeopath who forgot to take his medicine and died of an overdose.

Here is the full text of my letter:

There is an interesting and ongoing debate about the ethics of using placebos in medicine, so I was disappointed that the response to Unethical aspects of homeopathic dentistry has focused instead on disputing the overwhelming scientific consensus that homeopathy is baseless and unproven.

It is well known that people are prone to trust experiences and evidence that support their preconceptions. It is therefore inappropriate to challenge such an established consensus in the letters and opinion pages, particularly by citing personal experiences, individual studies and one’s own website. To make a convincing case, a large, unbiased systematic review is needed. The Cochrane Collaboration has already done this for several conditions, but has yet to find compelling evidence of any benefit. Usually, few or no well-conducted trials exist.

In the absence of evidence that homeopathy works, one is forced to estimate its priori plausibility as the homeopaths do – by comparing it to experience. The two founding principles of homeopathy are that a patient presenting with a given symptom is best cured by a substance known to cause that symptom, and that diluting medicine makes it stronger – including well beyond the point where no medicine remains. I wonder how your readers’ clinical experiences compare to these principles.

References are included on the journal’s webpage, including links to all the silly pro-homeopathy letters and articles. Here are links to the other responses:

Quackery Risk, R Levy (quoted above)
Ethically Unacceptable, K G Icaacson (“Lest your readers begin to think that there may be possible benefits of homeopathy…”)
A Substantial Gap, G Chapman (“Any ethical practice involving homeopathy must necessarily begin by telling the patient that it is scientifically implausible…”)

There was also one letter praising the silly pro-homeopathy article.

I like this kind of thing. You do see pseudoscience sneaking into places where it really shouldn’t be, but for all that nonsense does seem to be everywhere, people are actually pleasingly good at spotting and laughing at it.

Can AV elect a second- or third-placed candidate?

Under First Past The Post (FPTP), everyone picks their favourite option, and the one with most votes wins. Simple, right? Under the Alternative Vote (AV), the one with fewest votes is eliminated, and this process repeats until one candidate has a majority — or until only one candidate is left; it makes no difference. No2AV will tell you that this can hand victory to the second- or third-placed candidate. But what does that mean? The winner is by definition the first-placed candidate — what they mean is that AV sometimes returns a winner who would not have won under FPTP. The Yes campaign will tell you that that is sort of the point — if AV always returned the same winner as FPTP, nobody would care which we used.

The thing is, elections are so engrained in the public consciousness that it’s hard to dislodge the idea that after one there’s a very clear “best candidate”, and any system that doesn’t elect that person is fundamentally broken. But is that true?

And what if there was more than just FPTP and AV on the ballot this May?

What if we added Runoff Voting? This is the system used in France, whereby the two most popular candidates after round one face a second round, the winner of which is duly elected. We could also add Approval Voting. This is essentially exactly the same as FPTP, but you’re allowed to vote for as many options as you like. Lastly, we could add Range Voting. With Range Voting, you can score each option, usually out of 10 or 100, and the one with the maximum score is the winner. There are still more options, but let’s stick to these five for now.

Let’s ask an imaginary population to vote on which system they want. The population is made up of:

40 “Reds”. The reds like FPTP, but would settle for Approval. They’ll have no truck with numbers or multiple rounds.
21 “Yellows”. The yellows would like Runoff voting, and would be happy enough with AV. They would settle for Approval, but not Range or FPTP.
20 “Greens”. The greens would like AV, and would be happy enough with Runoff. They too would settle for Approval, but not Range or FPTP.
19 “Blues”. The blues are desperate to have Range voting, but otherwise have identical preferences to the greens.

Let’s assume nobody votes tactically.

Clearly, FPTP has most votes. So that should win, right?

Well, let’s not get ahead of ourselves. “Having the most votes” is only how you win under FPTP. If we used Runoff voting, the red and yellow vote would get FPTP and Runoff into the second round, where the blues and greens would side with Runoff, granting it a 60:40 win.

This graph shows an interesting fact: even though FPTP got the most votes in the general ballot, the population as a whole prefers Runoff. But what would happen under AV?

In round one, Approval voting would be eliminated, as nobody voted for it. In round two, Range voting, with only the 19-strong blue vote to its name, would also be eliminated. In round three, with Range gone, the blues vote like the greens, taking AV up to 39 votes. This puts Runoff into last place, and it is eliminated. Round four, therefore, is between AV and FPTP. The reds stay loyal to FPTP, but the yellows, greens and blues all prefer AV, which wins with a 60:40 majority.

Again, it is clear that the population as a whole prefers AV to FPTP. In fact, the population as a whole also prefers Approval to FPTP. This is a serious flaw with FPTP: it is possible for the “Condorcet Loser” to win. The Condorcet Loser is any candidate who would lose a two-horse race against each of the other candidates. (In our hypothetical example, however, Range Voting is the Condorcet Loser and so FPTP has not messed up as badly as it might.) AV and Runoff voting are immune from electing the Condorcet Loser, because such a candidate would by definition be defeated in the final round (if they got there).

Let’s have a look at the less well-known systems now. Range Voting is the only system to measure strength of feeling. I mentioned earlier than the blues are desperate for Range to win — so they rate it a 10. Then, they rate AV 3, Runoff 2 and Approval 1 — as do the greens. The greens have opted not to give the full 10 points to any system. That is their right. The yellows vote in a similar way, giving Runoff 3 points, AV 2 and Approval 1. The reds give 3 points to FPTP, and two to Approval. Anything I haven’t mentioned gets zero points.

In total FPTP has 120 points, all from the reds. Approval has 140 points, from everyone. Runoff slightly beats it with 141, from the yellow, green and blues. AV does a little better with 159, from the same ballot papers. But the sheer strength of feeling from the blues is enough to hand Range Voting the win, with 190 points.

Even though the majority would rather have anything but Range Voting (i.e., it is the Condorcet Loser), the theory goes that Range would make the most total happiness, albeit concentrated in a small minority of voters.

Lastly, what would happen under Approval Voting?

Only the reds approve of FPTP. Everyone else approves of AV and Runoff voting. Range gets only 19 votes, from the blues, but since everybody would be happy with Approval voting, it scores an easy win.

All five systems have elected themselves — and yet the voters’ preferences are the same in each case. Obviously this population was engineered so this would happen, and it’s very unlikely in real life. But it should illustrate some of the problems with the current system, and indeed with all systems. It is mathematically impossible to design a system without tactical voting.

Here is the issue: looking at the voter’s preferences, I don’t think there’s a single system that clearly ought to win. No2AV would tell you FPTP should win because it “came first” in terms of first-preference votes, but 60% of the population would rather have Alternative, Runoff or Approval Voting, and it’s hard to argue that democracy dictates that they mustn’t.

This makes designing a fair system almost impossible. For example, a criterion mathematicians use to judge voting systems is “independence of irrelevant alternatives”. Range and Approval Voting satisfy this criterion: if any losing system were removed from the ballot, it would not affect the scores for the remaining ones and the result would be the same. Under FPTP, removing Runoff Voting from the ballot would hand the victory to AV. In this example, AV would elect itself with any combination of opponents, but this is not generally the case.

Even though all five systems elect different winners, none of them is inherently biased. Nor do any of them give any voters more than their fair share of sway over the result, whatever No2AV might tell you. Nor, whatever Yes To Fairer Votes may have you believe, are any of them a panacea that will automatically usher in a “new politics”. We simply have to agree a system beforehand, so that going into the election everyone knows the rules. That way, while the result may not be popular, at least everybody can accept that it is fair, and knows that they have a chance to do something about it the next time around.

Can AV rob a rightful winner of victory? Yes. But so can FPTP, and and at least AV will never elect the single least popular candidate.

Pass me the Brain Protector™.

Manchester University publish a magazine thing called “UniLife”. I unsubscribed from the print version because I find that sort of thing rather distracting, but I still read the online version. And the latest issue has an article (page ten, left column) entitled

Colour purple sees off Alzheimer’s

Well, that explains the new logo.

This struck me immediately as bullshit. Quack nutritionists have been extolling the virtues of brightly coloured fruit ever since that weekend when everyone thought anti-oxidants were really good for you, and whenever new research apparently confirms something that a quack just made up my default reaction is both scepticism and skepticism. Sometimes even skeptikism. But I read on.

Eating purple fruits such as blueberries and drinking green tea can help ward off diseases including Alzheimer’s, Multiple Sclerosis and Parkinson’s, a University of Manchester report claims.

Claims about more than one disease are another classic quack hallmark.

Ground-breaking research from Professor Douglas Kell has found that the majority of debilitating illnesses are in part caused by poorly-bound iron…

This is going to be about anti-oxidants, I thought, isn’t it?

…which causes the production of dangerous toxins that can react with the components of living systems.

“Toxins”? This article has more canards than Sarkozy’s duck island. It still may be totally legitimate, of course, but my finely-tuned bullshit-detection heuristics are all lighting up like the BBC switchboard after Jan Moir forgets to Sky+ something and assumes it must have been offensive.

In order to protect the body from these dangerous varieties of poorly-bound iron, it is vital to take on nutrients, known as iron chelators, which can bind the iron tightly.

This caught me off-guard. I was expecting anti-oxidants, and suddenly here are chelators. Chelation therapy is already an established form of quackery, and already touted to treat Alzheimer’s disease. It’s also very dangerous (in its more common, non fruit-based form).

I guess there must have been a press-release, because the same article appears word-for word on PhysOrg, with approving comments from a chelation quack, and on HealthNewsTrack. Byron J Richards’ Wellness Resources, a website that sets off more of my alarm bells than anything in UniLife ever could, has a copy of the article, adding

I believe that Dr. Kell has hit upon a basic principle of health with far reaching application for prevention of diseases that are at epidemic levels. … I have recently highlighted the importance of blueberries for the protection of your brain and this is another important angle on this issue.

Richards’ website also sells a $80/month dietary supplement called “Brain Protector™” which, he says, “contains key nutrients to enhance protection of the brain and nervous system. Includes blueberry and more!” Radio nutritionist and chiropractor Alan Pressman has also written about the article, in almost so glowing terms as those he reserves for Deepak Chopra. Apparently nobody that Google News indexes picked up the story.

UniLife didn’t link to the original research, but the version of the article on the chemistry department website did. It’s a review article. Well, two review articles. They’re both very long, and packed with something like two thousand references each (along with some weirdly amateurish diagrams, “figures” that are really just paragraphs of text, and dodgy-looking underlined sections, often including only the “anti” in “antioxidants” — journals really need to stop trusting scientists to format articles). I profess no knowledge at all about biochemistry, but I came away from them with the impression that yeah, maybe this was a process present in quite a few diseases, and if so then slowing it down could help a bit, but nothing that would justify the claims in the UniLife article. A review article — even two review articles — isn’t really “ground-breaking research” to me. It’s a way of pointing out a connection you’ve spotted. It hypothesises. It should be followed with a systematic review, a meta-analysis, to show if there is evidence for the hypothesis. A regular review has no explicit inclusion or exclusion criteria for the papers it cites, and shouldn’t be seen as an unbiased snapshot of the evidence.

I think promoting early research findings too heavily is a risky game, especially where there are already dangerous quacks trading on the same ideas. You can’t retract from these people — even after he was struck off, they still cling to Andrew Wakefield’s debunked and retracted ‘study’. I think the claims in this press release are hugely overblown, as is the overall impact of the research, and I think singling out Alzheimer’s disease in the headline misrepresents the original articles. And even if it’s a totally legitimate bit of well-conducted research discovering something genuinely novel and useful, writing it up in the form of cargo-cult quackery probably wasn’t a great idea.

So is this article proper research? I hardly even care. If it’s quackery, I’m appalled. But if genuine science is now indistinguishable from quackery even to seasoned veterans, well surely that’s even worse?

The Interesting Views Of Lord Monckton

My New Star Sign

The attention astrology receives amazes me. I’m told that around a quarter of people genuinely believe that the positions of celestial bodies can be used to divine the future, but nobody actually acts like they believe it. They just say they do in surveys. I’m therefore surprised when, say, a dating website works out my starsign from my date of birth and sticks it up on my profile as if someone will care. It does presumably filter out eleven twelfths of girls thick enough to base their romantic endeavours on the zodiac, so that’s arguably useful, and I suppose it also allows me to search for partners with birthdays that don’t clash with mine or with Christmas, but basically I’d turn it off given the choice.

Late last year LiveScience published an article about how the precession of the Earth causes the constellations to drift around the sky over thousands of years, and this meant that the signs of the zodiac were now out of step with the constellations whose names they carry. For example, the usual “tropical” zodiac makes me an Aries, but here’s (roughly) where the sun was when I was born:

Picture from http://my.execpc.com/60/B3/culp/astronomy/Fall/Zodiac.html

The Star Tribune published a comment, which included a calendar so you could work out your new starsign without looking up the sky map. This included the constellation Ophiuchus (or Serpentarius) which was dropped by the Babylonians because dividing a circle into 13 bits is too hard (or something; I don’t know how the Babylonians rolled). This went viral, because people love a good ‘they changed it’ story. It’s like when ‘they’ got rid of Pluto, or when ‘they’ abolished limbo — “they” in these cases being the International Astronomical Union and the Catholic Church respectively (although the latter story is mostly false even before you start questioning whether the Pope gets to choose what hypothetical planes of reality exist). However, according to a delightfully pointless article in Toronto’s Star entitled “Relax, says astrologer. You’re still a Virgo”:

It’s all in the planets, and the names of the signs are just a handy convenience that is a lot easier than saying, “when the sun is in the second house,” [Toronto astrologer Milada Sakic] told the Star on Friday.

Sakic, an astrologer for 20 years and teacher with the Canadian Association for Astrological Education, explains, “Astrology does not look at the relationship between the constellations and the Earth. We study the relationship with the Sun and the Earth and the planets.

The constellations, she reminds us, are “many light years away, illusions in our night sky. Their effect on us is very secondary.”

Quite.

So not only has nothing changed, but astrology has no governing body any more than English does, so there is no ‘they’ who can change it in the first place. Nobody can tell Milada Sakic she’s doing it wrong except by pointing out that her predictions might as well be based on the positions of the stars she sees when you wallop her on the head with an encyclopædia of actual facts for all that they have anything to do with future events. That both Sakic and Jacqueline Bigar, the Star’s resident astrologer, will have to do 8.3% more work if Serpentarius catches on is not mentioned.

But I still don’t really feel like Pisces either truly represents me or actually looks very much like any number of fish, so I took another look at the sky map:

Then I started adding my own lines:

And suddenly I realised what my new, true starsign is:

Here, therefore, is my horoscope:

Dawkins (March 14th — April 28th) — You are a British ethologist and evolutionary biologist. You are an emeritus fellow of New College, Oxford, and were the University of Oxford’s Professor for Public Understanding of Science from 1995 until 2008.

You came to prominence with your 1976 book The Selfish Gene, which popularised the gene-centered view of evolution and introduced the term ‘meme’. In 1982, you introduced into evolutionary biology an influential concept, presented in your book The Extended Phenotype, that the phenotypic effects of a gene are not necessarily limited to an organism’s body, but can stretch far into the environment, including the bodies of other organisms.

You are an atheist and humanist, a Vice President of the British Humanist Association and supporter of the Brights movement. You are well-known for your criticism of creationism and intelligent design.

Notable Dawkinses include Professor Richard Dawkins (born 26 March 1941).

It’s not bang-on, but it’s better than anything the old system gave. And it does work very well for at least one person.

The Mass Libel Reform Blog — Fight for Free Speech

I don’t normally do this sort of mass-blog thing, but I’ll make an exception for this: today is the first anniversary of the report Free Speech is Not for Sale, which highlighted the oppressive nature of English libel law. In short, the law is extremely hostile to writers, while being unreasonably friendly towards powerful corporations and individuals who want to silence critics.

The cost of a libel case is staggering, even compared to fighting other sorts of court case and especially compared to libel cases in other countries. In fact, the cost can be 140 times that of a similar case in Europe. This wouldn’t be so bad, but (in part because the onus in an English libel case is on the defendant to prove what they say is true) a libel case brought against you is very difficult to win, and even if you do then you’re only likely to recover 70% of your costs. Simon Singh’s recent libel victory cost him personally £60,000, and prevented him from working for a full year. Most bloggers, who aren’t backed by publishers, simply can’t afford to win these cases, much less lose them, and can therefore be effectively silenced by the mere threat of a libel case — even when they’re obviously and demonstrably right. There is no right to free speech under these laws, and large organisations both know and exploit that fact.

This worries me as an English blogger, but the internet allows bloggers to reach a global audience, which gives the High Court in London a global reach. Anyone can be sued in London if their writing is available in England, and they routinely are — why bring a libel case anywhere but the most claimant-friendly court available? Had president Obama not signed the SPEECH Act into law, blocking oppressive foreign libel rulings from being enforced in the US, American newspapers would probably have blocked British readers from their websites rather than risk being sued here.

You can read more about the peculiar and grossly unfair nature of English libel law at the website of the Libel Reform Campaign. You will see that the campaign is not calling for the removal of libel law, but for a libel law that is fair and which would allow writers a reasonable opportunity to express their opinion and then defend it.

The good news is that the British Government has made a commitment to draft a bill that will reform libel, but it is essential that bloggers and their readers send a strong signal to politicians so that they follow through on this promise. You can do this by joining me and over 50,000 others who have signed the libel reform petition at http://www.libelreform.org/sign.

Remember, you can sign the petition whatever your nationality and wherever you live. Indeed, signatories from overseas remind British politicians that the English libel law is out of step with the rest of the free world.

The above is adapted from, and contains big, unedited chunks of, a template sent to bloggers a few days ago.

If bees are that smart and still disappearing we should probably start checking the skies for Vogons

Right now, the second most read story on the Guardian is that

Bees can solve complex mathematical problems which keep computers busy for days, research has shown.

The insects learn to fly the shortest route between flowers discovered in random order, effectively solving the “travelling salesman problem”, said scientists at Royal Holloway, University of London. … Computers solve the problem by comparing the length of all possible routes and choosing the one that is shortest.

Bees manage to reach the same solution using a brain the size of a grass seed.

No, they don’t.

The travelling salesman problem is NP complete, meaning that it belongs to a class of problems for which no efficient solution has been found. Mathematicians haven’t yet proven that no such solution exists, but crucially nor have bees. NP problems are considered difficult because they get impractical very quickly when they get big. A brute-force solution to Travelling Salesman requires you to check N! routes, where N is the number of places you need to visit. If there were 22 flowers then a computer could spend the age of the universe doing that, but the research being reported here involved only four flower patches, and a computer could do that in the blink of an eye. What will take a long time to find the fastest route between four flowers is a bee.

The bees in this research took about two hours to settle on the best route around the four patches, putting them a long, long way behind even the very worst computers. Which is fair enough, because as the Guardian was at pains to point out, they have brains the size of grass seeds. A bee can no more calculate the fastest route between a large number of flowers than it can play Minesweeper. This should be obvious, and indeed is when you think about it for (0.3 to 0.4) × blink of an eye. What a bee can do is to come up with a pretty good stab at the fastest route between flowers. That’s still impressive. It’s one of many deeply impressive things that bees do. But it’s not better than a computer.

As well as being interesting in and of itself, bee behaviour may be able to inform the next generation of smarter algorithms, which solve large-scale NP problems in manageable times, at the cost of potentially missing the absolute best solution. For example, you can get to within 0.02% of the optimal solution much faster if you use a “bee colony optimisation” algorithm. This name is not entirely a coincidence: bees’ behaviour has been used as a model for route optimisation for some time. That’s not to slag off this research, but the claim that “bees can solve complex mathematical problems” is only true the same sense that a goalkeeper can solve differential equations in his head.

To me, and to the editors of Natural News, the opening line of the Guardian piece strongly implies that bees can solve the problem better than a computer, which is simply false. But that line, along with most of the story, is lifted wholesale from the journal’s ‘forthcoming articles’ page. Presumably there was a press release, since the same research was reported even earlier by the BBC.

This is interesting stuff that should be in newspapers and the public eye, but scientific journals shouldn’t get it there by misrepresenting it.

Zooming Mandelbrot Screensaver v2

The sad death of Benoit Mandelbrot this week has prompted me to update my zooming Mandelbrot Set screensaver. The old version was pretty enough to look at but annoying as hell to have set as your actual screensaver, so I rewrote it from the ground up yesterday afternoon, with all the processing optimised and tucked away in a background thread, where it won’t interfere with checking for mouse movements and suchlike.

It also has multiple-monitor support, although I only have one screen and therefore haven’t tested it. Essentially, a separate copy of the screensaver is run for each monitor, so I’ve no idea how it will cope with that. It also has a cursory explanation of the maths in the settings page so that non-nerds can at least have a stab at setting good options for their system.

What it doesn’t have is any kind of framerate limiter, so if it runs too fast you should just increase the quality of the images until it’s sensible. Likewise if it’s too slow, decrease the quality of the images.

Download (11 kB so you may as well) — to install, just copy into C:\Windows or whatever your Windows directory is, and it should appear in the screen saver settings page.

Requires zip support and the .NET framework 4; most recent Windows versions should have these installed.

There does seem to be the odd weird fleck within the Set, as if there’s a calculation glitch. I don’t know what’s causing that. Other than that, let me know if there are any problems with it.

So what if they say it’s “chemical free”?

I saw a Twitter exchange today about a Starbucks employee and a homeopath selling their respective brands of instant coffee (just add water) and water (just add medicine) by saying “there were absolutely no chemicals in it”. The former predictably led to the skeptic “pointing to everything around him and saying it was all made of chemicals”. I think this approach is unhelpful. (I don’t think he’s a dick.)

Sure, to a scientist anything made of atoms (i.e., everything) is either a chemical or a combination of chemicals, but to a layperson a chemical is an artificial substance, especially a harmful one. Correcting errors is laudable, but once a usage has established itself it’s futile and counter-productive to insist that it’s wrong. In this case, upon hearing that instant coffee contains “absolutely no chemicals”, nobody will think it is made of neutronium.

The lay-meaning can be problematic, certainly – it’s a fuzzy definition, tainted by connotations that can be exploited, deliberately or otherwise, by marketers — but that’s true of most words. A word’s definition is an attempt to draw a series of straight lines around the fractal set of concepts we instinctively understand when we hear it. You know if something is a religion, or a cult, alive, or human, but the definitions are a little hazy around Buddhism, Catholicism, Alcoholics Anonymous, viruses, fœtuses and missing evolutionary links. You know what a chicken egg is, but was the first one laid by the first chicken or did the first chicken hatch from it? (Annoyingly, a good example of a fuzzy definition is the distinction between a common error and an established usage.) Equally, loads of words have connotations that can be exploited. L-ascorbic acid can be the additive E301 or the ingredient vitamin C. It’s part of why English is a rich language.

My point is that the phrasing is irrelevant. Had the barista said “buy our instant coffee; it’s all-natural”, my skeptic friend might have asked “isn’t that exploiting and reinforcing misconceptions about what’s healthy and environmentally sound?” This would also have been a better response to the original phrasing, because if Starbucks now change their script then we’ll still have to make this second case and they’ll be quite entitled to ask why we didn’t say so in the first place.

Which isn’t to say there’s no skeptical angle to such semantic quibbles. The government arbitrarily redefining “a drinking binge” to mean “slightly more than two pints”, and then claiming that lots of women are “binge drinking” is disingenuous, as is the Soil Association promoting the word “organic” as meaning “grown without chemicals” while maintaining a far longer and less widely-known list of even less evidence-based criteria. In both cases people think they understand what they’re being told but the person telling them is really saying something subtly different.

But when the alternative usage is so common as to be predominant, as it is in the case of ‘chemical’, you’re not standing up for science, honesty and skepticism, you’re ignoring them to dictate how people should use their own native tongue.

Fun with the Mercator Projection

On my brother’s wall is a map of Earth with the South Pole at the top. It uses an equal-area projection to show the true relative sizes of, say, Africa and Greenland. It aims to make a point about the more common North-up, Mercator projection. Mercator projection maps enlarge areas near the poles so that shapes are preserved. I like the sentiment but I think switching to an equal area projection and printing it upside-down is a little unimaginative. So I created this:

This is a map of Earth created using a Mercator projection, but with the magnified poles moved to the Atlantic and Pacific oceans. I created it using the Blue Marble map from Wikipedia. The dashed lines are the conventional latitude and longitude lines. You get a much better feel for the shape of Antarctica than normal Mercator maps give you, instead exaggerating Kamchatka.

Because, the Mercator projection doesn’t need to make Africa small and Greenland big. It can do anything you want it to. So for example, here is my little rebellion against Mercator’s underplaying of Africa’s troubles: a Mercator map in which the continent is infinite in area:

I’ve cropped the image here. In principle a Mercator projection can be continued infinitely in the vertical direction, and in this case the ‘north’ pole is in Africa, so the map would be Africa all the way up. The level of detail would, source image notwithstanding, get bigger and bigger until eventually sub-atomic particles started to appear. Theoretically, you could exploit this to produce a map where Britain opened out as Africa has at the top, and extend the map up to include a road map of England, including a large-scale street map of Manchester, eventually opening out to provide a floor-plan of one particular building, then room, and eventually the layout of one table. This, however, seems like it would be very difficult so I haven’t bothered.

Here’s another Mercator map, this time with the poles near Australia and Africa again:

And here’s one with poles in Asia and South America, creating a world with one central ocean:

Here’s the maths in case you want to make some maps yourself. Feel free to stop reading here if you wrongly find maths boring. I haven’t worried about sign conventions or being especially rigorous, though. It’s just enough to make some nice pictures. I treated the Earth as a sphere of unit radius centred at the origin.

Converting between (x, y) and (latitude, longitude) is fairly trivial if you read Wikipedia, although I did have to kludge the formula a bit for ‘negative’ (south) latitudes. To reorient the map, I converted the latitudes φ and longitudes λ to three-dimensional Cartesian coordinates using the following formulae:

x = cos φ × cos λ
y = cos φ × sin λ
z = sin φ

I’d planned to rotate these and then convert them back into spherical coordinates, but in the event I found it easier to go directly into rotated spherical coordinates. They’re defined by two points, called North Pole N and Greenwich G, each chosen at random from the surface of the sphere, and described by a set of (x, y, z) coordinates. So the new latitude and longitude are given as follows:

φ′ = ½π − cos⁻¹ P⋅N
= ½π − cos⁻¹ (x x_N + y y_N + z z_N)
λ′ = tan⁻¹ (P⋅G′ ÷ P⋅G″)

where G′ = N × G ÷ |N × G| and G″ = N × G′. Using the cross product this way generates two points on the new equator separated by 90°. In fact the conversion to new-longitude is more complex than this, because you have to mess around with quadrants, but I used the atan2 function to do that for me so I’ve not bothered working out all the steps.

I don’t know if this is the best way of doing this but welcome to stream-of-conciousness mathematics.