Defending tau

The Daily Mail today has a piece about the proposed alternative to the number pi, tau (τ), equal to 2 x π.*

Basically, the reason we might want to change to tau is simple:

Pi is defined as the circumference of a circle divided by its diameter. We came up with this definition thousands of years ago, before modern geometry had taken off, and it’s a perfectly good number.

However, we now know that the diameter of a circle – the distance all the way across – isn’t really the best way to measure a circle. Instead, it’s better to measure the distance the radius of the circle – the distance from the centre to the edge. For example, if you’re working out a planet’s orbit, the distance from the planet to the Sun (radius) is a nice logical thing to measure, but the distance from the planet to the other side (diameter) of its orbit doesn’t really mean anything important.

Since the radius is half the diameter, circumference divided by radius is twice as big as circumference divided by diameter, which is why tau is twice as big as pi.

The definition of 1 radian. The red line (radius) and blue line (arc length) are the same length. By Stannered, CC-SA-BY-3.0

Tau is also nice because it makes working with angles a bit easier. The most natural way of measuring angles – the way you have to do it if you’re doing maths or physics – is to measure them in radians. The radian is defined in terms of the circle’s radius – one radian is the angle you get if you walk around the circle for a distance of one radius. If you travel 2π radians, you travel a distance which is 2 times pi times the radius, or pi times the diameter; in other words, you travel the length of a full circle, 360°.

However, that 2 in there is a bit of a pain. It basically means that, in physics or maths, whenever you’re dealing with a circle or a wave you get annoying factors of 2 popping up in your equations, and unless you’re very careful, it’s easy to forget a factor here or put an extra one in there, making your sums completely wrong. For example, to switch between ordinary frequency and angular frequency (effectively, switching from revolutions per second to radians per second), you multiply the frequency by 2π. This is a change we need to make a lot when working with waves, and it’s so easy to lose a factor of 2 when you’re working with a bulky equation.

That factor of 2 is only in there in the first place because we made the mistake of basing pi on the diameter instead of the radius. If we replaced pi with tau, 1 circle would be τ radians, and that factor of two would disappear. It wouldn’t be a groundbreaking change, but it would still be quite nice, and it would make the mathematics of circles a bit easier to understand. Of course, the hassle of teaching people to use tau instead of pi is probably greater than the benefits, so its unlikely we’ll ever give up pi.

Unfortunately, the Mail is quoting from a (paywalled) interview in The Times with University of Leeds lecturer Kevin Houston, whose and they seem to cut down what he’s said to just:

‘Mathematicians don’t measure angles in degrees, we measure them in radians, and there are 2pi radians in a circle,’ Dr Houston said.

“That leads to all sorts of unnecessary confusion. If you take a quarter of a circle, it has a quarter of 2pi radians, or half pi. For the number of radians in three quarters of a circle, you have to think about it. It doesn’t come naturally.

‘How much simpler it would be if we just used tau instead of pi,’ Dr Houston added. ‘The circle would have tau radians, a semicircle would have half tau, a quarter of a circle a quarter tau, and so on. You don’t have to think.’

Trimmed like that, it sounds like Houston is only promoting tau because he’s too stupid to work out three quarters of two (his Youtube video, linked by the Mail, explaining the mathematics of tau, shows this is not the case). So perhaps naturally, Daily Mail commenters have leapt at the chance to prove that they’re smarter than a mathematician, because really, what do mathematicians know about mathematics?!

If an ‘A’ level Maths student has trouble with the difference between Tau and Pi, then they should be on another course. Any excuse to dumb down the kids and stop them thinking for themselves!

Alan, Frankfurt, Germany (ex pat)

Right on! How dare we dumb maths down by making it slightly clearer!

‘The circle would have tau radians, a semicircle would have half tau, a quarter of a circle a quarter tau, and so on. You don’t have to think.’ …..It may be appropriate for DM readers who like not to think, but I’m afraid mathematicians do think, and aren’t interested in tau, thank you very much.

rupert, Yors4hire, UK

Except for all the mathematicians and scientists who support tau I daresay because they think.

This tau thing is clearly aimed at mindless rote learners but to those of us who actually understand maths, pi expresses something meaningful which is precisely why we refer to it so much.

Vincent, Glasgow

Pi and tau express more or less the exact same thing – the shape of a circle – tau just expresses it in a slightly more logical way.

As a retired Maths teacher this idea lacks intuitive sense. The whole point of the exercise is to relate the circumference of a circle to its diameter – and the ratio is the irrational number 3.14159… The ratio is NOT 6.whatever! That’s the basic theory. Then there are various formulae needed at basic school level: for example, the area of a circle is pi time radious squared. The new formula would be tau times radius squared divided by four: an extra calculation step. Or the volume of a cone pi r squared h…. and so on. And in times of austerity how many schools could afford a new set of textbooks filled with tau? (And all authored by Leeds people, no doubt!) And then there are the millions of calculators with the pi button built in…. No! This idea is like trying to say that from 2012 we’ll all drive on the right – indeed the idea of tau is worse than that because there are regions of the world already driving on the right….. This is a big UM and No No

Andrew, Cwmbran

No, but the ratio of the circumference of a circle to its radius IS 6.whatever (6.28318… in fact). The fact that the area of the circle becomes a half times tau r squared (0.5 τr² – there’s no dividing by four involved) is one of the downsides to tau, but as Kevin explains in the video, there is a deeper mathematical reason why we would expect that half to be there, so while it’s annoying it does make sense.

This is the way you can waste a lot of time while avoiding doing anything of real interest or value to anyone. What a waste of brain cells!! Is this supported by public funds? There are real problems in the world today that are worthy of serious consideration, but when academics waste their time on things like this, they prove that they have no value to society at large and should be dismissed.

Samuel, Dubuque, Iowa

One mathematician making a 5 minute Youtube video in his spare time? How dare he! He should devote every second of his life to curing cancer!

Big Wow… Is “2pi” the only mathematical innovation Leeds University can come up with, it’s best 21st Century contribution to the advancement of mathematics? Talented Maths kids doing their 5 science A levels must be crossing Leeds off as even a 5th choice.

Russ H, Bucks

Again, this is one guy working in his spare time. This is not the only thing the University of Leeds Maths department does! (Incidentally, 55% of Leeds pure maths research is “world leading” or “internationally excellent”, and a further 40% is “internationally recognised”.)

BUT….e^([pi]i)]=-1 and this does not work for tau. There are uses for Pi beyond circles!

Miles, Australia

(This refers to Euler’s identity, e^πi = -1, which gives us a nice way of representing imaginary numbers using angles. And despite what Miles says, it’s all about circles.)

That’s true, it doesn’t. Instead, e^τi = 1. This is even better than Euler’s original version, since we no longer have that minus sign (and all that minus sign told us is that pi radians = half a circle, something we already know). Quite a few comments are like this – “OH NO EULER’S FORMULA IS BEAUTIFUL, WE CAN’T CHANGE PI COS THAT WILL BREAK IT” – and yet no-one actually bothers working out what it would look like with tau.

Professional jealousy. Eienstein acceepted Pi and I can assure you he was more intelligent than this egghead. This man just wants to make a name for himself. Change all the books indeed. He is “daft” !

Ruckus, myrtle Beach SC (ex pat)

And why do we bother speaking English? If German was good enough for Einstein, it’s good enough for us!

All those comments have been upvoted, by the way, unlike this comment, which currently has a score of -1:

I am in full favour of this proposition. Unfortunately, a large number of these comments seem be be from people with only a basic understanding of mathematics. Using Tau in place of Pi would reduce the need for a constant in a plethora of standard calculations involving circles and spheres, the like of which children will be schooled in. Furthermore, the simplicity of the equations using Tau may increase understanding and encourage children to pursue an interest into further mathematics. From my experience of mathematics lessons, the majority of students didn’t dislike mathematics, but rather found it too complicated to enjoy. Once provided with a topic they were able to grasp, students began to enjoy working the problems. Once children have achieved a satisfactory grasp of simple circle equations, the transition to understanding the calculus is a much easier one. I think that should the readers have been taught using Tau the comments here would be better informed.

Pep, Manchester

Good old Daily Mail comments, eh?

* They claim the idea was invented by Kevin Houston at the University of Leeds. In fact, it’s much older than that – Bob Palais first came up with it in 2001 – and Houston is just promoting it. Also, as a conflict of interest thing, I guess I should point out that Kevin taught me a few years ago, and is a thoroughly nice guy.

Hard-hitting research from Cravendale

(Hat tip to @TomChivers)

Look at me, boldly breaking press embargoes to bring you the FACTS. What IS the formula for the perfect cup of tea?

You might have thought that George Orwell answered that fairly conclusively years ago, but if you did then you are clearly an idiot. After all, George Orwell was just an author, journalist and political campaigner. What the hell do writers know about tea?! No, to answer this question we need the help of cargo cult science!

Good thing Cravendale’s taken the hit and hired scientists from Northumbria University to tell us the secret of good tea. Spoiler warning: the answer’s “Cravendale”.

Some of the more sceptical among you might be wondering how they tested this, and how awfully convenient it was that this research, which appears to have been funded by Cravendale, just happened to prove that Cravendale made the best cup of tea. Well, how can you argue with this experimental methodology:

Following the brewing process, teabags were removed and varied amounts of semi-skimmed Cravendale milk [0ml, 5ml, 10ml] were added to samples for our sensory advisory panel.

The panel’s results reveal that the ideal amount of milk to be added is 10ml.

Yep! They compared Cravendale milk to no milk at all, and shockingly found that Cravendale tastes better than nothing. This must prove Cravendale is the best milk ever, QED.

Incidentally, 10ml isn’t much milk at all, really – it’s less than half a single measure of spirits, after all. You’d think that if you were trying to test the ideal amount of milk to put in tea, you’d try everything from “no milk at all” to “nothing but milk”, but to be fair I guess the scientists involved had more important things to do than indulge the bizarre PR-driven whims of a milk filtration company.

There’s no mention of sugar in the paper, and certainly no mention of anything more exotic – a spoonful of honey, a dollop of cream, or a splash of lemon, for instance. I assume Cravendale hasn’t figured out how to filter sugar or lemon juice yet. They have however found time to make some truly groundbreaking progress in the field of thermodynamics, however:

The optimum temperature to drink tea at is 60°C. With the addition of Cravendale milk, our brews were able to reach the optimum temperature after just six minutes, two minutes faster than regular black tea.

Yes, adding a cold liquid to a hot liquid will in fact cool the hot liquid down! Of course, this only works with Cravendale milk – as everyone knows, if you add regular supermarket own-brand milk to tea, the tea just keeps getting hotter and hotter!

Still, all this stuff about “things cooling down over time” is pretty state of the art – I mean, Isaac Newton only figured out his law of cooling 300 years ago. Thank goodness Cravendale is on the cutting edge.

Anyway, based on all this research, the scientists at Northumbria have come up with a formula for the perfect cup of tea. Are you ready for this piece of extremely rigorous mathematics? Here goes:

TB + (H₂O @ 100°c) 2minsBT + C(10ml) 6minsBT = PC (@ OT60°c)

where TB = teabag, BT = brewing time, C = Cravendale milk, OT = optimum temperature and PC = perfect cuppa.

Look how science-y that is! There are letters and numbers and plus signs all over the place! And they say “H₂O” instead of “water” – only a true scientist would do that! Also according to that formula you should keep your tea brewing for 6 more minutes after you add the milk which sounds like a one-way trip to astringency-town to me, but then I’ve never written my tea making instructions down in the form of algebra, so what do I know?

This “research” has been embargoed until tomorrow morning – let’s see if any of the papers are stupid enough to run with it.

Anyway, this was quite a long post, so you should probably treat yourself to a nice cup of tea now. Just remember to use own-brand milk.

Edit: So far, the Mail, Telegraph, Express and Metro have all swallowed Cravendale’s PR rubbish. Sigh.

Littlejohn and renewable energy – both fueled by wind

I don’t normally blog about stupid Littlejohn columns – it’s a bit “dog bites man” – but today he’s written a piece about wind farms, and he’s really outdone himself on this one.

Here’s a quick rundown of the highlights.

At midday yesterday, wind power was contributing just 2.2 per cent of all the electricity in the National Grid. You might think that’s a pretty poor return on the billions of pounds spent already on Britain’s standing army of windmills.

In fact, for the amount of energy produced, onshore wind power is only slightly more expensive than coal, and less expensive than nuclear. Offshore wind is quite a bit more expensive, but hopefully this will come down as production gets more organised.

The Sun lies about disability benefits, what a shock

GOT hay fever? Why not throw a sickie?

Even better, take the rest of your working life off.

Well, this is going to go well.

New figures show that under Labour the state was happy to pay your way, no questions asked.

Those claiming Disability Living Allowance soared from 2.1million in 2000 to 3.1million last year. The annual cost is now £12billion.

So, 3.1 million have “taken the rest of their working lives off” on Disability Living Allowance, and the state is “paying their way”? Well, no.

Disability Living Allowance is a supplementary payment, given to people with disabilities, which helps cover their care and mobility costs – in The Sun‘s case, they seem to be talking solely about the part of the DLA that covers care, since that’s where the 3.1 million figure comes from. There are different levels of DLA, depending on how severe the disability is, but even in the most severe case – someone who requires 24 hour care – the recipient would only get £73.60 a week, or about £3,800 a year, and on average, people only receive about £46.30 a week, or £2,400 a year (and 500,000 of that 3.1 million get nothing at all). No-one has “taken the rest of their working lives off” to live on £2,400 a year.

Incidentally, that part about the annual cost being £12 billion does seem to be including the cost of mobility allowance as well – the cost of the care part of the DLA is only £6.4 billion a year. It sounds like a lot, but like I say, it only actually works out at about £46 per person per week – not very much at all when you think about the cost of a private carer, or the earnings lost by a friend or family member who takes time off work to provide care.

Clearly The Sun must realise this – they complain that “Many of those handed up to £73.60 a week are laid low with ailments such as “alcohol abuse” or allergies“, clearly hoping that we won’t realise that £73.60 is not all that much money. There maybe people on DLA because of alcohol abuse or allergies, but in that case, it will be because their condition is so serious that they need part-or-full-time care. To qualify for even the lowest rate, you need to be either physically unable to cook for yourself or require care for part of the day. That’s more than just “someone who cannot get out of bed because their hangover is so bad“.

The Sun also says that “The vast majority of claimants have never been medically assessed“, which also isn’t true. Most people aren’t assessed by the Department of Work and Pensions, true, but in order to qualify for DLA, you need to have been diagnosed by your doctor. Everyone who is on DLA was assessed by their doctor.

Now at last the Government plans to order regular assessments to weed out the workshy.

It should make the economy look healthier by a few billion pounds a year.

That’s something not to be sneezed at.

Unpaid carers are worth about £87 billion to the economy per year, by reducing the strain on the NHS. Making it even harder for them is hardly going to make the economy any healthier.

Edit: The Express’s coverage is more or less the same, but with TPA quotes and the added bonus that they express incredulity that people with back pain might have trouble moving around. WHO’D HAVE THOUGHT?

(The Sun discards its “Sun Says” columns each day. I’ve preserved this one beneath the fold)

Greek Island Labs and the Express – lazy churnalism or something deeper?

A little while back, I wrote a blog post about a nonsense story in the Express promoting an arthritis cream, Joint Mud, made by a company called Greek Island Labs.

A number of things struck me as odd about the story:

1. It quoted a doctor called Mark Binette, praising the efficiency of the product. Mark Binette also happens to work for Greek Island Labs, and was the creator of the cream – hardly the most independent guy you could quote!
2. The article claimed there had been incredible advance sales of the product, and went into great detail about how much the product cost and where it could be bought.
3. It also claimed that famous people had used the product – in this case, Premier League footballers.
4. Best of all, a number of comments appeared praising this product as soon as it went online, and the accounts that created those comments were newly created that day.

Well, today The Express has another story about a Greek Island Labs product, Adonia Hair Remover, under the lovely balanced headline “Rush to buy Adonia Hair Reducer cream that cuts down on shaving“. Here’s a few quotes from the article – maybe you can see a connection:

1. An independent American physician, Dr Mark Binette, said: “It’s safe, natural and works on even the most challenging cases.”
2. It sold out overnight when it launched in America last month, and will be available in London, at Harrods, from next month. It costs £29.99 for 1oz and more than 10,000 people have already put their names on a waiting list.
3. Its sales in America were boosted last month after American Pie actress Shannon Elizabeth, 37, was spotted buying some.
4. Picked this up when I was in the states last month. I’ve only been using it for a few weeks but it’s working well for me so far… • Posted by: JessMarwick at 12:45 AM

Digging into the Express archives, I also found this article from 2009 about another Greek Island Labs product, Adonia LegTone, which supposedly removes cellulite. Although this is at least marked as a review instead of a piece of news, it’s just as glowing as the other articles.  Again, they quote Binette (at least this time, it’s clearer that he works for Greek Island Labs – they don’t have the nerve to claim he’s “independent”). Again they gush about how many preorders it’s had and how much it will cost. Again they quote scientists who point out that the claims it makes are rubbish… then ignore them. Again, there’s a very positive comment from someone who’s never commented before or since on the Express website, which the moderators haven’t removed despite it clearly being spam.

Three articles, all rather similar, all praising a Greek Island Labs product. Have GIL simply realised that the Express is a soft touch for churning press releases into news, or is there a deeper connection here?

Mobile phones and cancer. Again

(H/T @DickMandrake)

This will be all over the papers today, so here’s a quick run down of what’s actually happened.

Last year the World Health Organisation released its Interphone report (PDF) into the link between brain cancer and mobile phones. For the most part, they found “no increase in risk of glioma or meningioma [types of brain cancer] was observed with use of mobile phones”. However, for very heavy users (the top 10% of the population), there was a statistically significant increase in the odds of developing a form of brain cancer known as glioma but “but biases and error prevent a causal interpretation” – there is too much uncertainty in the data to know whether heavy mobile phone use caused cancer or whether something else was to blame. On the one hand, they found there appeared to be a connection between which side of the brain the tumour developed in and which hand users held the phone in – a sign that phones might cause cancer – but on the other hand, while extreme users experienced a big increase in brain tumours, people who used their phones even slightly less saw no change in brain cancer – a sign that mobile phones might not cause cancer.

Fast forward a year, and the WHO’s International Agency for Research on Cancer (IARC) has issued a press release (PDF) about an upcoming report which, based on the Interphone study and some other papers, will classify the electromagnetic (EM) fields from mobile phones as “possibly carcinogenic to humans” meaning there is “limited evidence” that they may cause cancer. In other words, it’s the exact same conclusion as the Interphone report, but this time the report is focusing on the negative – heavy doses may cause cancer – rather than the positive – light doses probably don’t cause cancer.

That in itself is very reasonable – even if mobile phones carry a very slight cancer risk, the IARC still need to know what that risk is. The problem comes when the media, reporting this in the usual shades of black and white, ignore all this nuance in favour of scares:

The Mail screams “Mobile phones CAN increase risk of cancer: Doctors reveal shock results of major study into effect on the brain“, magically turning “possibly causes cancer” into “CAN cause cancer”. It’s not really a shock result either, since it came out a year ago. The Mail also brings up the recent Council of Europe draft report that suggested banning mobiles from around schools, ignoring the fact it was pseudoscientific rubbish based on research from quacks.

The Express goes for the similar “Shock cancer warning over mobile phone use“, claiming “MOBILE phones have been officially linked to cancer for the very first time by a team of world experts”. Again, the report is a year old, it’s not a shock! They also manage to get in some ridiculous guilt-by-association:

But they classified mobile phones in the same danger category as the pesticide DDT and petrol engine exhaust, meaning they are possibly carcinogenic to humans.

Petrol exhaust and DDT are pretty dangerous, but that’s not necessarily because they cause cancer! Nor does being in the same group as these mean it carries the same risk of cancer, either. It just means that we have the same level of evidence for a cancer risk, which isn’t the same thing. For example, DDT is now banned worldwide, with most developed countries banning it in the 70s. This means that people aren’t being exposed to it any more (a good thing, of course!), so we can’t study its effects and work out exactly how dangerous it is.

The Guardian meanwhile goes for a tenuous connection to the risks of mobile phone base stations and wi-fi, even though the exposure to EM from these is much lower than from holding a phone to your ear – and we know even then it’s only the heaviest exposures that may cause cancer.

The Telegraph also makes the ridiculous DDT comparison, but at least they quote the great Ed Yong, head of health information at Cancer Research UK, who sensibly sums up the evidence:

“The risk of brain cancer is similar in people who use mobile phones compared to those who don’t, and rates of this cancer have not gone up in recent years despite a dramatic rise in phone use during the 1980s.

“However, not enough is known to totally rule out a risk, and there has been very little research on the long-term effects of using phones.”

(They also quote a professor of Medical Physics and Director of the Mobile Operators Association, but to be honest, they seem to have gone into damage control mode before there’s been any damage to control. Pro-tip, guys – if you ever end a paragraph about the health risks of something by saying “The social and technological benefits also need to be emphasised“, then you sound like a cyberpunk bad guy, even if you truly mean it.)

Anyway, for a little perspective, there’s a great interview with Ed Yong about the news here. Read it!

Edit: There are also great pieces out there on this from Pharyngula, Respectful Insolence, Cancer Research UK, Tom Chivers, Only That In You, Scientific American and Bad Astronomer. Good old blogosphere!

Light bulb lies

Whose bright idea was this? asks Daily Mail Reporter today. New ‘green’ light bulbs will cost you $50 EACH (and you’ll HAVE to buy them after 100-watt bulbs are banned)

How terrible! If, that is:

1. You’re American. This article is about the upcoming phase-out of 100 W bulbs in the United States – 100 W light bulbs have been banned in Europe for nearly two years, and shockingly we’re not paying $50 (£30) per bulb.
2. You light your house exclusively with 100 W bulbs.
3. You refuse to buy the normal energy saving compact fluorescent light (CFL) bulb, which you can get for about £1, for some reason.
4. You decide to buy the first of a new sort of light bulb – the LED light bulb – and in particular, you buy the smallest and brightest bulb available, which costs about $45 (£27) at the moment, and you buy it right away instead of waiting for the price to come down.

Better headline: One particular light bulb might cost $45 EACH at first (but you won’t have to buy them even if you’re American because they’re stupidly powerful and there are cheaper bulbs available which are just as bright)

Hmm, perhaps it’s a good thing I’m not a newspaper sub.

PS: Oh, and why is “green” in “scare quotes”?

Election chicken recipe

Not a proper blog post at all, this, just a recipe for something I’ve decided to call election chicken. It’s basically coronation chicken crossed with chicken tikka masala (supposedly the most popular dish in the country and therefore the one with democratic mandate) crossed with various bits and bobs around the kitchen. Oh, and it tastes AMAZING.

Ingredients (fills about 4 to 6 sandwiches):

• about 400 g of diced chicken (you can use the precooked stuff from the supermarket, or just chop up some chicken breasts and fry them with some curry spices)
• one small pot (250 ml) of creme fraiche
• about the same amount of mayonnaise
• 3 heaped teaspoons tandoori masala
• 1 heaped teaspoon paprika
• splash of red wine
• splash of Worcestershire sauce
• a decent handful of grated cheese (I used double Gloucester for a quite creamy taste, but you could use mature cheddar for a stronger kick)
• 2 teaspoons of tomato puree

Basically just mix it all together in a big bowl (if you’re cooking the chicken yourself, let it cool a little first), stand for a few minutes, and serve with salad in a sandwich!

Easy!

Black holes and revelations in the Mail

Edit: After a quick email conversation with the author, the article has now been rewritten (direct link). Although it still has a slightly cringeworthy headline, it no longer contains misunderstandings about the paper and is all in all pretty good! I’ve hidden the original post behind the cut.

Another cancer cure that’s not a cure

The Daily Express today has a piece titled “The 20p ‘sunshine pill’ helps cancer patients live longer“. That’s a bold, unambiguous claim. Taking this pill will improve cancer survival rates, right?

A British oncologist will next week tell a conference that many of his skin cancer patients have unusually low ­levels of vitamin D.

He now gives them supplements and says they often survive for longer than expected.

Although he has not yet done a clinical trial to back up his theory, he believes that vitamin D pills at just 20p a day could eventually prove as ­effective as cancer treatments that cost thousands of pounds per patient.

Oh. So it might help cancer patients live longer one day, according to a doctor who hasn’t tested his theory yet.

There is some reasonably good evidence, summarised at Wikipedia, that low levels of vitamin D (a chemical produced when the skin is exposed to sunlight which controls bone growth and the life cycle of cells) are linked to an increased risk of certain types of cancer and faster cancer growth, but the cause and effect aren’t clear – are low vitamin D levels causing cancer, or is cancer causing low vitamin D levels? – and, as Cancer Research UK says:

We strongly advise cancer patients to talk to their doctor if they are concerned, before considering taking supplements – especially since there’s evidence that some vitamin supplements can have unintended consequences.   Moreover, vitamin D from supplements doesn’t appear to be regulated in the body as tightly as vitamin D from the sun – and there’s still a lot of uncertainty over what the ‘best’ dose is.

And we certainly don’t recommend that patients go overboard on the beach or sunbeds to top up their vitamin D. We know that we all need a bit of sunshine in our lives.  But we also know that excessive UV exposure (from the sun or sunbeds) and  sunburn are major risk factors for melanoma.

That’s some very good advice, which makes it all the more galling that the Daily Express is encouraging cancer patients to start taking vitamin D pills without visiting their doctor.