Mountains – Ben Craven

Why hillwalkers should love the Comte de Buffon

Part of Beinn a Bhuird, Cairngorms, Scotland.

Wandering the mountains of the UK has been a big part of my life. You won’t be surprised that before I start a long walk I like to know roughly what I’m letting myself in for. One part of this is estimating how far I’ll be walking.

Several decades ago my fellow student and hillwalking friend David told me of a quick and simple way to estimate the length of a walk. It uses the grid of kilometre squares that is printed on the Ordnance Survey maps that UK hillwalkers use.

To estimate the length of the walk, count the number of grid lines that the route crosses, and divide by two. This gives you an estimate of the length of the walk in miles.

Yes, miles, even though the grid lines are spaced at kilometre intervals. On the right you can see a made-up example. The route crosses 22 grid lines, so we estimate its length as 11 miles.

Is this rule practically useful? Clearly, the longer a walk, the more grid lines the route is likely to cross, so counting grid lines will definitely give us some kind of measure of how long the walk is. But how good is this measure, and why do we get an estimate in miles when the grid lines are a kilometre apart?

I’ve investigated the maths involved, and here are the headlines. They are valid for walks of typical wiggliness; the rule isn’t reliable for a walk that unswervingly follows a single compass bearing.

On average, the estimated distance is close to the actual distance: in the long run the rule overestimates the lengths of walks by 2.4%.
There is, of course, some random variation from walk to walk. For walks of about 10 miles, about two-thirds of the time the estimated length of the walk will be within 7% of the actual distance.
The rule works because 1 mile just happens to be very close to $\frac{\pi}{2}$ kilometres.

The long-run overestimation of 2.4% is tiny: it amounts to only a quarter-mile in a 10-mile walk. The variability is more serious: about a third of the time the estimate will be more than 7% out. But other imponderables (such as the nature of the ground, or getting lost) will have a bigger effect than this on the effort or time needed to complete a walk, so I don’t think it’s a big deal.

In conclusion, for a rule that is so quick and easy to use, this rule is good enough for me. Which is just as well, because I’ve been using it unquestioningly for the past 35 years.

And the Comte de Buffon?

George-Louis Leclerc, Comte de Buffon, (1707-1788) was a French intellectual with no recorded interest in hillwalking. But he did consider the following problem, known as Buffon’s Needle:

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?

Let’s recast that question a little and ask: if the lines are spaced 1 unit apart, and we drop the needle many times, what’s the average number of lines crossed by the dropped needle? It turns out that it is

$\dfrac{2l}{\pi}$

where l is the length of the needle. Now add another set of lines at right angles (as if the floor were made of square blocks rather than strips). The average number of lines crossed by the dropped needle doubles to

$\dfrac{4l}{\pi}$

Can you see the connection with the distance-estimating rule? The cracks in the floor become the grid lines, and the needle becomes a segment of the walk. A straight segment of a walk will cross, on average, $\frac{4}{\pi}$ grid lines per kilometre of its length. Now a mile is 1.609 kilometres, so a segment of the walk will, on average, cross $\frac{4}{\pi} \times 1.609 = 2.0486...$ grid lines per mile, which is very close to 2 grid lines per mile, as our rule assumes. If a mile were $\frac{\pi}{2}$ km (1.570… km), we’d average exactly 2 grid lines per mile.

So the fact that using a kilometre grid gives us a close measure of distance in miles is just good luck. It’s because a mile is very close to $\frac{\pi}{2}$ kilometres.

In a future post, I’ll explore the maths further. We’ll see where the results above come from, and look in more detail at walk-to-walk variability. We’ll also see why results that apply to straight lines also apply to curved lines (like walks), and in doing so discover that not only did the Comte de Buffon have a needle, he also had a noodle.

Mathematical typesetting by QuickLaTeX.

Faces in the snow

A hollow impression of my face in the snow.

The hollow face illusion is a wonderful visual effect in which a hollow mask of a face appears to be convex, like the face itself. Making a hollow mould of your face (for example using plaster) is difficult and potentially dangerous. However, last weekend my attention was drawn to an easier and safer way.

I was walking down from Coire an Lochain in the Scottish Highlands with a group from the Red Rope club, when I saw my friend Maia standing on the path ahead, chuckling. She’d been making face imprints in a steep snowdrift, and they showed the hollow face illusion beautifully.

The procedure needs no explanation (see right). The snow needs to be fresh and soft; you’d be surprised how hard it is to push your face into what feels to your hand like very soft snow. The tip of my nose is noticeably flattened in the picture above.

Ben and Matthew on skyline — Near Coire an Lochain on the day in question. (Readers familiar with Highland place names will realise that I’m not giving much away here.)

How high am I ?

Last week I was on holiday in Wales. It wasn’t the driest of weeks, and while I was inside not climbing mountains, I finally got round to doing some mountain-related geometry that I’ve been putting off for the last 30 years or so. It’s about knowing how high you are.

Hellaval and Skye from Askival — Am I higher or lower than the summit in the middle distance? (The peak of Hellaval on the island of Rum, taken from the flanks of Askival, with Skye in the background)

If you’re climbing or descending a mountain, you sometimes want to know roughly how far (vertically) there is still to go. One way to get an idea of your altitude is to use nearby peaks or other points of known altitude as reference points. But how do you judge whether you are above or below another point? It’s not always obvious, and without some sort of rule, the worry is that you’ll make optimistic judgements, leading to disappointment in the long run.

My friend Malcolm once told me that, as a rule of thumb, you should look at the reference point relative to the distant skyline. If the point appears to be above the skyline, you are lower than it, and if it appears to be below the skyline, you are higher than it. I’ve used this skyline-rule ever since, but I’ve never checked how accurate it is.

The fact that there’s any doubt about the rule is because the Earth is not flat. If it was flat, then your line of sight to the (infinitely distant) sea-level horizon would be exactly horizontal, and the rule would work perfectly. If the skyline was made up of mountains, the rule would work perfectly as long as they were as high as your reference point.

But the Earth isn’t flat: it’s a big ball. How does this affect the accuracy of the rule? I used a wet Welsh Wednesday afternoon to find out.

It turns out that the rule is good enough for general hillwalking purposes as long as the reference point is no more than two or three kilometres away (as it usually will be). The errors are smaller if the skyline is distant mountains rather than the horizon at sea level. The rule consistently underestimates your altitude, which, in ascent at least, is probably better than the alternative. Continue reading How high am I ?

On the fate of my food

Recently I spent a few days walking and wild camping among the mountains of the English Lake District. I was moving on every day and carrying everything I needed for the trip on my back. My rucksack got lighter and lighter as I worked my way through my food supply, and as there was no evidence that my body was getting any heavier, I started wondering where all that mass goes.

The food was largely very dry, so I’m not going to concern myself with water (though some water will be generated as the food is metabolised). Of the dry mass, I will excrete some as faeces, and some in my urine. I’ll secrete some skin oils and will also shed some skin. Then there’s hair, toenails and fingernails, and we mustn’t forget the odd bit of snot and earwax. Quite a trail of debris, really.

But what I started to wonder as I walked was: how much of this dry mass do I breathe out? I must lose some mass with each breath, because outbreaths are poorer in oxygen and richer in carbon dioxide than inbreaths, and carbon dioxide is heavier than oxygen. Can we put some numbers to this loss of mass?

It turns out that we can, and the result surprised me: by a big margin, the most important exit route for carbon appears not to be my bottom, but my lungs.

The sums

The volume of gas exchanged on each breath in ordinary breathing is about 0.5 litres. On the way in, the concentration of carbon dioxide is practically nothing (about 0.04%), but on the way out, it’s about 5%. So every 20 breaths results in me breathing out the equivalent of half a litre of pure CO₂, which means 40 breaths to breathe out a litre.

Assuming that for every molecule of carbon dioxide breathed out, there is one molecule of oxygen breathed in, then the extra (non-water) mass breathed out is just the mass of the carbon in the carbon dioxide; we can ignore the oxygen.

How much carbon is there in one litre of CO₂? One mole of carbon is 12 grams, and one mole of a perfect gas occupies roughly 24 litres in everyday conditions. So one litre of carbon dioxide contains one-24th of a mole of carbon, or about 0.5 grams.

So every 40 breaths, I breathe out 0.5 grams of carbon that originally entered my body as food. At rest, I breathe about once every 4 seconds, so it takes 2 × 40 × 4 = 320 seconds to breathe out a gram of carbon. There are 86400 seconds in a day, so over the course of a day, I’ll breathe out 86400/320 = 270 grams of carbon.

I’ve made all sorts of approximations, so let’s say that I breathe out 200-300 grams of carbon daily.

What about the other exit routes for carbon?

This is a much bigger number than I expected. How does it compare to the other output routes for carbon, and do the numbers stack up when we consider how much carbon I ingest?

From various internet sources, fat is something like 75% carbon, carbohydrate is about 40% carbon, and protein is about 50% carbon. On this basis, I’m going to estimate that the dry matter of faeces is in the region of 50% carbon. A typical person produces about 130 grams of faeces a day, of which 75% is water. So the amount of carbon excreted daily in faeces is about 130 × 0.25 × 0.5 = 16 grams.

Wikipedia tells me that we typically produce about 1.4 litres of urine a day, with 6.9 grams per litre of carbon, which gives about 10 grams of carbon excreted by this route per day.

Several internet sources (which may be equally wrong) suggest that we shed about 10 grams of dead skin every day. I don’t know what the moisture content of dead skin is, but it looks like we’re not going to lose more than 5g of carbon by this route every day.

Neglecting the other minor carbon loss processes, this gives a total of about 30 grams of carbon leaving the body daily by routes other than the lungs, compared to 200-300 grams via the lungs.

If these sums are right, I must ingest an equal amount of carbon in my food. This isn’t easy to estimate at all precisely. My diet is largely a mixture of fat, protein, and carbohydrate, but I don’t know what the balance is, and it’ll vary a lot between people. So let’s go to an extreme and suppose that I meet a daily energy requirement of 2500 kilocalories by eating fat only. At 9 kilocalories per gram I’d need to eat 280 grams of grease a day. At 75% carbon, this would be 210 grams of carbon. Doing a similar calculation assuming a diet of undiluted protein or carbohydrate gives a daily carbon intake of 310 grams or 250 grams respectively. So maybe I eat about 250 grams of carbon per day, which tallies reasonably well with the total figure for carbon output that I calculated earlier.

Not the Lake District

The picture at the top of the post wasn’t actually taken in the Lake District. It was taken in Glen Feshie, in the Cairngorms. To keep the weight down, I didn’t take my camera on the trip to the Lakes.