# Sitewide page header for website of Eric P Smith

## The puzzle

The geometry of the Earth lends itself to mathematical puzzles. Here is one that I composed.

Some years ago, the story goes, I was umpiring an adult league cricket match on a clear sunny day. As I was about to place the bails on top of the stumps at the start of the match, I perceived that the shadow of the middle stump just reached the popping crease. Some hours later, immediately after removing the bails for the lunch interval, I again perceived that the shadow of the middle stump just reached the popping crease. Some hours later again, immediately after removing the bails at the end of the match, I perceived for the third time that the shadow of the middle stump just reached the popping crease. The stumps were, of course, perfectly vertical. All you have to do is answer the following four questions.

• In what compass direction did the popping crease run?
• At which end of the pitch was I umpiring?
• What was the date in the year, give or take a day or so?
• Where in the world did the match take place?

If you are unfamiliar with cricket, you need a picture. Even if you are thoroughly knowledgeable about cricket, it may be that you could still do with some measurements. See Figure 1.

Following the convention for puzzles of this kind, you may make use of real‐world facts in the public domain, for example that adult league cricket is played only on land and not in the middle of oceans.

## The solution

Consider the top of the middle stump – not the top section, but the very highest point. (Ok, so because of the bail groove, the middle stump has two highest points: consider either of them.) Through any given day, its shadow traces out a locus – a path – on the ground.

For simplicity let us first ignore the slow movement of the sun through the year against the background of the celestial sphere. Then the apparent path of the sun during the day, relative to the surface of the Earth, is a circle in the sky. That means that the shadow of the top of the middle stump traces out a cone. Its locus on the ground is the intersection of that cone with the (locally flat) ground. The intersection of a cone with a plane is a curve known as a conic section. A conic section may be of any of four types: an ellipse, or a parabola, or a hyperbola, or a pair of straight lines. The locus of the shadow while the sun is above the horizon will therefore be an ellipse, or a parabola, or one of the two branches of a hyperbola, or a straight line. This is not the place to prove these things, but they are standard elementary geometry and they have been known since the time of the Ancient Greeks.

We know that the locus, whatever shape it is, intersects the popping crease at least three times, namely on the three occasions when the shadow of the middle stump was observed to just reach the popping crease. That rules out an ellipse, a parabola, and a branch of a hyperbola, because none of these curves can intersect a straight line three times. The only remaining possibility is that the locus is itself a straight line. That means that the sun is describing a great circle in the sky; it is at equinox; it rises due east and sets due west; the locus of the shadow is a straight line running due east⁠–⁠west; it follows the popping crease exactly. It wouldn’t have mattered at what time of day I looked at the shadow of the top of the middle stump, because it ran along the popping crease all day. As the popping crease runs east⁠–⁠west, the pitch runs north⁠–⁠south.

It is now an easy matter to work out the latitude of the place on the Earth where the match took place. At noon the sun was at its highest, due south (if the match was in the northern hemisphere) or due north (if the match was in the southern hemisphere). The middle stump, 28 inches high, cast a shadow which ran straight down the pitch and just reached the popping crease, a distance of 48 inches. So the sun was at an elevation of arctan(2848) = 30° 15′. This was at noon at equinox, and so the latitude was 90° − 30° 15′ = 59° 45′ north or south of the equator. (In this solution, “noon” means solar noon, and all angles are given to the nearest arc minute.)

The crease marking is a white line which may be up to about an inch wide. The popping crease itself, however, is the back edge of the crease marking, and it is that back edge which is 48 inches from the centres of the bases of the stumps.

We will now introduce three refinements to the above analysis. Surprisingly, they have no net effect on the conclusion.

The first refinement is to take into account the slow movement of the sun through the year round the celestial sphere. When we take that into account, we find that the locus is not precisely a conic section. But it is so close to a conic section that, on days other than equinox, our conclusion that it cannot intersect a straight line three times is still correct. On a day on which equinox falls, if the equinox is between sunrise and sunset, then the locus is an elongated and very nearly straight S shape with a point of inflexion just where the shadow falls at the moment of equinox. The locus is not precisely a straight line, but because it flexes it can intersect a straight line at three points. Any straight line that it intersects at three points (and in particular the popping crease) must run east⁠–⁠west within a margin of a few arc minutes. For all practical purposes therefore we can take the popping crease as running due east⁠–⁠west, and the pitch as running due north⁠–⁠south, and we will do so from now on.

The second refinement is to take into account the thickness of the stump. The Laws of Cricket say that except for its top, the stump is cylindrical, with a diameter not less than 1.375 inches and not more than 1.5 inches, and that the top is dome‐shaped except for the bail groove. The shape is thus not precisely defined, but Figure 2 shows a scale drawing of a typical shape.

Figure 2 shows a cross section of the top inch or two of the stump by a vertical plane running north and south in the direction of the pitch through the centre of the stump, and it shows the sun due south or due north, with such an elevation that the shadow of the stump just reaches the popping crease. The red spot shows the point on the surface of the stump whose shadow lies farthest to the right and lies on the popping crease. The red line, tangential to the stump at that point, points to the sun and to the popping crease. The horizontal distance from the red spot to the popping crease is 48 − 0.6 = 47.4 inches. The vertical distance from the red spot to the ground is 28 − 0.13 = 27.87 inches. The elevation of the sun is arctan(27.8747.4) = 30° 27′ instead of 30° 15′. The sun is in that position at noon at equinox, and so the latitude is 90° − 30° 27′ = 59° 33′ instead of 59° 45′, north or south of the equator. You can verify the consistency of these measurements, with enough accuracy for all practical purposes, by measuring Figure 2 on your screen with a physical ruler. (Or if you are determined and you have the computing skills, you could verify it exactly by examining the SVG markup of Figure 2.)

The third refinement is to take into account the width of the sun’s disc. Because the sun’s disc has width and is not a single‐point light source, the shadows that it casts are not perfectly sharp. The shadow of a sharply defined object consists of the umbra (the full shadow), and the penumbra (the partly illuminated border of the shadow). You might expect that the perceived edge of the shadow would be half way across the penumbra, but this is not so. The human visual system is highly nonlinear, and in clear sunlight the perceived edge of a shadow is close to the umbra, and only one‐eighth of the way across the penumbra. In other words, the perceived edge is displaced from the centre of the penumbra by a distance of three‐eighths of the width of the penumbra. The width of the penumbra corresponds to the width of the sun’s disc, which is 0° 32′, and so three‐eighths of the width of the penumbra translates to an angular distance of 38 × 0° 32′ = 0° 12′ across the sun’s disc. Before we took into account the width of the sun’s disc, we computed that when the sun is due north or due south it needs an altitude of 30° 27′ for the shadow to just reach the popping crease. In the case where the sun’s disc has nonzero width, the geometry is identical but is now interpreted as meaning that the centre of the sun’s disc needs an altitude of 30° 27′ for the middle of the penumbra to just reach the popping crease. But it is the perceived shadow, not just the middle of the penumbra, that we want to reach the popping crease, and that means that the centre of the sun’s disc has to be 0° 12′ lower, at 30° 15′. Again the sun is in that position at noon at equinox, and so the latitude is 90° − 30° 15′ = 59° 45′.

This correction of three‐eighths of the width of the penumbra, 0° 12′ or 0.20°, is common knowledge among those who make sundials. See for example:

Before we made corrections for the thickness of the stump and the width of the sun’s disc, we computed a latitude of 59° 45′. Now that we have made these two corrections we are back where we started, with a latitude of 59° 45′ once more. By sheer coincidence, the two corrections have proved to be equal and opposite, and so the conclusion is unchanged: the popping crease ran east⁠–⁠west, and the match took place at equinox, at latitude 59° 45′, north or south of the equator.

So where in the world did the match take place? We have a problem here. Cricket is an outdoor sport, and it relies on a reasonably warm outdoor temperature because a fieldsman may be inactive for up to six or seven minutes at a time. Almost no place on earth with such a high latitude is warm enough at equinox (around 20 March or 22 September) to play cricket. The only exception is northwest Europe, where the North Atlantic Current and the winds associated with it guarantee a climate considerably warmer than other places of the same latitude. In northwest Europe at such a high latitude it is not warm enough to play cricket at the March equinox, but it may be warm enough at the September equinox.

The 59° 45′ N line of latitude does not strike land in the British Isles: it runs through the sea channel separating Fair Isle from mainland Shetland. Following the line eastwards, it strikes land in Norway, where the cricket season runs to the end of September. Further east in Sweden the effect of the North Atlantic Current is slightly less, and the cricket season ends before the September equinox. Further east still – in Estonia and then in Russia – the temperature problem is increasingly limiting. So the match can only have taken place in Norway.

Where about in Norway? Cricket is very much a minority sport in Norway, and there are just five adult league cricket grounds in the whole country. One of them, beside the former harness racecourse in the city of Drammen, incredibly lies just 400 metres, or 13 arc seconds, off the 59° 45′ N line of latitude. See Figure 3. All of the four others have latitudes of 59° 54′ N or more. At a latitude of 59° 54′ N instead of 59° 45′ N, the perceived edge of the shadow of the middle stump at noon at equinox would overshoot the popping crease by about 0.3 inches instead of just reaching it. Even the umbra would overshoot the popping crease by nearly 0.2 inches (5 mm), and so the overshoot would be discernible to the human eye.

It follows, therefore, that the match took place at the cricket ground beside the former harness racecourse in Drammen, Norway, on the autumn equinox (22 or 23 September). The popping crease ran east⁠–⁠west and so the pitch ran north⁠–⁠south. I was umpiring at the south end. Well I wasn’t of course, but that’s the story.

League matches take place at Drammen regularly throughout the season from the beginning of May to the end of September, in all five adult divisions.