Geographical Association

Figure 1

The basic principle is that the globe is the only true representation of our planet. Even a cheap blow-up globe is far more accurate than the 'best' flat world map. So we should use a globe to teach about the world as often as possible. In this approach, we keep '3-D' as far as possible: the globe is the centre of our attention. The globe is the only objective truth among the many competing maps of the world.

Start with a very familiar visual aid - the unpeeling of an orange or a tangerine. This action establishes the key problem of 'flattening out' the earth's surface. It can't be done. There has to be compromise: either cutting, and/or stretching, and/or squeezing the world to show it on a flat surface.

One approach is to look at a segment of an orange and then analyse a 30º segment of the globe (Figure 2). Lines of latitude are marked at 0º, 30º and 60º.

Figure 2

Figure 3 shows the northern half of our segment in more detail. Use a piece of 'food-wrap' plastic sheet and a fibre-pen to trace off the segment: a quick and an effective way of discovering some of the problems of transferring from 3D to 2D. For example, on the globe, all intersections are at right-angles; when we flatten out our 'food-wrap’ map, the true right-angles may be lost.

Figure 3

We now have nine lines to study. No fewer than seven of these nine lines are identical in length (marked = on the diagram). These seven lines are easy to recognise and remember. They create three important and memorable shapes, which we will call 'sectors'.

1. The mainly Arctic sector (60º to 90º North) is a narrow triangle. Its base is at 60º North. This base is half the length of the other two sides.
   
   
2. The Temperate sector (30º to 60º North) is a trapezium, with its northern side half the length of its two sides (as with the Arctic section).
   
   
3. The mainly Tropical sector (0º to 30º North) is almost a square - a trapezium with three equal sides. The northern side is slightly shorter (by about 12%, or an eighth) than the other three sides.

Using these three shapes helps develop our understanding of world maps quickly and easily. Both the shape and the relative sizes of these three sectors are crucial for evaluating world maps. These three simple shapes give us a better means of assessing the strengths and weaknesses of every map projection.

If we can keep the sphere central to our thinking - and our three simple 30º x 30º shapes - then the projections used to create world maps become easy and straightforward to understand.

Every flat world map has to make compromises. The simplest way to express this fact in mathematical terms is by using this diagram (Figure 4). It is a 'triangle' with three right-angles (= 270º) - yet we all know that angles in a triangle should add up to 180º. Better still would be an invitation to pupils to discover this 'impossible triangle'. The diagram proves that big distortions are inevitable, even on a map of only one-eighth of the globe. Distortions will inevitably be massive on a whole-world map.

Figure 4

A few words about distance...

• The actual length on the ground of the seven equal lines on our key diagram (Figure 3) is very memorable: it is 3333 kilometres (to within 1%). The reason is simple: Napoleon decreed that 'Pole to Equator is 10,000 kilometres' - that is how a kilometre was defined. That calculation is still true to within 1%. One-third of 10,000 kilometres produces the unforgettable figure of 3333.33 km.
• The 30ºN line of latitude is a bit narrower than the equator - only by about an eighth, or 12%.
• The 60ºN line is much more memorable and recognisable: it is exactly half the length of the line at the equator - and nearly as easy to remember: 1666 km

The relative sizes of different parts of the world are of major significance to geography, and our three sectors diagram (Figure 3) can help us to discover this. This theme is important in its own right, and also very important in evaluating the quality of different map projections.

The mainly Tropical sector is roughly three-and-a-half times the size of the mainly Arctic sector. This is a very important discovery: such a lot of the world is Tropical, and so little is Arctic. But it is also a key means of assessing whether maps are 'equal area' or not. Compare the size of the equivalent Arctic and Tropical sectors from a world map. Immediately, Mercator and Gall are seen as very, very wrong. Even so, the biggest distortion cannot appear: neither Mercator nor Gall even reach 90º North.

When the lines of longitude are curved on a flat world map it is less easy to judge exact size - but a comparison is still important to make, and still straightforward. Maps with an 'oversized' Arctic are easily spotted.

Using the three sectors model we can identify some of the strengths and weaknesses of some well-known world maps:

• A 30º segment of Mercator's Projection shows why that map is out of fashion: it is quite a good shape in the Tropical sector, but in the Arctic sector it is far too big: too long and too fat
• A 30º segment of Peters Map is far too long and thin in the Tropical sector, and too short and fat in the Arctic sector
• Eckert IV is too long and thin in the tropics. It is not as bad as Peters, but still the wrong shape
• The Mollweide and Sanson-Flamsteed Projections are quite good in the centre, but 'bent' at the extremes
• Any other world map can be evaluated by looking at some 30º segments near the centre and at the edges.

Now go to page 2 - Mercator's Map