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The Mercator projection increases scale along with the latitude. Greenland seems larger than Australia, when in fact it is much smaller. The equal-area projections however stretch shapes away from the Equator. I suppose that there exist no projection that preserves size and shape at the same time. But for the purpose of comparing countries it is in fact not needed to preserve any properties exactly. The distortions should be only roughly the same in any point of the map. What is the map projection that accomplishes it most closely?

Comparison between Greenland and Australia

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For quickly conveying the distortion any projection introduces see Creating an accurate Tissot Indicatrix and the FlexProjector software –  matt wilkie May 15 '12 at 21:01

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Projections are like pushing string. As you try to preserve one aspect, you get distortions in some other parameter (e.g. distance or bearing). To preserve both shape and area you may need to consider an interrupted projection like Goodes Homolosine or the off-beat Buckminster-Fuller 'Dymaxion' projection. In these projections, the distortions are present but minimised because the interruptions effectively 'reset' the projection. However, you lose sensible bearings and distances with these projections so they would be useless for navigation.

Because you mention Australia and Greenland in the same breath, the presumption in your question is that you want a global projection. Local projections are best for locally preserving area and shape simultaneously of course... and there are a tediously huge number of these!

Just to be pedantic, a globe probably is a projection because globes tend to be perfectly spherical unlike the Earth... but that's getting into the realms of arguing about how many fairies could dance on the head of a pin :)

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Interrupted projections are ok as I only want to compare countries visually. The question that remains is which one gives the most uniform distortions. I think that in terms of mathematics we should be looking for a projection that maps a sphere onto a solid figure with the lowest standard deviation of an error. –  mithy Mar 5 '12 at 12:46
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Mithy, you can bring the SD of the error arbitrarily close to zero by interrupting the map along lots of lines. Think of peeling an orange into segments: when the segments are narrow, there is little distortion. Although the cuts across segments introduce "infinite" terms into the SD, because the cuts have measure zero, they don't affect the SD (averaged over all points of the globe). Thus, you need to be more careful about your criteria for a good projection. –  whuber Mar 5 '12 at 17:50
    
+1 You're quite right; mathematically, mapping the earth onto a globe is a projection. But (outside of mathematics) the term "projection" is usually reserved for projections onto developable surfaces: that is, flat map sheets. –  whuber Mar 5 '12 at 17:52
    
Exactly, hence the pedantry alert :). <Pedantry Alert #2> By 'flat' I presume you mean after unwrapping from some other shape such as a cone, cylinder, icosahedron etc. because projections to flat surfaces are less common than projections to an 'easily unwrapped' 3D shape, like the cone and cylinder especially.</Pedantry Alert #2> –  MappaGnosis Mar 6 '12 at 10:07
    
@whuber yes you are right, but if you cut the map too much you won't be able to compare much :). I think I outlined pretty precisely what I want to achieve. –  mithy Mar 6 '12 at 12:17

The best projection that does not distort is a globe. All the others are compromises to project the objects onto a piece of paper. The attempt to do that "projection" distorts something. It can distort distance, angles, shapes, area. Some of these properties are preserved by various projections. But no projection can preserve them all.

If you want to compare country size use a globe, or even better, use a table

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The globe is not a projection. –  R.K. Mar 4 '12 at 7:56
    
In light of comments the OP makes to another reply, a globe is looking more and more like the best solution: distortion is roughly the same everywhere and it can be "bought in a shop" :-). –  whuber Mar 6 '12 at 16:55

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