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The sort of world map you are describing is in effect a series of maps, joined together.
Distances are proportional to reality throughout each central meridian - ie throughout each north-south line on the map which ends in a point on the map - and along all parallels.
Shapes are more distorted the nearer to the edge of each triangle of map you get, with the length of meridians being exaggerated so that countries nearer the poles appearing more stretched north to south than they should be. Hence the more triangles you have, the less the distortion (since the edge of each triangle will be closer to its central meridian, where distances are represented accurately) though it can never be eliminated because of the problem of representing a 3-D shape (the globe) on a 2-D surface (the map).
As Loosehead points out, the empty triangles are places where there is nothing, though the edges of each empty triangle are in reality next to each other.
These maps are apparently called "Interrupted sinusoidal projections".
Wikipedia gives an explanation at
As most have pointed out - there is a problem representing a 3D, spherical (or geoid / oblate spheroid) on a 2D surface.
This is a problem as old as they have known the Earth to be 'round' and attempted to map it.
In small portions, the errors are negligible - over greater areas - the problems are great.
Probably one of the most familiar 'World maps' is the Mercator projection (projection being the representation of the globe in two dimensions). This shows the continents and oceans sure enough, but the scale is not constant:- land (or sea) areas are exaggerated and relative distances are altered, depending on where you are on the map.
For example - in terms of surface area, the British Isles appear much bigger than countries of equal area close to the equator.
Similarly, Antarctica (and some of the northern polar land masses) appear as almost continuous strips of land along the top and bottom of the map.
By the same virtue, it is difficult to see why the Titanic was up towards the Arctic Circle whilst sailing from Cobh to New York - on a map it seems a straight-enough journey "straight across" the Atlantic (New York is about the same latitude as Madrid) � but the shortest route on a globe is via a Great Circle.
These are all problems associated with trying to 'map' a 3D figure. The Mercator projection puts undue prominence on those countries of the mid-latitudes - hence the "British Empire" map, when, apart from half the map being painted pink, the British Isles were an enormous pair of islands somewhere around 50�N.
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The modern Peters Projection attempts to portray an equal-area projection, so that, say, a small, northern hemisphere country does not appear the same size on the map as the entire Indian Sub-Continent.
True, it has been used in a �PC� context to counter the undue prominence of certain �Western� nations in temperate latitudes � but as a lover of maps � it is interesting nonetheless.
I will leave it to the likes of madein78 and Lonnie to hark to the times when �Britain� (ie. England � so it�s tough if you aren�t English) was �Great� and half the globe was pink. (though madein will probably take this to mean �they was poof-tahs�).
'Half the globe' only appeared to be under the British Empire because of the flattering projection.
Here is an interesting site which discusses the various projections:
http://www.diversophy.com/petersmap.htm
i believe in ye olden days before mercator maps became popular, seafarers would use maps which accurately portrayed the distance between bodies but the shape of the bodies would be distorded. as a result, sailing from uk to india for example may take as long as you planned but you may land in a different place than you planned, or if your destination was a small island you may miss it altogether. thank you very much gps
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