432 Mr. G. Darwin on Maps of the World. 



icosahedron are made to hinge together, so that the whole can 

 be packed flat in the form of a half-hexagon. Such a globe 

 was exhibited at the British-Association Meeting at Bradford. 

 When mounted, the icosahedron circumscribed a sphere of 

 25 inches diameter. This form of globe might doubtless be 

 constructed much cheaper than a truly spherical one, because 

 the framework would be ordinary carpentry, and the twenty map- 

 sheets might be printed flat like ordinary maps. 



In 1872 I showed the above described maps and globe to 

 General Strachey ; and he suggested that by cutting down the 

 iscosahedron in some way, a still more satisfactory projection 

 might be attained. It then occurred to us that by truncating 

 the solid angles of the icosahedron, a solid figure of 32 faces 

 would be obtained, viz. 20 hexagons and 12 pentagons. 



If the truncation be carried on by slices until the truncating 

 planes touch the sphere enclosed in the icosahedron, these 

 hexagons are not regular, but have two sets of three sides equal 

 to one another ; a long side is always opposite to a short side. 

 If unity is the radius of the sphere, the long sides and short 

 sides are respectively "4913 and *3401. The pentagons are 

 always regular ; and at this particular degree of truncation the 

 side of the pentagon is *4913, and a pentagon is therefore 

 always contiguous to the long side of a hexagon ; whilst hexagons 

 are always contiguous along their short sides*. 



This projection was utilized by having a sort of umbrella-like 

 stand, with a pentagonal face in the middle, surrounded by 

 five hexagons; or else with a hexagon in the middle, surrounded 

 by three pentagons and by three hexagons. The maps were 

 drawn on 32 separate sheets ; and the sheets required to represent 

 any part of the world were mounted on the umbrella. 



By these means about one fifth of the globe is shown at 

 once; and thus the equivalent of a very large globe might be 

 used in a room of ordinary size. The sheets may also be 

 conveniently kept, since they are all flat, and will lie one on 

 another. 



The figures 1, 2, 3, 4 (Plate I.) show the forms of the various 

 map-sheets, together with the figures required for laying out the 

 meridians and parallels of latitude. Besides those kinds shown 

 in the figures, there are two pentagons which close in the two 

 poles ; but it is so easy to lay them out, that it does not seem 



* This leads me to observe that if the angles of any one of the regular 

 solids be truncated in this way, another one is ultimately produced. The 

 20-hedron and 12-hedron, the 8-hedron and cube, and the tetrahedron and 

 tetrahedron are thus correlated. This property is of course due to the fact 

 that the polar reciprocal of any regular solid is itself a regular solid. It is 

 curious to observe the transitional forms as the slices are cut off the angles. 



