428 On the Angle of Dock Gates and the Bee^s Cell. 



The partition of another regular solid body, the tetrahedron, 

 effected by cutting off four smaller tetrahedrons, of half the 

 length of the base, will leave the platonic or regular octahedron, 

 whose eight faces are equilateral triangles — and these faces we 

 find incline to each other at an angle of 109^ 38' 16", thus arri- 

 ving at the same angle, although we make use of two very differ- 

 ent simple solids — the cube, and tetrahedron. There are other 

 curious interchanges, as in the partition of the dodecahedron, 

 the trihedral summit forms one-fourth part of a tetrahedron. 



The laws of nature are always simple; w'e might therefore be 

 led to expect that the same angle which is best for the dock 

 gates, should be precisely the same as that of the trihedral roof 

 of the bee's cell. The mode of arriving at the angle of the bee's 

 cell I have shown in a letter which is inserted in ' The Literary 

 Gazette' of the 9th of July, and which I beg to annex at foot*, 



* " On the Partition of the Cube, and the Construction of the Bee's Cell. 



(To the Editor of the Literary Gazette.) 



" Sir, — Many years ago you inserted in your journal a paper of mine on 

 the subject of an approximate geometrical quadrature of the circle. That 

 ajiproximation I afterwards succeeded in obtaining to within the ij^^Vyth 

 part of the side of the square sought; and the Koyal Society, on the 10th 

 of May, 1855, so far relaxed the rule adopted v.ith reference to questions 

 of this" description as to admit the paper to be read ; and a short account 

 was inserted in their ' Proceedings.' 



" I am now anxious to announce that I have succeeded in dividing the 

 cube into several geometrical solids, with which many definite and regular 

 geometrical bodies may be constructed. 



" Perhaps one of the most curious is that of the bee's cell, which is iu 

 fact an elongated dodecahedron ; and consequently the angles of the tri- 

 hedral roof and base, respecting which so many learned investigations have 

 been made, can be no other than those of the true geometrical solid. 



" Without the aid of diagrams it is not easy to make the forms of solids 

 clear to the mind in a popular way. 



"A cube may be divided into six equal and uniform bodies in two differ- 

 ent ways : — 



" 1 St. By lines from the centre to the eight angles of the cube, which will 

 give six 4-sided pyramids. 



"2ndly. By lines from one of the upper angles of the cube, drawn dia- 

 gonally to the three opposite angles, dividing the cube into three equal and 

 uniform solids. Each of these solids being halved, forms a left- and a 

 rio-ht-handed sohd. These six solids, though equal in solidity, difl'er so far 

 in shape, as three are left-handed and three right-handed, in the same way 

 as the hands of the human body. 



" Each of the six bodies obtained by the second mode of partition may 

 be divided into two of equal solidity and of similar shape. Two of these 

 bodies, each being one-twelfth of the cube, may be so united as to produce 

 the pyramid obtained by the first mude of ])artition. Six of these bodies, 

 each beiug one-twelfth part of a cube, may be so arranged as to form the 

 oblique rhomboid. 



" For the present investigation we will not proceed further than the 

 solid thus obtained, being the one-twelfth jjart of the cube. By this body. 



