TRANSACTIONS OF THE SECTIONS. 7 



On the Partition of the Cube, and some of the Combinations of its parts. By 

 Charles M. Willich, late Actmrij and Secretanj to the University Life 

 Assui'ance Society. 



A cube maj^ be divided iuto equal aud uuiforin bodies in various ways. 

 1st, B}' lines from the centre to the eight angles of the cube, which will give six 

 four-sided pyramids (B). 



2nd. By lines from one of the upper angles of the cube drawn diagonally to the 

 opposite angles, dividing the cube into three equal and uniform solids. Each of these 

 solids being^halved, forms a left- and right-handed solid. These six bodies pro- 

 duced though equal in mass, differ so far in shape, as three may be termed left- 

 handed and three right-handed, in the same way as the hands of the human body. 

 .3rd. By lines drawn from the centre to four angles of the cube, and continued on 

 each facej will produce four equal and similar bodies(G), each composed of two three- 

 sided pyramids united at their base— the one having the same angle as the trihedi-al 

 roof of 'the Bee's cell, viz. 109° 28' 16", the other 9(P. These bodies rearranged 

 produce the half of a dodecahedron with rhomboidal faces. 



4th. Another division of the cube may be made producing the tetrahedron and 

 octahedron, viz., by diagonal lines from two of the upper angles of the cube, con- 

 tinued on the other faces, will cut off' four three-sided pyi-amids, leaving m the 

 centre a tetrahedron. The four three-sided pyramids cut off" may be so aiTanged as 

 to produce the half of the true octahedron. 



The four-sided p;sTamid obtained by the first mode of division being cut into 

 two portions by a diagonal line will produce a body which I have assumed as a 

 unit (A) for the construction of many geometrical and ciystalline bodies. The 

 models laid before the Association show some of the forms produced. The rhom- 

 boidal cube (J), and the rhomboidal dodecahedi'on (I.) with pyramidal faces (con- 

 tainino- in mass one-half of the cube from which it is derived), may be considered 

 interesting ; but the various ciystalline figures which may be formed by a combi- 

 nation of my unit (A) I cannot even estimate— though probably all geometrical 

 solids and even many, if not all, crystalline bodies may be included, if we use sec- 

 tions of bodies produced by a partition of the cube. 



It may be observed that the pyramid (B), or one-sixth of the cube, which con- 

 tains two units, may itself be divided juto four bodies by sections parallel to the sides, 

 each of which is one-third of a cube containing one-eighth of the mass of the cube 

 from which it was derived ; so that, in fact, we may go on dividing and reproducing 

 bodies of a similar shape, and still retaining the same angles as in the portion from 

 the original cube, flow far this subdivision may be earned in nature, or how 

 niuch fiirther than our powers of vision will reach, I will not venture an opinion. 

 We can imagine that the commencing atoms may be infinitely small when we 

 remember the wonders revealed by the microscope. 



I entertain a sanguine hope that, should the attention of philosophers be drawn 

 to this subject, the further development may perhaps be the means of throwing 

 some unexpected light as to the shape of an atom. I incline, however, to think 

 that atoms may differ in shape in the three kingdoms of nature— mineral, vegetable, 



and animal. , i ^ j j. j. 



As to the practical use in education, I am of opinion that the study ot geometiy 

 would be simplified by the use of models showing the relative value as to the 

 solidity of geometrical bodies, and thus convey knowledge to the youthful mind 

 by means of the eye more readily than by any description, as when convinced by 

 the sight the mind" would understand with greater facility. 



Lid of models which accompanied the above jnqyer. 



A. The wiit or ^l part of the cube having a side of 1 inch. , . , . 



B. 1st union of two units, forming a low foiu:-sided pyramid of which six make 



up a cube. , . , , .,.,.,. , 



C. 2nd union of two units, forming a high four-sided pyramid of which six also 



I). 3rd union of two units, forming right-handed solid, being J of cube. 

 E. ith union of two units, forming left-handed solid, being I of cube. 



