712 PROFESSOR CRUM BROWN ON THE 



cube, and the fourth face the equilateral triangle whose side is the face-diagonal of the 

 cube. The characteristic pair of opposite edges of this tetrahedron consists of the body- 

 diagonal and the face-diagonal of the cube, the other two pairs of opposite edges consist 

 each of a face-diagonal and an edge of the cube. We may call the corner which differs 

 in sign from the others, the singular corner. At the singular corner a body-diagonal 

 and two edp-es of the cube meet. 



4 and 5. A pair of enantiomorph tetrahedra, AABC and AACB ; each having for 

 two faces the scalene triangles ; and for the other two, half-faces of the cube. Their 

 characteristic pair of opposite edges consists of a body-diagonal and an edge of the cube, 

 the other two pairs are two face-diagonals and two edges of the cube respectively. 



These five forms are shown in figs. 1-5. Fig. 6 shows the 4th and 5th forms placed 

 so as to indicate their enantiomorphism. 



As it will be convenient to have a symbol for each of the five tetrahedra independent 

 of its position in the cube, the following will be used : — For ABCD, &c, ; for ABCD, 

 &c, A ; for AABC, &c, I ; for AABC, &c, L ; for AACB, &c, I\* 



That these five are all the tetrahedra that can be cut out of a cube without making 

 new corners can be proved as follows. There are seventy ways in which the eight 

 corners can be taken four at a time. Of these, six correspond to the faces and six to 

 sections of the cube through opposite face-diagonals, so that there remain fifty-eight 

 corresponding to tetrahedra. Now can occur in two positions, ABCD and ABCD ; 

 A can occur in eight positions, because its singular corner, that in which three edges of 

 the cube meet (A in fig. 2), can be at any one of the eight corners of the cube ; I can 

 occur in twenty-four positions, six for each body-diagonal, thus, for the body-diagonal 

 AA, we have AABC, AACD, AADB, AACB, AADC, AADB ; L and T can occur in 

 twelve positions each, three for each body-diagonal, thus, for the body-diagonal AA, we 

 have AABC, AACD, AADB, and AACB, AADC, AABD. We have thus in all fifty- 

 eight positions of the five tetrahedra, as we have fifty-eight groups of four corners of the 

 cube corresponding to tetrahedra. 



The volume of Q is one-third of the volume of the cube, the volume of each of the 

 other four tetrahedra is one-sixth of the volume of the cube, each of them being a pyramid 

 with one-half of the face of the cube as base, and as vertical height the edge of the cube. 



Having ascertained what our bricks are, we have now to find out in how many ways 

 wc can build a cube with them. 



We shall first look at the complex, for there is only one, in which ® occurs. & has a 

 volume equal to one-third of the cube. But a little consideration shows that only one Q 

 can have a place in a cube. If we put two O's together in the most compact way, that 



* Tin: letters A, I, L, and r have been chosen because they contain 3, 1, 2 ami 2 straight lines respectively, as the 

 corresponding tetrahedra contain 3, 1, 2 and 2 half-faces of the cube respectively. 0, which contains no straight line, 

 nii^ht have been chosen for the regular tetrahedron, as it contains no part of the surface of the cube, but, as has been 

 sometimes used to symbolise the regular octahedron, n was selected, perhaps partly because this tetrahedron has twice 

 the volume of any of the others. 



