( 385 ) 



the heighl is respectively 2| 2. .*> ! 2. 4j 2 and the fractions above 



2 3 4 .1 2 



(o the lelt are . , , tin* the fractions below to the right 0. 



8 8 8 4 4 



and for the sections of C\q "' an edge, a cube covered at two 

 opposite faces by square pyramids, a square double-pyramid. 



I). Inclined cells. In the live cases corresponding to the fractions 



1 2 



i (2) 



0, — , -,•—,■ the section of the circumscribed C% is a rectan- 



8 8 8 8 



gnlar parallelopipedon, the base of which is a square with side two, 



1 3 



with a height 0, \ 2, y 2, - \ 2, 2 \ 2 successively. I he sections 



Ci (J 



of the inscribed 6i6 " represented in the first, the third an<l the tit'lli 

 of the live figures are equal to those of the preceding column and 

 in the second and the fourth intermediate forms between these ; in 

 general the section can be characterized as a rectangular parallelo- 

 pipedon with a square as base and upperplane, covered at these two 

 faces by square pyramids, the faces of which have a determined 

 inclination. 



Sections normal to OK s . 



a. Erect cells. Here too, a difference presents itself as to the 

 fractions, according as the diameter normal to the intersecting space 

 is considered either as an axis OF lt or as an axis OK s . If the point 



of intersection of the intersecting space with the axis OK % of C& 

 describes that axis completely, the base of the prismatic section, the 

 height of which remains four, transforms itself' in the same manner 

 as the section of a cube with edge four by a plane normal to a 

 diagonal, and now at the moment that this base is increased to a 



triangle with side 4^2 a face of the inscribed Gi6 " appears in the 



4 5 6 



intersecting space. So, to the fractions — , — , above to the left, 



i ~ i — i _ 



1 2 

 correspond the fractions 0, , below to the right ; so we find 



4 4 



in the first diagram a triangle in a triangular prism, in the third 

 a regular hexagonal double-pyramid in a regular hexagonal prism, 

 in the second a form 12, 24, 14) bounded by two equilateral trian- 

 gles, six isosceles triangles, si.\ isosceles trapezia in a semiregular 

 hexagonal prism regular as to the angles. 



