Dr Burnside, Convex Solids in Higher Space 441 



polyhedra each of which is bounded by all the six 3-spreads. Two 



of these have eight vertices, viz. 



12 23 25 26 14 34 45 46 

 12 14 15 16 23 34 35 36 



It is obvious that two of the 3-dimensional faces of these are 

 tetrahedra and the other four polyhedra with five faces and six 

 vertices. 



The remaining one has nine vertices, viz. 



12 14 16 23 25 34 36 45 56 

 Its 3-dimensional faces are all polyhedra with six vertices and five 

 faces. 



A similar examination of the next scheme shows that seven 

 4-spreads in 5-dimensional space bound just four distinct convex 

 polyhedra. The vertices are 



23 24 37 47 12 17 25 26 57 67 



23 24 37 47 13 14 35 45 36 46 



13, 14 16 23 24 26 37 47 67 15 25 57 



13 14 16 23 24 26 37 47 67 35 45 56 



