DERIVED KROM THE MGÜLAÈ POLYTOPES. 63 



with exception of one of the permutable units, we get successively 

 the three quadruples of vertices 



_ [1 + k% — 1 + |/2] 0, —, (1, — 1) i []/%, yz] —, (l, - 1) (j/S, — 1/2) 



lying at minimum distance \/6 from 0'. These 12 points form 

 the vertices of a prism P with octahedral base ; each of the three 

 quadruples just found lies in a plane passing through the axis 

 of the prism and consists of a pair of opposite vertices of each of 

 the two limiting octahedra. The equations of the three planes are 



x s = 0, x k = — , a?j -j- #2 = 0, #3 = #4 — , x + x 2 = 0, #3 + x k = . 



So the axis of the prism is represented by x 3 = , x k = , 



#1 + ^2 = 0. 



Moreover it is easily verified that the three quadrangles are 

 rectangles with sides 2V/2 and 4. As we can unite the second 

 and third symbols the P can be represented by the two symbols 

 |[1 + V/2,— 1 + V/2] 0,0 and (1 , — 1) [V/2 , V<2]. 



Vertex gap poly tope. Finally the three net symbols are, in the 

 simplest form, 



. 4 



[4,2,0,0], (6 -f- 2 \/2) ^ , o 2 , a 3 , % , 2^ odd, 



£[3,3,1,1], (3+ V/2) 2^+1, 2« 2 +l, 2a 3 +l, 2a 4 +l, E^ even, 



i 



. 4 



— ■£■[3,3,1,1], (3+ V/2) 2^+1, 2^+1,203+1,204+1,2^ odd. 



4 



By taking for a u a 2 , a 3 , a k in the first symbol [1,0,0,0], in the 

 second either 0,0,0,0 or ( — 1, — 1,0,0) or — 1, — 1, — 1, — 1, 

 in the third either ( — 1,0,0,0) or ( — 1, — 1, — 1,0), and by 

 assigning to the permutable digits the sign which decreases the 

 absolute value of the coordinate, we find the three sets of 48 

 points represented by the symbols 



[2 + )/2, 2, 0, 0], _ [2 + ]/<_, 2 + j/2, J/2, V^\ — _ [2 + 1/2, 2 + */2, 1/2, \/%\ 

 which can be reduced to 



[2 + 2V/2, 2, 0, 0], [2 + V/2, 2 + V/2, V/2, V/2]. 



These 144 points prove to be the vertices of the polytope e s C_ with 

 the characteristic numbers (144, 576, 672, 240). 



^l^^^^We)- Extension number 6 + 3V/2, three net symbols 



