Geometrical deduction of semiregular from regular 

 polytopes and space fillings 



BY 



Mrs. A. BOOLE STOTT. 



Introduction. 



1. The object of this memoir is to give a method by which 

 bodies having a certain kind of semiregularity may be derived from 

 regular bodies in an Euclidean space of any number of dimensions; 

 and space fillings of the former from space fillings of the latter. 



These space fillings or nets for threedimensional space have been 

 given in a paper entitled „Sulle reti di poliedri regolari e semire- 

 golari e sulle corrispondenti reti correlative" by Mr. A. Andreini 1 ), 

 who deduced them by means of the angles of the different poly- 

 hedra. Photographs prepared for the stereoscope, taken from that 

 paper, representing the various semiregular space fillings were sent 

 to me by Prof. Sohoute to whom I desire to record here my thanks 

 for the generous help he has given me during the whole course 

 of this investigation. These photographs suggested a method by 

 which at once the semiregular bodies and the manner in which 

 they combine to fill fourdimensional space could be derived from 

 regular polytopes and nets in that space. It will be seen that this 

 method can be applied to spaces of any other number of dimensions. 



The semiregularity considered here is that in which there is 

 one kind of vertex and one length of edge 2 ), and the symbols used 



l ) Memorie délia Società italiana della Scienze (detta dei XL), serie 3a, tomo XIV. 

 *) So the greater part of the forms called semiregular here will have a degree of 

 regularity less than \ in the scale of Mr. E. L. Elte. 



A 1* 



