( 63 ) 



the semiregular polvtopes derived bv Mrs. A. Boole Stott ') from 

 the regidar tivecell b}' means of the geometrical operations of ex- 

 pansion and contraction can be constructed. 



But it will be useful to develop first some general laws. 



2. We consider the projection of the tivecell S{5) more closely 

 which leads us to the following remarks : 



a. In pentagonal projection the ten edges of *S'/5) present themselves 

 in five directions only, any diagonal of the [)entagon being parallel 

 to one of its sides. 



/;. Though all the edges of >S\5) have the same length we 11 nd 

 in projection two different lengths, with the proportion s : d, where 

 s and d indicate side and diagonal of the pentagon. 



If we wish to take into consideration the length of the edge itself 

 we can use a very well known rectangular triangle of plane geometry 

 saying that when r is the radius of any circle and s^^ and 0^5 denote 

 the sides of the regular decagon and pentagon described in it, s^ is 

 the length of the edge itself, s^^ and r being the projections. 



It goes without saying that the diffei-ence in length of i)rojection 

 is a consequence of difference in inclination ; five edges of >S'(5' make 

 with the plane of projection an angle (p for which tgip = h (1^5 — 1), 

 the five others the complementary angle with h (l/5-(-l) as tangent. 



c. In projection the ten equilateral faces of >S(5) split up into two 

 quintuples of isosceles triangles, one group (2.^, d) with an obtuse, 

 one group (.5, 2d) with an acute \ ertex angle. 



(/. In projection the five limiting tetrahedva present the same Irape- 

 zoidal form (fig. 2). We show tiiat this is of great importance witii 

 respect to our aim l>y saying that a rotation of the projection (2345) 

 of the tetrahedron in the sense of the hands of a watch around the 

 centre C indicated in fig. 1" to an amount of one, two, three, four 

 times 72° bring this projection succesively into coincidence with the 

 projections (345J), (4512), (5123), (1234) of the other four limiting 

 tetrahedra. 



In order to give some relief to the single tetrahedron of fig. 2 we 

 have dotted one of the two diagonals of tlie trapezoid ; by doing 



senting therefore llie lines at infinity of the piiuies ©(A'oA'g), O(A'jA'i) to be u.sed; 

 unless any plane thiongh m (or m') makes with OiXiX^) two equal angles and 

 the lines J, I', m, m' form a hyperboloidical quadruple, in which case the planes 

 OCA'gA';^) , 0{X^X\) may he selected from a singly infinite system. 



^) "tieomelrical deduction of semiregular from regular polytopes and space fillings" 

 (this Academy, Verhandelingen, vol. 0, n". 1). In the following we suppose the 

 results obtained there to be known. 



