114 



can be discriminated as a plane, endless pattern', the original figure, 

 by the repetition of which the pattern is produced, may be called 

 the repeat or the motif of it. The repeat is the essential unit of the 

 endless pattern, and the special nature of the latter is determined as 

 well by the shape of this unit, as by the specific mode of its repetition. 

 From what we have seen in the preceding chapters, we can conclude 

 that "symmetrical" arrangements of a repeat have an essentially 

 "periodical" character. 



Moreover, if the arrangement be such that every repeat of it be 

 surrounded by all others in the same way as every other motif is 

 by the remaining, then we say that the pattern is homogeneous. 

 The homogeneity of the pattern involves, that its aspect will always 

 be the same to an observer, if placed at any point whatsoever of 

 the infinitely extended system. 



In jig. 102 a portion of a pattern is reproduced which shows 

 clearly what is meant by this homogeneity ; of course, the pattern 

 must be imagined to be infinitely continued in all directions of the 

 plane. If A j be a given point of the repeat and A z the corresponding 

 point in the next figure, the line joining A l and A 2 will be parallel 

 and equal to a number of other lines joining two corresponding 

 points B l and # 2 , Q and C 2 , in both pattern-units considered. The 

 points A l and A 2 , B and B 2 , C 1 and C 2 , etc., are said to be homologous 

 points of the pattern ; round such homologous points the distribution 

 of all other points in every pattern-unit is the same as in all other 

 units of the pattern. The lines A^A^ B^B 2 , C-^C^, are evidently 

 equal and parallel to the translation FF' which brings the original 

 motif F into the position of the next parallel figure F f . However, 

 it is easily seen that there are a number of other translations by 

 which the original repeat can be made to coincide with the surrounding 

 figures F" , F'", etc., if it be shifted along various directions of the 

 plane, such as A^A^, B^B^, C^ etc. If we do not consider the 

 special shape of the repeat F and simply take one of its points P, - 

 for instance its geometrical centre, we can describe the situations 

 of all corresponding figures F' , F" , F'" by fixing only the final 

 situations of the points P' ', P" , P'" , which are the homologues of P, 

 i.e. in the case considered: the geometrical centres of the figures F' t 

 F", F'", etc. All these homologous points form together a plane 

 system of homogeneously and regularly distributed points which, 

 on closer examination, appear to be situated like the knots of a net- 

 work with parallelogrammatic, rectangular, or quadratic meshes. 



