The idea of symmetry resulted originally from the study of 

 geometrical forms and the observation of natural objects. Now, 

 after its mathematical formulation has been elaborated, and all 

 the conclusions drawn from it have been systematically gathered 

 into what is commonly named "the general doctrine of sym- 

 metry", - - the said principle finds application in those fields of 

 natural science also, such as e.g. physics and chemistry, where 

 geometrical properties are taken into account commonly in cases 

 in which the dimensions considered will most probably for ever 

 remain beyond the scope of direct observation. 



2. By the use of the word "symmetry" in the study of forms 



and figures it is intended to 

 draw attention to some geome- 

 trical regularity, to a certain 

 process of repetition l ) which 

 manifests itself in the external 

 habit of the figure considered. 

 It is well known, that there is 

 implied a powerful aesthetic fac- 

 tor 2 ) in the mere repetition of 

 a visual impression, and it is 

 in this, that symmetry became 

 an important aesthetic principle 

 in decorative art and architec- 

 ture. 



In Fig. i. a perfectly arbitrary complex of large and small black 

 dots is represented. The figure formed by them does not make any 

 particular impression on the observer. If, however, this meaningless 

 figure is repeated several times by rotating it round an axis A, 

 perpendicular to the plane of the figure, each time through an 



*) The Greek word a\tp.p.irpise signifies: right proportion, evenness, measura- 

 bleness, dimension; cbfj.psTpos is: uniform, suitable, proportional, measured -off ; and 

 likewise the adverb svpfj.iTfw? has the meaning: keeping the due medium be- 

 tween. Democritos e. g. speaks of: vpptTpti) icu for: the equableness of life; etc, 



2 ) The significance of symmetry as an aesthetic principle was already brought 

 into the fore by the ancients. Thus e. g. Plotinos in his famous book on Aes- 

 thetics, in the first Ennead, liber 6, cap. 1, says: "Now almost by all persons 

 it is maintained, that it is the symmetry of the different parts with respect 

 to each other, and the beautiful colour, which produce beauty for visual 

 observation ; and for those as well for the common intellect beauty is identical 

 with symmetry and being shaped after fixed proportions." 



