## 3 Fuzzy System of EmPatGen  
Human reasoning and some other phenomena cannot be accurately described by two-valued logic. This idea led [PERSON] to the creation of fuzzy logic [34]. Fuzzy logic is an extension of the classical two-valued logic, hence, in fuzzy logic, the conditions can be not only true or false, but it is also possible to provide values between them. In the same way, the fuzzy set theory is an extension of the classical set theory [35; 36]. In the fuzzy set theory, an element can belong to a fuzzy set by a membership value. Figure 9 depicts this situation by providing a graphical representation of a standard set (crisp set) and a fuzzy set. Fuzzy logic is able to handle numerical data and linguistic knowledge at the same time [37; 38]. Thus, fuzzy logic was used in order to connect the attributes of geometry and emotions.  
Figure 10 shows the generic structure of a fuzzy system. Basically, it has input data, a fuzzy rule base, an inference mechanism, and output data. The fuzzification and the defuzzification actions mean the transformation of crisp to fuzzy and fuzzy to crisp data [39].  
The fuzzy rule base contains fuzzy rules. The general form of a fuzzy rule with one input and one output dimension is as follows:  
\(R\)**: If \(x\) is \(A\), then \(y\) is \(B\),  
where \(x\in X\) is the input and \(y\in Y\) is the output variable, \(X\) is the universe of discourse for the input and \(Y\) is the universe of discourse for the output variable, and \(A\) and \(B\) are linguistic labels that are expressed by fuzzy sets. Fuzzy set \(A\) is the antecedent while fuzzy set \(B\) is the consequent of rule \(R\). The general form of a fuzzy rule with multiple inputs and one output dimension can be written in the following, so called Mamdani-form [40].  
\(R\)**: If \(x_{1}\) is \(A_{1}\) and \(\ldots\) and \(x_{n}\) is \(A_{n}\) then \(y\) is \(B\),  
where \(x=(x_{1},\ldots,x_{n})\) is the input vector, \(x_{j}\in X_{j},\ X=X_{1}\times\ldots\times X_{n}\) is the \(n\)-dimensional universe, \(A=(A_{1},\ldots,A_{n})\) is the antecedent vector, \(y\in Y\) is the output variable, \(Y\) is the universe for the output, and \(B\) is the consequent set.  
The fuzzy system works as follows. The inference mechanism compares the given observation with the antecedent parts of the fuzzy rules in the fuzzy rule base. Based on these comparisons, it will be known what the degree of matching of each rule will be for the observation. These degrees are used by the inference mechanism when taking into account the consequent parts of the fuzzy rules. In the  
Figure 10: Generic structure of a fuzzy system.  
Figure 9: Graphical representation of a conventional set and a fuzzy set.  
case of Mamdani inference [40], each rule's consequent fuzzy set is cut by using the rule's degree of matching providing a sub-conclusion for the rule. The conclusion of the given observation for the whole rule base can be computed by taking the union of the previously calculated sub-conclusions. This conclusion is a fuzzy set; however, in most of the cases, the expected conclusion is not a fuzzy set, but a crisp value. Hence, the crisp value needs to be determined, which describes the conclusion fuzzy set in the best way. This procedure is called defuzzification.  
The application of the method in the present task is appropriate, as other researchers have also successfully applied fuzzy logic to link mathematics and art [41]. Nevertheless, while it was used there to shape geometries, in this system, the pattern is formed by emotion-based inputs.  
For our task, the following structure is used. The three inputs of the fuzzy system are the dynamics, style, orientation; the 11 outputs cover the four basic geometry and seven pattern space variables. The fuzzy rule base contains eight rules created to cover all ranges of the inputs. The following subsections explain the details of this structure.