Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are F p = {0, 1,..., p - 1} produces a 1-Lipschitz map f A from the space Z p of p -adic integers to Z p . The map f A can naturally be plotted in a unit real square I 2 ⊂ R 2 : To an m -letter non-empty word v = γ m -1 γ m -2 ... γ0 there corresponds a number 0. v ∈ R with base- p expansion 0.γ m -1 γ m -2 ... γ0; so to every m -letter input word w = α m -1 α m -2 ··· α0 of A and to the respective m -letter output word a( w ) = β m -1 β m -2 ··· β0 of A there corresponds a point (0. w ; 0.a( w )) ∈ R 2 . Denote P (A) a closure of the point set (0. w ; 0.a( w )) where w ranges over all non-empty words.We prove that once some points of P (A) constitute a C 2 -smooth curve in R 2 , the curve is a segment of a straight line with a rational slope. Moreover, when identifying P (A) with a subset of a 2-dimensional torus T 2 ∈ R 3 , the smooth curves from P (A) constitute a collection of torus windings which can be ascribed to complex-valued functions ψ ( x, t ) = e i ( Ax-2πBt ) ( x, t ∈ R), i.e., to matter waves. As automata are causal discrete systems, the main result may serve a mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter.