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1. Introduction


There is an infinite number of ways in using numbers for describing a visual experience that human eye can have. A group of sophisticated methods which is based on continuous electromagnetic waves that are digitalized using of different criteria and subtle mathematical methods has been developed. The reversal procedure, the conversion of numerical data array into a picture is then usually well defined. But if one wants to define both simple and effective criteria in order to turn a number into a picture, one can run up against a difficult task. In this paper, two methods of visualizing real numbers are suggested. What is common for both approaches is that they are based on a regular continued fraction expansion of  xÎ R ,  i.e.

                                                                                        1

x   =   a0 ¾¾¾¾¾¾¾¾¾
                        1
                 a1 + ¾¾¾¾¾¾¾¾
                                     1
                          a2 + ¾¾¾¾¾

                                    a3 +   . . .

where  aiÎR ,  with the usual shorter notation  x  =  [a0 a1, a2 a3 ,  ... ] .  If x is a rational number, then there exists  k ³ 1 such that ak+i  = 0,  i = 1, 2, 3,...  i.e. the continued fraction expansion is finite. Irrational numbers have infinite expansions with the peculiarity that quadratic irrationals (roots of second degree algebraic equation with real coefficients) have periodic expansions (Euler-Lagrange theorem, see for example [2]), like Ö2 = [1, 2, 2, 2, 2, ...] or shorter, Ö2 = [1, á 2 ñ ] with period 1. The expansion  Ö7 = [2,  á 1, 1, 1, 4 ñ] has period 4, while, for example, period of Ö2001 is 94.

The sequence  a0 a1, a2, a3, ... from the continued fraction expansion [a0 a1a2 a3 , ... ]  can then be used to obtain a graphical diagram that will represent our number.

 In Section 2, the sequence {ai} is used to create a kind of  graphics that resembles  the "random walk"  trajectory. The terms of the sequence are used  to code  specific directions of movement in the plane. The trajectory obtained can be  periodic or aperiodic. Periodic trajectories yield either closed  geometric shapes (squares, triangles) or  different kinds of friezes. Aperiodic trajectories are usually wild and behave randomly.  The inverse problem of getting the number that corresponds to a particular graph is also treated. The shortcoming of the method is caused by the fact that  the mapping number ® graph is not a bijective mapping. So, infinitely many numbers can have the same "portrait".

The second method, described in Section 3, produces a symmetric graph  using the same sequence {ai} of the continued fraction expansion coefficients. The graphical output looks aesthetically more pleasant and heavily depends on the number of  terms of {ai} used.

Both methods are tested by suitable programs made with the program system Mathematica (see Appendix), and the diagrams obtained are used to present examples in this paper.

Some of the patterns obtained in our experiments may be found aesthetically pleasant. So, the described methods  can be used both for investigating the relationship between quantities and their visual counterparts and for playing  building beautiful motifs. For some different trials in producing interesting shapes see [5].


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