# The Design and Simulation of an Astronomical Clock
## 6 Approximations of Astronomical Data  
It is important to emphasize that the astronomical clock whose design is presented in this work determines and displays geocentrically not the apparent or true position but the mean position of the Sun, the Moon, the lunar nodes the zodiac circle and their motion during the year. All irregularities of the Sun, the Moon and the lunar node celestial motions are neglected for this mechanical model. In other words, it is presumed that the observed movements of the aforementioned objects across the celestial sphere are circular and uniform. These uniform circular motions are realized by a set of clock gear trains with meticulously calculated gear ratios. Since the number of gear teeth is an integer, the gear ratios must be approximated by rational numbers. The proper gear ratios of the mechanism gear trains are determined by the method of continued fractions since this calculation technique was mathematically proven [11] to produce the best possible rational approximation of a real number [12]. (Given a number x, its best rational approximations of the first kind are those fractions p/q such that x is closer to p/q than to any fraction with a smaller denominator. Given a number x, its best rational approximations of the second kind are those fractions p/q such that for any fraction p\({}^{\prime}\)/q\({}^{\prime}\) with q\({}^{\prime}\) < p, we have \(\mid\)q\({}^{\prime}\)x - p\({}^{\prime}\)\(\mid\) > \(\mid\)q\({}_{\rm X}\) - p\({}^{\prime}\)\(\mid\).) However, for this objective, the method of continued fractions is not sufficient. Since the diameter of the gear cannot be arbitrarily large nor can the gear modulus be arbitrarily small, the number of gear teeth is always limited by the clock dimensions and by the method of gear production. Thus, it is necessary to introduce an additional criterion which will determine the practically acceptable rational approximation obtained by the method of continued fractions. Basically, this precaution can be formulated as the limitation of the number of gear teeth and depends on gear production technology, gear modulus and dimensions of the clock. In this case, we are planning to produce gears by ABS filament 3D printing and since the clock face diameter is approximately 400 mm, the gear diameter should not be greater than 150 mm and the modulus not less than 1 mm. Briefly, the previous considerations can be summarized as follows:  
* The proper gear ratio of the particular gear train must be determined by the method of continued fractions. This method generates the set of convergents or semi-convergents of the continued fraction representation of the real ratio [12; 13]. * The error of the chosen rational approximation should be sufficiently small. * The rational approximation should be selected in such a way that prime factors of their numerators and denominators are less than 150. The operational scheme of the astronomical clock mechanism, which consists of a set of gear trains, is given in Figure 8. In this figure, C is the input hour shaft of the ordinary clock, \(S\) is the Sun, \(M\) is the Moon, \(Z\) is the zodiac, \(Ph\) is the lunar phases and \(N\) represents lunar nodes. In the same figure, \(R_{CS}=2:1\) is the ratio between angular velocities of the clock hour shaft and the astronomical clock's Sun pointer. \(R_{SM}\) is the ratio between angular velocities of the Sun and the Moon pointers, \(R_{MZ}\) is the Moon and the zodiac, \(R_{SPH}\) is the Sun and the lunar phases and \(R_{SN}\) is the ratio between angular velocities of the Sun and the lunar node pointers on the same astronomical dial. These ratios are approximated by the set of continued fractions convergents (C) and semi-convergents (SC) and the prime factors of their numerators and denominators are determined. To demonstrate these calculations more concisely, not all convergents and semi-convergents are presented, but only the range from the first to the one which is accepted as a sufficiently correct approximation.