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Antikythera mechanism – A compound epicyclic gearing for Venus
Digital Applications in Archaeology and Cultural Heritage 12 (2019) e00089
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Digital Applications in Archaeology and Cultural Heritage journal homepage: www.elsevier.com/locate/daach
Antikythera mechanism – A compound epicyclic gearing for Venus ⁎
Ioannis S. Diolatzis , Gerasimos Pavlogeorgatos1 Department of Cultural Technology and Communication, University of the Aegean, University Hill, Building of Geography, off. 2.14, 81100 Mytilene, Lesvos, Greece
A R T I C L E I N F O
A BS T RAC T
Keywords: Anikythera mechanism Epicyclic theory Epicyclic gearing Gears Epitrochoid Geocentric Sun gear Satellite gear GeoGebra Venus apparent trajectory Retrograde motion
One undeniable characteristic of Antikythera mechanism is the complex gear arrangements that generously have been used by the constructor. He succeeded the best combination of integer numbers to reach the maximum accuracy, no matter the number of the gears he used for it. One representative example is the “moon train” consisted of eleven meshing gears which he used to simulate the non-uniform lunar motion. Someone could say that the constructor preferred the complexity for the sake of accuracy. Many Scholars, believe that the mechanism could be a kind of planetarium. In that case, there is a possibility of a more complex mechanical subsystem comprised of more imaginative and specialized gear arrangements. The latest decoded inscriptions on the mechanism's surfaces, are referred to planets (Inferiors and Superiors) and show records of astronomical events. The lack of physical evidence of this planetary construction prevents its conception, but on the other hand, challenges the researchers to discover the inimitable thought of its creator. Nevertheless, fitting this excess gearing in a limited space of a compact construction like Antikythera mechanism is really a puzzler. The planetary subsystem which will complete the mechanism should stand out for the maximum possible measurement accuracy and the best space management. This research proposes a feasible epicyclic gearing specialized for a compact planetary construction achieving the highest possible accuracy.
1. Introduction A deciphered text on the back face of the Antikythera mechanism (BCI2) reveals many details about the characteristics and the functional capabilities of this amazing gearwork machine. This decoded inscription (lines 15–25 Freeth and Jones, 2012) is referred to the five known planets of that era, with an ascending order of their distances from the Sun. Unfinished words inside fragmentary expressions, show planets' names starting from Mercury (EPMHΣ ) which is the closest planet to the Sun and ending to Saturn(KPONOΣ ) which is the farthest planet from the Sun. A particular impression causes the repeated words (ΣΦAIPION ) “little sphere”, one for every planet and (KΥ KΛOΣ ) “circle” in which the little sphere possibly belongs to. All the above references show that possibly the Antikythera mechanism could have on its top a constructive formulation like a kind of planetarium. This compartment is completely absent from the
⁎
remains of the mechanism found in a shipwreck in 1900. Speculated models of this planetary synthesis have already been made and they are all embed in the available area between the sun gear b13 and the top of the mechanism (see Fig. 1) The construction of this imaginary geocentric planetarium presupposes the knowledge of the “epicyclic theory” which was well formulated by Hipparchus and Apollonius. Using this theory the two famous Greek astronomers managed to explain the weird planetary motion on the sky. According to Safronov (2016), Antikythera mechanism is estimated to be constructed between 100 and 150 BCE, a long time after the invention of epicyclic theory (225 B.C Margolis, 1993) 2. The epicyclic theory as an essential tool The epicyclic theory was expressed by Hipparchus4 and Apollonius5 and later was implemented in Almagest by Ptolemy.6 According to this
Corresponding author. E-mail addresses: [email protected] (I.S. Diolatzis), [email protected] (G. Pavlogeorgatos). 1 Co-author. 2 Back Cover Inscription. 3 The biggest gear in Antikythera mechanism with 224 teeth which according to Freeth and Jones (2012) used to support the lost planetary subsystem. 4 Hipparchus of Nicaea (ca. 190 BCE - ca. 120 BCE) is known not only because contributed significantly to trigonometry but also as an astronomer with a unique talent. Unfortunately, all his astronomical memoirs have been lost so access to his theories could be accessed only through later Greek astronomers such as Ptolemy. 5 Apollonius was a Greek Geometer and astronomer who lived from about 262 B.C. to approximately 190 B.C. 6 Claudius Ptolemy 100–170 CE was a Greek mathematician, astronomer, geographer and astrologer. https://doi.org/10.1016/j.daach.2018.e00089 Received 19 August 2018; Accepted 30 December 2018 2212-0548/ © 2019 Elsevier Ltd. All rights reserved.
Digital Applications in Archaeology and Cultural Heritage 12 (2019) e00089
I.S. Diolatzis, G. Pavlogeorgatos
Fig. 1. The lost planetary subsystem – Source: (Freeth and Jones, 2012).
theory the apparent motion of a planet as it is observed from a fixed point of Earth is consisted of two independent motions
• • •
thinks that he is the center of the universe (geocentricism) and the Sun is orbiting him dragging the rotating planets with it. The apparent trajectory of the observed planet is called epitrochoid19 and can be expressed by a parametric equation.7
The first motion is a rotation of the planet around a center. This imaginary circle was called from the inventors EΠ IKΥ KΛOΣ or epicycle (red circle) as is widely known today (see Fig. 2). The second motion is a rotation of epicycle's center around the Earth forming an imaginary circle which was called Φ EPΩ N KΥ KΛOΣ or deferent (blue circle) as it is known today(see Fig. 2). The epicycle is consider to be fixed with the concentric y-circle (black circle) which is rolling on circumference of an immobile xcircle (black circle) concentric with the deferent (see Fig. 2).
2.1. Retrograde motion There are points on the planet's apparent orbit where the planet seems to stop and then starts again moving to the opposite direction. This backward motion is called retrograde motion which lasts for awhile until the planet stops again and starts moving to its initial direction. These points are called stations8. The station where the planet turns to go backward is called retrograde or first station while the station where turns to resume forward motion is called direct or second station. The question that could be raised here is why the planet appears to stop and then starts moving backward. The key to understand this weird motion is to isolate the velocity and the direction of observation. As long as the planet travels along the observation direction it appears to stop. This is because the vertical velocity which gives the sense of motion, is zero. These points are referred in the BCI3 as Stations (Σ THPIΓ MOI ). According to Jones (2017), p. 192, Apollonius showed that the ratio between the radii of x-circle and y-circle could determine the time between two identical positions (in respect to fixed stars) in the sky which is called synodic period. As it will be analyzed below, the choice of gears' teeth numbers in a simple epicyclic gearing (like the shown in Fig. 2) is based on the synodic period of the planet which represents. So the first priority of the constructor should be the building of a reliable astronomical database. According to Freeth (2012) the exclusive suppliers of this knowledge in that era were the Babylonians astronomers who recorded with remarkable precision the synodic phenomena of all planets. For example, according to Neugebauer (2012) they recorded for Venus, 720
So,the planet is participating in two independent circular motions. One is the rotation of the planet around epicycle's center, which could be represented by the Sun, and the other is the rotation of epicycle center around the Earth. This motion is conceived by an Earthy observer who
7 Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. 8 Station in retrograde motion is the point where the planet appears to stop moving in the sky.
Fig. 2. Epicycle – deferent model - Source: Author, 2018.
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Fig. 5. Venus static gear train - Source: (Evans et al., 2010).
•
• Fig. 3. Venus gears train (colorized grey) - Source: (Wright, 2013).
occurrences in 1151 years. In other words, they were aware about the synodic period of Venus which is:
TS =
1151 = 1.5986111111 years / synod 720
(1)
T.Freeth and A. Jones model. In this model T.Freeth and A. Jones used a simple form of epicycle-deferent model (see Section 2) to simulate the Inferior planets’ motion. Fig. 4 shows the Venus epicyclic gearing which consisted of the fixed gear x of 40 teeth at 64 8 b14 axis and the satellite gear y of 64 teeth obtaining a ratio of 40 = 5 (Freeth and Jones, 2012) which simulates the synodic period of Venus with an error of 0.086%. James Evans, Carman and Allan Thorndike model. A different proposal is coming from James Evans, Carman and Allan Thorndike. In their model they proposed five fixed sub-dials one for every planet which are evenly distributed around the circumference of the front dial. In each separate dial a pointer performs one rotation every synodic cycle. The gear trains they proposed simply calculate the synodic period of the planets without any planetary motion sense. For example they proposed for Venus gear train, consisted of six gears (included b1), the following 29 63 98 sequence: 224 · 25 · 20 = 1.598625. Evans et al. (2010) which simulates a synodic period of Venus with a negligible error of 8.68·10−4% (see Fig. 5)
3. Basic models of planetary motion All above proposed planetary apparatuses are accommodated between the Sun gear b14 and the top of the mechanism.
Scholars have already presented speculated planetary subsystems of Antikythera mechanism. The most representative of them are listed below.
•
3.1. Remarks on the proposed models
M. Wright's model. In this model M. Wright used compound gear trains trying to achieve the best accuracy. A characteristic example is the six gear train which he used for Venus. Specifically the Venus 51 70 34 train is 45 · 56 · 77 leading to ratio 462 . Wright (2013) which simulates
We express our skepticism about the third proposal because:
•
289
−3
the synodic period of Venus with a negligible error of 3·10 % (See Fig. 3)
•
The planetary motion is completely absent. So if we accept this configuration then its like neglecting all the graphical representation capabilities which the constructor implies in the inscriptions (retrograde stations or conjunctions etc) This stating image is incompatible with the epicycle - deferent model (Section 2) which is essential for the planetary motion demonstration.
The second proposal uses an epicyclic gearing couple and thus it reduces the number of used gears, at least for the Inferiors planets. This though, restricts the tooth number choices because of the limited available space. As a result, the approach to ideal value isn't desirable. Specifically in the case of Venus the discrepancy error is about two and a half days on every eight year cycle! (This will be analyzed below). We believe that the first proposal meets the requirements to be considered as the closest to the philosophy of the constructor. It does use a required gear number to eliminate the reading error indication. Also it implements the epicycle-deferent model as it should do. But as its seen in the Fig. 3, the Venus epicyclic gearing occupies a large portion of b14 and we think that the limited space over b14 must be used with parsimony. 4. Basic characteristics of spur gears As it is well known Antikythera mechanism was a gearwork device. So before of any study associated with whatever part of the mechanism,
Fig. 4. Venus epicyclic gearing on the 1 o'clock spoke - Source: (Freeth and Jones, 2012).
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(Black and Kohser, 2017). other The pitch circle is the imaginary circle which rolls without slip. In two meshing gears the pitch circles have one common point contact. In later sections pitch circles will substitute the corresponding gears as representative schemes of contact. 5. The geocentric orbit of Venus In this section will be verified that the ancient epicyclic theory and the theory of relative motion are the faces of the same coin. The motion of the planet could be studied:
• •
Fig. 6. 27 teeth spur gear with radius 27 mm and module 2 mm/teeth - Source: Author, 2018.
The trajectory of a moving object (i.e. a planet as moving in the space) is defined by its location as it varies spatially and temporally. The position of the planet with respect to the geocentric coordinate system is determined by its position vector with respect to the Earth which starts from the Earth and ends at the planet. According to Galilean Coordinate transformations, the position vector of a planet with respect to a relative coordinate system is given by the following equation
some elementary characteristics about the gears should be mentioned. According to Maitra (1994)
• •
pitch circle This circle intersects the teeth of the gear at the points where its teeth mesh with the teeth of another gear module m If N is the teeth number of the gear and D is the Pitch circle diameter in mm then the module m of the gear in metric system is defined by the equation
m=
D N
→ → ⎯→ ⎯ (3) r′ = r − R ⎯→ ⎯ → where r and R are the positions vectors of the planet and → Earth with respect to the Sun. The vector r′ performs a complex motion with respect to the real world, as it is rotating and translating simultaneously (see Fig. 7). If there is a way to freeze the start point of this vector then its end will move on the planet's apparent orbit. With this technique it is manageable to move from the heliocentric to geocentric coordinate system. The above mentioned procedure could be easily manipulated with the use of an interactive-dynamic mathematics software known as
(2)
(Maitra, 1994) (see Fig. 6) 4.1. Meshing two spur gears In order two gears to be meshed properly:
• •
in respect to a reference coordinate system fixed in a immobile star. This system is called absolute coordinate system. If our Sun takes the place of this stationary star then the coordinate system bound with it, is the heliocentric system. in respect to a moving coordinate system. This system is called relative coordinate system like the geocentric system which is bound to the Earth.
they must have the same module m (Spitas and Spitas, 2006) the pitch circles of the meshing gears must be tangential to each
Fig. 7. A snapshot just after a Venusian year has passed, showing Earth and Venus animated motion in heliocentric system (left) and the Venus in geocentric system (right) – Source: Author, 2018.
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GeoGebra.9 Below would be presented briefly this process which is applied in the case of Venus. 5.1. Heliocentric to geocentric transformation – The apparent trajectory of Venus According to Williams (2005) the eccentricity of Venus is 0.0068 which could be considered as negligible and so the planet's orbit can be accepted as circular without a significant error. The same assumption could be done for the Earth whose orbit's eccentricity is e = 0.0167. “… Earth's eccentricity is 0.0167 which is close to circular, but Mercury's is 0.205 …” (Strom and Sprague, 2003). The procedure is described by the following steps:
•
•
The construction in GeoGebra of a solar system consisted only from Sun, Earth and Venus. In this heliocentric system the Earth and Venus will be considered that orbit the Sun in circular orbits with radii equal to their mean distances from the Sun. Their mean sidereal period also are required to simulate their heliocentric motions. All these data are available in NASA planetary fact sheets. Specifically, the mean distances from the Sun are: for Earth 1 AU and for Venus 0.723 AU10 (Ragnarsson, 1995). Their mean sidereal periods are: for Earth approximately 1 year and for Venus 0.615 years (Williams, 2005). The orbit parametric equation is for: 2π
Fig. 8. Simulation in GeoGebra based on Epicyclic theory-Two successive superior conjunctions of planet P - Source: Author, 2018.
circle) of radius RE which is fixed on the small black circle of radius y (y-circle). Since the y-circle is rolling the big black circle of radius x (xcircle), the planet P is participating in two motions which are:
2π
1. the point which represents Earth (sin( t ), cos( t )) 2πt
2πt
2. the point which represents Venus (rp sin( Tp ), rpcos( Tp ))
•
•
•
• •
Now a slider is connected with the variable t and the animation begins with the two points Earth and Venus to follow the circular paths around the Sun with angular velocities respectively 2π and 2π ·t / Tp (see left part of Fig. 7) where Tp = 0.615 years, rp = 0.723 AU , are the mean sidereal period of Venus, the mean distance Venus Sun and t is the time animating variable. Since the reference system origin must changed from Sun to Earth, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ the two vectors EarthSun and EarthVenus must be translated parallel until the point Earth coincides with the point Sun. The two vectors ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ EarthSun and EarthVenus continue to change spatially and temporally and their ends now represent, the Sun as seen from Earth (SunFromEarth) and the Venus as seen from Earth (VenusFromEarth) (see right part of Fig. 7). The trace of the point VenusFromEarth gives the apparent Venus trajectory as seen from Earth. It would be convenient in GeoGebra to plot the parametric curve of the apparent Venus trajectory for a long period of time. In the right part of Fig. 7-geocentric it is shown the Venus trajectory in a nine Earth year interval in faint gray color. This parametric curve is easy to be plotted in GeoGebra with the use of the following equation
Its rotation around the center E The rotation of E around D
(see Fig. 8) As it will be analyzed just below, the ratio y determines the x synodic period11 of the planet and the greatest elongation12 (angle λ ) determines the epicycle radius. 5.2.1. Determination of ratio y x Now lets name ωP the angular velocity of the rolling y-circle which is also the angular velocity of point P (fixed on epicycle) and ωE the angular velocity of the center E of epicycle around the center D. It is obvious that the point P is participating in a compound motion x consisted from a local rotation around E with angular velocity y ·ωE and a rotation around D with angular velocity ωE . So for the real world the angular velocity of P will be:
ωP =
x+y ωE y
(5)
Now the point E could be considered as the Sun orbiting the D which could be considered as the Earth. On the other hand, the point P 2π 2π represents the planet P. Therefore ωE = T and ωP = T , where E P TE = 1 year is the mean sidereal period of Earth around the Sun (equal to the time that it takes for the Sun to make a complete circuit on the ecliptic13) and TP is the mean sidereal period14 of the planet P around the Sun. So ωE = 2π rad/year and the Eq. (5) will be transformed into:
Curve((r (u )·sin(2πu − ϕ(u )), − r (u )·cos(2πu − ϕ(u )))u , 0, 9) (4) where ϕ is the angle with its vertex at Earth and their sides on the ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ two vectors EarthSun and EarthVenus respectively. As it can be observed during the animation, the Venus follows the path of the previously referred parametric curve.
x+y 1 = TP y
(6)
In Fig. 8 the planet P appears in two successive superior conjunctions
5.2. How the theoretical epicyclic model works As it was discussed in Section 2 the planet P belongs to epicycle (red
11 Synodic period of a planet is the time interval needed for the planet to return to same position relative to the Sun as seen from Earth. 12 The angular distance between Sun (center of epicycle) and the planet P. 13 The ecliptic is the circular path on the celestial sphere that the Sun follows over the course of a year. 14 Sidereal period of a planet is the time interval needed for the planet to return to same position relative to the fixed stars as seen from a fixed point outside the system.
9 GeoGebra is an interactive mathematics software program which can be used to demonstrate various geometrical schemes or graphs that can be changed dynamically. 10 1 Astronomic Unit (AU) is exactly 149,597,870,700 m and it is approximately the average distance between Sun and Earth
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(P, P0 ).15 The time interval between these two successive superior conjunctions is the synodic period16 of the planet. So, since the planet P which belongs to the epicycle with center E, is rotating around E in one synodic period, at (2π + θs ) rad then it yields that:
θS + 2π = ωP TS
(7)
where TS is the synodic period of planet P. Since in TS the radius DE rotates at θS rad, then:
2π TS + 2π = ωP TS
(8)
or
TP =
TS TS + 1
(9)
From the Eqs. (6) and (9) it is concluded that:
y = TS x
(10)
The last equation shows that the optimum selection of radii x and y determines the synodic period of a inferior planet. 5.2.2. The epicycle radius As it can be seen from Fig. 8 the greatest elongation of a planet as it is observed from Earth (D) happens when the line of observation is tangent to epicycle. Then from ▵(DBE ) it is concluded that:
RE = sinλ ·(x + y )
Fig. 9. Simulation in GeoGebra of Venus apparent trajectory using the Babylonians astronomical records - Source: Author, 2018.
(11)
where s is a free variable which simulates the animated time. The trace of point P during the animation gives the Venus epitrochoid18
5.2.3. Conclusion The appropriate selection of x and y presupposes the knowledge of synodic period and on the other hand, the determination of epicycle needs the greatest elongation of the planet. Since the maximum elongation of a planet, varies during its apparent motion it is convenient to use the mean value of greatest elongation, especially in the case of planets with negligible eccentricity like Venus. Because the synodic period of a planet and the mean greatest elongation could be defined from historical astronomical records (like the Babylonian ephemerides17) it is concluded that the epicycle model meets all the requirements to reproduce the apparent trajectory of a inferior planet like the Venus and this will be the subject of the next section.
5.4. The implementation in praxis According to Freeth (2012) there are remains of pillars on the four spokes of the gear b14 showing that might be used for supporting plates where could be hosted more complex epicyclic arrangements. Since the space above b14 is limited by its radius (65 mm) (Freeth and Jones, 2012), the teeth number of the involving gears is expected to be restricted. According to Section 4 the x-circle and y-circle represent the pitch circles of two meshing gears (x-gear, y-gear) with Nx and Ny teeth respectively. Since the two meshing gears have the same module then by considering the Eq. (2) it yields that:
5.3. The epicyclic model lined up with the Babylonians
2y 2x = Nx Ny
Looking at the Eqs. (1) and (11) it is clearly that if x = 72 and y = 115.1 are respectively the radii of x and y-circle then a temporally simulation of Venus apparent motion could be established and the determination of epicycle radius could specify its geocentric orbit. According to Fitzpatrick (2010) the mean greatest elongation of Venus is 46.3° and from Eq. (11) RE = 187.1·sin 46.3° = 135.2733. It must be mentioned here that using the Babylonians astronomical records, the sidereal period of the planet according to Eq. (9) is 0.615179 years while from NASA VENUS FACT SHEET is 0.615197 years (Williams, 2005). The difference is 0.00648°/year which is impressively negligible. In Fig. 9 are shown the pitch circles of radius x = 72 and y = 115.1 (black )and the epicycle of radius Re= 135.27 (red). A point P in the circumference of epicycle represents the Venus. The simulation of Venus apparent trajectory is achieved by setting the parametric curve of the point P which is: 2·(x + y )πs 2·(x + y )πs ((x + y )·sin(2πs ) − Resin( y ), − (x + y )cos(2πs ) + Recos( y ))
(12)
or
y Ny = x Nx
(13)
Because Nx and Ny are teeth numbers they must be integers. Also Nx, Ny could be common multipliers of x and y respectively except in the case that y is an irreducible fraction. Then forcedly Nx=x and Ny=y. This x
happens in Venus because the fraction 1151 is irreducible. But this 720 combination of teeth number (Nx=720, Ny=1151) is completely inapplicable in the limited space of b1. This difficulty can be overcome by approaching the ratio 1151 with another one which is applicable. In their 720 model Freeth and Jones (2012) used for the fixed (steady) gear Nx= 40 Ny 8 teeth and for satellite gear Ny= 64 teeth with result Nx = 5 = 1.6 . “…We have chosen (40, 64) since this means that the epicycle of Venus is in the right position to use the attachment area on the 1 o′clock spoke…” (Freeth and Jones, 2012). With this selection they managed to fit the Venus epicycle in the attachment area but on the other hand they created a temporally mismatch which will be analyzed below.
15 Superior conjunction occurs when a inferior planet passes behind the Sun as viewed from the Earth. 16 Synodic period of a planet is the time interval needed for the planet to return to same position relative to the Sun as seen from Earth. 17 In astronomy an ephemeris gives the positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times.
18 Epitrochoid is a curve traced out by a point on the extended radius of a circle that is rolling externally on another circle.
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Fig. 10. Simulated Venus trajectory in GeoGebra using ratio y:x = 8:5 (left) and using ratio y:x = 1151:720 (right), for a period of 16 years - Source: Author, 2018. Ny
8
of Venus according to Babylonians. The elements of the third column are the multiplication product of the first two columns and represents the potential teeth numbers Ny. The forth column contains the integer part of the previous column's values because Nx and Ny must be integers. The fifth column gives the synodic period which can be produced from the achievable values Nx and Ny and in the sixth column is calculated the deviation between the ideal and practical synodic period. The sorting order of rows in the Table 1 is the ascending order of the last column's elements and thus the column (Ts) start from ratios (Ny:Nx) closer to ideal(1151:720). It must be mentioned that the radius of gear b1 sets an upper limit in the sum Nx+Ny. According to Freeth et al. (2006), p. 7 the module of the thirty recognized gears of Antikythera mechanism varies from 0.42 to 0.6 mm/teeth. So, the lowest limit of module gives the maximum total number of teeth Nx+Ny. To be more specific according to the table in Freeth et al. (2006), p. 7 the radius of b1 is estimated at 65 mm and 2·65 the Ny+Nx must be less than Ny + Nx = 0.42 = 309.5 ≈ 310 . (See Section 4) Lets focus on the top rows of Table 1. Using these couples (Nx, Ny) the ideal ratio could be approached closely. But the numbers of Nx, Ny continues to be out of limits except the last line which leads to the problematic ratio 8:5 and so is excluded from the list. May be its time to investigate another epicyclic gear arrangement which is the most appropriate in this case.
5.5. The problematic ratio Nx = 5 In the left part of Fig. 10 is shown the simulated Venus apparent trajectory using the theoretical epicyclic model in Section 5.2 with y = 64 and x = 40 (like the choice of Freeth and Jones, 2012). This integer ratio 8:5 creates a repetition of 5 synodic arcs every 8 years. That means that after 8 years have passed, begins a new cycle of 5 synods and this makes unchanged the image of Venus apparent trajectory, over the time. In the right part of image as ratio y:x has been selected the integer ratio which is compatible with the astronomical Babylonians records which is 1151:720 or y = 115.1 and x = 72. 720 Now it is observed that every 8 year are completed 1151 ·8 = 5.004344 synods and not 5. So there is a difference of 0.004344 synods·1.5986111111 year / synod = 0.00694436 years = 2.53 days This causes the image of apparent Venus trajectory to be shifted by 2.53 days, at every 8-year cycle. In the right part of the image has been selected 16 years and it is seems clearly the first sift of 2.53 days. The choice of ratio 8:5 gives a discrepancy of 2.53 days which is added at the end of every 8-year cycle and increases constantly the error on the stations9 determination. 5.6. Approaching the Babylonians ratio as much as possible In this section there will be an effort to approach the ideal ratio 1151:720 as close as possible. The investigation starts in a Excel spreadsheet. The first column is filled by integer numbers of Nx teeth. All the cells of the second column are equal to ideal ratio 1151:720 which represents the synodic period
6. Epicyclic gearing 3 × 2 This gear arrangement is consisted from six gears which are distributed in three levels. This gearing can reproduce rotational frequencies which are unreachable from the classic epicyclic gearing couple. Moreover this compound epicyclic is more adaptable in the restricted area of the Sun gear b14 (see Fig. 11). The analysis of this compound epicyclic gearing is the following:
Table 1 The best fit Ny:Nx. Nx
1151:720
Ny
720 1.598611 1151.00 431 1.598611 689.00 289 1.598611 462.00 578 1.598611 924.00 573 1.598611 916.00 715 1.598611 1143.01 436 1.598611 696.99 583 1.598611 931.99 142 1.598611 227.00 284 1.598611 454.01 426 1.598611 681.01 568 1.598611 908.01 710 1.598611 1135.01 147 1.598611 235.00 ………………………………………… 40 1.598611 63.94
int(Ny)
Ts
deviation
1151 689 462 924 916 1143 697 932 227 454 681 908 1135 235
1.5986111 1.5986079 1.5986159 1.5986159 1.5986038 1.5986014 1.5986239 1.5986278 1.5985915 1.5985915 1.5985915 1.5985915 1.5985915 1.5986395
0.00000E+ 00 2.01580E− 06 3.00626E− 06 3.00626E− 06 4.54874E− 06 6.07559E− 06 7.97073E− 06 1.04317E− 05 1.22368E− 05 1.22368E− 05 1.22368E− 05 1.22368E− 05 1.22368E− 05 1.77308E− 05
64
1.6000000
1.38889E− 01
• • •
7
The first level is consisted from a steady gear (radius:Rx1, teeth:Nx1) (Sun gear) whose axis coincides with the main axis of b1 and another gear (Ry1, Ny1)(satellite gear) meshed with it, whose axis is fixed on one spoke of b1 The second level is consisted from a gear (Ry2, Ny2) which is connected with the (Ry1, Ny1) and thus is rotating with the same frequency. The (Ry2, Ny2) gear forces an identical meshed gear (Rx2, Nx2) (Rx2 = Ry2, Nx2 = Ny2) to rotate around the main axis of b1. The third level is consisted from (Rx3, Nx3) which is connected with the (Rx2, Nx2) and thus is rotating with the same frequency. The (Rx3, Nx3) forces the gear (Ry3, Ny3) to rotate around its axis which is the common axis of (Ry1, Ny1) and (Ry2, Ny2)
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Fig. 11. A special compound epicyclic gear arrangement in front view consisted of three levels, with two gears on each level - Source: Author, 2018.
concluded that Nx1 + Ny1 = Nx 2 + Ny2 = Nx 3 + Ny3. So a possible combination could be:
Let's consider that the rotation frequency of the Sun gear b1 is 1 rot/ year and a reference system bound on the spoke, where the common axis of (Ny1, Ny2, Ny3) is fixed. For this moving reference system, the rotation frequency of (Rx1, Nx1) is −1 rot/year while the rotation frequency of (Ry1, Ny1) is Nx1 . This is also the rotation frequency of
• • •
Ny1
Nx1
(Ry2, Ny2) which forces the (Rx2, Nx2) to rotate with frequency − Ny1 . The last frequency is adopted by the connected gear (Rx3,Nx3) which Nx 3 Nx1 forces the (Ry3,Ny3) to rotate with frequency Ny3 · Ny1 . So the frequency of (Ry3, Ny3) for a stationary system reference will be
Now if κ =
f=1+
and λ =
Ny3 Nx 3
The numbers of teeth were chosen so that the total teeth number of each couple, to be the same in every level. To sum up 1. Nx1 = 49, Ny1 = 47, Nx1 + Ny1 = 96 2. Nx 2 = 48, Ny2 = 48, Nx 2 + Ny2 = 96 3. Nx 3 = 36, Ny3 = 60, Nx 3 + Ny3 = 96
Nx1 Ny1 Nx1 Nx 3 · =1+ f=1+ Ny3 Ny1 Ny3 Nx 3 Nx1 Ny1
(14) Over the gear Ny3 was fixed an epicycle disc with radius Re = (R x3 + R y )·0.723 = 96·0.723 = 69.408 according to (11)
it is concluded that
3
κ λ
(15)
6.2. A similar gear arrangement in the Antikythera mechanism The transfer of the frequency in this compound gearing has many commons with an existing gear train in Antikythera mechanism. For example in the moon train19 there are six gears used to transmit the rotation to the lunar pointer. Specifically this arrangement starts from gear e2 which is rotating with uniform motion and ends to b3 which is rotating with non-uniform motion in the following order: e2 → e5 → e6 → k1 → k 2 → e1 → b3 (see Fig. 12) “The train is then e5-k1 + k2-e6 +e1-b3 and through to the lunar pointer and phase mechanism on the Front Dial” (Freeth et al., 2006) As it can be seen in this figure there is an effort for the best possible space management from the constructor.
6.1. Implementation of epicyclic 3×2 in the case of Venus Now it is quite interesting to see how this compound epicyclic gearing can reproduce a frequency which corresponds to a ratio Ny:Nx where Ny and Nx are big enough integers making this frequency unapproachable from the classic epicyclic couple. If theoretically accept this frequency f to be reproduced from the classic epicyclic couple, in respect to a steady reference system then it would be:
f=1+
Nx Ny
(16)
From Eqs. (15) and (16) it is concluded that
κ Nx = λ Ny Since κ and λ are ratios (κ = Nx Ny
Nx1 = 49, Ny1 = 47 Nx 3 = 3·12 = 36, Ny3 = 5·12 = 60 Nx 2 = 48, Ny2 = 48 (identical couple)
6.3. Regarding the accuracy using 3 × 2 epicyclic gearing The above combination (Sum of teeth/level=96) can simulate ratios of epicyclic couples with much larger teeth numbers (i.e. Sum of teeth / level=147 +235 =382). The deviation from the ideal ratio that can be achieved by using this epicyclic compound 3 × 2, is 50 times less than the classic epicyclic couple which is used on model of Freeth and Jones (2012) (see last row of Table 1). Also the compact dimensions of this compound epicyclic gearing makes it more usable especially in
(17) Nx1 , Ny1
λ=
Ny3 ) Nx 3
it is obvious that the fraction
must not be irreducible. Searching the Table 1 for the optimum ratio
with the minimum deviation which is simultaneously compliant with the latest requirement it was chosen the seventh combination Nx = 147 and Ny = 235. The terms of the fraction 147 can be factorized as 235 147 = 49·3 and 235 = 47·5. From Fig. 11 it is concluded that: Rx1 + Ry1 = Rx 2 + Ry2 = Rx 3 + Ry3. and since the meshed gears of the compound epicyclic must have the same module (see Eq. (2)) it is
19 A gear train consisted form eleven meshed gears which was used from the Antikythera mechanism constructor to simulate the non-uniform motion of the moon pointer.
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x=
m·Ny m·Nx ,y= 2 2
(18)
2·(x + y ) 2·27.1 mm = ≈ 0.5625 Nx + Ny 96
(19)
and
m=
where m is the module of gears (see Section 4) This is an acceptable module according to Freeth et al. (2006). 6.4.1. 3d simulation of apparent Venus trajectory in GeoGebra The 3d preview ability of GeoGebra10 was exploited in order the three levels of epicyclic gearing to be distinguishable. In this phase were used the pitch circles of the six gears and for their rotation were used the frequencies as they determined in Section 6. The result of this 3d simulation is shown in Fig. 13
Fig. 12. A similar gear arrangement to epicyclic 3×2 inside the moon train - Source: An exploded view using the 3d Interactive Simulation of the Antikythera Mechanism (Diolatzis and Pavlogeorgatos, 2017).
Fig. 13. 3d simulation of Venus epitrochoid by compound epicyclic “3×2” utilizing the ratio 235:147 - Source: Author, 2018.
restricted compartments like the available in the Antikythera mechanism. It must be mention here that this combination (147,235) although is closer to theoretical (720,1151) it is rejected from T. Freeth and A. Jones as unreachable using their limited options. “…Mathematical analysis shows that this list is comprehensive, except for the period relations (147, 235) for Venus and (96, −205) for Mars. Since the large numbers 235 and 205 are unsuitable for our application, we have discarded these possibilities… (Freeth and Jones, 2012)
6.5. Mechanical simulation of Venus apparent epitrochoid In this subsection there is a description of how the above simulation gained a more tangible form. GeoGebra10 has the ability to manipulate inserted images as objects. Every inserted image could embed with any geometrical object. The idea was to connect images of all involved gears with their pitch circles. The animation repeated and the result is shown in the Fig. 14. It must be mentioned here that during the animation it was not observed any overlapping between teeth and the gears were engaged smoothly.
6.4. Simulation in GeoGebra of Venus apparent trajectory with the use of the epicyclic “3 × 2”
7. Completion of the proposal
This compound epicyclic gearing could be implemented to simulate the apparent trajectory of Venus in Antikythera mechanism. The scope of this research is not a strict reconstruction of the lost planetary subsystem in Antikythera mechanism but a suggestion of a more reliable epicyclic gearing in order to eliminate the indications errors. For that reason, the axis of satellite gears Ny1, Ny2, Ny3 could be located in a position which have already been proposed …In the 4 o′clock position, there is a prominent hole, which looks like the remains of a bearing. Its outer diameter is 9.7 mm and its inner diameter is 6.6 mm. It is 27.1 mm from the central axis. … (Freeth and Jones, 2012). Now if this bearing is accepted for the satellite gears axis then:
According to “…For Mars, the inscription's 349 days between conjunction and the stations and 82 days for the retrogradation between the stations agrees well with the accurate values, 354 and 72 days …” (Jones, 2017), a detailed orbital view of all planets, could be projected on the top of Antikythera mechanism. 7.1. Characteristic orbital positions As seen from a planet that is superior, if an inferior planet is on the opposite side of the Sun, it is in superior conjunction with the Sun. An inferior conjunction occurs when the two planets lie in a line 9
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Fig. 14. Mechanical simulation of Venus epitrochoid by compound epicyclic “3 × 2” utilizing the ratio 235:147 in 16 years time depth - Source: Author, 2018.
Fig. 15. 3d modeling of Venus orbit simulated in UNITY3D22 by compound epicyclic “3 × 2” utilizing the ratio 235:147 in 9 years time depth (isolated Venus epicyclic gearing) - Source: Author, 2018.
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Fig. 16. 3d Venus modulation in UNITY3D22 compatible with the 3 × 2 epicyclic gearing - Up: a superior conjunction - Down: In inferior conjunction - Source: Author, 2018.
environment. A very important effect which UNITY3D22 offers is the trail renderer. If this feature is added to a planet (little sphere) then its path is traced. This could be functions as an additional evaluation criterion because the form of the imprinted trajectory could be compared with the theoretical one (see Fig. 15)
on the same side of the Sun. In an inferior conjunction, the superior planet is “in opposition” to the Sun as seen from the inferior planet. All above referred positions occur when the planet, Earth and Sun lie in line. Also there are two stationary points the first stationary when the planet enters the retrograde motion and the second stationary when exit from retrograde motion. In these positions the planet is moving across the observing line and appears as stationary.
8. Epilogue Antikythera mechanism is characterized as a complicated gearing machine with an advanced technology far enough from its era. In many parts of the mechanism there is an excellent balance between the complexity and the precision. Our proposed (3d modeled) Venus epicyclic gearing gives a compatible image (little sphere moving on a circle) according to BCI3 and we believe that this configuration (sphere with slotted pointer see Fig. 15) must also be implemented for the rest of the planets. The advantages of using this combination (sphere slotted arm pointer) are:
7.2. 3d modeling of the proposal All above could be well indicated if the planets are represented by little spheres connected with slotted pointers. This imaginary figure, guide us to make a 3d modulation in CINEMA 4D20 and a further editing in UNITY3D22. Regarding the 3d modeling in CINEMA 4D21 we used a technique known as Interactive Simulation of Rigid Body Dynamics where the gears are simulated as rigid bodies. So the gears rotation is the result of their physical interaction. In UNITY3D21 the animated model gained interactivity in an attractive photo realistic
• • • •
20 3D modeling, animation and rendering application developed by MAXON Computer GmbH. 21 Unity 3d is a multi-platform game development tool that allows textures compression and customization of resolution settings.
11
Graphical display of geocentric planets orbits Recording of astronomical events like Superior-Inferior conjunctions (See Fig. 16) Compatibility with the BCI3 inscriptions (planets presented by little spheres moving on circles) Closer approach to Babylonians time records
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• •
Zafeiropoulou, M., Hadland, R., Bate, D., Ramsey, A., et al., (2006). Decoding the antikythera mechanism: Investigation of an ancient astronomical calculator 2 supplementary notes. Freeth, T., 2012. Building the cosmos in the antikythera mechanism. In: From Antikythera to the Square Kilometre Array: Lessons from the Ancients. Jones, A., 2017. A Portable Cosmos: Revealing the Antikythera Mechanism, Scientific Wonder of the Ancient World. Oxford University Press. Maitra, G.M., 1994. Handbook of Gear Design. Tata McGraw-Hill Education. Margolis, H., 1993. Paradigms and Barriers: How Habits of Mind Govern Scientific Beliefs. University of Chicago Press. Neugebauer, O., 2012. A History of Ancient Mathematical Astronomy 1. Springer Science & Business Media. Ragnarsson, S.-I., 1995. Planetary distances: a new simplified model. Astron. Astrophys. 301, 609. Safronov, A., 2016. Antikythera mechanism and the ancient world. J. Archaeol. 2016. Spitas, C.A., Spitas, V.A., 2006. Generating interchangeable 200 spur gear sets with circular fillets to increase load carrying capacity. In: Proceedings of the International Conference on Gears, volume 1, pp. 927–941. Strom, R.G., Sprague, A.L., 2003. Exploring Mercury: The Iron Planet. Springer Science & Business Media. D. R. Williams, Planetary fact sheets, National Space Science DataCenter (NSSDC) 6 (2005). Wright, M., 2013. The antikythera mechanism: compound gear-trains for planetary indications. Almagest 4, 4–31.
Better space management High accuracy in the chronological determination of astronomical events
Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at doi:10.1016/j.daach.2018.e00089. References Black, J.T., Kohser, R.A., 2017. DeGarmo's Materials and Processes in Manufacturing. John Wiley & Sons. Diolatzis, I.S., Pavlogeorgatos, G., 2017. Deepening to antikythera mechanism via its interactivity. Digit. Appl. Archaeol. Cult. Herit.. Evans, J., Carman, C.C., Thorndike, A.S., 2010. Solar anomaly and planetary displays in the antikythera mechanism. J. Hist. Astron. 41, 1–39. Fitzpatrick, R., 2010. Determination of conjunction and greatest elongation dates. Freeth, T., Jones, A., 2012. The cosmos in the Antikythera mechanism. Inst. Study Anc. World. Freeth, T., Bitsakis, Y., Moussas, X., Seiradakis, J., Tselikas, A., Magkou, E.,
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Antikythera mechanism – A compound epicyclic gearing for Venus ⁎
Ioannis S. Diolatzis , Gerasimos Pavlogeorgatos1 Department of Cultural Technology and Communication, University of the Aegean, University Hill, Building of Geography, off. 2.14, 81100 Mytilene, Lesvos, Greece
A R T I C L E I N F O
A BS T RAC T
Keywords: Anikythera mechanism Epicyclic theory Epicyclic gearing Gears Epitrochoid Geocentric Sun gear Satellite gear GeoGebra Venus apparent trajectory Retrograde motion
One undeniable characteristic of Antikythera mechanism is the complex gear arrangements that generously have been used by the constructor. He succeeded the best combination of integer numbers to reach the maximum accuracy, no matter the number of the gears he used for it. One representative example is the “moon train” consisted of eleven meshing gears which he used to simulate the non-uniform lunar motion. Someone could say that the constructor preferred the complexity for the sake of accuracy. Many Scholars, believe that the mechanism could be a kind of planetarium. In that case, there is a possibility of a more complex mechanical subsystem comprised of more imaginative and specialized gear arrangements. The latest decoded inscriptions on the mechanism's surfaces, are referred to planets (Inferiors and Superiors) and show records of astronomical events. The lack of physical evidence of this planetary construction prevents its conception, but on the other hand, challenges the researchers to discover the inimitable thought of its creator. Nevertheless, fitting this excess gearing in a limited space of a compact construction like Antikythera mechanism is really a puzzler. The planetary subsystem which will complete the mechanism should stand out for the maximum possible measurement accuracy and the best space management. This research proposes a feasible epicyclic gearing specialized for a compact planetary construction achieving the highest possible accuracy.
1. Introduction A deciphered text on the back face of the Antikythera mechanism (BCI2) reveals many details about the characteristics and the functional capabilities of this amazing gearwork machine. This decoded inscription (lines 15–25 Freeth and Jones, 2012) is referred to the five known planets of that era, with an ascending order of their distances from the Sun. Unfinished words inside fragmentary expressions, show planets' names starting from Mercury (EPMHΣ ) which is the closest planet to the Sun and ending to Saturn(KPONOΣ ) which is the farthest planet from the Sun. A particular impression causes the repeated words (ΣΦAIPION ) “little sphere”, one for every planet and (KΥ KΛOΣ ) “circle” in which the little sphere possibly belongs to. All the above references show that possibly the Antikythera mechanism could have on its top a constructive formulation like a kind of planetarium. This compartment is completely absent from the
⁎
remains of the mechanism found in a shipwreck in 1900. Speculated models of this planetary synthesis have already been made and they are all embed in the available area between the sun gear b13 and the top of the mechanism (see Fig. 1) The construction of this imaginary geocentric planetarium presupposes the knowledge of the “epicyclic theory” which was well formulated by Hipparchus and Apollonius. Using this theory the two famous Greek astronomers managed to explain the weird planetary motion on the sky. According to Safronov (2016), Antikythera mechanism is estimated to be constructed between 100 and 150 BCE, a long time after the invention of epicyclic theory (225 B.C Margolis, 1993) 2. The epicyclic theory as an essential tool The epicyclic theory was expressed by Hipparchus4 and Apollonius5 and later was implemented in Almagest by Ptolemy.6 According to this
Corresponding author. E-mail addresses: [email protected] (I.S. Diolatzis), [email protected] (G. Pavlogeorgatos). 1 Co-author. 2 Back Cover Inscription. 3 The biggest gear in Antikythera mechanism with 224 teeth which according to Freeth and Jones (2012) used to support the lost planetary subsystem. 4 Hipparchus of Nicaea (ca. 190 BCE - ca. 120 BCE) is known not only because contributed significantly to trigonometry but also as an astronomer with a unique talent. Unfortunately, all his astronomical memoirs have been lost so access to his theories could be accessed only through later Greek astronomers such as Ptolemy. 5 Apollonius was a Greek Geometer and astronomer who lived from about 262 B.C. to approximately 190 B.C. 6 Claudius Ptolemy 100–170 CE was a Greek mathematician, astronomer, geographer and astrologer. https://doi.org/10.1016/j.daach.2018.e00089 Received 19 August 2018; Accepted 30 December 2018 2212-0548/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. The lost planetary subsystem – Source: (Freeth and Jones, 2012).
theory the apparent motion of a planet as it is observed from a fixed point of Earth is consisted of two independent motions
• • •
thinks that he is the center of the universe (geocentricism) and the Sun is orbiting him dragging the rotating planets with it. The apparent trajectory of the observed planet is called epitrochoid19 and can be expressed by a parametric equation.7
The first motion is a rotation of the planet around a center. This imaginary circle was called from the inventors EΠ IKΥ KΛOΣ or epicycle (red circle) as is widely known today (see Fig. 2). The second motion is a rotation of epicycle's center around the Earth forming an imaginary circle which was called Φ EPΩ N KΥ KΛOΣ or deferent (blue circle) as it is known today(see Fig. 2). The epicycle is consider to be fixed with the concentric y-circle (black circle) which is rolling on circumference of an immobile xcircle (black circle) concentric with the deferent (see Fig. 2).
2.1. Retrograde motion There are points on the planet's apparent orbit where the planet seems to stop and then starts again moving to the opposite direction. This backward motion is called retrograde motion which lasts for awhile until the planet stops again and starts moving to its initial direction. These points are called stations8. The station where the planet turns to go backward is called retrograde or first station while the station where turns to resume forward motion is called direct or second station. The question that could be raised here is why the planet appears to stop and then starts moving backward. The key to understand this weird motion is to isolate the velocity and the direction of observation. As long as the planet travels along the observation direction it appears to stop. This is because the vertical velocity which gives the sense of motion, is zero. These points are referred in the BCI3 as Stations (Σ THPIΓ MOI ). According to Jones (2017), p. 192, Apollonius showed that the ratio between the radii of x-circle and y-circle could determine the time between two identical positions (in respect to fixed stars) in the sky which is called synodic period. As it will be analyzed below, the choice of gears' teeth numbers in a simple epicyclic gearing (like the shown in Fig. 2) is based on the synodic period of the planet which represents. So the first priority of the constructor should be the building of a reliable astronomical database. According to Freeth (2012) the exclusive suppliers of this knowledge in that era were the Babylonians astronomers who recorded with remarkable precision the synodic phenomena of all planets. For example, according to Neugebauer (2012) they recorded for Venus, 720
So,the planet is participating in two independent circular motions. One is the rotation of the planet around epicycle's center, which could be represented by the Sun, and the other is the rotation of epicycle center around the Earth. This motion is conceived by an Earthy observer who
7 Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. 8 Station in retrograde motion is the point where the planet appears to stop moving in the sky.
Fig. 2. Epicycle – deferent model - Source: Author, 2018.
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Fig. 5. Venus static gear train - Source: (Evans et al., 2010).
•
• Fig. 3. Venus gears train (colorized grey) - Source: (Wright, 2013).
occurrences in 1151 years. In other words, they were aware about the synodic period of Venus which is:
TS =
1151 = 1.5986111111 years / synod 720
(1)
T.Freeth and A. Jones model. In this model T.Freeth and A. Jones used a simple form of epicycle-deferent model (see Section 2) to simulate the Inferior planets’ motion. Fig. 4 shows the Venus epicyclic gearing which consisted of the fixed gear x of 40 teeth at 64 8 b14 axis and the satellite gear y of 64 teeth obtaining a ratio of 40 = 5 (Freeth and Jones, 2012) which simulates the synodic period of Venus with an error of 0.086%. James Evans, Carman and Allan Thorndike model. A different proposal is coming from James Evans, Carman and Allan Thorndike. In their model they proposed five fixed sub-dials one for every planet which are evenly distributed around the circumference of the front dial. In each separate dial a pointer performs one rotation every synodic cycle. The gear trains they proposed simply calculate the synodic period of the planets without any planetary motion sense. For example they proposed for Venus gear train, consisted of six gears (included b1), the following 29 63 98 sequence: 224 · 25 · 20 = 1.598625. Evans et al. (2010) which simulates a synodic period of Venus with a negligible error of 8.68·10−4% (see Fig. 5)
3. Basic models of planetary motion All above proposed planetary apparatuses are accommodated between the Sun gear b14 and the top of the mechanism.
Scholars have already presented speculated planetary subsystems of Antikythera mechanism. The most representative of them are listed below.
•
3.1. Remarks on the proposed models
M. Wright's model. In this model M. Wright used compound gear trains trying to achieve the best accuracy. A characteristic example is the six gear train which he used for Venus. Specifically the Venus 51 70 34 train is 45 · 56 · 77 leading to ratio 462 . Wright (2013) which simulates
We express our skepticism about the third proposal because:
•
289
−3
the synodic period of Venus with a negligible error of 3·10 % (See Fig. 3)
•
The planetary motion is completely absent. So if we accept this configuration then its like neglecting all the graphical representation capabilities which the constructor implies in the inscriptions (retrograde stations or conjunctions etc) This stating image is incompatible with the epicycle - deferent model (Section 2) which is essential for the planetary motion demonstration.
The second proposal uses an epicyclic gearing couple and thus it reduces the number of used gears, at least for the Inferiors planets. This though, restricts the tooth number choices because of the limited available space. As a result, the approach to ideal value isn't desirable. Specifically in the case of Venus the discrepancy error is about two and a half days on every eight year cycle! (This will be analyzed below). We believe that the first proposal meets the requirements to be considered as the closest to the philosophy of the constructor. It does use a required gear number to eliminate the reading error indication. Also it implements the epicycle-deferent model as it should do. But as its seen in the Fig. 3, the Venus epicyclic gearing occupies a large portion of b14 and we think that the limited space over b14 must be used with parsimony. 4. Basic characteristics of spur gears As it is well known Antikythera mechanism was a gearwork device. So before of any study associated with whatever part of the mechanism,
Fig. 4. Venus epicyclic gearing on the 1 o'clock spoke - Source: (Freeth and Jones, 2012).
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(Black and Kohser, 2017). other The pitch circle is the imaginary circle which rolls without slip. In two meshing gears the pitch circles have one common point contact. In later sections pitch circles will substitute the corresponding gears as representative schemes of contact. 5. The geocentric orbit of Venus In this section will be verified that the ancient epicyclic theory and the theory of relative motion are the faces of the same coin. The motion of the planet could be studied:
• •
Fig. 6. 27 teeth spur gear with radius 27 mm and module 2 mm/teeth - Source: Author, 2018.
The trajectory of a moving object (i.e. a planet as moving in the space) is defined by its location as it varies spatially and temporally. The position of the planet with respect to the geocentric coordinate system is determined by its position vector with respect to the Earth which starts from the Earth and ends at the planet. According to Galilean Coordinate transformations, the position vector of a planet with respect to a relative coordinate system is given by the following equation
some elementary characteristics about the gears should be mentioned. According to Maitra (1994)
• •
pitch circle This circle intersects the teeth of the gear at the points where its teeth mesh with the teeth of another gear module m If N is the teeth number of the gear and D is the Pitch circle diameter in mm then the module m of the gear in metric system is defined by the equation
m=
D N
→ → ⎯→ ⎯ (3) r′ = r − R ⎯→ ⎯ → where r and R are the positions vectors of the planet and → Earth with respect to the Sun. The vector r′ performs a complex motion with respect to the real world, as it is rotating and translating simultaneously (see Fig. 7). If there is a way to freeze the start point of this vector then its end will move on the planet's apparent orbit. With this technique it is manageable to move from the heliocentric to geocentric coordinate system. The above mentioned procedure could be easily manipulated with the use of an interactive-dynamic mathematics software known as
(2)
(Maitra, 1994) (see Fig. 6) 4.1. Meshing two spur gears In order two gears to be meshed properly:
• •
in respect to a reference coordinate system fixed in a immobile star. This system is called absolute coordinate system. If our Sun takes the place of this stationary star then the coordinate system bound with it, is the heliocentric system. in respect to a moving coordinate system. This system is called relative coordinate system like the geocentric system which is bound to the Earth.
they must have the same module m (Spitas and Spitas, 2006) the pitch circles of the meshing gears must be tangential to each
Fig. 7. A snapshot just after a Venusian year has passed, showing Earth and Venus animated motion in heliocentric system (left) and the Venus in geocentric system (right) – Source: Author, 2018.
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GeoGebra.9 Below would be presented briefly this process which is applied in the case of Venus. 5.1. Heliocentric to geocentric transformation – The apparent trajectory of Venus According to Williams (2005) the eccentricity of Venus is 0.0068 which could be considered as negligible and so the planet's orbit can be accepted as circular without a significant error. The same assumption could be done for the Earth whose orbit's eccentricity is e = 0.0167. “… Earth's eccentricity is 0.0167 which is close to circular, but Mercury's is 0.205 …” (Strom and Sprague, 2003). The procedure is described by the following steps:
•
•
The construction in GeoGebra of a solar system consisted only from Sun, Earth and Venus. In this heliocentric system the Earth and Venus will be considered that orbit the Sun in circular orbits with radii equal to their mean distances from the Sun. Their mean sidereal period also are required to simulate their heliocentric motions. All these data are available in NASA planetary fact sheets. Specifically, the mean distances from the Sun are: for Earth 1 AU and for Venus 0.723 AU10 (Ragnarsson, 1995). Their mean sidereal periods are: for Earth approximately 1 year and for Venus 0.615 years (Williams, 2005). The orbit parametric equation is for: 2π
Fig. 8. Simulation in GeoGebra based on Epicyclic theory-Two successive superior conjunctions of planet P - Source: Author, 2018.
circle) of radius RE which is fixed on the small black circle of radius y (y-circle). Since the y-circle is rolling the big black circle of radius x (xcircle), the planet P is participating in two motions which are:
2π
1. the point which represents Earth (sin( t ), cos( t )) 2πt
2πt
2. the point which represents Venus (rp sin( Tp ), rpcos( Tp ))
•
•
•
• •
Now a slider is connected with the variable t and the animation begins with the two points Earth and Venus to follow the circular paths around the Sun with angular velocities respectively 2π and 2π ·t / Tp (see left part of Fig. 7) where Tp = 0.615 years, rp = 0.723 AU , are the mean sidereal period of Venus, the mean distance Venus Sun and t is the time animating variable. Since the reference system origin must changed from Sun to Earth, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ the two vectors EarthSun and EarthVenus must be translated parallel until the point Earth coincides with the point Sun. The two vectors ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ EarthSun and EarthVenus continue to change spatially and temporally and their ends now represent, the Sun as seen from Earth (SunFromEarth) and the Venus as seen from Earth (VenusFromEarth) (see right part of Fig. 7). The trace of the point VenusFromEarth gives the apparent Venus trajectory as seen from Earth. It would be convenient in GeoGebra to plot the parametric curve of the apparent Venus trajectory for a long period of time. In the right part of Fig. 7-geocentric it is shown the Venus trajectory in a nine Earth year interval in faint gray color. This parametric curve is easy to be plotted in GeoGebra with the use of the following equation
Its rotation around the center E The rotation of E around D
(see Fig. 8) As it will be analyzed just below, the ratio y determines the x synodic period11 of the planet and the greatest elongation12 (angle λ ) determines the epicycle radius. 5.2.1. Determination of ratio y x Now lets name ωP the angular velocity of the rolling y-circle which is also the angular velocity of point P (fixed on epicycle) and ωE the angular velocity of the center E of epicycle around the center D. It is obvious that the point P is participating in a compound motion x consisted from a local rotation around E with angular velocity y ·ωE and a rotation around D with angular velocity ωE . So for the real world the angular velocity of P will be:
ωP =
x+y ωE y
(5)
Now the point E could be considered as the Sun orbiting the D which could be considered as the Earth. On the other hand, the point P 2π 2π represents the planet P. Therefore ωE = T and ωP = T , where E P TE = 1 year is the mean sidereal period of Earth around the Sun (equal to the time that it takes for the Sun to make a complete circuit on the ecliptic13) and TP is the mean sidereal period14 of the planet P around the Sun. So ωE = 2π rad/year and the Eq. (5) will be transformed into:
Curve((r (u )·sin(2πu − ϕ(u )), − r (u )·cos(2πu − ϕ(u )))u , 0, 9) (4) where ϕ is the angle with its vertex at Earth and their sides on the ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ two vectors EarthSun and EarthVenus respectively. As it can be observed during the animation, the Venus follows the path of the previously referred parametric curve.
x+y 1 = TP y
(6)
In Fig. 8 the planet P appears in two successive superior conjunctions
5.2. How the theoretical epicyclic model works As it was discussed in Section 2 the planet P belongs to epicycle (red
11 Synodic period of a planet is the time interval needed for the planet to return to same position relative to the Sun as seen from Earth. 12 The angular distance between Sun (center of epicycle) and the planet P. 13 The ecliptic is the circular path on the celestial sphere that the Sun follows over the course of a year. 14 Sidereal period of a planet is the time interval needed for the planet to return to same position relative to the fixed stars as seen from a fixed point outside the system.
9 GeoGebra is an interactive mathematics software program which can be used to demonstrate various geometrical schemes or graphs that can be changed dynamically. 10 1 Astronomic Unit (AU) is exactly 149,597,870,700 m and it is approximately the average distance between Sun and Earth
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(P, P0 ).15 The time interval between these two successive superior conjunctions is the synodic period16 of the planet. So, since the planet P which belongs to the epicycle with center E, is rotating around E in one synodic period, at (2π + θs ) rad then it yields that:
θS + 2π = ωP TS
(7)
where TS is the synodic period of planet P. Since in TS the radius DE rotates at θS rad, then:
2π TS + 2π = ωP TS
(8)
or
TP =
TS TS + 1
(9)
From the Eqs. (6) and (9) it is concluded that:
y = TS x
(10)
The last equation shows that the optimum selection of radii x and y determines the synodic period of a inferior planet. 5.2.2. The epicycle radius As it can be seen from Fig. 8 the greatest elongation of a planet as it is observed from Earth (D) happens when the line of observation is tangent to epicycle. Then from ▵(DBE ) it is concluded that:
RE = sinλ ·(x + y )
Fig. 9. Simulation in GeoGebra of Venus apparent trajectory using the Babylonians astronomical records - Source: Author, 2018.
(11)
where s is a free variable which simulates the animated time. The trace of point P during the animation gives the Venus epitrochoid18
5.2.3. Conclusion The appropriate selection of x and y presupposes the knowledge of synodic period and on the other hand, the determination of epicycle needs the greatest elongation of the planet. Since the maximum elongation of a planet, varies during its apparent motion it is convenient to use the mean value of greatest elongation, especially in the case of planets with negligible eccentricity like Venus. Because the synodic period of a planet and the mean greatest elongation could be defined from historical astronomical records (like the Babylonian ephemerides17) it is concluded that the epicycle model meets all the requirements to reproduce the apparent trajectory of a inferior planet like the Venus and this will be the subject of the next section.
5.4. The implementation in praxis According to Freeth (2012) there are remains of pillars on the four spokes of the gear b14 showing that might be used for supporting plates where could be hosted more complex epicyclic arrangements. Since the space above b14 is limited by its radius (65 mm) (Freeth and Jones, 2012), the teeth number of the involving gears is expected to be restricted. According to Section 4 the x-circle and y-circle represent the pitch circles of two meshing gears (x-gear, y-gear) with Nx and Ny teeth respectively. Since the two meshing gears have the same module then by considering the Eq. (2) it yields that:
5.3. The epicyclic model lined up with the Babylonians
2y 2x = Nx Ny
Looking at the Eqs. (1) and (11) it is clearly that if x = 72 and y = 115.1 are respectively the radii of x and y-circle then a temporally simulation of Venus apparent motion could be established and the determination of epicycle radius could specify its geocentric orbit. According to Fitzpatrick (2010) the mean greatest elongation of Venus is 46.3° and from Eq. (11) RE = 187.1·sin 46.3° = 135.2733. It must be mentioned here that using the Babylonians astronomical records, the sidereal period of the planet according to Eq. (9) is 0.615179 years while from NASA VENUS FACT SHEET is 0.615197 years (Williams, 2005). The difference is 0.00648°/year which is impressively negligible. In Fig. 9 are shown the pitch circles of radius x = 72 and y = 115.1 (black )and the epicycle of radius Re= 135.27 (red). A point P in the circumference of epicycle represents the Venus. The simulation of Venus apparent trajectory is achieved by setting the parametric curve of the point P which is: 2·(x + y )πs 2·(x + y )πs ((x + y )·sin(2πs ) − Resin( y ), − (x + y )cos(2πs ) + Recos( y ))
(12)
or
y Ny = x Nx
(13)
Because Nx and Ny are teeth numbers they must be integers. Also Nx, Ny could be common multipliers of x and y respectively except in the case that y is an irreducible fraction. Then forcedly Nx=x and Ny=y. This x
happens in Venus because the fraction 1151 is irreducible. But this 720 combination of teeth number (Nx=720, Ny=1151) is completely inapplicable in the limited space of b1. This difficulty can be overcome by approaching the ratio 1151 with another one which is applicable. In their 720 model Freeth and Jones (2012) used for the fixed (steady) gear Nx= 40 Ny 8 teeth and for satellite gear Ny= 64 teeth with result Nx = 5 = 1.6 . “…We have chosen (40, 64) since this means that the epicycle of Venus is in the right position to use the attachment area on the 1 o′clock spoke…” (Freeth and Jones, 2012). With this selection they managed to fit the Venus epicycle in the attachment area but on the other hand they created a temporally mismatch which will be analyzed below.
15 Superior conjunction occurs when a inferior planet passes behind the Sun as viewed from the Earth. 16 Synodic period of a planet is the time interval needed for the planet to return to same position relative to the Sun as seen from Earth. 17 In astronomy an ephemeris gives the positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times.
18 Epitrochoid is a curve traced out by a point on the extended radius of a circle that is rolling externally on another circle.
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Fig. 10. Simulated Venus trajectory in GeoGebra using ratio y:x = 8:5 (left) and using ratio y:x = 1151:720 (right), for a period of 16 years - Source: Author, 2018. Ny
8
of Venus according to Babylonians. The elements of the third column are the multiplication product of the first two columns and represents the potential teeth numbers Ny. The forth column contains the integer part of the previous column's values because Nx and Ny must be integers. The fifth column gives the synodic period which can be produced from the achievable values Nx and Ny and in the sixth column is calculated the deviation between the ideal and practical synodic period. The sorting order of rows in the Table 1 is the ascending order of the last column's elements and thus the column (Ts) start from ratios (Ny:Nx) closer to ideal(1151:720). It must be mentioned that the radius of gear b1 sets an upper limit in the sum Nx+Ny. According to Freeth et al. (2006), p. 7 the module of the thirty recognized gears of Antikythera mechanism varies from 0.42 to 0.6 mm/teeth. So, the lowest limit of module gives the maximum total number of teeth Nx+Ny. To be more specific according to the table in Freeth et al. (2006), p. 7 the radius of b1 is estimated at 65 mm and 2·65 the Ny+Nx must be less than Ny + Nx = 0.42 = 309.5 ≈ 310 . (See Section 4) Lets focus on the top rows of Table 1. Using these couples (Nx, Ny) the ideal ratio could be approached closely. But the numbers of Nx, Ny continues to be out of limits except the last line which leads to the problematic ratio 8:5 and so is excluded from the list. May be its time to investigate another epicyclic gear arrangement which is the most appropriate in this case.
5.5. The problematic ratio Nx = 5 In the left part of Fig. 10 is shown the simulated Venus apparent trajectory using the theoretical epicyclic model in Section 5.2 with y = 64 and x = 40 (like the choice of Freeth and Jones, 2012). This integer ratio 8:5 creates a repetition of 5 synodic arcs every 8 years. That means that after 8 years have passed, begins a new cycle of 5 synods and this makes unchanged the image of Venus apparent trajectory, over the time. In the right part of image as ratio y:x has been selected the integer ratio which is compatible with the astronomical Babylonians records which is 1151:720 or y = 115.1 and x = 72. 720 Now it is observed that every 8 year are completed 1151 ·8 = 5.004344 synods and not 5. So there is a difference of 0.004344 synods·1.5986111111 year / synod = 0.00694436 years = 2.53 days This causes the image of apparent Venus trajectory to be shifted by 2.53 days, at every 8-year cycle. In the right part of the image has been selected 16 years and it is seems clearly the first sift of 2.53 days. The choice of ratio 8:5 gives a discrepancy of 2.53 days which is added at the end of every 8-year cycle and increases constantly the error on the stations9 determination. 5.6. Approaching the Babylonians ratio as much as possible In this section there will be an effort to approach the ideal ratio 1151:720 as close as possible. The investigation starts in a Excel spreadsheet. The first column is filled by integer numbers of Nx teeth. All the cells of the second column are equal to ideal ratio 1151:720 which represents the synodic period
6. Epicyclic gearing 3 × 2 This gear arrangement is consisted from six gears which are distributed in three levels. This gearing can reproduce rotational frequencies which are unreachable from the classic epicyclic gearing couple. Moreover this compound epicyclic is more adaptable in the restricted area of the Sun gear b14 (see Fig. 11). The analysis of this compound epicyclic gearing is the following:
Table 1 The best fit Ny:Nx. Nx
1151:720
Ny
720 1.598611 1151.00 431 1.598611 689.00 289 1.598611 462.00 578 1.598611 924.00 573 1.598611 916.00 715 1.598611 1143.01 436 1.598611 696.99 583 1.598611 931.99 142 1.598611 227.00 284 1.598611 454.01 426 1.598611 681.01 568 1.598611 908.01 710 1.598611 1135.01 147 1.598611 235.00 ………………………………………… 40 1.598611 63.94
int(Ny)
Ts
deviation
1151 689 462 924 916 1143 697 932 227 454 681 908 1135 235
1.5986111 1.5986079 1.5986159 1.5986159 1.5986038 1.5986014 1.5986239 1.5986278 1.5985915 1.5985915 1.5985915 1.5985915 1.5985915 1.5986395
0.00000E+ 00 2.01580E− 06 3.00626E− 06 3.00626E− 06 4.54874E− 06 6.07559E− 06 7.97073E− 06 1.04317E− 05 1.22368E− 05 1.22368E− 05 1.22368E− 05 1.22368E− 05 1.22368E− 05 1.77308E− 05
64
1.6000000
1.38889E− 01
• • •
7
The first level is consisted from a steady gear (radius:Rx1, teeth:Nx1) (Sun gear) whose axis coincides with the main axis of b1 and another gear (Ry1, Ny1)(satellite gear) meshed with it, whose axis is fixed on one spoke of b1 The second level is consisted from a gear (Ry2, Ny2) which is connected with the (Ry1, Ny1) and thus is rotating with the same frequency. The (Ry2, Ny2) gear forces an identical meshed gear (Rx2, Nx2) (Rx2 = Ry2, Nx2 = Ny2) to rotate around the main axis of b1. The third level is consisted from (Rx3, Nx3) which is connected with the (Rx2, Nx2) and thus is rotating with the same frequency. The (Rx3, Nx3) forces the gear (Ry3, Ny3) to rotate around its axis which is the common axis of (Ry1, Ny1) and (Ry2, Ny2)
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Fig. 11. A special compound epicyclic gear arrangement in front view consisted of three levels, with two gears on each level - Source: Author, 2018.
concluded that Nx1 + Ny1 = Nx 2 + Ny2 = Nx 3 + Ny3. So a possible combination could be:
Let's consider that the rotation frequency of the Sun gear b1 is 1 rot/ year and a reference system bound on the spoke, where the common axis of (Ny1, Ny2, Ny3) is fixed. For this moving reference system, the rotation frequency of (Rx1, Nx1) is −1 rot/year while the rotation frequency of (Ry1, Ny1) is Nx1 . This is also the rotation frequency of
• • •
Ny1
Nx1
(Ry2, Ny2) which forces the (Rx2, Nx2) to rotate with frequency − Ny1 . The last frequency is adopted by the connected gear (Rx3,Nx3) which Nx 3 Nx1 forces the (Ry3,Ny3) to rotate with frequency Ny3 · Ny1 . So the frequency of (Ry3, Ny3) for a stationary system reference will be
Now if κ =
f=1+
and λ =
Ny3 Nx 3
The numbers of teeth were chosen so that the total teeth number of each couple, to be the same in every level. To sum up 1. Nx1 = 49, Ny1 = 47, Nx1 + Ny1 = 96 2. Nx 2 = 48, Ny2 = 48, Nx 2 + Ny2 = 96 3. Nx 3 = 36, Ny3 = 60, Nx 3 + Ny3 = 96
Nx1 Ny1 Nx1 Nx 3 · =1+ f=1+ Ny3 Ny1 Ny3 Nx 3 Nx1 Ny1
(14) Over the gear Ny3 was fixed an epicycle disc with radius Re = (R x3 + R y )·0.723 = 96·0.723 = 69.408 according to (11)
it is concluded that
3
κ λ
(15)
6.2. A similar gear arrangement in the Antikythera mechanism The transfer of the frequency in this compound gearing has many commons with an existing gear train in Antikythera mechanism. For example in the moon train19 there are six gears used to transmit the rotation to the lunar pointer. Specifically this arrangement starts from gear e2 which is rotating with uniform motion and ends to b3 which is rotating with non-uniform motion in the following order: e2 → e5 → e6 → k1 → k 2 → e1 → b3 (see Fig. 12) “The train is then e5-k1 + k2-e6 +e1-b3 and through to the lunar pointer and phase mechanism on the Front Dial” (Freeth et al., 2006) As it can be seen in this figure there is an effort for the best possible space management from the constructor.
6.1. Implementation of epicyclic 3×2 in the case of Venus Now it is quite interesting to see how this compound epicyclic gearing can reproduce a frequency which corresponds to a ratio Ny:Nx where Ny and Nx are big enough integers making this frequency unapproachable from the classic epicyclic couple. If theoretically accept this frequency f to be reproduced from the classic epicyclic couple, in respect to a steady reference system then it would be:
f=1+
Nx Ny
(16)
From Eqs. (15) and (16) it is concluded that
κ Nx = λ Ny Since κ and λ are ratios (κ = Nx Ny
Nx1 = 49, Ny1 = 47 Nx 3 = 3·12 = 36, Ny3 = 5·12 = 60 Nx 2 = 48, Ny2 = 48 (identical couple)
6.3. Regarding the accuracy using 3 × 2 epicyclic gearing The above combination (Sum of teeth/level=96) can simulate ratios of epicyclic couples with much larger teeth numbers (i.e. Sum of teeth / level=147 +235 =382). The deviation from the ideal ratio that can be achieved by using this epicyclic compound 3 × 2, is 50 times less than the classic epicyclic couple which is used on model of Freeth and Jones (2012) (see last row of Table 1). Also the compact dimensions of this compound epicyclic gearing makes it more usable especially in
(17) Nx1 , Ny1
λ=
Ny3 ) Nx 3
it is obvious that the fraction
must not be irreducible. Searching the Table 1 for the optimum ratio
with the minimum deviation which is simultaneously compliant with the latest requirement it was chosen the seventh combination Nx = 147 and Ny = 235. The terms of the fraction 147 can be factorized as 235 147 = 49·3 and 235 = 47·5. From Fig. 11 it is concluded that: Rx1 + Ry1 = Rx 2 + Ry2 = Rx 3 + Ry3. and since the meshed gears of the compound epicyclic must have the same module (see Eq. (2)) it is
19 A gear train consisted form eleven meshed gears which was used from the Antikythera mechanism constructor to simulate the non-uniform motion of the moon pointer.
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x=
m·Ny m·Nx ,y= 2 2
(18)
2·(x + y ) 2·27.1 mm = ≈ 0.5625 Nx + Ny 96
(19)
and
m=
where m is the module of gears (see Section 4) This is an acceptable module according to Freeth et al. (2006). 6.4.1. 3d simulation of apparent Venus trajectory in GeoGebra The 3d preview ability of GeoGebra10 was exploited in order the three levels of epicyclic gearing to be distinguishable. In this phase were used the pitch circles of the six gears and for their rotation were used the frequencies as they determined in Section 6. The result of this 3d simulation is shown in Fig. 13
Fig. 12. A similar gear arrangement to epicyclic 3×2 inside the moon train - Source: An exploded view using the 3d Interactive Simulation of the Antikythera Mechanism (Diolatzis and Pavlogeorgatos, 2017).
Fig. 13. 3d simulation of Venus epitrochoid by compound epicyclic “3×2” utilizing the ratio 235:147 - Source: Author, 2018.
restricted compartments like the available in the Antikythera mechanism. It must be mention here that this combination (147,235) although is closer to theoretical (720,1151) it is rejected from T. Freeth and A. Jones as unreachable using their limited options. “…Mathematical analysis shows that this list is comprehensive, except for the period relations (147, 235) for Venus and (96, −205) for Mars. Since the large numbers 235 and 205 are unsuitable for our application, we have discarded these possibilities… (Freeth and Jones, 2012)
6.5. Mechanical simulation of Venus apparent epitrochoid In this subsection there is a description of how the above simulation gained a more tangible form. GeoGebra10 has the ability to manipulate inserted images as objects. Every inserted image could embed with any geometrical object. The idea was to connect images of all involved gears with their pitch circles. The animation repeated and the result is shown in the Fig. 14. It must be mentioned here that during the animation it was not observed any overlapping between teeth and the gears were engaged smoothly.
6.4. Simulation in GeoGebra of Venus apparent trajectory with the use of the epicyclic “3 × 2”
7. Completion of the proposal
This compound epicyclic gearing could be implemented to simulate the apparent trajectory of Venus in Antikythera mechanism. The scope of this research is not a strict reconstruction of the lost planetary subsystem in Antikythera mechanism but a suggestion of a more reliable epicyclic gearing in order to eliminate the indications errors. For that reason, the axis of satellite gears Ny1, Ny2, Ny3 could be located in a position which have already been proposed …In the 4 o′clock position, there is a prominent hole, which looks like the remains of a bearing. Its outer diameter is 9.7 mm and its inner diameter is 6.6 mm. It is 27.1 mm from the central axis. … (Freeth and Jones, 2012). Now if this bearing is accepted for the satellite gears axis then:
According to “…For Mars, the inscription's 349 days between conjunction and the stations and 82 days for the retrogradation between the stations agrees well with the accurate values, 354 and 72 days …” (Jones, 2017), a detailed orbital view of all planets, could be projected on the top of Antikythera mechanism. 7.1. Characteristic orbital positions As seen from a planet that is superior, if an inferior planet is on the opposite side of the Sun, it is in superior conjunction with the Sun. An inferior conjunction occurs when the two planets lie in a line 9
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Fig. 14. Mechanical simulation of Venus epitrochoid by compound epicyclic “3 × 2” utilizing the ratio 235:147 in 16 years time depth - Source: Author, 2018.
Fig. 15. 3d modeling of Venus orbit simulated in UNITY3D22 by compound epicyclic “3 × 2” utilizing the ratio 235:147 in 9 years time depth (isolated Venus epicyclic gearing) - Source: Author, 2018.
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Fig. 16. 3d Venus modulation in UNITY3D22 compatible with the 3 × 2 epicyclic gearing - Up: a superior conjunction - Down: In inferior conjunction - Source: Author, 2018.
environment. A very important effect which UNITY3D22 offers is the trail renderer. If this feature is added to a planet (little sphere) then its path is traced. This could be functions as an additional evaluation criterion because the form of the imprinted trajectory could be compared with the theoretical one (see Fig. 15)
on the same side of the Sun. In an inferior conjunction, the superior planet is “in opposition” to the Sun as seen from the inferior planet. All above referred positions occur when the planet, Earth and Sun lie in line. Also there are two stationary points the first stationary when the planet enters the retrograde motion and the second stationary when exit from retrograde motion. In these positions the planet is moving across the observing line and appears as stationary.
8. Epilogue Antikythera mechanism is characterized as a complicated gearing machine with an advanced technology far enough from its era. In many parts of the mechanism there is an excellent balance between the complexity and the precision. Our proposed (3d modeled) Venus epicyclic gearing gives a compatible image (little sphere moving on a circle) according to BCI3 and we believe that this configuration (sphere with slotted pointer see Fig. 15) must also be implemented for the rest of the planets. The advantages of using this combination (sphere slotted arm pointer) are:
7.2. 3d modeling of the proposal All above could be well indicated if the planets are represented by little spheres connected with slotted pointers. This imaginary figure, guide us to make a 3d modulation in CINEMA 4D20 and a further editing in UNITY3D22. Regarding the 3d modeling in CINEMA 4D21 we used a technique known as Interactive Simulation of Rigid Body Dynamics where the gears are simulated as rigid bodies. So the gears rotation is the result of their physical interaction. In UNITY3D21 the animated model gained interactivity in an attractive photo realistic
• • • •
20 3D modeling, animation and rendering application developed by MAXON Computer GmbH. 21 Unity 3d is a multi-platform game development tool that allows textures compression and customization of resolution settings.
11
Graphical display of geocentric planets orbits Recording of astronomical events like Superior-Inferior conjunctions (See Fig. 16) Compatibility with the BCI3 inscriptions (planets presented by little spheres moving on circles) Closer approach to Babylonians time records
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• •
Zafeiropoulou, M., Hadland, R., Bate, D., Ramsey, A., et al., (2006). Decoding the antikythera mechanism: Investigation of an ancient astronomical calculator 2 supplementary notes. Freeth, T., 2012. Building the cosmos in the antikythera mechanism. In: From Antikythera to the Square Kilometre Array: Lessons from the Ancients. Jones, A., 2017. A Portable Cosmos: Revealing the Antikythera Mechanism, Scientific Wonder of the Ancient World. Oxford University Press. Maitra, G.M., 1994. Handbook of Gear Design. Tata McGraw-Hill Education. Margolis, H., 1993. Paradigms and Barriers: How Habits of Mind Govern Scientific Beliefs. University of Chicago Press. Neugebauer, O., 2012. A History of Ancient Mathematical Astronomy 1. Springer Science & Business Media. Ragnarsson, S.-I., 1995. Planetary distances: a new simplified model. Astron. Astrophys. 301, 609. Safronov, A., 2016. Antikythera mechanism and the ancient world. J. Archaeol. 2016. Spitas, C.A., Spitas, V.A., 2006. Generating interchangeable 200 spur gear sets with circular fillets to increase load carrying capacity. In: Proceedings of the International Conference on Gears, volume 1, pp. 927–941. Strom, R.G., Sprague, A.L., 2003. Exploring Mercury: The Iron Planet. Springer Science & Business Media. D. R. Williams, Planetary fact sheets, National Space Science DataCenter (NSSDC) 6 (2005). Wright, M., 2013. The antikythera mechanism: compound gear-trains for planetary indications. Almagest 4, 4–31.
Better space management High accuracy in the chronological determination of astronomical events
Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at doi:10.1016/j.daach.2018.e00089. References Black, J.T., Kohser, R.A., 2017. DeGarmo's Materials and Processes in Manufacturing. John Wiley & Sons. Diolatzis, I.S., Pavlogeorgatos, G., 2017. Deepening to antikythera mechanism via its interactivity. Digit. Appl. Archaeol. Cult. Herit.. Evans, J., Carman, C.C., Thorndike, A.S., 2010. Solar anomaly and planetary displays in the antikythera mechanism. J. Hist. Astron. 41, 1–39. Fitzpatrick, R., 2010. Determination of conjunction and greatest elongation dates. Freeth, T., Jones, A., 2012. The cosmos in the Antikythera mechanism. Inst. Study Anc. World. Freeth, T., Bitsakis, Y., Moussas, X., Seiradakis, J., Tselikas, A., Magkou, E.,
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