Editing Antikythera mechanism (section)

===Gearing===
The mechanism is remarkable for the level of miniaturisation and the complexity of its parts, which is comparable to that of 14th-century [[astronomical clock]]s. It has at least 30 gears, although mechanism expert Michael Wright has suggested the Greeks of this period were capable of implementing a system with many more gears.<ref name=freeth-09/>

There is debate as to whether the mechanism had indicators for all five of the planets known to the ancient Greeks. No gearing for such a planetary display survives and all gears are accounted for—with the exception of one 63-toothed gear (r1) otherwise unaccounted for in fragment D.<ref name=freeth-12/>

Fragment D is a small quasi-circular constriction that, according to Xenophon Moussas, has a gear inside a somewhat larger hollow gear. The inner gear moves inside the outer gear reproducing an epicyclical motion that, with a pointer, gives the position of planet Jupiter.<ref name="auto"/> The inner gear is numbered 45, "ME" in Greek, and the same number is written on two surfaces of this small cylindrical box.

The purpose of the front face was to position astronomical bodies with respect to the [[celestial sphere]] along the ecliptic, in reference to the observer's position on the Earth. That is irrelevant to the question of whether that position was computed using a heliocentric or geocentric view of the [[Solar System]]; either computational method should, and does, result in the same position (ignoring ellipticity), within the error factors of the mechanism. The epicyclic Solar System of [[Ptolemy]] ({{circa|100 AD}}–{{circa|170 AD}})—hundreds of years after the apparent construction date of the mechanism—carried forward with more epicycles, and was more accurate predicting the positions of planets than the view of [[Nicolaus Copernicus|Copernicus]] (1473–1543), until [[Johannes Kepler|Kepler]] (1571–1630) introduced the possibility that orbits are ellipses.<ref name=amrp-07-2/>

Evans et al. suggest that to display the mean positions of the five [[classical planets]] would require only 17 further gears that could be positioned in front of the large driving gear and indicated using individual circular dials on the face.<ref name=cte-10/>

Freeth and Jones modelled and published details of a version using gear trains mechanically similar to the lunar anomaly system, allowing for indication of the positions of the planets, as well as synthesis of the Sun anomaly. Their system, they claim, is more authentic than Wright's model, as it uses the known skills of the Greeks and does not add excessive complexity or internal stresses to the machine.<ref name=freeth-12/>

The gear teeth were in the form of [[equilateral triangle]]s with an average circular pitch of 1.6&nbsp;mm, an average wheel thickness of 1.4&nbsp;mm and an average air gap between gears of 1.2&nbsp;mm. The teeth were probably created from a blank bronze round using hand tools; this is evident because not all of them are even.<ref name="freeth-12"/> Due to advances in imaging and [[X-ray computed tomography|X-ray]] technology, it is now possible to know the precise number of teeth and size of the gears within the located fragments. Thus the basic operation of the device is no longer a mystery and has been replicated accurately. The major unknown remains the question of the presence and nature of any planet indicators.<ref name=freeth-06-1 />{{rp|8}}

A table of the gears, their teeth, and the expected and computed rotations of important gears follows. The gear functions come from Freeth et al. (2008)<ref name=freeth-08/> and for the lower half of the table from Freeth et al. (2012).<ref name=freeth-12/> The computed values start with 1 year per revolution for the b1 gear, and the remainder are computed directly from gear teeth ratios. The gears marked with an asterisk (*) are missing, or have predecessors missing, from the known mechanism; these gears have been calculated with reasonable gear teeth counts.<ref name=freeth-08/><ref name=freeth-06-1/> (Lengths in days are calculated assuming the year to be 365.2425 days.)

{| class="wikitable"
|+ The Antikythera Mechanism: known and proposed gears and accuracy of computation
|-
! Gear name<ref group=table>Change from traditional naming: X is the main year axis, turns once per year with gear B1. The B axis is the axis with gears B3 and B6, while the E axis is the axis with gears E3 and E4. Other axes on E (E1/E6 and E2/E5) are irrelevant to this table.</ref>
! Function of the gear/pointer
! Length of time for a full circular revolution
! Mechanism formula<ref group=table>"Time" is the interval represented by one complete revolution of the gear.</ref>
! Computed interval
! Gear direction<ref group=table>As viewed from the front of the Mechanism.  The "natural" view is viewing the side of the Mechanism the dial/pointer in question is actually displayed on.</ref>
|-
! x
| Year gear
| 1 tropical year
| 1 (by definition)
| 1 year (presumed)
| clockwise<ref group=table>The Greeks, being in the northern hemisphere, assumed proper daily motion of the stars was from east to west, anticlockwise when the ecliptic and zodiac is viewed to the south. As viewed on the front of the Mechanism.</ref>
|-
! b
| The Moon's orbit
| 1 sidereal month (27.321661 days)
| Time(b) = Time(x) * (c1/b2) * (d1/c2) * (e2/d2) * (k1/e5) * (e6/k2) * (b3/e1)
| 27.321 days<ref group=table name=epicycle>On average, due to epicyclic gearing causing accelerations and decelerations.</ref>
| clockwise
|-
! r
| Lunar phase display
| 1 synodic month (29.530589 days)
| Time(r) = 1 / (1 / Time(b2: ''mean sun'' or sun3: ''true sun'')) – (1 / Time(b)))
| 29.530 days<ref group=table name=epicycle/>
|
|-
! n*
| Metonic pointer
| Metonic cycle / 5 turns = 1387.94 days
| Time(n) = Time(x) * (l1/b2) * (m1/l2) * (n1/m2)
| 1387.9 days
| anticlockwise<ref group=table name=boxside>Being on the reverse side of the box, the "natural" rotation is the opposite</ref>
|-
! o*
| Games dial pointer
| 4 years (5551.8 days)
| Time(o) = Time(n) * (o1/n2)
| 4.00 years
| clockwise<ref group=table name=boxside/><ref group=table>This was the only visual pointer naturally travelling in the anticlockwise direction.</ref>
|-
! q*
| Callippic pointer
| 27758.8 days
| Time(q) = Time(n) * (p1/n3) * (q1/p2)
| 27758 days
| anticlockwise<ref group=table name=boxside/>
|-
! e*
| Lunar orbit precession
| 8.88 years (3244.37 days)
| Time(e) = Time(x) * (l1/b2) * (m1/l2) * (e3/m3)
| 8.8826 years
| anticlockwise<ref group=table>Internal and not visible.</ref>
|-
! g*
| Saros cycle
| Saros time / 4 turns = 1646.33 days
| Time(g) = Time(e) * (f1/e4) * (g1/f2)
| 1646.3 days
| anticlockwise<ref group=table name=boxside/>
|-
! i*
| Exeligmos pointer
| 19755.8 days
| Time(i) = Time(g) * (h1/g2) * (i1/h2)
| 19756 days
| anticlockwise<ref group=table name=boxside/>
|-
! colspan="6" style="text-align: center;" |The following are proposed gearing from the 2012 Freeth and Jones reconstruction:
|-
! sun3*
| True sun pointer
| 1 mean year
| Time(sun3) = Time(x) * (sun3/sun1) * (sun2/sun3)
| 1 mean year<ref group=table name=epicycle/>
| clockwise<ref group=table name=retro>Prograde motion; retrograde is obviously the opposite direction.</ref>
|-
! mer2*
| Mercury pointer
| 115.88 days (synodic period)
| Time(mer2) = Time(x) * (mer2/mer1)
| 115.89 days<ref group=table name=epicycle/>
| clockwise<ref group=table name=retro/>
|-
! ven2*
| Venus pointer
| 583.93 days (synodic period)
| Time(ven) = Time(x) * (ven1/sun1)
| 584.39 days<ref group=table name=epicycle/>
| clockwise<ref group=table name=retro/>
|-
! mars4*
| Mars pointer
| 779.96 days (synodic period)
| Time(mars) = Time(x) * (mars2/mars1) * (mars4/mars3)
| 779.84 days<ref group=table name=epicycle/>
| clockwise<ref group=table name=retro/>
|-
! jup4*
| Jupiter pointer
| 398.88 days (synodic period)
| Time(jup) = Time(x) * (jup2/jup1) * (jup4/jup3)
| 398.88 days<ref group=table name=epicycle/>
| clockwise<ref group=table name=retro/>
|-
! sat4*
| Saturn pointer
| 378.09 days (synodic period)
| Time(sat) = Time(x) * (sat2/sat1) * (sat4/sat3)
| 378.06 days<ref group=table name=epicycle/>
| clockwise<ref group=table name=retro/>
|}
 
''Table notes:''
{{reflist|group=table|30em}}

There are several gear ratios for each planet that result in close matches to the correct values for synodic periods of the planets and the Sun. Those chosen above seem accurate, with reasonable tooth counts, but the specific gears actually used are unknown.<ref name=freeth-12/>

==== Known gear scheme ====
[[file:AntikytheraMechanismSchematic-Freeth12.png|thumb|upright=1.4|A hypothetical schematic representation of the gearing of the Antikythera Mechanism, including the 2012 published interpretation of existing gearing, gearing added to complete known functions, and proposed gearing to accomplish additional functions, namely true sun pointer and pointers for the five then-known planets, as proposed by Freeth and Jones, 2012.<ref name=freeth-12/> Based also upon similar drawing in the Freeth 2006 Supplement<ref name=freeth-06-1/> and Wright 2005, Epicycles Part 2.<ref name="wright-05-1"/> Proposed (as opposed to known from the arte<!-- 'e' - this article uses British spelling -->fact) gearing crosshatched.]]

It is very probable there were planetary dials, as the complicated motions and periodicities of all planets are mentioned in the manual of the mechanism. The exact position and mechanisms for the gears of the planets is unknown. There is no coaxial system except for the Moon. Fragment D that is an epicycloidal system, is considered as a planetary gear for Jupiter (Moussas, 2011, 2012, 2014) or a gear for the motion of the Sun (University of Thessaloniki group).

The Sun gear is operated from the hand-operated crank (connected to gear a1, driving the large four-spoked mean Sun gear, b1) and in turn drives the rest of the gear sets. The Sun gear is b1/b2 and b2 has 64 teeth. It directly drives the date/mean sun pointer (there may have been a second, "true sun" pointer that displayed the Sun's elliptical anomaly; it is discussed below in the Freeth reconstruction). In this discussion, reference is to modelled rotational period of various pointers and indicators; they all assume the input rotation of the b1 gear of 360 degrees, corresponding with one tropical year, and are computed solely on the basis of the gear ratios of the gears named.<ref name=freeth-06/><ref name=freeth-08/><ref name=ieeecomp1/>

The Moon train starts with gear b1 and proceeds through c1, c2, d1, d2, e2, e5, k1, k2, e6, e1, and b3 to the Moon pointer on the front face. The gears k1 and k2 form an [[epicyclic gearing|epicyclic gear system]]; they are an identical pair of gears that do not mesh, but rather, they operate face-to-face, with a short pin on k1 inserted into a slot in k2. The two gears have different centres of rotation, so the pin must move back and forth in the slot. That increases and decreases the radius at which k2 is driven, also necessarily varying its angular velocity (presuming the velocity of k1 is even) faster in some parts of the rotation than others. Over an entire revolution the average velocities are the same, but the fast-slow variation models the effects of the elliptical orbit of the Moon, in consequence of [[Kepler's laws of planetary motion#Position as a function of time|Kepler's second and third laws]]. The modelled rotational period of the Moon pointer (averaged over a year) is 27.321 days, compared to the modern length of a lunar sidereal month of 27.321661 days. The pin/slot driving of the k1/k2 gears varies the displacement over a year's time, and the mounting of those two gears on the e3 gear supplies a precessional advancement to the ellipticity modelling with a period of 8.8826 years, compared with the current value of precession period of the moon of 8.85 years.<ref name=freeth-06/><ref name=freeth-08/><ref name=ieeecomp1/>

The system also models the [[lunar phase|phases of the Moon]]. The Moon pointer holds a shaft along its length, on which is mounted a small gear named r, which meshes to the Sun pointer at B0 (the connection between B0 and the rest of B is not visible in the original mechanism, so whether b0 is the current date/mean Sun pointer or a hypothetical true Sun pointer is unknown). The gear rides around the dial with the Moon, but is also geared to the Sun—the effect is to perform a [[differential gear]] operation, so the gear turns at the synodic month period, measuring in effect, the angle of the difference between the Sun and Moon pointers. The gear drives a small ball that appears through an opening in the Moon pointer's face, painted longitudinally half white and half black, displaying the phases pictorially. It turns with a modelled rotational period of 29.53 days; the modern value for the synodic month is 29.530589 days.<ref name=freeth-06/><ref name=freeth-08/><ref name=ieeecomp1/>

The Metonic train is driven by the drive train b1, b2, l1, l2, m1, m2, and n1, which is connected to the pointer. The modelled rotational period of the pointer is the length of the 6939.5 days (over the whole five-rotation spiral), while the modern value for the [[Metonic cycle]] is 6939.69 days.<ref name=freeth-06/><ref name=freeth-08/><ref name=ieeecomp1/>

The [[Olympiad]] train is driven by b1, b2, l1, l2, m1, m2, n1, n2, and o1, which mounts the pointer. It has a computed modelled rotational period of exactly four years, as expected. It is the only pointer on the mechanism that rotates anticlockwise; all of the others rotate clockwise.<ref name=freeth-06/><ref name=freeth-08/><ref name=ieeecomp1/>

The Callippic train is driven by b1, b2, l1, l2, m1, m2, n1, n3, p1, p2, and q1, which mounts the pointer. It has a computed modelled rotational period of 27758 days, while the modern value is 27758.8 days.<ref name=freeth-06/><ref name=freeth-08/><ref name=ieeecomp1/>

The Saros train is driven by b1, b2, l1, l2, m1, m3, e3, e4, f1, f2, and g1, which mounts the pointer. The modelled rotational period of the Saros pointer is 1646.3 days (in four rotations along the spiral pointer track); the modern value is 1646.33 days.<ref name=freeth-06/><ref name=freeth-08/><ref name=ieeecomp1/>

The Exeligmos train is driven by b1, b2, l1, l2, m1, m3, e3, e4, f1, f2, g1, g2, h1, h2, and i1, which mounts the pointer. The modelled rotational period of the exeligmos pointer is 19,756 days; the modern value is 19755.96 days.<ref name=freeth-06/><ref name=freeth-08/><ref name=ieeecomp1/>

It appears gears m3, n1-3, p1-2, and q1 did not survive in the wreckage. The functions of the pointers were deduced from the remains of the dials on the back face, and reasonable, appropriate gearage to fulfill the functions was proposed and is generally accepted.<ref name=freeth-06/><ref name=freeth-08/><ref name=ieeecomp1/>