A Model of the Cosmos in the ancient Greek Antikythera Mechanism

The Antikythera Mechanism, an ancient Greek astronomical calculator, has challenged researchers since its discovery in 1901. Now split into 82 fragments, only a third of the original survives, including 30 corroded bronze gearwheels. Microfocus X-ray Computed Tomography (X-ray CT) in 2005 decoded the structure of the rear of the machine but the front remained largely unresolved. X-ray CT also revealed inscriptions describing the motions of the Sun, Moon and all five planets known in antiquity and how they were displayed at the front as an ancient Greek Cosmos. Inscriptions specifying complex planetary periods forced new thinking on the mechanization of this Cosmos, but no previous reconstruction has come close to matching the data. Our discoveries lead to a new model, satisfying and explaining the evidence. Solving this complex 3D puzzle reveals a creation of genius—combining cycles from Babylonian astronomy, mathematics from Plato’s Academy and ancient Greek astronomical theories.


Scientific Reports
| (2021) 11:5821 | https://doi.org/10.1038/s41598-021-84310-w www.nature.com/scientificreports/ all the data [1][2][3]6,9 . The evidence defines a framework for an epicyclic system at the front 9 , but the spaces available for the gears are extremely limited. There were also unexplained components in Fragment D, revealed by the X-ray CT, and technical difficulties calculating the phase of the Moon 9 . Then came the discovery in the tomography of surprisingly complex periods for the planets Venus and Saturn, making the task very much harder 12 .

Discussion and results
We wanted to determine the cycles for all the planets in this Cosmos (not just the cycles discovered for Venus and Saturn); to incorporate these cycles into highly compact mechanisms, conforming to the physical evidence; and to interleave them so their outputs correspond to the customary cosmological order (CCO), described below.
Here we show how we have created gearing and a display that respects the inscriptional evidence: a ring system with nine outputs-Moon, Nodes, Mercury, Venus, Sun, Mars, Jupiter, Saturn and Date-carried by nested tubes with arms supporting the rings. The result is a radical new model that matches all the data and culminates in an elegant display of the ancient Greek Cosmos. With so much missing, we ensure the integrity of our model with a strict set of Reconstruction Principles (Supplementary Discussion S1) and we assess the strength of data that validates each element-discussed in Supplementary Discussion S1. The loss of evidence might suggest many options for a model. What has struck us forcefully in making the present model is just how few these options are: the constraints created by the surviving evidence are stringent and very difficult to meet.

Inscriptional evidence
Reconstructing the Cosmos at the front of the Antikythera Mechanism begins with analysing some remarkable inscriptions. Figure 1 shows the Front and Back Cover Inscriptions (FCI & BCI) 9,12,13 , which are critical for understanding this Cosmos. For previous analysis 13 and our own line-by-line exploration of the BCI, see Supplementary Discussion S2. The BCI describes the front display as a Planetarium 9,13 : a Cosmos arranged in rings, with planets marked by "little spheres" and the Sun as a "little golden sphere" with "ray" and "pointer" (Fig. 1c, Supplementary Table S1, Supplementary Figs. S2, S3). The FCI lists the synodic cycles of the planets (cycles relative to the Sun) 12 . This is a systematic list, itemizing the synodic events and the intervals in days between them. The planets are written in the same geocentric order as the BCI. Adding Moon and Sun gives the customary cosmological order (CCO): Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn ( Supplementary Fig. S4), whose origins are discussed in Supplementary Discussion S2.

Period relations and ancient Greek theories
Ancient astronomers were fascinated by the motions of the planets. As seen from Earth, they exhibit periodic reversals of motion against the stars 14 . In Babylonian astronomy these synodic cycles were the basis of planetary prediction 15 , utilizing period relations, such as 5 synodic cycles in 8 years for Venus, which we denote by (5,8).
The FCI describes synodic events, such as stationary points, and intervals between these events ( Apollonios of Perga (third-second century BC) created elegant (albeit inaccurate) epicyclic theories to explain these anomalous movements as the sum of two uniform circular motions, their periods defined by period relations-the deferent and epicycle models 15 (Supplementary Discussion S3, Supplementary Figs. S6, S7, S8). Such theories were certainly employed in the Antikythera Mechanism, given that the Moon was mechanized using a similar epicyclic theory 7 . The true Sun-the Sun with its variable motion-was also explained in ancient Greece by eccentric and equivalent epicyclic models 14 (Supplementary Discussion S3).
Babylonian texts list planetary periods and their errors: shorter, less accurate periods in Goal-Year Texts (GYT) and longer, more accurate periods in later Astronomical Cuneiform Texts (ACT) 15 (Supplementary Tables S3, S4). The GYT periods could have been derived from observations, but not the longer ACT periods, such as (720, 1151) for Venus (Supplementary Discussion S3). To understand what period relations were built into the Antikythera Mechanism, the tough problem was to discover their derivation. For Venus the original designer faced a dilemma: the known period relation (5,8) was very inaccurate, whereas the accurate (720, 1151) was not mechanizable because 1151 is a prime number, requiring a gear with 1151 teeth. Then came a notable discovery in 2016 in the FCI 12 : unexpected numbers ϒΞΒ (462) in the Venus section of the FCI and ϒMΒ (442) in the Saturn section, translating into highly accurate period relations: for Venus (289, 462) and Saturn (427, 442) (Fig. 1b, Supplementary Fig. S4). Crucially, these are factorizable, meaning they can be mechanized with moderate-sized gears, with tooth counts incorporating the prime factors of the period relations. To fit the geometry of the epicyclic system, mechanisms must have gears with < 100 teeth: period relations must have prime factors < 100 (Supplementary Discussion S3. There are few such accurate period relations for the planets (Supplementary Tables S5, S6).
The fact that the new period relations for Venus and Saturn from the FCI are factorizable strongly reinforces the idea that they were incorporated into planetary mechanisms in the Antikythera Mechanism 16 . The periods for the other planets are unreadable (in missing or damaged areas of the FCI). To build our model, it was essential to discover the period relations embodied in all the planetary mechanisms. Previous publications 12,16 derived the Venus period relation (289, 462) as an iterative approximation to the known Babylonian (720, 1151) period relation, using a number of equivalent processes: continued fractions, anthyphairesis or the Euclidean algorithm 17,18 . No similar method for deriving the (427, 442) period relation for Saturn could be found, so this type of iterative approximation was almost certainly not the route to the original discoveries of these periods by the ancient Greeks.

Discovering cycles in the Antikythera Mechanism
The newly-discovered periods for Venus and Saturn are unknown from studies of Babylonian astronomy. Figure 2 explores how these periods might have been derived. Clues came from the Babylonian use of linear combinations of periods designed to cancel out observed errors 14 . Figure 2a shows how this might generate the periods  www.nature.com/scientificreports/ for Venus and Saturn, but choosing the correct linear combinations essentially uses knowledge about errors in known period relations relative to the true value. The lack of fine error-estimates from antiquity excludes these methods for our model: errors like < 1° in 100 years for (720, 1151) were beyond the naked-eye astronomy of the Hellenistic age. We have developed a new theory about how the Venus and Saturn periods were discovered and apply this to restore the missing planetary periods. A dialogue of Plato 19 (fifth-fourth century BC) was named after the philosopher Parmenides of Elea (sixth-fifth century BC). This describes Parmenides Proposition 17,18 : In approximating θ, suppose rationals, p/q and r/s, satisfy p/q < θ < r/s. Then (p + r)/(q + s) is a new estimate between p/q and r/s: If it is an underestimate, it is a better underestimate than p/q. If it is an overestimate, it is a better overestimate than r/s.
Assuming it is a better underestimate, the next stage combines this with the original overestimate to create (p + 2r)/(q + 2s). This would be tested against q and the process repeated. Thus, from two seed ratios we can generate increasingly accurate linear combinations that converge to θ. The Parmenides process is facilitated and constrained by knowledge of θ to determine whether each new estimate is an under-or over-estimate. Figure 2b shows how a conventional Parmenides Process can generate our target periods, but again this relies on unavailable knowledge about errors. The key step for discovering the missing cycles is to modify the Parmenides Process, so it is not constrained by knowledge of errors-an Unconstrained Parmenides Process (UPP). Here we clarify how we believe the process was used. The designer would have generated linear combinations using the UPP. At each stage, these possible period relations would have been examined to see if they met the designer's criteria of accuracy, factorizability and economy. Factorizability would have been an easy criterion to assess. Accuracy is more problematic, since we do not believe that ancient astronomers had the ability to make very accurate astronomical observations, as is witnessed by the Babylonian records (Supplementary Tables S3,  S4). Economy must be examined in relationship with the period relations generated for the other inferior or superior planets to identify shared prime factors.
Venus is a good example. The ancient Babylonians knew that the (5, 8) period for Venus was very inaccurate and they had derived the unfactorizable (720, 1151) from observation of an error in the 8-year cycle (Supplementary Discussion S3). Such periods were often described in the ancient world as "exact periods", though of course in modern terms this is not the case. When the factorizable period (289, 462) was discovered from the UPP, it would have been easy to calculate that it is in fact very close to the "exact period" (720, 1151). Thus, the designer would have been confident that it was an accurate period. (289, 462) would then have been compared with (1513, 480) for Mercury to discover that they shared the common factor 17 in the number of synodic cycles-meaning that they were suitable for use in a shared-gear design to satisfy the criterion of economy. When the designer had discovered period relations that matched all the criteria, the process would have been stopped, since further iterations would likely have lead to solutions of greater complexity.
The UPP, combined with our three criteria, leads to remarkably simple derivations of the Venus and Saturn period relations. For Venus, Fig. 2d shows that the first factorizable period relation is (1445, 2310) = 5 × (289, 462) ≡ (289, 462) = (17 2 , 2 × 3 × 7 × 11), as found in the FCI. For Saturn, it is (427, 442) = (7 × 61, 2 × 13 × 17), again from the FCI. This discovery enables derivations of the missing planetary periods. To ensure our third criterion of economy, some of the prime factors of the synodic cycles must be incorporated into the first fixed gear of a planetary train (Supplementary Discussion S4). For Mercury, we are looking for a factor of 17 in the number of synodic cycles to share with Venus. The first factorizable iteration is (1513, 480) = (17 × 89, 2 5 × 3 × 5)-sharing the prime factor 17 with (289, 462) for Venus-so, a very good choice. Multiplying by integers to obtain viable gears leads to economical designs with a single fixed 51-tooth gear shared between Mercury and Venus (Fig. 3c, e) 16 . For the superior planets, Mars and Jupiter, we are looking for synodic periods that share the factor 7 with Saturn ( Fig. 3d, f). Just a few iterations yield suitable synodic periods-leading to very economical designs with a single 56-tooth fixed gear for all three superior planets and the true Sun.
From Supplementary Table S5,S6, in Supplementary Discussion S3 we establish that the missing periods for Mercury and Mars are uniquely determined by our process. There are two additional options for Jupiter that share the prime 7 in the number of synodic cycles (Supplementary Table S6). In Supplementary Discussion S3 we show how one of these is not possible and the other is very unlikely. The UPP, combined with criteria of accuracy, factorizability and economy, explains the Venus and Saturn periods and (almost) uniquely generates the missing period relations.

Theoretical mechanisms for our model
The calculation of the Moon's position in the Zodiac and its phase are defined by surviving physical evidence 7,10 . Since the evidence is missing for the Sun and planets, we need to develop theoretical mechanisms, based on our identified period relations. Figure 3 shows theoretical gear trains for the mean Sun, Nodes and the Planets.
Geometrical parameters for the planetary mechanisms in Fig. 3c The way that the Saros Dial on the Back Plate predicts eclipses essentially involves the lunar nodes, but they are not described in the extant inscriptions. With their integral role in eclipses, a display of the nodes is a logical inclusion, unifying Front and Back Dials. To maximise the displayed information, we created a mechanism for a hypothetical Dragon Hand to indicate the Line of Nodes of the Moon, as included in many later astronomical clocks 20 (Supplementary Fig. S2). We should emphasize that there is no direct physical evidence for an indication of the Line of Nodes of the Moon. We have added this feature as a hypothetical element for the thematic reasons already explained and because it is easily mechanized to good accuracy with a simple 4-gear epicyclic system on Spoke B of b1. It is an interesting option for the reader to consider and it coincides with the designer's . Epicyclic Mechanisms for the Cosmos. Fixed gears are underlined; blue gears calculate synodic cycles; red gears calculate years; black gears are idler gears: all designated by their tooth counts. " ~ " means "meshes with"; " + " means "fixed to the same arbor"; "⊕" means "with a pin-and-follower, turning on the central axis" or "with a pin-and-slot on eccentric axes"-creating variable motion (turquoise). Followers are slotted rods that follow a pin on the epicyclic gear and turn on the central axis. For each mechanism, there is a fixed gear at the centre, meshing with the first epicyclic gear, which is forced to rotate by the rotation of b1 or the CP. www.nature.com/scientificreports/ apparent ambition to create an astronomical compendium, displaying most of the astronomical parameters that preoccupied Hellenistic astronomy. All the Cosmos mechanisms must output in the CCO, so that they are consistent with the description in the BCI. At the centre of this Cosmos is the Earth, then the Moon's position in the Zodiac and lunar phase. The Moon's position is carried by the central arbor linked to the (mostly) surviving epicyclic system that calculates the Moon's variable motion at the back of the Mechanism (Supplementary Fig. S1) 7 . We follow the original proposal 10 for the Moon phase device as a simple differential, which subtracts the motion of the Sun from that of the Moon to calculate the phase, displayed on a small black and white sphere.
A rotation of − 12 223 for the Line of Nodes, derived from the Metonic and Saros cycles 9 , could not be mechanized because of the large prime 223. We show that a simpler ratio − 5 93 , with a more accurate period of 18.6 years 14 , can be calculated by a 4-gear epicyclic train (Fig. 3a, Supplementary Figs. S21, S22). This turns a hypothetical double-ended Dragon Hand 20 , whose Head shows the ascending node of the Moon and Tail the descending node.
Using our identified period relations for all the planets, we have devised new theoretical planetary mechanisms expressing the epicyclic theories, which fit the physical evidence. For the inferior planets, previous 2-gear mechanisms 3,9,21 are inadequate for more complex period relations because the gears would be too large. Twostage compound trains with idler gears are necessary, leading to new 5-gear mechanisms with pin-and-slotted followers for the variable motions 7,9,21 (Fig. 3c). For the superior planets, earlier models 3,16 used direct mechanisms, directly reflecting epicyclic theories with pin-and-slotted followers. Here we propose novel 7-gear indirect mechanisms with pin-and-slot devices 7,9 for variable motions (Fig. 3d), analogous to the subtle mechanism that drives the lunar anomaly 7 . Compared to direct mechanisms, these are more economical; a better match for the evidence; and incorporate period relations exactly for higher accuracy. The crucial advantages of indirect mechanisms are expanded in Supplementary Discussion S4. Without these compact systems that can all be mounted on the same plate, it would have been impossible to fit the gearing into the available spaces. Proofs that the mechanisms in Fig. 3 correctly calculate the ancient Greek epicyclic theories are included in Supplementary Discussion S4.
The key question: could we match our theoretical mechanisms to the physical data? Fig. 4 shows some of the challenging evidence from Fragment A 7,22 (Supplementary Figs. S9, S10, S11, S12) and Fragment D 1-3,7,9 (Supplementary Figs. S13, S14). Any model must be consistent with these data (Supplementary Discussion S5) as well as conform to horological/engineering principles from the rest of the Mechanism (Supplementary Figs. S15, S16).

Fragment A: essential framework for reconstruction
The Main Drive Wheel, b1, has four spokes with prominent holes, flattened areas and damaged pillars on its periphery ( Fig. 4a-h, Supplementary Figs. S11, S12)-definitive evidence of a complex epicyclic system [1][2][3]9 . In the original Mechanism, there were four short and four long pillars with shoulders and holes for retaining pins, as shown in Fig. 4a-d by the X-ray CT evidence. These imply that the pillars carried plates: a rectangular plate on the short pillars, the Strap, and a circular plate on the long pillars, the Circular Plate (CP) (Fig. 4m-o) 9 . This is the essential framework for any faithful reconstruction, with the four spokes advocating four different functions (Fig. 4e-h). First, we reconstruct the mechanisms between b1 and the Strap.  Supplementary Fig. S13 show evidence of the crucial components in Fragment D. Earlier studies 2,4,5 suggested that there are two gears in Fragment D, but this is an illusion created because the arbor has split 7,9 , as established in Supplementary Discussion S5 and Supplementary Fig. S13. The original tooth count can be reliably determined as 63 teeth, given all but three of the teeth survive 5,7,9 . The basic components of Fragment D are a disk, gear and plate, referred to here as the D-plate, and an arbor linking all three elements. The disk and gear are riveted together and have square holes at their centre matching squared sections on one end of the arbor. Inside the thickness of the gear, the arbor changes from square to round, where it emerges into the plate. With no space for any other bearing on this arbor, it must have pivoted in the D-plate, which also serves as a spacer to bring the epicyclic components to the correct level in the output hierarchy and as a bearing for an idler gear 26 in the Venus train.

Fragment D: epicyclic components for venus
No other surviving gear in the Mechanism has a disk attached. In an inferior planet mechanism, the pinand-slotted follower requires a pin attached to the epicyclic gear but beyond its edge 3,21 : the attached disk is the right size to carry the pin at the correct distance from the centre to model the maximum elongation of Venus. It is surely the epicycle for Venus, as previously suggested 9 and strongly reinforced here. The width of the D-Plate is commensurate with the width of the Strap, based on the separation of the short pillars. Figure 5, Supplementary Fig. S22 show how the mean Sun, nodes and inferior planets are intricately constructed within the 15.0 mm space 9 between b1 and the Strap in nine closely-packed layers-matching the evidence and the layer density of the surviving gears (Supplementary Figs. S16, S20, S21, Supplementary Video S1). The mechanisms are interleaved so that their output tubes are nested in the CCO, with the lunar output on the central arbor.

Mechanisms between b1 and the strap
The Moon phase device 10 needs access to adjacent lunar and solar rotations, since the phase is the difference between these rotations: a ring output system appears to require calculating the true Sun twice 9 -once for input to the Moon phase and once for the true Sun ring, which is the third output in the ring system-so mechanically separated from the central lunar arbor. Here we solve this key problem with a mean Sun output, feeding into the Moon phase device as the first output tube adjacent to the central lunar arbor. b1 carries the mean Sun rotation, but it is not possible to attach a mean Sun output at its centre because the central fixed gears prevent this: an attachment half-way along Spoke D is necessary to bridge the fixed gears. This is why the mean Sun output is attached via a bar to the previously-mysterious pierced block on Spoke D (Fig. 4e)  www.nature.com/scientificreports/ a ring display for the Cosmos, with a single true Sun output for the solar ring. The small approximation inherent in using a mean Sun rather than a true Sun input to the Moon phase is negligible at the scale of the 6 mm diameter Moon phase sphere. There are great advantages in a ring system of outputs as opposed to a pointer system. It coincides far better with the description of the output display in the BCI. It eliminates the severe parallax inherent in a pointer system with nine outputs. It greatly enhances the astronomical outputs, by enabling the synodic phases of the planets to be described by index-linked inscriptions, as we discuss later (Fig. 8). It leads to a robust and elegant display.
The close match between our proposed mechanisms and the data is shown in Fig. 4. The four spokes of b1 suggest four different functions (Supplementary Fig. S12). The mean Sun and inferior planets take up three of these. What is the function of the prominent bearing on Spoke B (Fig. 4f)? Fig. 5b shows a solution: the bearing enables a four-gear epicyclic system that calculates the lunar nodes. Our proposed tooth counts for the gears  Discussion S4) mean that the bearing is in exactly the right place on Spoke B. No other use has previously been found for this bearing. The complex deductions that lead to unique reconstructions of the Venus & Mercury gear trains are described in Supplementary Discussion S5. We argue that Fragment D includes epicyclic components for Venus (Fig. 4, Supplementary Figs. S13, S14), that the gear trains follow our 5-gear design (Fig. 3) and that all must fit into the framework created by the pillars (Fig. 4). The prime factors in the period relations combined with the physical evidence then determine the gear trains (Fig. 3c, e, Fig. 4g, h, Fig. 5c, d). In particular, we show that the astronomical meaning of 63 is that it shares the prime numbers 3 × 7 with the period relation for Venus, (289, 462) = (17 2 , 2 × 3 × 7 × 11). The Strap is inclined to the spokes at just the correct angle of 11° to accommodate the epicyclic gears for Mercury and Venus-explaining the angle of the short pillars relative to b1. For the first time, the features on b1 and the components of Fragment D are fully explained (Figs. 4, 5, Supplementary Fig. S21, Supplementary Discussion S5, Supplementary Video S1). We conclude that our Venus and Mercury gear trains are strongly indicated by the evidence.

Mechanisms between the strap and the CP
There is no surviving direct evidence for the gearing systems that calculated the true Sun and the superior planets. Inevitably this means choices, though the space available strongly limits these choices, since very compact systems are necessary to calculate the advanced period relations. Figure 6a-e show how most of the gears for the true Sun and superior planets are reconstructed within the 9.7 mm space between the Strap and the CP.
The initial gears for these systems are in front of the CP (Fig. 6a)-alleviating the space problem and creating a robust mechanical design with no need for brackets to support the mechanisms as in a previous model 9 . A fixed gear 56 at the centre engages with a compound epicyclic train on the CP, calculating the synodic rotation of the Sun/planet relative to the CP. The arbors of the three gears 52, 56 and 64 go through the CP to drive the mechanisms at the back.
The mechanisms are arranged with their outputs in the CCO and are aligned on cardinal axes to facilitate calibration. The planetary periods and gear trains are listed in Fig. 3f and a schematic diagram is shown in Supplementary Fig. S23. Since the tooth counts must include the prime factors of the period relations, there are few viable options. The true Sun mechanism is a simple 3-gear system, previously proposed 3 (Fig. 3b), calculating the ancient Greek epicyclic theory of the true Sun. It shares the fixed gear 56 with all the superior planets and it shares 52 with the Saturn mechanism. Hence it only needs one additional gear 56. The superior planets, Mars, Jupiter and Saturn, are arranged clockwise from the top in Fig. 6b. All their mechanisms share a fixed gear 56 and follow the same economical 7-gear design shown in Fig. 3d.
The exploded diagram in Fig. 6f illustrates how all the Cosmos gearing fits together. We reconstruct 34 gears in front of b1 for the Cosmos system. Extant systems account for 35 gears behind b1 (Supplementary Table S8  The Dragon Hand indicates eclipses by its closeness to the true Sun pointer at New or Full Moon. Closenessto-node defines the sophisticated eclipse prediction scheme on the Antikythera Mechanism 8,23 , with symmetrical limits for lunar eclipses; and asymmetrical limits for solar eclipses, according to whether the Moon is North or South of the node 8,23 . These wider and narrower limits are indicated by triangles on the true Sun ring. When the Dragon Hand is within the relevant limits, an eclipse prediction glyph can be found on the Saros Dial, with eclipse characteristics listed in the eclipse inscriptions 8,23,24 . If the Dragon Hand is within the wider limits, an eclipse season 23 is in progress-occurring twice each eclipse year, shown by a full rotation of the Sun relative to the Dragon Hand. As a User Manual, the BCI (Fig. 1c) may have described these functions in the missing area above the planets (Fig. 1a).

Indexing of synodic events to planetary rings
As a rule, formulaic and repetitive inscriptions in the Antikythera Mechanism are indexed to their dials: for example, Parapegma inscriptions to the Zodiac Dial 1,2,7,25 and eclipse inscriptions to the Saros Dial 8,12,23 . For each planet, its synodic events-maximum elongations, stationary points, conjunctions and oppositions-occur when the planet is at a characteristic angle from the Sun. By turning the Mechanism, we can note the Sun's position on the planet's ring for each synodic event (Fig. 7). We propose that the planetary rings were engraved with scale marks for these events read by the Sun pointer, with associated index letters beside the scale marks. Figure 8 shows how the index letters would have referenced the formulaic and repetitive events in the FCI.
Though this indexing scheme is not provable, as the beginning of the lines are lost (Fig. 1b, Supplementary  Fig. S4), it makes such good sense in enhancing the astronomy on the Cosmos Display and it fits exactly with the line-by-line structure of the FCI. It is striking that the synodic events in the FCI are only those observable on the planetary rings: the customary appearances and disappearances of the planets are omitted, strengthening the   www.nature.com/scientificreports/ indexing hypothesis. It is difficult to understand how the information in the FCI could have been easily accessed by the user without such an indexing system, which in turn justifies our ring system of outputs. The FCI 9,12 enumerates intervals in days between synodic events-probably calculated from epicyclic models, not observations, since the actual intervals are so variable (Fig. 1b). The embryonic trigonometry of the Hellenistic age 26 would have made calculating these difficult. Here we propose that the Antikythera Mechanism itself calculated these synodic intervals by counting days on the Calendar Dial between synodic events indicated by the synodic scale marks on the planetary rings-entirely without trigonometry. Because of the loss of evidence, we cannot claim that our model is a replica of the original, but our solution to this convoluted 3D puzzle draws powerful support from the logic of our model and its exact match to the surviving evidence. The Antikythera Mechanism was a computational instrument for mathematical astronomy, incorporating cycles from Babylonian astronomy and the Greek flair for geometry. It calculated the ecliptic longitudes of the Moon 7 , Sun 3 and planets [1][2][3]9,11 ; the phase of the Moon 10

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.