Department of Mechanical Engineering,National Cheng Kung University,
1 University Road,Tainan 70101, Taiwan
Reconstruction Synthesis of theCalendrical Subsystem ofAntikythera MechanismThe damaged excavation of the Antikythera mechanism presents the oldest astronomicalanalog computer in ancient Greece. Its interior mechanism is a complicated gear trainwith many subsystems in which some are unclear, such as the calendrical subsystem. Thiswork focuses on the reconstruction synthesis of the calendrical subsystem and provides asystematic approach to generate all feasible designs. Based on the studies of historicalliteratures and existing designs, the required design constraints are concluded. Then,according to the concepts of generalization and specialization of mechanisms, two fea-sible designs and 14 results of teeth counting, including the existing one by Freeth et al.(2002, “The Antikythera Mechanism: 1. Challenging the Classic Research,” Mediterra-nean Archaeology & Archaeometry, 2, pp. 21–35; 2002, “The Antikythera Mechanism: 2.Is It Posidonius Orrery?,” Mediterranean Archaeology & Archaeometry, 2, pp. 45–58;2006, “Decoding the Ancient Greek Astronomical Calculator Known as the AntikytheraMechanism,” Nature (London), 444, pp. 587–591; 2008, “Calendars With OlympiadDisplay and Eclipse Prediction on the Antikythera Mechanism,” Nature (London), 454,pp. 614–617; 2009, “Decoding an Ancient Computer,” Sci. Am., 301(6), pp. 76–83),which are in consistent with the science theories and techniques of the subject’s timeperiod, are synthesized. �DOI: 10.1115/1.4003185�
Keywords: Antikythera mechanism, reconstruction design, calendrical subsystem, syn-thesis of mechanism, ancient machines
IntroductionThe destroyed fragments of the Antikythera mechanism, the
ldest known Greek astronomical machine, were accidentally dis-overed at the site of an antique shipwreck in 1900. In the lastundred years, with the help of image technology, some recon-truction designs have gradually resolved the secret of the Anti-ythera mechanism. It is confirmed that the Antikythera mecha-ism is a geared mechanism with several functions: It displays theotions of heavenly bodies, calculates between different calendar
ystems, and records astronomical phenomena.From 1902, scholars, such as Stais, Svoronos, and Rediadis,
egan to study the Antikythera mechanism. They understood theeep significance of the mechanism but were unable to present aoherent explanation. Afterward, Rehm and Theophanidis alsouccessively studied this mechanism. Around 1905, Rehm was therst man to define that the Antikythera mechanism was an astro-omical calculator. Because of the limitation of technology at thatime, the previous reconstructions were incorrect in many ways.n 1974, de Solla Price produced the first complete reconstructionesign �Fig. 1� �1,2�. However, he misunderstood several fragmentmages and designed a differential gear train to display the lunar
otion. In 2000, based on de Solla Price’s work, Edmunds andorgan provided the gear trains with a pin-in-slot device to ex-
ress the motions of Venus and Mars �Fig. 2� �3�. From 2002 to007, Wright presented a new reconstruction design in a series oftudies, including the gear trains to display the functions of backials and a gear train with a pin-in-slot device to express the lunarotion �4–14� �Fig. 3�. From 2006 to 2008, the Antikythera re-
earch project by Freeth et al. decoded numerous inscriptions on
1Corresponding author.Contributed by the Mechanisms and Robotics Committee of ASME for publica-
ion in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 18, 2010; finalanuscript received November 4, 2010; published online January 24, 2011. Assoc.
the exterior and explained the transmission of interior gear trains.Furthermore, he discovered the existence of a dial to display theOlympiad cycle �15–19�. In 2008, Koetsier presented the histori-cal development of reconstructing the Antikythera mechanismthrough the analysis of each previous design and Greek astronomy�20�.
Based on the reconstruction design by Freeth et al. shown inFig. 4, the Antikythera mechanism can be divided into the follow-ing five subsystems: the date subsystem, the calendrical sub-system, the lunar subsystem, the eclipse prediction subsystem, andthe lost subsystem. While previous studies have confirmed themechanisms of the date subsystem, the lunar subsystem and theeclipse prediction subsystem, there is still much debate on theunclear calendrical subsystem and the lost components of the in-terior mechanisms.
This study presents an approach for the reconstruction synthesisof the mechanism contained in the unclear calendrical subsystemof the Antikythera mechanism. Through this approach, all feasibledesigns of the calendrical mechanism that agree with the scienceand technology standards of the subject’s time period can be sys-tematically synthesized.
2 Historical BackgroundFigure 5 shows the excavated Antikythera mechanism �15–18�.
Its deterioration underwater is the primary reason for the difficultyof the reconstruction. In order to accurately reconstruct the Anti-kythera mechanism, the first work is to understand and define theproblems, including the origins and applications. This can be ac-complished through the study of the existing literatures and his-torical background.
From the decoding of inscriptions, the Antikythera mechanismis thought to be an ancient machine that existed between 150 B.C.
and 100 B.C. Due to the lack of historical records, people knewnothing about the Antikythera mechanism before its discovery inthe year of 1900. Therefore, only direct evidences from the sur-
viving fragments can be used to support the understanding of this
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evice. Since 1902, countless scholars have involved the recon-truction work. Through the image technology, the photos of frag-ents provide help to recombine this device. It is confirmed that
he interior of the Antikythera mechanism is composed of gearsnd links, and the exterior is covered with dials and inscriptions.hrough the decoding of inscriptions, the use and functions of theevice become clear. Based on the applications of astronomicalheories and the time origins of the device, some argued that therchitect of the Antikythera mechanism might be Posidonius ofhodes. Others contend that the creator might be Hipparchus.hese viewpoints were overturned until 2008. Through decoding
he month names of the calendar in this mechanism, Freeth et al.trongly confirmed that the mechanism’s calendar is identical tohe Corinthian calendar coming from the calendar of Tauromenionn Sicily. It also explained that the Antikythera mechanism shoulde from Corinthian colonies, and a certain workshop adopted fromrchimedes seemed to be likely the origin �18�.It is believed that the technology of the Antikythera mechanism
as spread by both cultural communications and wars. Ancientachines were copied by ancient cultures in other countries.herefore, ancient machines with similar functions can providedditional study references in addition to the surviving objects andistorical archives. The Islamic calendrical sundial designed byl-Biruni dating from 1000 A.D. is such an example; it functions
s a modeling cyclic astronomical phenomenon �Fig. 6� �20,21�.his geared instrument consisting of eight gears is driven byands. It displays the phases of the moon and its age in days, asell as the movements of the moon and sun around the zodiac in
hree separated axes. Interestingly, in Europe, technological arti-acts of similar complexity did not reappear until the 14th century,hen mechanical astronomical clocks appeared.
Fig. 1 Reconstruction
ig. 2 Reconstruction design of the Venus display by Ed-
unds and Morgan
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Due to a similar application of this device, it is suggested thatthe technology of the Antikythera mechanism may continue to beactive and have influenced the Islamic tradition. Since the calen-drical subsystem of the Antikythera mechanism is unclear, themechanism of Islamic calendrical sundial may supply the knowl-edge to reconstruct the unclear interior mechanism.
3 Existing Reconstruction DesignsThe Antikythera mechanism, approximately 315�190
�100 mm3 in size, is a bronze geared mechanism �Fig. 7�. Its
sign by de Solla Price
de
Fig. 3 Reconstruction design by Wright
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xterior shows the dials that cover a large number of inscriptions.ts interior consists of gear trains that drive the display of pointersn those dials.
3.1 Decoding of Exterior Dials. The front plate of the Anti-ythera mechanism contains two concentric dials �Fig. 7�a��. Thenner dial shows the Egyptian calendar while the outer dial showshe Zodiac. The pointers can display, respectively, the date, the
otions of the sun, the moon, and five planets on the front dials.he Antikythera mechanism also displays the moon phase in aonth.The back plate includes two parts �Fig. 7�b��. The upper part is
esigned to express the calculations between different calendarystems and the records of celebrations in ancient Greece. Thepper part of the back plate includes one large spiral dial and twoubsidiary dials. The large spiral dial displays the Metonic period,n approximate common multiple of the synodic month and theropical year. The Greek astronomer Meton observed that the pe-iod of 19 tropical years, approximately 6940 days, is almost
ig. 5 Surviving fragments of the Antikythera mechanism
Fig. 4 Reconstruction design by Freeth et al.
15–18‡
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equal to that of 235 synodic months. Two subsidiary dials show,respectively, the Olympiad period, 4 years in one turn, and theCallippic period with four times the Metonic period. The lowerpart of the back plate that serves to predict the eclipse of the mooncontains one large spiral dial and one subsidiary dial. The largespiral dial displays the Saro period, i.e., a period of 18 years. Thesubsidiary dial depicts the Exeligmos period that is three times theSaro period, i.e., a period of 54 years.
In summary, the exterior functions of the Antikythera mecha-nism display the motions of the heavenly bodies and record im-portant astronomical events and cultural activities.
3.2 Mechanism Analysis. The calendrical subsystem on theupper back dials of the Antikythera mechanism displays the cyclicrecords both of astronomical phenomena and festivities. As theparts of the calendrical mechanism are unclear or lost, the interiormechanism remains unknown until the decoding of the exteriordials. In 2003, Wright provided the corresponding gear trains forthe Meton period and the Callippic period �4–14�. In 2007, Freethet al. further collected previous designs and revealed the existenceof a dial displaying the Olympiad period �15–18�.
Based on the existing design by Freeth et al. �Fig. 8�a��, it canbe concluded that the topological characteristics of the calendricalsubsystem are as follows.
1. It has six members including a ground link �member 1, KF�,an input link �member 2, KI�, a Meton cycle link �member 3,KM�, an Olympiad cycle link �member 4, KO�, a Callippiccycle link �member 5, KC�, and a transmission link �member6, KT�.
2. It has nine joints including five revolute pairs �joints a, b, c,d, and e; JR� and four gear pairs �joints f, g, h, and i; JG�.
3. It is a simple gear train formed exclusively by externalgears.
4. It has one degree of freedom.
Fig. 6 Islamic sun dial †20‡
Fig. 7 A reconstruction model of the Antikythera mechanism
†19‡
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Based on mobility analysis, the following expression is true forplanar calendrical mechanism with one degree of freedom, NLembers, NJR revolute pairs, and NJG gear pairs:
2NJR + NJG − 3NL + 4 = 0 �1�
t is obvious that the following expression is true for a gear trainith NJ joints:
NJR + NJG − NJ = 0 �2�
ince every gear in a gear train must at least have a revolute joint,he following expression is true:
NJR − NL + 1 = 0 �3�
y solving Eqs. �1�–�3� in terms of the number of links, the fol-owing relations for the numbers of joints are concluded:
NJ = 2NL − 3 �4�
NJR = NL − 1 �5�
NJG = NL − 2 �6�It is confirmed that the calendrical subsystem includes three
ials to display different periods. In the reconstruction design ofhe mechanism, the number of gear pairs is three at least in ordero achieve the transformations of rotation rates. Based on Eq. �5�,he number of links in the calendrical subsystem of the Anti-ythera mechanism should be at least five. Therefore, two simpleesigns are concluded. For the mechanism of a calendrical sub-ystem with one degree of freedom and five members, sevenoints consisting of four revolute pairs and three gear pairs areecessary. For the mechanism of a calendrical subsystem with oneegree of freedom and six members, nine joints consisting of fiveevolute pairs and four gear pairs are necessary.
Based on the decoding of the inscriptions on the exterior dials,he rotation rates and directions of links have been confirmed17,18�. The Meton period link rotates clockwise at 5/19 rotationser year. The Olympiad period link rotates counterclockwise at/4 rotations per year. The meanings of the inscriptions on theallippic period dial are still unknown. The only confirmed fact is
hat the Callippic period link must rotate at 1/76 rotations per yearithout defining its rotation direction. Based on the confirmed
xisting design, the input link rotates counterclockwise at3 / �19�3� rotations per year.
When a gear pair is incident to the input link �member 2� andhe Meton period link �member 3�, it is obvious that the followingxpression is true for its gear ratio:
�53/�19 � 3�� � N2/N3 = 5/19 ⇒ N2/N3 = 15/53
hen a gear pair is incident to the input link �member 2� and thelympiad period link �member 4�, the following expression is true
ig. 8 Mechanism of the calendrical subsystem by Freeth etl.
or its gear ratio:
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�53/�19 � 3�� � N2/N4 = 1/4 ⇒ N2/N4 = 57/212
When a gear pair is incident to the input link �link 2� and theCallippic period link �link 5�, the following expression is true forits gear ratio:
�53/�19 � 3�� � N2/N5 = 1/76 ⇒ N2/N5 = 3/212
When a gear pair is incident to the Meton period link �link 3� andthe Olympiad period link �link 4�, the following expression is truefor its gear ratio:
5/19 � N3/N4 = 1/4 ⇒ N3/N4 = 19/20
When a gear pair is incident to the Meton period link �link 3� andthe Callippic period link �link 5�, the following expression is truefor its gear ratio:
5/19 � N3/N5 = 1/76 ⇒ N3/N5 = 1/20
Finally, when a gear pair is incident to the Olympiad period link�link 4� and the Callippic period link �link 5�, the following ex-pression is true for its gear ratio:
1/4 � N4/N5 = 1/76 ⇒ N4/N5 = 1/19
Different gear ratios result in a difference in teeth number andgear size between two gears in mesh. It is suggested that theCallippic period link should not be adjacent to the input link, theMeton period link, or the Olympiad period link, to avoid generat-ing a too big gear. Therefore, there must be a link adjacent to theCallippic period link.
In conclusion, the topological structure of the calendrical sub-system should be a planar mechanism with one degree of free-dom, six links, and nine joints that include five revolute pairs andfour gear pairs.
4 Design ConstraintsFrom the decoding of the inscriptions on external dials �17,18�,
it can be concluded that each dial of the calendrical subsystem hasits designated rotation rate. The gear ratios contribute to analyzethe possibility for gear pairs incident to any two links. Accordingto the studies of existing designs and the analysis of the mecha-nism mentioned above, design constraints of the calendrical sub-system are concluded as follows:
1. It must generate the designated astronomical periods in or-der to demonstrate the Meton period, the Olympiad period,and the Callippic period on the exterior dials.
2. It must at least include the ground link, the input link, theMeton period link, the Olympiad period link, the Callippicperiod link and the transmission link.
• Ground link �member 1, KF�
a. There must be a ground link as the frame.b. A ground link must be a multiple link in order to serve as
the input and three confirmed functions displayed on theexterior dials.
• Input link �member 2, KI�
a. There must be a link as the input link for the powertransmission.
b. The input link must rotate counterclockwise at 53 / �19�3� rotations per year.
c. The input link must include a gear with 15 teeth. This isknown by distinguishing the image of the survivingmechanism.
d. Previous analysis shows that the gear ratio for the inputlink to the Olympiad period link is 57/212. This gear ratioconflicts with the existing evidence that the Olympiad
period link must have a gear with 60 teeth. This means
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that the teeth number of the input link is not an integer.Therefore, the input link �link 2� must not be adjacent tothe Olympiad period link �link 4�.
e. Based on previous analysis of gear pairs, the input link�link 2� must not be adjacent to the Callippic period link�link 5� with a gear pair, in order not to generate a largegear ratio.
• Meton period link �member 3, KM�
a. There must be a link as the Meton period link.b. The Meton period link rotates clockwise at 5/19 rotations
per year.c. The Meton period link is adjacent to the ground link with
a revolute pair.d. The Meton period link �link 3� must not be adjacent to
the Callippic period link �link 5� with a gear pair, in ordernot to generate a too large gear ratio.
• Olympiad period link �member 4, KO�
a. There must be a link as the Olympiad period link.b. The Olympiad period link rotates counterclockwise at 1/4
rotations per year.c. The Olympiad period link is adjacent to the ground link
with a revolute pair.d. The Olympiad period link must have one gear with 60
teeth. This is understood by distinguishing the image ofthe surviving mechanism.
e. The Olympiad period link �link 4� must not be adjacent tothe Callippic period link �link 5� with a gear pair, in ordernot to generate a too large gear ratio.
• Callippic period link �member 5, KC�
a. There must be a link as the Callippic period link.b. The Callippic period link rotates at 1/76 rotations per
year.c. The Callippic period link is adjacent to the ground link
with a revolute pair.
• Transmission link �member 6, KT�
a. There must be a link as the transmission link in order togenerate appropriate teeth numbers.
b. The transmission link is adjacent to the ground link witha revolute pair.
c. The transmission link must be at least a ternary link inorder to avoid generating a redundant structure.
d. When the transmission is adjacent to the input link andthe Callippic period link simultaneously, the followingexpression is true for its gear ratio:
�53/�19 � 3�� � N2/N5 � N5�/N6 = 1/76 ⇒ N2/N5
� N5�/N6 = 3/212
where N5 is the gear of the Callippic period link adjacentto the input link, and N5� is the gear of the Callippicperiod link adjacent to the transmission link.
This large gear ratio results in unsuitable gear sizes. Therefore,he transmission link �link 6� must not be adjacent to the input linklink 2� and the Callippic period link �link 5� simultaneously.
Reconstruction SynthesisThrough the mobility analysis mentioned above, it can be con-
luded that the mechanism of the calendrical subsystem includeshe ground link, the input link, the Meton period link, the Olym-iad period link, the Callippic period link, and the transmissionink. In order to synthesize all feasible topological structures of
he calendrical mechanism subject to the design constraints de-
ournal of Mechanical Design
scribed above, this work follows a design procedure consisting offour steps: design specifications, generalized kinematic chains,specialized chains, and reconstruction designs, as shown in Fig. 9�22–25�.
5.1.1 Step 1: Design Specifications. Design specifications arethe observed rules in the process of reconstruction design. Theydescribe the statement of a product and should be defined at thebeginning of the design process. After studying available litera-tures and historical records, the numbers of members and joints inthe existing design of the calendrical mechanism can be recog-nized. Once the characteristics of topological structure are identi-fied, the specifications, including the numbers and types of mem-bers and joints, can be defined.
For the calendrical subsystem of the Antikythera mechanism,the design specifications that satisfy these confirmed functions areconcluded as follows.
1. It is a planar six-bar mechanism with nine joints and onedegree of freedom.
2. The members are gears and links.3. The joints consist of five revolute joints and four external
gear joints.
5.1.2 Step 2: Generalized Kinematic Chains. The second stepin the methodology is to obtain the atlas of generalized kinematicchains. This is based on the concepts of generalization and num-ber synthesis. The purpose of generalization is to transform amechanism involving various types of members and joints into ageneralized kinematic chain with only generalized links and gen-eralized joints. A generalized joint is a joint in general; it can be arevolute pair, a prismatic pair, a spherical pair, a helical pair, orothers. A generalized link is a link with generalized joints; it canbe a binary link, a ternary link, a quaternary link, etc. For anyexisting design of the calendrical mechanism, the correspondingoriginal generalized kinematic chain can be obtained through theprocess of generalization. Based on the algorithm of number syn-thesis, all possible generalized kinematic chains with the samenumbers of members and joints as the original generalized kine-matic chain can be obtained.
Figure 8�b� shows the generalized kinematic chain with sixlinks and nine joints of the existing design by Freeth et al. Theground link �link 1� is generalized into a quinary link, the inputlink �link 2� is generalized into a binary link, the Meton periodlink �link 3� is generalized into a quaternary link, the Olympiadperiod link �link 4� is generalized into a binary link, the Callippicperiod link �link 5� is generalized into a binary link, and the trans-mission link �link 6� is generalized into a ternary link. Based onthe algorithm of number synthesis �24�, the atlas of generalized
Fig. 9 Procedure of reconstruction synthesis
kinematic chains with six links and nine joints can be generated.
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ccording to the concluded design constraints of the ground link,ach link of the calendrical subsystem must be adjacent to theround link with a revolute pair. Therefore, only three generalizedinematic chains with six links and nine joints are feasible, ashown in Fig. 10.
5.1.3 Step 3: Feasible Specialized Chains. Specialization ishe process of assigning specific types of members and joints inhe available atlas of generalized kinematic chains subject to theoncluded design constraints. After specialization, a generalizedinematic chain is called a specialized chain. A specialized chainubject to design constraints is called a feasible specialized chain.herefore, through the process of specialization, the atlas of fea-ible specialized chains is generated.
For the three feasible generalized chains shown in Fig. 10, alleasible specialized chains subject to the concluded design con-traints can be obtained through the following steps.
5.1.3.1 Ground link (link 1). Since there must be a multipleink with five revolute pairs as the frame, the ground link can bedentified as follows.
1. For the generalized kinematic chain shown in Fig. 10�a�, theassignment of the ground link generates one result �Fig.11�a��.
2. For the generalized kinematic chain shown in Fig. 10�b�, theassignment of the ground link generates one result �Fig.11�b��.
3. For the generalized kinematic chain shown in Fig. 10�c�, theassignment of the ground link generates no result �Fig.11�c��.
herefore, three specialized chains with identified ground link arevailable, as shown in Fig. 11.
5.1.3.2 Callippic period link (link 5). Since there must be aink as the Callippic period link, the Callippic link can be identi-ed as follows.
1. For the generalized kinematic chain shown in Fig. 11�a�, theassignment of the Callippic period link generates three re-sults �Figs. 12�a�–12�c��.
2. For the generalized kinematic chain shown in Fig. 11�b�, theassignment of the Callippic period link generates four results�Figs. 12�d�–12�g��.
3. For the generalized kinematic chain shown in Fig. 11�c�, theassignment of the Callippic period link generates two results�Figs. 12�h� and 12�i��.
ig. 10 Qualified generalized kinematic chains with six linksnd nine joints for specialization
Fig. 11 Specialized chains with identified ground link
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Therefore, nine specialized chains with identified ground link andCallippic period link are available, as shown in Fig. 12.
5.1.3.3 Olympiad period link (link 4). Since there must be alink as the Olympiad period link to display the Olympiad period,the Olympiad period link can be identified for each result, asshown in Fig. 12. Therefore, after the assignment of the Olympiadperiod link subject to the required design constraints, 14 special-ized chains with identified ground link, Callippic period link andOlympiad period link, are available, as shown in Figs.13�a�–13�n�.
5.1.3.4 Input link (link 2). Since there must be an input linkthat at least includes a gear pair and a revolute pair, the input linkcan be identified for each result, as shown in Fig. 13. Therefore,after the assignment of the input link subject to the required de-sign constraints, ten specialized chains with identified ground link,Callippic period link, Olympiad period link, and input link, areavailable, as shown in Figs. 14�a�–14�j�.
5.1.3.5 Meton period link (link 3). Since there must be a linkas the Meton period link to generate the confirmed gear ratio, i.e.,the Meton period, the Meton period link can be identified for eachresult, as shown in Fig. 14. Therefore, after the assignment of theMeton period link subject to the required design constraints, eightspecialized chains with identified ground link, Callippic periodlink, Olympiad period link, input link, and Meton period link, areavailable, as shown in Figs. 15�a�–15�h�.
5.1.3.6 Transmission link (link 6). Finally, since there must bea transmission link at least incident to a gear pair and a revolutepair, the transmission link can be identified for each result, asshown in Fig. 15. Therefore, after the assignment of the transmis-sion link subject to the required design constraints, three special-ized chains with identified ground link, Callippic period link,Olympiad period link, input link, Meton period link, and transmis-sion link, are available, as shown in Fig. 16. Because the Olym-piad period link in Fig. 16�b� would rotate clockwise, it did not
Fig. 12 Specialized chains with identified ground link and Cal-lippic period link
satisfy the constraints of rotation directions of links mentioned
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bove. Therefore, only Figs. 16�a� and 16�c� are the feasible re-onstruction designs.
According to present study, the inscriptions on the Callippicial was not decoded completely, and the existence of the Callip-ic dial was speculated �18�. The rotation direction of the Callip-ic period link is unconfirmed. Therefore, through this design pro-edure, each possible origin design of the calendrical subsystemould be discussed. Figure 16�c� is the existing design, and theallippic dial displays rotate clockwise. Figure 16�a� is a possiblelternative origin design of the calendrical subsystem; the Callip-ic dial displays counterclockwise.
5.1.4 Step 4: Reconstruction Designs. After identifying alleasible topological structures of the calendrical subsystem, theeeth counting for each feasible design is an important issue.
ig. 13 Specialized chains with identified ground link, Callip-ic period link, and Olympiad period link
ig. 14 Specialized chains with identified ground link, Callip-
ic period link, Olympiad period link, and input link
ournal of Mechanical Design
Known situations include the confirmed gear ratios as well as twogears, respectively, with 60 teeth and 15 teeth. However, these donot adequately generate an accurate teeth count for each uncleargear in the feasible reconstruction designs. For calculating theteeth number for each feasible design, this work, referring to thedirect evidences and the existing design by Freeth et al., hypoth-esizes the following constraints for sizing the gears:
1. The gear ratio of a gear pair should be at most five.2. The speculative teeth numbers should appear in the existing
discovery by Freeth et al. and their minimum teeth numberis ten and the maximum is hundred.
To express clearly, this work presents the following representa-tion for a gear:
Ngear on the assigned linkgear pair between two adjacent links
where the superscript denotes the gear pair between two adjacentlinks and the subscript denotes the gear on the assigned link.
According to the known gear ratios, the direct evidences, andthe size constraint, the teeth numbers of two feasible reconstruc-tion designs could be analyzed as follows.
5.1.4.1 Feasible reconstruction design, Fig. 16(a). Based onthe relationship between the input link and the Meton period linkand the direct evidence, the following expression is true:
� 53
19 � 3� �
N223
N323 =
5
19⇒ �N2
23,N323� = �15,53�
Based on the relationship between the Meton period link and theOlympiad period link and the direct evidence, the following ex-pression is true:
5
19�
N334
N434 =
1
4⇒
N334
N434 =
19
20⇒ �N3
34,N434� = �57,60�
Based on the relationship between the Olympiad period link andthe Callippic period link, the following expression is true:
Fig. 15 Specialized chains with identified ground link, Callip-pic period link, Olympiad period link, input link, and Meton pe-riod link
Fig. 16 Atlas of feasible specialized chains
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N646 �
N656
N556 =
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hrough the size constraint mentioned above, the possible resultsre generated as follows:
�N446,N6
46,N556,N6
56� = �12,60,57,15�, �N446,N6
46,N556,N6
56�
= �15,57,60,12�
�N446,N6
46,N556,N6
56� = �12,48,57,12�, �N446,N6
46,N556,N6
56�
= �12,57,48,12�
�N446,N6
46,N556,N6
56� = �15,60,57,12�, �N446,N6
46,N556,N6
56�
= �12,57,60,12�
�N446,N6
46,N556,N6
56� = �24,96,57,12�, �N446,N6
46,N556,N6
56�
= �12,57,96,24�herefore, eight possible results of teeth number are generated. In
hese results, all teeth numbers could be found from the existingiscovery by Freeth et al.
5.1.4.2 Feasible reconstruction design, Fig. 16(c). Throughhe same analysis procedure mentioned above, the possible results as follows:
�N223,N3
23� = �15,53�, �N334,N4
34� = �57,60�ased on the relationship between the Meton period link and theallippic period link, the following expression is true:
5
19�
N336
N636 �
N656
N556 =
1
76
hen, through the size constraint mentioned above, the possibleesults are generated as follows:
�N336,N6
36,N556,N6
56� = �12,48,60,12�, �N336,N6
36,N556,N6
56�
= �15,60,60,12�
�N336,N6
36,N556,N6
56� = �24,96,60,12�, �N336,N6
36,N556,N6
56�
= �12,60,48,12�
�N336,N6
36,N556,N6
56� = �12,60,60,15�, �N336,N6
36,N556,N6
56�
= �12,60,96,24�herefore, for this existing design, six feasible results including
he original teeth combination are generated.
Fig. 17 Atlas of re
In these results, all teeth numbers could be found from the
21004-8 / Vol. 133, FEBRUARY 2011
existing information by Freeth et al. The evaluation of the feasibleteeth number mentioned above is related to the constraints onsizes. Once a new discovery from the surviving evidence is ap-peared, a confirmation of teeth number or more possible resultscan be provided. Moreover, according to the geometric constraintsidentified by the computer tomography �CT� images taken for theAntikythera mechanism, each design in the atlas of feasible spe-cialized chains is further particularized into its correspondingschematic diagram, as shown in Fig. 17.
6 ConclusionsThe lack of historical records and the destruction of surviving
unearthed artifact make any attempt difficult to create a recon-struction design for the calendrical subsystem of the Antikytheramechanism. Through mobility analysis of the mechanism, thiswork analyzes the required minimum numbers of links. Based ondial functions and the geometrical constraints of surviving frag-ments, the mechanism of the calendrical subsystem is a planarmechanism with one degree of freedom, six links, and nine jointsincluding five revolute pairs and four gear pairs. Furthermore, thiswork presents a procedure to systematically reconstruct all fea-sible designs of the calendrical subsystem. Based on the study ofhistorical archives and existing designs, required design con-straints are concluded. Finally, according to the concepts of thegeneralization and specialization of mechanisms subject to theconcluded design constraints, two feasible reconstruction designsand 14 corresponding results of teeth analysis, including the ex-isting design by Freeth et al., are generated. As a result, one of thereconstruction designs should be the original mechanism of thecalendrical subsystem.
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