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Home Albrecht Dürer Home Aléria Dodecahedron Dodecahedron loudspeaker (modern) A dodecahedron is a Platonic solid, a regular polyhedron with twelve Escher's Dodecahedron Etruscan Dodecahedron pentagonal faces. Five solids meet those criteria, and each is named after its Il dodecaedro di Luca number of faces. They are Platonic, because was the Greek philosopher Plato Pacioli Platonic solid who theorized in his Timaeus, that the classical elements, Earth, Water, Wind Pseudoglobes (modern) and Fire, were made of these regular solids. Replica Roman Dodecahedron Ancient and modern objects have the dodecahedral shape; a list of them (on Roman Dodecahedron the left) is here collected. Sundials Sitemap A.C. Sparavigna
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Home Albrecht Dürer Albrecht Dürer Aléria Dodecahedron Dodecahedron loudspeaker (modern) From the book by Albrecht Dürer on geometry. Opera Alberti Dureri, das ist alle Bücher d. weitberhümbten Escher's Dodecahedron u. künstreichen Mathematici u. Mahlers Albrechten Durers von Nürenberg so viel deren von ihm selbst in Etruscan Dodecahedron An. 1525 u. 1528 kurtz vor u. gleich nach seinem todt in Truck geben ; jetzo denselben zum besten aus Il dodecaedro di Luca mangel d. Exemplarien mit sein selbst eigenen gemachten u. geschnittenen zugehörigen Fig. von newem Pacioli auffgelegt und getruckt. Platonic solid Pseudoglobes (modern) Replica Roman Dodecahedron Roman Dodecahedron Sundials Sitemap
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Home Albrecht Dürer Aléria Dodecahedron Aléria Dodecahedron Dodecahedron loudspeaker (modern) A rock crystal dodecahedron from Aléria, Corsica Escher's Dodecahedron http://stretchingtheboundaries.blogspot.it/2012/07/aleria-dodecahedron.html Etruscan Dodecahedron Il dodecaedro di Luca Pacioli Platonic solid Pseudoglobes (modern) Replica Roman Dodecahedron Roman Dodecahedron Sundials Sitemap
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Home Albrecht Dürer Dodecahedron loudspeaker (modern) Aléria Dodecahedron Dodecahedron loudspeaker (modern) The Dodecahedron is used to obtain a sound source which is omni-directional. Escher's Dodecahedron See for instance Qom, http://www.qsources.be/qom-dodecahedron-sound-source Etruscan Dodecahedron and the Look Line, http://www.lookline.com Il dodecaedro di Luca Pacioli Platonic solid Pseudoglobes (modern) Replica Roman Dodecahedron Roman Dodecahedron Commenti Sundials Sitemap Non disponi dell'autorizzazione necessaria per aggiungere commenti.
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Home Albrecht Dürer Escher's Dodecahedron Aléria Dodecahedron Dodecahedron loudspeaker (modern) Escher's Dodecahedron Source: Wikipedia Etruscan Dodecahedron Il dodecaedro di Luca Reptiles is a work by M. C. Escher first printed in March 1943. "It depicts a Pacioli desk on which is a drawing of a tessellated pattern of reptiles. The reptiles at Platonic solid Pseudoglobes (modern) one edge of the drawing come to life and crawl around the desk and over the Replica Roman objects on it to eventually re-enter the drawing at its opposite edge. The desk is Dodecahedron Roman Dodecahedron littered with ordinary objects, as well as a metal dodecahedron that the reptiles Sundials climb over." Sitemap
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Home Albrecht Dürer Etruscan Dodecahedron Aléria Dodecahedron Dodecahedron loudspeaker (modern) An Etruscan Dodecahedron, by Amelia Carolina Sparavigna Escher's Dodecahedron Department of Applied Science and Technology Etruscan Dodecahedron Politecnico di Torino, C.so Duca degli Abruzzi 24, Torino, Italy Il dodecaedro di Luca Pacioli published on arXiv, http://arxiv.org/abs/1205.0706 Platonic solid Pseudoglobes (modern) Replica Roman Dodecahedron How old is the knowledge of Platonic solids? Were they already known to the ancients, before Plato? If we Roman Dodecahedron consider Wikipedia [1], the item on Platonic solids is telling that there are some objects, created by the late Sundials Neolithic people, which can be considered as evidence of knowledge of these solids. It seems therefore Sitemap that it was known, may be a millennium before Plato, that there were exactly five and only five perfect bodies. These perfect bodies are the regular tetrahedron, cube, octahedron, dodecahedron and icosahedron. In his book on regular polytopes [2], Harold Scott Macdonald Coxeter, writes "The early history of these polyhedra is lost in the shadows of antiquity. To ask who first constructed them is almost as futile to ask who first used fire. The tetrahedron, cube and octahedron occur in nature as crystals. ... The two more complicated regular solids cannot form crystals, but need the spark of life for their natural occurrence. Haeckel (Ernst Haeckel's 1904, Kunstformen der Natur.) observed them as skeletons of microscopic sea animals called radiolaria, the most perfect examples being the Circogonia icosaedra and Circorrhegma dodecahedra. Turning now to mankind, excavations on Monte Loffa, near Padua, have revealed an Etruscan dodecahedron which shows that this figure was enjoyed as a toy at least 2500 years ago." Traduci Before Plato, Timaeus of Locri, a philosopher among the earliest Pythagoreans, invented a mystical correspondence between the four easily constructed solids (tetrahedron, icosahedron, octahedron and cube), and the four natural elements (fire, air, water and earth). “Undeterred by the occurrence of a fifth solid, he regarded the dodecahedron as a shape that envelops the whole universe.” [2]. It is interesting that Donald Coxeter is reporting the existence of an Etruscan dodecahedron, that is, an object having the shape of a Platonic solid found in Italy, not of Greek origin. In Refs.3 and 4 too, it is told that there exists an Etruscan dodecahedron made of soapstone found near Padua and believed to date from before 500 BC. Another book referring to this dodecahedron is Ref.5, is that written by György Darvas. György Darvas discusses in [5] the Platonic solids and their use as dice. He tells that the best known of them is the cube. We use it in gambling, “because of its symmetries, it is equally likely to fall on any of its sides. … In truth, any regular body satisfies this condition of falling on any side with the same probability, not just the six-sided cube, that we in contemporary Europe are accustomed to call dice in this context.“. The author continues telling that etymologically, “the noun dice does not even refer to a cube. This is the plural of the noun die, here meaning a surface with a relieved design forming one of the facets of polyhedron. In principle, any of the five regular polyhedra can be used as a die. “There is an evidence to suggest that in Italy of old, dodecahedra were used in games, while in Etruscan cultures, they can have a religious significance (Figure 6.9a)". This is what reference 5 is telling. In fact, this figure 6.9a of Ref.5 (Fig.1 shows a snapshot of what we can see by means of Google Books) is showing a Roman dodecahedron, not an Etruscan dodecahedron as the caption is telling. The book continues: "In Japan, for example - where the number five is considered a lucky mascot - a dodecahedron delimited by regular pentagons is still used for this purpose to this day. Sometimes it is customary to write the digits from one to twelve on its faces, sometimes the names of the twelve months.”.
Fig.1 The image shows a snapshot of a page of Ref.4 that we can see by means of Google Book.
Fig.1 shows that Figure 6.9a of Ref.[4] can be misleading. This is a Roman dodecahedron of the second or third century AC (see Ref.6), having probably a use quite different from that of dice. What was then the shape of the Etruscan dodecahedron? Let us report the original discussion and illustration of the researcher that found it. He was Stefano De' Stefani. In the Proceedings of the Royal Venetian Institute of Sciences, Arts and Letters ([7], 1885), the author tells where the dodecahedron was found and reports about the existence of an icosahedron in Turin. The paper is entitled “On an almost regular dodecahedron of stone, with pentagonal faces carved with figures, discovered in the ancient stone huts of Monte Loffa“. The place of discovery belongs to Sant'Anna del Faedo village of Breonio, in the region of the western Lessini Mountains, called by the ancient historians as the region of Reti and Euganei, who were destroyed and scattered by the Gauls. De’ Stefani is in agreement with several ancient writers, who considered Reti an ancient Italic people of Etruscan origin, that under the Gauls pressure had to find refuge on Alps [8]. The author continues telling that Gauls, “people of wild and fierce aspect”, leaved in the same huts of Monte Loffa the manifest evidence of their presence, shown by tools, weapons and ornaments. “This village or encampment of prehistoric times shows objects of human industry that are represented by flint tools and weapons from the Neolithic period, of Etruscan bronzes type or Euganeo and Gaulish coins and other objects. The paper has an illustration showing the dodecahedron (see Fig.2).
Fig.2 Etruscan dodecahedron from Monte Loffa (from ref.7).
The paper continues with a deep discussion of the nature and use of the dodecahedron in Fig.2. Several scholars were interviewed by De’ Stefani, and he came to the conclusion that this dodecahedron was a die. The paper [7] reports the opinion of Ariodante Fabretti [9], that De’ Stefani received in a letter written by Carlo Cipolla [10]. Fabretti says that it is a die. The signs are conventional, perhaps a sort of numerals. In this case, this specimen is interesting because it seems to show a mixture of dots, as in our modern dice, and Etruscan numbers, adapted from the Greek numerals. On one of the face we can see “IV”, may be, for “four”. There is also another interesting fact. Fabretti showed to Cipolla an icosahedron that could had some link with this dodecahedron. The icosahedron was made of blue-glazed earthenware. On each face there were impressed some Greek letters. Cipolla asked Fabretti if he knew anything about the origin of the icosahedron. He replied that it was owned by the city of Turin, before coming to the Museum of Antiquities, on occasion of an exchange. It was therefore supposed that this object was found in Piedmont. It seems that in 1885, the existence of the icosahedron was unpublished. Unfortunately, I do not know where the Turin icosahedron is. Probably it is like that shown in Fig.3, from the second century AD, sold at auction for about $18,000 [11]. In my opinion, the Turin icosahedron could be older.
Fig3. The icosahedron die of Ref.11.
We could conclude that the ancient people in Italy, trading with Greeks, imported some numerals, and, among the first applications, used them on dice for gambling. In any case, they developed their own numeral system that evolved in the Roman numeral system.
References 1. http://en.wikipedia.org/wiki/Platonic_solid 2. Regular Polytopes, by Harold Scott Macdonald Coxeter, 1973, Dover, ISBN 0-486-61480-8. 3. The number of things: Pythagoras, geometry and humming strings, by Evans G. Valens, Methuen, 1964. 4. A History of Mathematics, by Carl B. Boyer, Uta C. Merzbach, 1968, John Wiley and Sons 5. Symmetry: Cultural-historical and ontological Aspects of Science-arts relations; the natural and man- made world in an interdisciplinary approach, by György Darvas, 2007, Birkhäuser Verlag, Basel, Switzerland 6. A Roman Dodecahedron for measuring distance, A.C. Sparavigna, 2012, arXiv, arXiv:1204.6497v1 [physics.pop-ph], http://arxiv.org/abs/1204.6497 7. Intorno un dodecaedro quasi regolare di pietra a facce pentagonali scolpite con cifre, scoperto nelle antichissime capanne di pietra del Monte Loffa, Stefano De' Stefani, Atti del Reale Istituto veneto di scienze, lettere ed arti (1885). 8. Reti http://it.wikipedia.org/wiki/Reti; Euganei, http://it.wikipedia.org/wiki/Euganei,http://en.wikipedia.org/wiki/Euganei 9. Ariodante Fabretti (1816 - 1894) was an Italian politician and historian. He was senator of the Kingdom of Italy in the sixteenth legislature. In 1860 he became professor of archaeology at the University of Turin. From 1871 to 1893 he was director of the Egyptian Museum of Turin. In 1876 he became Emeritus Member of the Accademia dei Lincei. 10. Carlo Cipolla (1854 - 1916) was an Italian historian, Professor of Modern History at the University of Turin, 1882-1906, and later at the Institute of Higher Studies in Florence. 11. http://www.georgehart.com/virtual-polyhedra/dice.html
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Home Albrecht Dürer Il dodecaedro di Luca Pacioli Aléria Dodecahedron Dodecahedron loudspeaker (modern) Un dettaglio del ritratto di Luca Pacioli (1495), attribuito a Jacopo de' Barbari, Museo Nazionale di Escher's Dodecahedron Capodimonte. Etruscan Dodecahedron Il dodecaedro di Luca Pacioli Platonic solid Pseudoglobes (modern) Replica Roman Dodecahedron Roman Dodecahedron Sundials Sitemap
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Home Albrecht Dürer Platonic solid Aléria Dodecahedron Dodecahedron loudspeaker (modern) From Wikipedia Escher's Dodecahedron In Euclidean geometry, a Platonic solid is a regular, convex polyhedron. Etruscan Dodecahedron The faces are congruent, regular polygons, with the same number of faces meeting at each vertex. There are Il dodecaedro di Luca exactly five solids which meet those criteria; each is named according to its number of faces. Pacioli Platonic solid Pseudoglobes (modern) Tetrahedron Cube or hexahedron Octahedron Dodecahedron Icosahedron Replica Roman (four faces) (six faces) (eight faces) (twelve faces) (twenty faces) Dodecahedron POV-Ray- Roman Dodecahedron Dodecahedron.svg Sundials (Animation) Sitemap
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Home Albrecht Dürer Pseudoglobes (modern) Aléria Dodecahedron Dodecahedron loudspeaker (modern) Very interesting web site on Escher's Dodecahedron Dodecahedral Pseudoglobes at Etruscan Dodecahedron http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/Foldout/Dodecahedron/dodecahedron.html Il dodecaedro di Luca Pacioli Platonic solid Pseudoglobes (modern) Replica Roman Dodecahedron Roman Dodecahedron Sundials Sitemap
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Home Albrecht Dürer Replica Roman Dodecahedron Aléria Dodecahedron Dodecahedron loudspeaker (modern) Here you can find the link to some web pages containing images you can print, cut, fold and glue, Escher's Dodecahedron in order to create you replica of a Roman Dodecahedron. Etruscan Dodecahedron Il dodecaedro di Luca Pacioli Platonic solid Pseudoglobes (modern) Replica Roman Dodecahedron Roman Dodecahedron Sundials Sitemap
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Home Albrecht Dürer Roman Dodecahedron Aléria Dodecahedron Dodecahedron loudspeaker (modern) Escher's Dodecahedron Etruscan Dodecahedron Il dodecaedro di Luca Pacioli Platonic solid Pseudoglobes (modern) Replica Roman Dodecahedron Roman Dodecahedron Sundials Sitemap
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A discussion on the Roman Dodecahedron, by A.C. Sparavigna A very interesting paper entitled "The magic dodecahedron of Gauls, that saved Roman legions: Mirror of Universe and gauge of seasons." by Cinzia di Cianni was published on La Stampa, in Italian (Il dodecaedro magico dei Galli che salvò le legioni romane: Specchio dell’Universo e misura delle stagioni:, July 28, 2010). This article starts with the following questions. Was is it, "a sacred symbol for Druids or the tip of a Traduci scepter? A gauge or a candlestick? Nobody knows what it is really, in spite of the fact that in museums and private collections, we find over than a hundred of them. It is a small hollow object of metal, dating from the fourth century and having a Gallo-Roman origin." The object exists in a variety of designs and sizes, always consists of 12 regular pentagons and this is known as "Roman dodecahedron". All the found "Roman dodecahedra" have a diameter between 4 and 11 cm. and have at the center of the 12 faces holes of different sizes. Each of the 20 vertices is surmounted by one or three knobs, may be to fit them on some surfaces. "The Roman dodecahedron is a simple object, actually a "time capsule", containing an incredible density of history and myth. By itself it does not reveal anything relevant because it has no inscriptions on patterned surfaces. No document speaks about it." The article continues telling that, in fact, there are 27 theories about its use, ranging from a game for divination to surveying or military purposes. Scholars gave up probably until some new finds: however, some amateur archaeologists, among them Sjra Wagemans, continue to study this mystery, that is "what was it used for?" Cinzia di Cianni tells that the first description of the geometrical volume of this object is in the "Timaeus" by Plato. It is a solid as the tetrahedron, octahedron, cube and icosahedron, that is, one of the five Platonic solids. Before Plato, it was also described in the fifth century BC by the Pythagorean Hippasus of Metaponto. "Harmony of proportions and mathematical properties, has continued to captivate artists and scientists, from Euclid to Poincare, from Leonardo to Luca Pacioli, to Escher. So, during several centuries, the dodecahedron had accumulated magic and symbolic features, from Greeks to Celts, from Renaissance to modern times." Di Cianni continues reporting the interest on dodecahedra by Francesco Maurolico, a Greek mathematician and astronomer of Sicily, who lived in the XVI century, and the contemporary astronomer Jean-Pierre Luminet, who works with data provided by the scientific probe "WMAP" (Wilkinson Microwave Anisotropy Probe), used to observe the cosmic background radiation in the microwave range. For what concerns the Roman dodecahedra the article tells that all of them, collected in several European museums, always came from Gaul and the lands of the Celts: Great Britain, Belgium, Holland, Germany, Switzerland, Austria and Eastern Europe. Recently Sjra Wagemans, of the Dutch multinational DSM Research and amateur archeology, proposed a theory which assigns an astronomical feature to these objects. Sjra used a bronze copy of a dodecahedron to see that it is possbile to determine the equinoxes of spring and autumn. "The dodecahedron is therefore linked to the agricultural cycle, both sophisticated and simple at the same time, to determine without a calendar, the most suitable period of time during the autumn for sowing wheat." And crops were of vital importance for the Roman legions. In fact, on the measurement of time, we know that Romans used gnomons (the Vitruvian equinoxial gnomons) to determine the latitude and that they had very good meridians. Vitruvius deeply describes in his De Archtectura, how to prepare the analemma (see for instance, Measuring times to determine positions, http://arxiv.org/abs/1202.2746). The roman dodecahedron is not the best instrument for measuring time. I have discussed its possible use for measuring distance. I used a dodecahedron, found at Jublains, the ancient Nouiodunum, dating from the 2nd or 3rd century AD, to create a model. Looking through the model, it is possible to test it for measurements of distance based on similar triangles. Therefore the Dodecahedron can be an ancient rangefinder for ballistics. It is simple to use. During the Middle Age, other instruments had been developed: probably, the use of the dodecahedron was lost after the collapse of the Roman Empire.
See my papers:
A Roman Dodecahedron for measuring distance, by Amelia Carolina Sparavigna, arXiv, 29 Apr 2012. Here I am discussing a possible use of a Roman Dodecahedron, a bronze artifact of gallo-roman origin, for measuring distance. A dodecahedron, found at Jublains, the ancient Nouiodunum, dating from the 2nd or 3rd century AD, is used to create a model. Looking through the model, it is possible to test it for measurements of distance based on similar triangles. Comments: Ancient surveying methods Subjects: Popular Physics (physics.pop-ph) Cite as: arXiv:1204.6497v1 [physics.pop-ph] http://arxiv.org/abs/1204.6497
Ancient and modern rangefinders, by Amelia Carolina Sparavigna, arXiv, 9 May 2012. Rangefinders are instruments used for ballistics and for surveying in general. Here a short discussion of some of them, ranging from the ancient Rome to the modern methods. Comments: Surveying Subjects: Popular Physics (physics.pop-ph) Cite as: arXiv:1205.2078v1 [physics.pop-ph] http://arxiv.org/abs/1205.2078
Sui Dodecaedri Romani sono state proposte molte teorie: porta-candele (ma all'epoca si usavano le lucerne), dadi per la divinazione (ma questi oggetti hanno le facce con masse diverse e quindi cadono sempre sulla faccia più pesante), strumenti astronomici (per stabilire gli equinozi, ma gli antichi usavano le meridiane) e altro ancora. Purtroppo non è facile districarsi tra le varie teorie, anche perché spesso non si trovano gli articoli originali da studiare. Quello che appare chiaro è che questi dodecaedri di bronzo sono cavi, con facce che hanno buchi perfettamente circolari di diametri diversi. Cercando in rete ho trovato l'articolo di alcuni archeologi che descrivono con precisione un dodecaedro di bronzo, trovato a Jublains, l'antica Nouiodunum, del secondo o terzo secolo DC. Usando i dati sui diametri dei fori ho creato un modello di cartone. Guardando attraverso il modello ho capito che poteva essere uno strumento ottico per misurare la distanze e funzionare come un telemetro. Come molti telemetri, la misura si basa sui triangoli simili. I dettagli del dodecaedro come strumento telemetrico sono pubblicati in un articolo pubblicato su arXiv http://arxiv.org/abs/1204.6497 (In italiano http://porto.polito.it/2497004/ ). Il dodecaedro ha quattro differenti range di misura. I dodecaedri cavi, aventi struttura con buchi di diverso diametro, erano strumenti militari per determinare le distanze in balistica. Ed in effetti, questi oggetti sono stati trovati lontano dall'Italia, nei paesi che sono stati al confine dell'Impero. Ho scritto anche un confronto con strumenti medievali e moderni su arXiv,http://arxiv.org/abs/1205.2078 Il dodecaedro era uno strumento facile da usare. Per cambiare il range di distanza, bastava ruotare il dodecaedro. Se si guardano le foto dei dodecaedri, si vede che avevano delle piccole sferette ai vertici. Esse rendevano più facile la presa dello strumento nelle operazioni veloci. Durante il Medioevo sono stati creati altri telemetri poiché l'uso del dodecaedro, che era uno strumento dell'esercito romano, si era perso col crollo dell'Impero. Questi oggetti medievali, conosciuti come "fore- staff", erano molto più ingombranti e con parti mobili da cambiare se cambiavano le distanze da valutare. Il dodecaedro romano è in fondo un comodo "telemetro a coincidenza". Gli archeologi che hanno trovato il dodecaedro a Jublains dicono che esso è un dado per la divinazione. I Romani, come gli Etruschi, avevano dei dadi dodecaedrici per il gioco o la divinazione, ma essi erano molto diversi. Si veda per esempio http://arxiv.org/abs/1205.0706 oppure la pagina di questo sito sul dodecaedro etrusco.
On Romam Dodecahedra many theories have been proposed: candlestick holders (but at that time people used oil lamps), dice for divination (these objects have faces of different masses and therefore they always fall on the heaviest face), astronomical instruments (to determine the equinoxes, but the ancients used sundials), and so on. Unfortunately it is not easy to dela with so many theories. What is clear is that these bronze dodecahedra are hollow, with faces that have perfectly circular holes of different diameters. Searching the Web I found the article written by some archaeologists that accurately describe a dodecahedron made of bronze, found at Jublains, the ancient Nouiodunum, dated to the second or third century AD. Using the data on the diameters of the holes I made a cardboard model. Looking through the model I found it could be an optical instrument for measuring distances, being therefore a rangefinder. Like several rangefinders, the measure is based on similar triangles. The details of the dodecahedron as an instrument for telemetry are published in arXiv http://arxiv.org/abs/1204.6497 (In Italian http://porto.polito.it/2497004/). The dodecahedron has four different measuring ranges. The dodecahedra, having a structure with holes of different diameters, were military devices to determine distance for ballistics. And in fact, these objects have been found outside of Italy, at the border of Roman Empire. I also wrote a comparison with medieval and modern instruments, arXiv,http://arxiv.org/abs/1205.2078 The dodecahedron was easy to use. To change the range of distance, it was enough to turn the dodecahedron. If you look at pics of dodecahedra, you can see the small knobs. They made easier to grip the tool during quick operations. During the Middle Ages, rangefinders have been created different from the dodecahedron, which was an instrument of the Roman army. Probably its use was lost with the collapse of the Empire. These medieval objects, known as "fore-staff," were bulkier and had the necessity to change and move some parts of it, for different ranges. The Roman dodecahedron is basically a "coincidence rangefinder." Archaeologists that have found the dodecahedron at Jublains say that it is a dice for divination. The Romans, like the Etruscans, had dodecahedra for the game of dice or divination, but they were quite different. See, for example http://arxiv.org/abs/1205.0706 or the page of this site on the Etruscan Dodecahedron.
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Dodecahedron With Sundial Faces at Palazzo Madama Traduci http://stretchingtheboundaries.blogspot.it/2012/09/dodecahedral-sundial.html
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