Benvenuti in queste pagine dedicate all'archeologia. Amelia Carolina Sparavigna

Friday, 18 May 2012

An ancient rangefinder (Roman Dodecahedron)

According to Wikipedia, "a rangefinder is a device that measures distance from the observer to a target, for the purposes of surveying, determining focus in photography, or accurately aiming a weapon. Some devices use active methods to measure (such as sonar, laser, or radar); others measure distance using trigonometry (stadiametric rangefinders and parallax, or coincidence rangefinders). These methodologies use a set of known information, usually distances or target sizes, to make the measurement, and have been in regular use since the 18th century".
It could be surprising, but probably the Roman Army had a rangefinder. It was the Roman Dodecahedron (I have already discussed it in some posts and papers: on arXiv, where I am explaining how it can be used for measuring distance as a rangefinder, http://arxiv.org/abs/1204.6497 , and in Italiano ahttp://porto.polito.it/2497004/ ).
For me, those dodecahedrons having a structure with holes of different sizes, are military instruments to evaluate distances for ballistics. It is simple to use. Of course, later, during the Middle Age, different instruments had been developed for surveying: the dodecahedron was of the Roman Army, and, probably, its use lost after the collpase of the Empire. See also "Ancient and Modern Rangefinders", arXiv, http://arxiv.org/abs/1205.2078

Antiquaries - Roman Dodecahedron

The Antiquaries Journal

Saturday, 5 May 2012


Nundinae, form an Article by Leonhard Schmitz, Rector of the High School of Edinburgh, in William Smith, A Dictionary of Greek  and Roman Antiquities, John Murray, London, 1875.

NUNDINAE is derived by all the ancient writers from novem and dies, so that it literally signifies the ninth day. In ancient Calendaria all the days of the year, beginning with the first of January, are divided into what we may call weeks, each containing eight days which are marked by the letters A, B, C, D, E, F, G, H. Now it is admitted on all hands that this division is made to mark the nundinae, for every eighth day, according to our mode of speaking, was a nundinae. There were thus always seven ordinary days between two nundinae. The Romans in their peculiar mode of reckoning added these two nundinae to the seven ordinary days, and
consequently said that the nundinae recurred every ninth day, and called them nundinae, as it were novemdinae. A similar mode of stating the number of days in the week is still customary in Germany, where, in common life, the expression eight days is used for a week, and the French and Italians in the same manner call a fortnight quinze jours and quindici giorni....
The nundinae were, according to Plutarch, sacred to Saturn....
The nundinae were originally market-days for the country-folk, on which they came to Rome to sell the produce of their labour, and on which the king settled the legal disputes among them. ... For the populus the nundinae were feriae, but they were real days of business (dies fasti or comitiales) for the plebeians, who on these occasions pleaded their causes with members of their own order, and held their public meetings (the ancient comitia of the plebeians).

Friday, 4 May 2012

An Etruscan Dodecahedron

An Etruscan Dodecahedron, by Amelia Carolina Sparavigna
Department of Applied Science and Technology
Politecnico di Torino, C.so Duca degli Abruzzi 24, Torino, Italy

published on arXiv, http://arxiv.org/abs/1205.0706

The paper is proposing a short discussion on the ancient knowledge of Platonic solids, in particular, by Italic people.

How old is the knowledge of Platonic solids? Were they already known to the ancients, before Plato? If we consider Wikipedia [1], the item on Platonic solids is telling that there are some objects, created by the late Neolithic people, which can be considered as evidence of knowledge of these solids. It seems therefore 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."
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.

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,
[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

Thursday, 3 May 2012

Dodecahedrons and Roman soldiers

In a recent post (April 2012) I have discussed about Roman Dodecahedra.
After preparing a model of a Roman Dodecahedron, I was able to investigate it as an optical instrument.
In the paper "A Roman Dodecahedron for measuring distance", published in arXiv, http://arxiv.org/abs/1204.6497 you can find how a Roman soldier could had used it to determine the distance of a target.The dodecahedron is quite simple to use and portable. Someone could tell (or is telling) that it is more complicated compared to a simple cross-staff. Well, a cross-staff is rather long. In the case it were of bronze, the instrument turned out to be too heavy. Moreover, it seems that the cross-staff had been developed during the 14th century, therefore it was an instrument of the Middle Age in Europe.