Dimensions for making a replica Roman dodecahedron (1)

Let us suppose somebody wants to make a replica of a Roman dodecahedron,
here a recipe. Consider the following image

Centered at the center of each face, draw a circle, which will be a hole. These circles  usually have different diameters. Print and cut the image.

If you want some figures, you can, for instance use the article at the following link:

http://www.scribd.com/doc/116123429/The-Roman-Dodecahedron-of-Kadath

Here I am reproducing a table.

Here we have not the exact potions of the holes, but we have the diameters of hole on the opposite faces.

According to the image of the dodecahedron, 1 is opposite 12, 2 opposite 11, 3 opposite 10, etc.

"Carved stone balls from Scotland are an enigmatic class of objects. They date to the late Neolithic or early Bronze Age, between 3200 and 1500 BC, and are made of various stones ranging from sandstone to granite. They ... are decorated with carved evenly-spaced patterns of circular bosses or knobs around the surface of the sphere. The designs vary with the majority being based around a series of six bosses, but the number of bosses varies from 3-160."

More at http://www.ashmolean.org/ash/britarch/highlights/stone-balls.html

When America discovered Europe

A very remarkable book this one
The American discovery of Europe,  By Jack D. Forbes
 Let me suggest the reading of it. To stimulate the curiosity, here some small abstracts

What is Cornelius Nepos telling? And Pliny?

Empresa destruye patrimonio cultural

Empresa destruye patrimonio cultural
"El observatorio más antiguo de América, es afectado por trabajos de empresa autorizada por el INC. Desconocen informes que dan cuenta de restos arqueológicos alrededor de área patrimonial."

Ancient solar observatories: Syrian desert

The Syrian desert is covered by ancient stone structures created during Neolithic times, as I discussed in the paper, Stone circles on the harraat of Syrian desert, published at http://arxiv.org/abs/1106.4665
Some of these stone structures are circular, with radial lines of stones. These circles have been also compared with Stonehenge. Therefore, we can ask ourselves, are these solar observatories too, as Stonehenge is.
The answer is probably so, see my paper at http://arxiv.org/abs/1208.2591
I used a software for the solar energy applications to see the direction of sun on sunrise and sunset of solstice. It seems that there is a good agreement. Therefore, the Syrian stone circles are probably the Stonehenges of Middle East.

This is one of the stone structure of Syrian Desert. The image shows directions of sun during the day. “The lines on the drawing show the direction and height (altitude) of the sun throughout the day. Thicker and shorter lines mean the sun is higher in the sky. Longer and thinner lines mean the sun is closer to the horizon”, according to Sollumis.com http://www.sollumis.com/. On the left, the site as it appears in the Google Maps. In the middle, the direction of the sun on the summer solstice, choosing the center of the circle for observation. We see that, at sunrise, the sun is passing near the dot. At the sunset the direction is that of a line. In the image on the right, we see the direction of the sun on winter solstice. At sunrise, the lines is passing between dots. The sunset has the direction of a radius (images have been obtained from original provided by Google Maps and sollumis.com).

Circles and lines!
Alignemnt at the sunrise on the winter solstice


At the site http://www.dp-news.com/en/detail.aspx?articleid=97188
it is told that "He (David Kennedy) explains that they come in a huge variety of forms, some being ‘kites’, structures that funnelled animals, some being seemingly random meandering lines of stone and others being rectangular. None are believed to be aligned with the stars, which has added to the mystery surrounding their purpose." Well, in my opinion it is not so.

Roman jewelry in Japan

TOKYO —"Glass jewelry believed to have been made by Roman craftsmen has been found in an ancient tomb in Japan, researchers said Friday, in a sign the empire’s influence may have reached the edge of Asia."
http://www.japantoday.com/category/national/view/roman-jewelry-found-in-ancient-tomb-near-kyoto

20,000 years old pottery in a Chinese cave

"BEIJING (AP) - Pottery fragments found in a south China cave have been confirmed to be 20,000 years old, making them the oldest known pottery in the world, archaeologists say."
New Jersey Herald - Pottery 20,000 years old found in a Chinese cave

Small oscillations ...

"Researchers Terry Hunt and Carl Lipo test a new theory that suggests how ancient Easter Islanders may have used ropes to “walk” the moai to their platforms. Mystery of Easter Island"

http://ngm.nationalgeographic.com/2012/07/easter-island/walking-statue-video

Aléria dodecahedron

This image is adapted from YouTube http://www.youtube.com/watch?v=x_2OrVzC46Y

It is showing a rock crystal dodecahedron found at Aléria Corisca

http://en.wikipedia.org/wiki/Al%C3%A9ria

Arles, Turin and Caesar

The heart of the ancient history of Arles, this is what is on show at the Louvre Museum of Paris, until 25 June 2012. The Museum is hosting an exhibition of spectacular Roman pieces recovered from the bottom of the Rhone! From March 9 to June 25, 2012, the Louvre Museum in Paris

On exhibition fifty of the most spectacular artifacts unearthed by archaeologists: columns and capitals, fragments of statues and reliefs, Roman jewelery, lamps, vases ... These pieces are coming from the museum of Arles, and other pieces from Avignon, Vienne and Turin.
Among the pieces from Turin, you can admire Julius Caesar's bust, unearthed in 1825, long considered unique, until the discovery in 2007 in Arles of another bust of the Roman politician. At the Louvre, these sculptures will be presented for the first time side by side.

http://www.france.fr/it/arti-e-cultura/evenement/arles-gli-scavi-del-rodano-un-fiume-memoria

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

• The Antiquaries Journal July 1924 4 : pp 273-274
• Copyright © The Society of Antiquaries of London 1924
• DOI: http://dx.doi.org/10.1017/S0003581500091459 (About DOI)
• Published online: January 201 Table of Contents - July 1924 - Volume 4, Issue 03  

Nundinae

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).

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.

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

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.

(http://en.wikipedia.org/wiki/Jacob's_staff)

Roman Dodecahedra

Recently I have read 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, 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). Here I shortly discuss this article.
It starts with the following questions. Was is it, "a sacred symbol for Druids or the tip of a 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. A defined scholar theory about their use is still lacking. 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. At the site www.romandodecahedron.com, Wagemans introduced the research and waiting for comments.


Courtesy: DieBuche, Wikipedia

I like very much the discussion by Cinzia di Cianni about this mistery of archaeology.
For what concerns 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. In fact, Vitruvius deeply describes in his De Archtectura, how to prepare the analemma. See my Measuring times to determine positions, http://arxiv.org/abs/1202.2746 It is possible that the dodecahedron was used to determine more precisely the time during the equinoctial period.
According to Cinzia, there are many proposal for their use.
In my opinion it is necessary to study how they can move, since they are biased structures, in order to understand whether they could have been used as dice for divination or bowls for simply playing with them. In a static use of them, the hypothesis for measuring time is quite interesting.

However, in my opinion it is possible to use as a rangefinder

Ale's stone boat

Ale's Stones (or Ales stenar) is a megalithic monument in Kaseberga,  southern Sweden. It is a boat of stones,  67 meters long formed. There are 59  boulders of sandstone,  up to 1.8 tonnes each. According to the local  lores,  King Ale lies buried there.
http://en.wikipedia.org/wiki/Ale's_Stones

Calendar Puzzles Deciphered in Ancient Statue

Calendar Puzzles Deciphered in Ancient Statue, by Rossella Lorenzi

on the Trundholm Sun Cariot and my proposal of a calendar

http://news.discovery.com/history/bronze-age-calendar-120330.html

 

A calendar of the Bronze Age

Ancient bronze disks, decorations and calendars, http://arxiv.org/abs/1203.2512,
by Amelia Carolina Sparavigna, Department of Applied Science and Technology, Politecnico di Torino, Italy, 

Some ancient bronze disks, found in burial places in Denmark are covered by amazing decorations. These decorations are composed from several concentric circles and spirals, and bands with zigzag lines. As told in Refs.[1,2], some disks can represent the Sun, which was the supreme power of the Bronze Age cosmology in Denmark. It seems that the religion was based on the daily journey of the Sun and on the progression of the year. It is therefore quite logical to discuss these disks as symbols of time progression and therefore as calendars. This is what is proposed in Ref.2, for some of these items, such as the Trundholn Sun Chariot, a bronze disk and a bronze statue of a horse placed on a device with spoked wheels, and the disk of Egtved [3]. Here I will discuss the disk of the Trundholm Sun Chariot, proposing a new interpretation of it, giving a calendar of 360 days.


Among the burial objects of the Early Bronze Age, the Trundholm Sun Chariot (Fig.1, [4]) is beautiful and amazing for the contrast between the fine decoration of the disk and the stylized shapes of chariot and horse. This artifact is also known as Solvognen. The sculpture was discovered in 1902 in a peat bog on the Trundholm moor and is now in the collection of the National Museum of Denmark in Copenhagen. It was cast by the lost wax method [1,5,6]: it means that this technique was known during the Bronze Age. The disk has a diameter of approximately 25 cm. In fact, it consists of two bronze disks, flanged by an outer bronze ring. One of the disks had been gilded on one of his side. The models of the disks had been probably decorated with some standard punches, because concentric circles and spirals seem to be identical in the decoration.

The two sides of the disk are considered as representations of the sun, on a chariot pulled by a horse across the heavens from East to West during the day, showing its bright side, the gilded one. During the night, it returns from West to East [1], showing his “dark side” to the Earth. The sculpture is dated to about 1400 BC [1]. However, the same reference is telling, “a model of a horse-drawn vehicle spoked wheels in Northern Europe at such an early time is surprising”. They were common in the Late Bronze Age, which ranges from 1100 BC to 550 BC. Ref.1 is suggesting a possible Danubian origin or influence, although the Museum supposes it of Nordic origin.

Let us consider the gilded side of the disk: it has the outer zone, which may represent the solar rays (Fig.2, [7]). There is an annulus (the region lying between two large concentric circles) decorated with small multiple concentric circles, linked by a looping band, which creates a “yin and yang” ornamental motif (see Fig.3 on the left) [8]. The image on the right of the same figure is reproducing the dark side of the sun. In Ref.2 the author is proposing that this side is a calendar. The author, Klaus Randsborg is considering the following calculation. Starting from the centre of the disk, we add the number of spirals in each annulus of the disk, multiplied by the order of the annulus where they are, that is (1x1 + 2x8 + 3x20 + 4x25). This results in a total of 177, a number very close to the number of days in six synodic months. In the Reference 2, the author is also proposing a calendar for the Egtved disk and other objects, supposing that the “spiral” symbol, that is, the figure formed by multiple concentric circles or by a true spiral, is representing the day. The annulus where the symbol is places provides the multiplication factor.

Here I propose another interpretation for the decoration in Fig.3, right panel, that is, of the side corresponding to the night. In the inner part of the disk (see the Figure 4), there are the days of a “week”, having therefore 8 days. For the moment, let us not consider the central large circle with many concentric circumferences. It could be a symbol for the cosmos as an ordered and harmonic system, as the cosmos was for the ancient Greeks. In the outer two annuli, there are the weeks of the year, which are 45. Then if we multiply the days in a week by the number of weeks, we obtain 360 days. That is: (8 days) x (45 weeks) = 360 days of the year. As in the ancient Egypt, the year has 360 days: the Egypt divided the year into 12 months of 30 days each, plus five extra days. Let us note that the weeks (see Fig.4) are grouped in two annuli: if we consider the winter solstice as the beginning of the year, the two groups of weeks could have the meaning that during the year there are two seasons, that of a “young sun” followed by the season of a “mature and then old” sun.

Let us note that weeks having eight days existed. The ancient Etruscans developed a week known as the nundinal cycle, around the 8th or 7th century BC. This system passed to Rome, no later than the 6th century BC. It seems that Rome had for a certain period of time a calendar based on two cycles, one having weeks of seven days and the other having eight-day weeks [9]. In any case, using two markers, a marker for the day in the central part and another marker for the week in the outer part, we can use the disk in Fig. 4 as a calendar for a nundinal system. Of course, we need a reference axis, as the black one in the figure. For the five extra days at the end of the year, we can use the circle at the centre of the disk. This is the centre of rotations: everything is turning about it. This figure contains both the end and the beginning of the year, able to “adjust” the circle of time, restoring the cosmic order.

For what concerns the other side of the Trundholm disk, the gilded day-side, I can only tell that, if we consider the total number of spirals (52), central included, and assume that each spiral is representing a week having seven days, we can obtain 364. The central “week” is larger because it contains one or two extra days, depending on years. Is it possible that the Trundholm disk is a calendar having two cycles? The answer is beyond the author’s knowledge. I consider more reliable the 360 days calendar, as in Fig.4, using the night-side of the disk.

Of course, the decorations in the disk could be simply a beautiful decoration. In any case, if we try to repeat it, we need to arrange in some manner the number of circles/spirals at specific relative distances. The two diagrams of Fig.5 are showing how the artist could have assembled the decoration, subdividing in some angular sectors the disk. It is probable that the artist possessed some specific knowledge of geometric rules. In my opinion, further studies of the decorations of ancient bronze artifacts can be useful to understand the progression of human knowledge of mathematics and geometry.

References
1. Trundholm sun chariot  http://en.wikipedia.org/wiki/Trundholm_sun_chariot
2. Klavs Randsborg, SPIRALS! Calendars in the Bronze Age in Denmark, 2010, Adoranten. Vol.2009,  http://www.ssfpa.se/pdf/2009/Randsborg.pdf
3. Egtved Girl,  http://en.wikipedia.org/wiki/Egtved_Girl
4. The image of the Trundholm sun chariot was created by Malene Thyssen, downloadable from Wikipedia, http://commons.wikimedia.org/wiki/User:Malene
5. Lost-wax casting  http://en.wikipedia.org/wiki/Lost-wax_casting
6. Molding (process),. http://en.wikipedia.org/wiki/Molding_(process)
7. Note the “rays” at the rim of the golden disk in the image adapted from a picture taken at the National Museum, Copenhagen Denmark, by Kim Bach.
8. The sketches in the Figures 3, 4 and 5 have been created according to the drawings reported in Reference 2. Please see this reference to see the details, which are amazing.
9. Week, http://en.wikipedia.org/wiki/Week, Nundinal cyle,
http://en.wikipedia.org/wiki/Roman_calendar#Nundinal_cycle

Fig.1 The Trundholm sun chariot

Fig.2 The gilded solar face of the disk.

Fig.3 Decorations of the front/day (left) and back/night (right) sides of the disk of the Trundholm sun chariot. Note on the left image, the circles linked by a “yin and yang” pattern.

Fig.4. In the inner part of the disk, there are the “days” of a week having 8 days. In the outer two annuli, the weeks of the year, that is 45, subdivided in two “seasons”. We have (8 days) x (45 weeks) = 360 days of the year. We can use the decoration as a calendar, using two markers: one (red) for the day and the other (blue) for the week. In the lower part of the image we can see two examples. We count clockwise from the vertical axis.  This calendar is working as a clock with two hands for days and weeks. For the days, the red marker turns on the first ring. The second marker, the blue, turns on the second ring for the first season, and on the third ring for the second. 

Fig.5. Two diagrams are showing how, probably, the artist had assembled the decoration, subdividing the disk in a few sectors. It seems that the artist knew some geometric rules.

Aristippus of Cyrene

from THE ARCHITECTURE, by MARCUS VITRUVIUS POLLIO.
BOOK THE SIXTH, INTRODUCTION.

ARISTIPPUS, the Socratic philosopher, shipwrecked on the coast of Rhodes, perceiving some geometrical diagrams thereon, is reported to have exclaimed to his companions, " Be of good courage, I see marks of civilization" and straightway making for the city of Rhodes, he arrived at the Gymnasium; where, disputing on philosophical subjects, he obtained such honours, that he not only provided for himself, but furnished clothing and food to his companions. When his companions had completed their arrangements for returning home, and asked what message he wished to send to his friends, he desired them to say that the possessions and provision to be made for children should be those which can be preserved in case of shipwreck; inasmuch as those things are the real supports of life which the chances of fortune, the changes of public affairs, and the devastation of war, cannot injure.