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.

A mound near Jefferson City

"Before the coming of white settlers, the region surrounding Jefferson City was home to an ancient group known as the Mound People. In fact, America's largest prehistoric city was located only 160 miles away at what is now Cahokia, Illinois. Why this civilization disappeared remains a mystery."
http://www.jeffcitymo.org/main/historyheritage.html
Information on the mounds near Jefferson City we can find in  the paper
published by The Project Gutenberg EBook of Scientific American Supplement, No. 841,
February 13, 1892, by Various
INVESTIGATION OF A MOUND NEAR JEFFERSON CITY, MO, By A.S. LOGAN.
Logan wrote
"Recently, a party consisting of engineers and employes of the Missouri River Improvement Commission began an exploration of one of the mounds, a work of a prehistoric race, situated on the bluff, which overlooks the Missouri River from an elevation of one hundred and fifty feet, located about six miles below Jefferson City.
This mound is one of about twenty embraced in a circle one quarter of a mile in diameter.
The above party selected the mound in question apparently at haphazard; all the mounds presenting nearly a uniform outline, differing only in size and mostly circular in form, and from twenty to twenty-four feet at the base, rising to a height of eight feet and under. A trench was cut on a level with the natural soil, penetrating the mound about eight feet. A stone wall was encountered which was built very substantially, making access in that direction difficult, in consequence of which the earth was removed from the top for the purpose of entering from that direction. The earth was removed for a depth of four feet, when the top of the wall was exposed. Further excavation brought to light human bones, some of them fairly well preserved, especially the bones of the legs. On the removal of these and a layer of clay, another layer of bones was exposed, but presenting a different appearance than the first, having evidently been burned or charred, a considerable quantity of charcoal being mixed with the bones. In this tier were found portions of several skulls, lying close together, as if they had been interred without regard to order. They were, in all probability, detached from the body when buried....
A few pieces of flint weapons were found in the upper layers, and nothing else of any significance....
At this juncture the diggers abandoned the search, and some days later the writer, desirous of seeing all that was to be seen, resumed the work and removed the earth and remains until the bottom of the vault was reached; several layers being thus removed. All of these had evidently been burned, as charcoal and ashes were mixed with the bones of each succeeding layer. The layers were about an inch in thickness, with from two to four inches of earth between, and small flat stones, about the size of a man's hand, spread on each different layer, as if to mark its division from the next above.
Between the bottom layers, mixed with charcoal, ashes and small portions of burned bones were found what gives value to the search, numbering about fifty tools and a smoking pipe.
The material of the tools is the same as the rock forming the vault, locally known as "cotton rock." I would consider it a species of sandstone.
Overlying the edge of "cotton rock" in the bluff is flint in great quantities, and in every conceivable shape, that these people could have resorted to had they been so disposed, and why they used the softer material I will leave to some archæologist to determine. The tools themselves are made after no pattern, but selected for their cutting qualities, as they all have a more or less keen edge which could be used for cutting purposes, and were no doubt highly prized, as they were found all in a pile in one corner of the vault and on top of which was found a stone pipe. The pipe is made bowl and stem together, and it is curious that people of such crude ideas of tools and weapons should manufacture such a perfect specimen of a pipe. It is composed of a very heavy stone, the nature of which would be difficult to determine, as it is considerably burned.
A description of the vault will be found interesting to many. The wall of the vault rests upon the natural surface of the ground, about three feet high and eight and a half feet square, the inside corners being slightly rounded; it is built in layers about four inches in thickness and varying in length upward to three feet, neither cement nor mortar being used in the joints; the corners formed a sort of recess as they were drawn inward to the top, in which many of the stones were found. The stone for constructing the vault was brought from a distance of about a quarter of a mile, as there is none in sight nearer.
I assume from all these circumstances that these people lived in this neighborhood anterior to the age of flint tools, as the more recent interments indicate that they were then entering upon the flint industry, and it may be that the "cotton rock" had become obsolete.
These people buried their dead on the highest ground, covering and protecting them with these great mounds, when it would seem much easier to bury as at the present day; but instead, they, with great labor, carried the rock from a great distance, and it is reasonable to suppose, also, that the earth was brought from a distance with which they are surrounded, and piled high above, as there is no trace of an immediate or local excavation....
My object is the hope of a more thorough investigation at some future time...."

Hic sunt Garamantes

"Satellites and aerial photographs have revealed evidence of over a hundred fortified farms, villages, and towns - many with castle-like structures - in the southwestern deserts of Libya. These structures date back to between 1 and 500 C.E., meaning they predate the rise of Islam. ...
Archaeologists from the University of Leicester have used all this aerial data to identify as much as they can about the so-called Garamantes people, who are otherwise known purely through Greek and Roman sources. They've already discovered "the mud brick remains of the castle-like complexes, with walls still standing up to four metres high, along with traces of dwellings, cairn cemeteries, associated field systems, wells and sophisticated irrigation systems." "
http://io9.com/5857958/satellites-reveal-libyas-lost-2000+year+old-cities

Master and Dog

Master and his dog in the ancient Egypt.

http://www.archaeology.org/1009/dogs/

Leonardo, Genio e Mito

Alla Venaria Reale (Torino) si apre la mostra su Leonardo da Vinci. Il titolo è "Leonardo, il Genio, il Mito".

http://www.lavenaria.it/mostre/ita/mostre/archivio/2011/leonardo.shtml

In esposizione vi è l'autoritratto della Biblioteca Reale e il Codice del Volo.

In una pagina del codice, sotto la scrittora di Leonardo, vi è un ritratto, forse un suo autoritratto da giovane.

Vedi http://staff.polito.it/amelia.sparavigna/da-vinci-portrait.html

Image processing della pagina del codice, A.C. Sparavigna

The Digital Restoration of Da Vinci's Sketches

Mela, Pomponius

"Pomponius Mela, who wrote around AD 43, was the earliest Roman geographer. He was born in Tingentera (Algeciras) and died c 45 CE. His short work (De situ orbis libri III.) occupies less than one hundred pages of ordinary print. It is laconic in style and deficient in method, but of pure Latinity, and occasionally relieved by pleasing word-pictures. Excepting the geographical parts of Pliny's Historia naturalis (where Mela is cited as an important authority) the De situ orbis is the only formal treatise on the subject in Classical Latin."

http://en.wikipedia.org/wiki/Pomponius_Mela

Munich Digitization Center
Digitised works/ Author: Mela, Pomponius
http://www.digital-collections.de/index.html?c=autoren_index&l=en&ab=Mela%2C+Pomponius

Indos

"Sed praeter physicos Homerumque qui universum orbem mari circumfusum esse dixerunt, Cornelius Nepos ut recentior, auctoritate sic certior; testem autem rei Quintum Metellum Celerem adicit, eumque ita rettulisse commemorat: cum Galliae pro consule praeesset, Indos quosdam a rege Botorum (Boiorum) dono sibi datos; unde in eas terras devenissent requirendo cognosse, vi tempestatium ex Indicis aequoribus abreptos, emensosque quae intererant, tandem in Germaniae litora exisse. Restat ergo pelagus, sed reliqua lateris eiusdem adsiduo gelu durantur et ideo deserta sunt."
Pomponius Mela
De chorographia

"Mais à l'autorité des anciens philosophes et d'Homère, qui ont prétendu que la terre était de tous côtés environnée par la mer, on peut ajouter celle de Cornélius Nepos, qui, étant plus moderne, est par conséquent plus sûr. Or, cet auteur rapporte, à l'appui de cette opinion , le témoignage de Q. Metellus Celer, auquel il fait dire qu'étant proconsul dans la Gaule, le roi des Bètes lui fit présent de quelques Indiens , et que, s'étant informé d'où ils étaient venus, il apprit que, les tempêtes les ayant emportés loin de la mer des Indes, ils avaient été jetés, après un long trajet, sur les rivages de la Germanie. Le reste de la côte asiatique est donc baigné au nord par une mer sans bornes; mais cette partie est couverte de glaces éternelles, et par conséquent déserte.
Louis Baudet translation

To the ancient philosophers and Homer, who claimed that the land was surrounded on all sides by the sea, we may add Cornelius Nepos, who, being more modern, is therefore more reliable. However, this author reports in supporting this fact, the witness of Q. Metellus Celer: when he was proconsul in Gaul, the king of Botos made ​​him a present of a few Indians, and that, being informed of where they came from, he learned that the storms have swept away them of the Indian Ocean. They were thrown, after a long journey, on the shores of Germania. The rest of the northern Asian shores are surrounded by the sea, but this part is covered with eternal ice, and therefore deserted.

King Arthur's round table

According to Telegraph, http://www.telegraph.co.uk/news/uknews/8724183/King-Arthurs-round-table-may-have-been-found-by-archaeologists-in-Scotland.html, 26 Aug 2011, King Arthur's round table may have been found by archaeologists in Scotland. In fact, the archaeologists  have found a "circular feature" beneath the historic King's Knot in Stirling. The King's Knot is a geometrical earthwork in the gardens below Stirling Castle. The design of the Knot that we see today dates from the 1620s, but its central mound could  be much older. It seems that some writers going back more than six centuries have linked the landmark to  King Arthur. Archaeologists from Glasgow University, working with the local Archaeological Society, found that the round feature on the site pre-dates the visible earthworks.

How to measure the stair angle

Use a protractor (available at most hardware stores) to determine this angle.

Lay a straightedge across a minimum of 3 stair nosings. Put your universal protractor on the straightedge and measure the angle.  Note that the more stair nosings you have under the straightedge, the more accurate the angle will be. In the case that the run of stairs is very long, measure the stair angle at 2 or 3 places along the run to check that the angle remains constant along the full run.

I am proposing that Kha the architect at Deir El-Medina, Egypt, supervisor of the projects of three kings of the 18th Dynasty used his protractor to check the inclination of stairs for instance. You can see it at the Egyptian Museum of Torino. Read more at http://stretchingtheboundaries.blogspot.com/2011/08/khas-protractor.html

A modern universal protractor for building  is that shown in the following figure.

Kha

Kha, the architect

Museo Egizio, Torino

Back to Kha's protractor

In the post "Kha's protractor", I have discussed an object found in the tomb of  an ancient egyptian architect, Kha.  It can be used to measure angles.Plums, levels, protractors , compasses and so on are instruments for surveying. See http://en.wikipedia.org/wiki/Surveying and the table http://en.wikipedia.org/wiki/File:Table_of_Surveying,_Cyclopaedia,_Volume_2.jpg
In this table there is the image of a compass rose. We can start from it and go back to the Kha's rose.