Dodecahedron

Il solido platonico che preferisco

The monk, the polyhedrons and the wardrobe

Fra Giovanni da Verona. Tarsia raffigurante un armadio con poliedri. Courtesy Laurom di Wikipedia in italiano. 

Fra Giovanni da Verona (1457 circa – 1525) è stato un intarsiatore, miniatore, scultore nonché architetto italiano, attivo tra la fine del XV secolo e l'inizio del XVI secolo. Artista poliedrico, è ricordato in particolare per la maestria nell'intarsio e nella prospettiva.

Sfere cinesi (puzzle spheres)

Sfera cinese, in avorio, fatta di diverse sfere concentriche (British Museum. Original photograph from Ged Carroll)

Wikipedia dice che questo tipo di lavori è diventato popolare in Europa grazie alla produzione Cinese del XIX secolo. Ma la creazione di questi rompicapo ha origini ben più lontane. Antikitera.net ci informa che i primi esemplari fecero la loro comparsa durante la Dinastia Song, intorno all'anno 1000 d.C.

http://www.antikitera.net/news.asp?ID=11753

Pierre MEYER and the puzzle dodecahedron

Pierre Meyer is an artist who works with ivory.

https://www.maitresdart.com/pierre_meyer-40/parcours_et_realisations.html

Great his puzzle dodecahedron!

A puzzle ball

Puzzle ball. This image is a courtesy of   " The Puzzle Museum http://puzzlemuseum.org". 

Chinese and European ivory puzzle balls

From Wikipedia: https://en.wikipedia.org/wiki/Ivory_carving

"By the 18th century China had a considerable market in items such as figures made for export to Europe, and from the Meiji Period Japan followed. Japanese ivory for the domestic market had traditionally mostly been small objects such as netsuke, for which ivory was used from the 17th century, or little inlays for sword-fittings and the like, but in the later 19th century, using African ivory, pieces became as large as the material would allow, and carved with virtuosic skill. A speciality was round puzzle balls of openwork that contained a series of smaller balls, freely rotating, inside them, a tribute to the patience of Asian craftsmen."

Usually, many of these balls have a decorated stand made of ivory too.

Chinese puzzle ball, with openwork and a series of twelve smaller balls, ivory, 19th century. British Museum. Original photograph from Ged Carroll

Form http://tinglefactor.typepad.com/thetinglefactorbox/2013/07/where-is-the-seam-chinese-puzzle-ball.html

"Originally, they (Chinese puzzle balls) were made almost exclusively from ivory, or the tusks of elephants and were the playthings of rich men because of the time and effort involved in making them. ... Usually, puzzle balls are symbols of good luck, and are decorated with a variety of feng shui symbols. The outermost layer often features the phoenix and dragon, symbols of yin and yang. The phoenix represents the wife while the dragon is the husband and emperor, and balls decorated with these symbols are thought to bring good luck and happiness to a marriage. In fact, almost all of the symbols most commonly associated with puzzle balls are associated with ensuring a long and happy marriage. Some balls even have different symbols on different layers, though the most common is a highly decorative outer ball and ‘latticed’ balls inside (with geometric patterns of holes)."

Detail of an ivory ball on show in the German Rautenstrauch-Joest Museum. It has 16 layers, which can spin. Courtesy Till Niermann , Wikipedia.

In the above image we see an example of Canton ivory carving. From Wikipedia (on the Lingnan culture or Cantonese culture). "Canton ivory woodcarving is another well-known product from Lingnan. With a history of 2000 years, it traditionally uses ivory as raw material to make sculptures, with the Canton-style renowned for being particularly delicate and detailed without being brittle. The Cantonese people have also successfully produced the legendary craft product - Ivory ball. After the 1980s, however, international ivory trade has been banned. This results in the Cantonese people now trying to find substitute materials - materials that look and feel like but are actually not ivory - in their attempt to pass on this ancient art."

From http://www.odditycentral.com/pics/chinese-puzzle-balls-the-rubiks-cube-of-the-ancient-world.html

"Chinese puzzle balls are ornate decorative items that consist of several concentric spheres, each of which rotates freely, carved from the same piece of material. ... These detailed works of art are usually made up of at least 3 to 7 layers, but the world’s largest puzzle ball is actually made of 42 concentric balls all enclosed one within the other. Although the inner balls can be manipulated to align all the holes, Chinese puzzle balls got their name from people who, through the ages, pondered the mystery of making such objects. So how exactly are puzzle balls made? .... Chinese masters rotate a solid ball on a lathe and start by drilling holes toward the center of the objects. Then, using special “L”-shaped tools, they begin to separate the innermost balls. ...  Because it is easier to work with, the exterior shell is the most elaborately carved, usually featuring an intertwined dragon and a phoenix."

Antikitera.net tells us that the first puzzle balls appeared during the Song Dynasty, around 1000 d.C.
http://www.antikitera.net/news.asp?ID=11753

After having shown the Chinese ivory balls, it seems that the puzzle balls became popular in Europe thanks to Chinese products of the later XIXth century. However, puzzle balls existed in Europe in XVI or XVII century. Here an example.

European puzzle ball, XVI-XVII Century (Image Courtesy: Maureen and Renato Bucci, Italy). It was exhibited with a rosary having the beads made in the same manner of the ball. 

The rosary, XVI-XVII Century (Image Courtesy: Maureen and Renato Bucci, Italy).  

The ball shown in the image is remarkable because it looks like a Roman Dodecahedron. Actually Renato Bucci was so kind to send me the picture because of this similarity. Probably, this was an object of a Wunderkammer (in italiano, camera delle meraviglie o gabinetto delle curiosità o delle meraviglie), encyclopedic collections of objects of the Reinassance Europe.  

From https://en.wikipedia.org/wiki/Cabinet_of_curiosities

"The Kunstkammer was regarded as a microcosm or theater of the world, and a memory theater. The Kunstkammer conveyed symbolically the patron's control of the world through its indoor, microscopic reproduction."

An example of Kunstkammer
http://wonder-cabinet.sites.gettysburg.edu/2017/cabinet/carved-ivory-puzzle-balls/

Besides the balls, we have also the polyhedra. Here the dodecahedra created by Egidius Lobenigk (1581 - 1584). From https://artsandculture.google.com/entity/m0t50361?categoryid=artist we can see them. 

These dodecahedra are at Staatliche Kunstsammlungen Dresden, (courtesy image from Jürgen Karpinski, photographer).

Let me conclude remarking that today puzzle balls are created too. Here the image of one of them, which is showing a contempoary "puzzle dodecahedra". The artist that created it is Pierre Meyer is an artist who works with ivory. https://www.maitresdart.com/pierre_meyer-40/parcours_et_realisations.html

Pierre Meyer's ivory "puzzle dodecahedron".

Also "new production of ornamental turning ivory of '600" is evidenced by the works of Andrea Pacciani, architect in Parma, by the web https://www.etsy.com/it/listing/225172225/tornitura-ornamentale-da-un-modello-in. A piece "is inspired by a piece of the museum's collection of Rosenborg in Denmark (*). Another piece is inspiered to the drawings of Grollier de Serviere, (1596–1689), French inventor and ornamental turner.
According to Andrea, "Thanks to the new generation of 3D technologies we could bring back the light of contemporary production about this object collection of great visual impact". That is, new technologies for creating objecs for our modern Wunderkammer.

(*) the reader can see the pieces at http://www.bobkatsjaunt.com/denmark.html.

A drawing from a book on the works of  Grollier de Serviere

Dodecahedral die

"In modern role-playing games, the dodecahedron is often used as a twelve-sided die, one of the more common polyhedral dice" http://en.wikipedia.org/wiki/Dodecahedron

http://en.wikipedia.org/wiki/Dice#Non-cubical_dice

Even in the past, the dodecahedron was used for dice.

Here an example form the site archéologique de la Cathédrale Saint-Pierre de Genève 

http://www.site-archeologique.ch/contenu.php?id-node=25&id-img=84

Copie d'un dé romain, en forme du dodécaèdre, datant du IVe siècle

Even older are the Etruscan dodecahedra:

More on the etruscan dodecahedron at sites:

https://sites.google.com/site/ancientdodecahedra/etruscan-dodecahedron

http://stretchingtheboundaries.blogspot.it/2012/05/etruscan-dodecahedron.html

CALCULI, COMPAS, DODECAEDRE

Very interesting page at http://www.alienor.org/Articles/ecriture/instrument3.htm
telling that "Les Gallo-Romains possédaient également différents instruments destinés aux disciplines scientifiques Pour compter, on utilisait des petits cailloux appelés calculi qui étaient placés dans des cases. On disposait aussi de compas se rapportant à la géométrie mais aussi au traçage des lignes d’écriture. ... Aujourd’hui, les avis convergent pour interpréter les dodécaèdres en bronze, creux, comme des instruments de géomètre."

Musée de Poitiers. Visit the site!

Les dodécaèdres gallo-romains (2)

Adapted from http://letarot.com/pages-vrac/pages/Truc-a-trous.html

Les dodécaèdres gallo-romains (1)

Adapted from http://letarot.com/pages-vrac/pages/Truc-a-trous.html

Vienne Roman Dodecahedron

Print! Cut! Fold! Glue!

Image obtained from some data in the paper:

Duval Paul-Marie. Comment décrire les dodécaèdres gallo-romains, en vue d'une étude

comparée. In: Gallia. Tome 39, fascicule 2, 1981. pp. 195-200, doi : 10.3406/galia.1981.1829

Bristol Roman Dodecahedron

Adapted from the paper:

Etwas Gewisses hievon zu bestimmen waere ein Gewagtes

260 Jahre Dodekaeder-Forschung, by Michael Guggenberger

Roman Dodecahedron replica

Print! Cut! Fold! Glue!

Dodecahedral sundial

Dodecahedron With Sundial Faces 

Palazzo Madama, Torino

End  XVII Century

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

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 ).
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", on arXiv, http://arxiv.org/abs/1205.2078 and on SCIRP  http://www.scirp.org/journal/PaperInformation.aspx?PaperID=23245

On Roman and Etruscan Dodecahedra again

This morning I found the following link: http://sculpture.net/community/showthread.php?t=10801 having a quite interesting discussion on the " Roman Dodecahedron Mystery "
Let me report what SteveW wrote:
"This object is an on-the-fly surveying device. Worn on a chain around the neck, a military officer could hold it out to arms length and "fit" an enemy soldier or catapult, ballista etc between the opposing holes and then multiply by 100 (or whatever factor/number of links of the chain) to guage his distance. It would have been accurate enough to place a man within a few feet up to roughly a mile away. "
This is a quite good  reference for my paper  A Roman Dodecahedron for measuring distance
http://arxiv.org/abs/1204.6497

A figure from my paper http://arxiv.org/abs/1204.6497

Somebody is considering that there are also dodecahedra of stones. These were tools for gambling, please see An Etruscan Dodecahedron http://arxiv.org/abs/1205.0706 or the previously written post.

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

A dodecahedron of a Roman soldier

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 there is the possibility to use it to measure distances as in the following approach: