Showing posts with label polyhedra. Show all posts
Showing posts with label polyhedra. Show all posts

Saturday, October 15, 2022

Chasing Ancient Pentagrams Part I: the Roman Dodecahedra

A "Roman Dodecahedron" found in the UK by Brian Campbell in 1987.

 
The Gallo-Roman Dodecahedra
_________________________________________________________
 
 
(This post was originally intended to be the first of the pentagonal phi series, and much of it was written early last year. It has been revised, however, and will contain new material. It may also run into two parts.)
 
"One August day in 1987, Brian Campbell was refilling the hole left by a tree stump in his yard in Romford, East London, when his shovel struck something metal. He leaned down and pulled the object from the soil, wondering at its strange shape. The object was small—smaller than a tennis ball—and caked with heavy clay. 'My first impressions,' Campbell tells Mental Floss, 'were it was beautifully and skillfully made … probably by a blacksmith as a measuring tool of sorts.'

Campbell placed the artifact on his kitchen windowsill, where it sat for the next 10 or so years. Then, he visited the Roman fort and archaeological park in Saalburg, Germany—and there, in a glass display case, was an almost identical object. He realized that his garden surprise was a Roman dodecahedron: a 12-sided metal mystery that has baffled archaeologists for centuries. Although dozens, and perhaps hundreds, of explanations have been offered to account for the dodecahedrons, no one is certain just what they were used for."

- Via an excellent article on Mental Floss found here. Inset right (above) is a Roman Dodecahedron exhibited in the Württembergisches Landesmuseum, Stuggart, Germany, and found in this article.


"A Roman dodecahedron or Gallo-Roman Dodecahedron is a small hollow object made of copper alloy which has been cast into a dodecahedral shape: twelve flat pentagonal faces, each face having a circular hole of varying diameter in the middle, the holes connecting to the hollow center. Roman dodecahedra date from the 2nd to 4th centuries AD.

Since the first dodecahedron was found in 1739, at least 116 similar objects have been found[1] from Wales to Hungary and Spain and to the east of Italy, with most found in Germany and France. Ranging from 4 to 11 centimetres (1.6 to 4.3 in) in size. A Roman icosahedron has also come to light after having long been misclassified as a dodecahedron. This icosahedron was excavated near Arloff in Germany and is currently on display in the Rheinisches Landesmuseum in Bonn.

No mention of them has been found in contemporary accounts or pictures of the time. Speculated uses include as a candlestick holder (wax was found inside two examples); dice; survey instruments for estimating distances to (or sizes of) distant objects; devices for determining the optimal sowing date for winter grain; gauges to calibrate water pipes, army standard bases, a coin measuring device for counterfeit detection. Use as a measuring instrument of any kind seems improbable since the dodecahedra were not standardized and come in many sizes and arrangements of their openings. It has also been suggested that they may have been religious artifacts, or even fortune-telling devices. This latter speculation is based on the fact that most of the examples have been found in Gallo-Roman sites. Several dodecahedra were found in coin hoards, providing evidence that their owners considered them valuable objects. Other suggestions include a knitting frame for creating gloves, supported by the fact that many are found in the northern range of the empire.

Smaller dodecahedra with the same features (holes and knobs) and made from gold have been found in South-East Asia along the Maritime Silk Road. They have been used for decorative purposes and the earliest items appear to be from the Roman epoch."


- Via the Wiki entry for Roman or Gallo-Roman dodecahedron. Inset left is a museum photograph of 2 Roman dodecahedrons and the icosahedron mentioned in the quote. In my eyes, the icosahedron, although a polyhedron and similar in size to the dodecahedrons, does not seem to be in the same family of objects as the former.


***

Did you know there is a geometry in running water... or that small objects of equal mass - specifically small and lightweight - tend to fall into groups of three? Regarding the latter, at one point this year, I noticed this so often during my waking hours that I began to document these geometrical events, snapping photos with my cell phone whenever they occurred. Is this madness? Precisely. But it's a sort of "magical" madness involving a particular phenomenon which very likely spawned numerous oracles in the past; everything from casting knuckle bones, dice, sticks, coins, tea leaves, etc. for the purposes of divining or interpreting the resulting patterns as a narrative of the future.

In reality, small objects very often fall into rudimentary geometric patterns or figures - a little understood phenomena Jungian psychologist, Marie-Louise von Franz, 1915–1998, once described as a form of synchronicity or "meaningful chance" especially in regards to divination and the role of natural numbers.

Inset right is one example I shot with my cell-phone September 8, 2020; an interesting triangle formed by the random distribution of red pepper seeds scattered during the course of making a salad! I didn't see the figure at first; the pattern had formed on the far side of the table on which I was working. It was during clean-up that I came across it, and I swear I felt like some mystified farmer who discovers a crop circle in his field one day; it was eerie.

The X-file factor of this event was that the triangle bore a strong resemblance to the triangle I'd been working with over a year (and you've been encountering on this blog for months): the pentagonal golden triangle. Had it fallen out of my head and imprinted itself on the table? Who can say? But, I saw it as an affirmation of an artist's journey well taken. And, for an artist who faithfully follows the muse down one rabbit hole after another, affirmation is a much coveted thing... because, what is a "rabbit hole" other than some strange anomaly stumbled across during an intellectual excursion which might potentially lead one to the greatest of epiphanies or the most confounding of delusions?

But, while it may have ended that way, my journey down this particular rabbit hole did not begin with a random distribution of pepper seeds. It began with an online article regarding that unusual metal object currently staring at you from the top of this post... its empty cyclopean eye revealing just about everything we know regarding its existence...which is pretty much nada.

Now, the general opinion among experts (and one must always use this term loosely) is that a lot is known - and verified - about the past. Even the long past. But, don't be fooled. The past is as elusive (and illusive) as the future. New discoveries keep popping up each day with the potential to completely overturn all previous determinations. There are, after all, numerous newsworthy items. And, no, I 'm not referring to those well-documented abominations of the world's daily affairs. I'm talking about those little, weird tangible things - the products of human ingenuity - which emerged in the days before "artificial intelligence" was even a bad dream in somebody's head. This is not so say that the days of which I speak - and, no, none of us witnessed those days in any memorable way - were benign or utopian, but, intriguingly enough, ancient humans were admirably capable of flummoxing the oh-so-sophisticated humans of the future (us),  producing artifacts that we - with all of our modern expertise... and some highly sophisticated calculators - are unable to identify.

Which brings us back to that cyclopean object resting above - the subject of this section - the Roman dodecahedron inset left. No one seems to have a precise explanation for its presence on the earth - although the general drift is towards some practical, utilitarian instrument (such as a candle holder or knitting device) or complicated measuring device (such as a military range-finder) - and, yet, at least a few ancient folk across a number of countries - and all living in the earlier centuries AD - created these objects for reasons of which we can only speculate. Over a hundred of them have been dug up in parts of Europe and one imagines more may be found. I find this oddly exhilarating; there are still things the experts, admittedly, can't explain. In any case, I'm sure that all intuitive readers have realized by now that this mysterious little artifact is quintessential rabbit-hole material... and, yes, it most surely was!

In any case, "Cyclops" is actually believed to have been created around 200 AD... somewhere within the vast Roman Empire. Which is how it got its name: The Roman dodecahedron. The thing is,  despite archaeologists finding more than a hundred of them, no written reference to them has ever been discovered. And this might be a our most important clue: perhaps the objects had no practical use whatsoever and were never intended for the general public.

What's more, the object is an early example of a polyhedron - and the dodecahedron is the most complex of the Platonic solids - artifacts which we rarely see in the ancient world (despite the contributions of Pythagoras and Plato). As it so happens, this blogger loves polyhedra - even documenting a set of my very own (inset right, above: the dodecahedron, the icosahedron, and the pentakis dodecahedron). Not very long ago, I posted about some unusual Scottish sundials (example below the jump)...

Monday, June 27, 2022

Albrecht Dürer and the Divine Ratio (Part I)

Melencolia I - copper engraving - 1514, Albrecht Dürer. Geometry: 2022, DS


"Melencolia I is a large 1514 engraving by the German Renaissance artist Albrecht Dürer. The print's central subject is an enigmatic and gloomy winged female figure thought to be a personification of melancholia – melancholy. Holding her head in her hand, she stares past the busy scene in front of her. The area is strewn with symbols and tools associated with craft and carpentry, including an hourglass, weighing scales, a hand plane, a claw hammer, and a saw. Other objects relate to alchemy, geometry or numerology. Behind the figure is a structure with an embedded magic square, and a ladder leading beyond the frame. The sky contains a rainbow, a comet or planet, and a bat-like creature bearing the text that has become the print's title.

The art historian Erwin Panofsky... wrote that 'the influence of Dürer's Melencolia I—the first representation in which the concept of melancholy was transplanted from the plane of scientific and pseudo-scientific folklore to the level of art—extended all over the European continent and lasted for more than three centuries.'"

- Excerpt from Wiki's entry for Dürer's Melencolia 1 (shown above). While many art historians seem to unanimously assume the robed, angelic figure is of the female gender, the figure is most assuredly male, and, judging by its facial expression, Dürer himself. Inset right: an early self-portrait (executed in 1498 at age 26) of the fashionable - but seriously introspective - young dude.

Regarding the central "ladder leading beyond the frame," note that it forms the apex of a large golden triangle.

Note: To give an example of how deeply this image continues to resonate over the years, Wiki mentions Peter-Klaus Schuster's 1991 publication, Melencolia I: Dürers Denkbild, an exhaustive history of the print's interpretation in two volumes.


"It should be noted that even Leonardo was unable to apply his own proportion and anatomical studies to his work given he painted little, or not at all, during the last decade of his life. Hence, the applicability of the study of proportion to practicing artists was still unclear. Dürer would spend nearly three decades working to remedy this ambiguity. He completed two treatises that would be the dominant basis for art theory in Renaissance Germany; their popularity and influence spreading with their subsequent translations. In 1525, Underweysung der Messung, or Four Books on Measurement, was published as a practical guide to geometric perspective for students of the arts; and, in 1528, Vier Bücher von Menschlicher, or Four Books on Human Proportion, appeared a few months after his death. Taken together, the studies illustrated the Renaissance belief that mathematics formed the firm basis and grounding for the arts."

- Excerpt from an commentary regarding Dürer's Vier Bücher von Menschlicher by Giovanni Paolo Gallucci found here. Inset left is one of Dürer's diagrams - a construction of a spiral - found in his Four Books on Measurement.


"There is much speculation as to why Dürer chose this construction rather than Euclid's construction which uses the 'golden ratio' proportion. The speculation stems from the fact that Dürer makes no mention of the golden ratio, although he was no doubt aware of its use in Italian art. It may be that Dürer simply did not feel comfortable with the precepts of the 'divine' ratio. The German architects had their own 'divine' ratio which was the vesica piscis ratio of  1:3. In addition Ptolemy's construction is simpler than Euclid's and these constructions were just a preliminary step in his program."

- Excerpt - along with Dürer's diagram (inset left) - from The Polygons of Albrecht Dürer by G.H. Hughes. (.pdf)

Regarding the diagram, the pentagon is constructed within the mason's "Sacred" tradition utilizing the Vesica Piscis as its generative source.


"Divine truth alone, and no other, contains the secret of what the most beautiful form and measure may be."

- Albrecht Dürer, from his essay Discourse on A
esthetics
 published as a conclusion to the Third Book of his proportion studies. (See the Giovanni Paolo Gallucci link for the full quote.)


"The greatest miracle that I have seen in all my days, happened in the year 1503, when crosses fell on many people, especially on children more than on other people. Among them all, I once saw one in the shape which I have drawn here; it fell on the linen blouse of Eyer's maid, who was in the Pirckheimer's back-house. And she was so upset about it that she cried and wailed; for she thought she was going to die of it.

Also, I saw a comet in the heavens."

 - Albrecht Dürer from the last page of his 1503 Gedenkbuch regarding an episode of a phenomenon known as Red (or Blood) RainDürer's drawing can be found on this page.

***

(Note: Originally, the title of this post and the title of the URL were one and the same. That is, till I realized that Melancolia I was one of three designated Master Prints. I am not quite sure who did the designation, but, after reviewing the two other prints involved, it seemed all three might have what I (now) refer to as hidden, occulted, or passive GTS. Unlike the more outrageously active spirals - e.g., those of Caravaggio, which seem as if they were deliberately designed - the passive spirals almost seem to creep into an image with the artist unaware. The thing is, it is logical to assume Albrecht Dürer did know about the golden ratio. Alas, the jury is still out.)

Albrecht Dürer (May 21,1471 - April 6 1528) is, in his own quiet way, possibly one of the most popular artists of the Renaissance period; certainly one of the most prominent. (You know you've arrived when there's a conspiracy blog written about your life!) After all, unlike many other artists, he kept a meticulously written record of his daily affairs. And, then, there were those self portraits... even while he was ill and nearing the end of his life, he sketched his ravaged body for posterity. Perhaps, he just desperately needed to be remembered. But, why is it that all of his self-exposure seems, in the end, superficial? Because, regardless of what we learn, Dürer remains as firmly screwed into his shell as the most resistant of mollusks; he is an enigma even unto himself. In fact, his vital nature seems very much like another cryptic element found in one of his most popular images: Melencolia I (below the jump). Observe...

Tuesday, April 30, 2019

A Journey to the Bottom of a Rabbit Hole (Interlude #3)

Rosslyn Chapel - window on north side. Photo credit ©: Rob Farrow.
(All images in this post can be clicked-on for original size.)

"The building of cathedrals was part of a colossal and cleverly devised plan which permitted the existence of entirely free philosophical and psychological schools in the rude, absurd, cruel, superstitious, bigoted and scholastic Middle Ages. These schools have left us an immense heritage, almost all of which we have already wasted without understanding its meaning and value."

- A quote attributed to P.D. Ouspensky. sourced from Tim Wallace Murphy's Enigma of the Freemasons: Their History and Mystical Connections (2006). Inset right is a column from the Cathedral of St. John the Divine in New York, featuring the Kabbalistic Tree of Life symbol.

"The Gothic cathedral, that sanctuary of tradition, science and Art, should not be regarded as a work dedicated solely to the glory of Christianity, but rather as a vast concretion of ideas, of tendencies, of popular beliefs, a perfect whole to which we can refer without fear, whenever we would penetrate the religious, secular, philosophic or social thoughts of our ancestors."

- Another quote sourced from Murphy's book which originated from Le Mystère des Cathédrales (1926) by a mysterious French alchemist known only as "Fulcanelli." Inset left is another - more enigmatic - column from the Cathedral of St. John the Divine.


From Chartres Cathedral in France: a mechanized astrological clock added in 1528.

***
What's this? A third Interlude?

Alas, yes. As you may have guessed, I've been tottering on the brink of one rabbit hole or another for months now, so, I suppose it was inevitable I'd lose my footing eventually. Originally, I intended to add 2 quotes I found recently (now posted above) to the Chinese Sewing Basket addendum - in which I mention Notre Dame de Paris - and, for all sensible reasons, that's where the operation should've ended.

But, no. Instead, I decided to Google the French alchemist Fulcanelli (who wrote the second quote above)... a man I was unfamiliar with. After all, a writer should know her subject matter, right?

Famous Last Words. For, at that very moment, although unbeknownst to myself, a massive black rabbit hole was silently opening directly beneath my feet like the mouth of a terrestrial Moby Dick.

Maybe it's the topic itself: alchemy - specifically hermeticism - the black hole of the esoteric world. The further you go, the less you know. And, as for Fulcanelli... ah, yes, Fulcanelli, now there's a name to conjure with. And, certainly many have. For some he's the man-who-never-was with an ongoing archive of potential identities (see the Fr. Wiki entry).

But, ultimately, this post isn't about Fulcanelli any more than it's about cathedrals. As it so happened, while ferreting out info about Fulcanelli I discovered his friend, Julien Champagne, also a French alchemist and an artist. It was he who created the frontispiece emblem (inset left) for Fulcanelli's Mystère des Cathédrales.

(Continued below the jump...)

Sunday, October 7, 2012

Sacred Geometry, Chirality and the Cyclohedra


Tetracyclohedron on a mirror - cast figure - DS 1988



"We live on a planet which is essentially a rotating sphere, in a system comprised of other rotating spheres. These revolve around a rotating ball of burning gases in orbits that roughly describe a series of concentric rings. This system, in turn, is rotating within a spiral galaxy, which, in itself is also rotating with a host of other spiral and spherical galaxies in what some hypothosize to be a circular universe. In view of this, how else can the "phenomena of life" behave? How could it possibly extricate itself from the "spiral urge"? Worlds turn, cells divide, and flowers bloom using rhythmic processes not wholly deciphered by mechanistic equations. Physical laws and physical life must, by necessity, share a common ground, and this "ground", this mysterious omphalos, appears to be round."


...

"In the end one cannot help but sympathize with old Archimedes who, while drawing circles in the sand, allegedly remarked to a passing Roman soldier - and, presumably these were his Famous Last Words - 'Don't step on my circles!'"


- two excerpts from the intro to Cyclosymmetrics, The Implicate Geometry of the Circle - 1993, Dia Sobin



***


Geometry confounds, but it never lies. And, when the going gets tough, the tough draw circles. Which is why I'm posting geometry today, despite its obvious departure from recent material.

Above is what almost seems like a Brancusian structure. What it really is, however, is a photograph of my first casting of a cyclohedron - specifically, a tetrahedron -  sitting on a mirror.  As for Constantine Brancusi, I discovered today he was Romanian, born in a similar place in the vicinity of the Carpathian mountains as both sets of my grandparents. He was a very spiritual man, and I find it interesting that geometry and the spiritual seem to intertwine in so many respects. Geometry is so subliminally present in so many aspects of life, it's not unusual that it was always, an still is, a "sacred" discipline. 

Re: cyclohedron. You won't find the word in Wiki, or anywhere else for that matter (but here, presently)*, because it's one I coined to describe a set - specifically, the Platonic cyclodhedra - which describe a regularly convoluted set of polyhedra - I inadvertently discovered in the 1980's during the course of a design project. I've tried to document them myself - the quotes above come from its introduction - but, as I have had no intensive mathematical training, I never attempted to publish my "treatise". I did have a web-site several years ago - "The Circle Zone"  (I've just up-loaded its home page graphic here...) -  but apart from one Chinese teacher (and new media artist) Zhang Yanxiang - and Petral, if you're out there, I am eternally grateful - it didn't attract a great deal of attention. Why those from the East might find the cyclohedra attractive, is not unusual. The figures emerge from the circle and its infinite symmetries, and the East has an intimate relationship with the circle, in ways the cruciform-fixated West could never quite comprehend. (see Mandala)


Page 65 from Contemporary Art of Science and Technology - Science Press - 2007
Funded by: China Association for Science and Technology, and the
National Philosophy & Social Innovation Base for Sci-Tech History & Sci-Tech Civilization


Specifically the figures literally enfold from an expansion of an ancient pattern called the "Flower of Life", or, as sacred geometer, Charles Gilchrist, refers to it: "Natures First Pattern."  There a number of correlations that are drawn - either metaphorically or demonstrably -  to this pattern and the natural world... but, allow me to add another one: quantum entanglement

Chirality, on the other hand, is a word most often used in physics and chemistry to describe symmetries that are applicable to those disciplines. But, chirality also describes what differentiates the cyclohedra from their rectilinear counterparts - the regular polyhedra - in that, two orientations of the planes are possible... a left-handed twist, and a right-handed one. The two "pinwheels" I created from the tetrahedron photo, for instance, are "spinning" in opposite directions. They're admittedly odd formations...almost alien really... and I often muse about an intelligent alien race - or perhaps just a parallel one - which developed along the devious, organic lines of a cyclohedron as opposed to those static, antiseptic rectilinear planes of the regular polyhedra, or Platonic solids, we know so well.




Rotational symmetry - (top) 6-fold - (bottom) 8-fold - DS 2012


Fractals, of course, are a visual example of organic geometry... the cyclohedra are another. Demonstrably, the circle is the mother of all geometrical figures, organic or inorganic - the dynamic of the material world... and whether you are an artist or a scientist, your inquiries inevitably resolve themselves in her domain. My geometrical muse is adamant about this, and I trust this muse implicitly. As I said, geometry never lies.





A second set of Platonic Cyclohedra cast in 1993 - DS


Cast tetracyclohedron & octacyclohedron (using 120 degree arcs) on a mirror
1993, DS





Vesica Piscis



*  Recently (2/28/14) I found this entry for the word "cyclohedron" on Wolfram. I have no idea when it appeared, when the word was coined, or, in fact, what it's referring to... however, I am continuing to use the word regarding the solids depicted here.