Tuesday, June 25, 2024

The Dürer Files: A Series Introduction

Nemesis or The Great Fortuna, Albrecht Dürer - copper-plate engraving - 1502.
Geometry: 2024, DS.

"Precariously balanced on an unsteady sphere, Albrecht Dürer’s nude female figure of Fortuna conveys a sense of the instability and unpredictability of fortune. The artist’s treatment of the subject derives from a Latin poem by the Italian Renaissance poet and humanist Angelo Poliziano (1454–1494), who describes fortune as the “power to crush the arrogant minds and triumphs of men and to confound their too ambitious plans.” With her bridle and cup, Dürer’s figure—in contrast to misogynistic portrayals of her by a later generation of artists—also embodies the virtues and rewards of temperance."

- A nice assessment of Albrecht Dürer’s Nemesis image (inset left & above) via this NY Public Library page. It seems cynical that Dürer should symbolically combine the word nemesis - meaning rival, enemy or punisher - with the idea of fortune (fortuna) together in one image. Was he referring to the idea of fate as karma? In reality, the goddess Nemesis and the goddess Fortuna are mythologically and symbolically connected and it is more than likely he created his own hybrid.

It actually took me several days to see the spiral in Nemesis, the Greater Fortuna... which is astounding considering what an awesome spiral it is. It's a dialing spiral but of a slightly different kind than the one I'm familiar with. You can create a small animation with it growing in size and spreading over the image as it revolves around one central point (in the vicinity of her elbow). In the fuller image - shown above with the phi-shell - the spiral continues into the landscape.

As for the image itself, the figure of Nemesis/Fortuna exhibits a female body type humanity rarely sees anymore. We probably stopped seeing her around the same time Dürer created her, at the turn of the 16th century. She is an older woman; you can tell by her diminishing face and thinning hair... unlike Durer's (1496) Little (or Lesser) Fortuna - inset right - the younger Fortuna (with a lesser spiral) and an enigmatic image the size (and dimensions) of a Tarot card.*

But, the older Fortuna - who has in her maturity taken on the role of Nemesis, a "formidable and usually victorious rival or opponent" - reveals a body with contours which, while not aesthetically correct in our times, has retained its hard-bodied youthfulness. She has powerful wings and holds a large chalice. Traditionally, Fortuna, is shown holding a cornucopia but, as mentioned, Dürer has created his own Fortuna hybrid. But, then, it seems he rarely sticks with the "tried and true," preferring his own innovations. In any case, Nemesis/Fortuna is a force to be reckoned with. She may be offering the chalice - the cup of plenty - but, we are reminded that she is also carrying a horse's bridle. In other words, while we may succeed  above and beyond our wildest dreams, our egotistical aspirations are kept in check by this goddess. Either that, or she represents a stereotypical negative female archetype.

Incidentally, Nemesis is much more massive than she appears here. An enlargement of the little Italian town which she overlooks (as she balances upon her sphere) can be found here.


By now, I should know better than to make promises regarding the future I can't keep... but, in regards to my previous projections made in my last post, when and where I envisioned one more addition to the Dürer series, I am now here to report that my plans have had to change.

Yes, while in the midst of feverishly writing the proposed post (Dürer Part III)  I initially had in mind, I was suddenly presented with far more bits of information than I could possibly process in one post.  An example would be the Nemesis image with it's astounding spiral (above). But, there were other images found as well, deserving some special consideration... such as Dürer's diagram of his 'Schneckenlinie' ('snail-line') or Logarithmic spiral - (inset left) - a spiral which, oddly, almost appears as if it's in perspective. Sadly, I cannot translate the text. (If you can, please inform us!)

In any case, after the shock wore off and my first attempts were put aside, I decided the new information required a new series... The Dürer Files, a new testament of sorts. Moreover, it would have to be broken up into bits... thereby requiring shorter posts and less time, or, seriously, I would never get any of it online. So, this is the Introduction. The series contents will (tentatively) include 5 sections, which are as follows:

The Dürer Files: 1. Soli Deo Gloria (Glory only to God)
The Dürer Files: 2. Maximillian & The Bauhütte
The Dürer Files: 3. Dürer & the Phi Goddess
The Dürer Files: 4. Dürer & the Black Madonna
The Dürer Files: 5. Dürer & the Whale

(Note: This Intro will also serve as the series Table of Contents page with links becoming active as the posts appear. The links will also appear in the Golden Series content post (click the Golden Snail on the sidebar.) (Know, however, that the titles and order in which they appear may change.)


(More below the jump break...)

Saturday, May 18, 2024

The Mirror of Venus: 5 Keys

The Mirror of Venus: The 5 Keys - 2023, DS.

"Moving now from the decad to its half - we meet one of the most brilliant personalities of the 'society of numbers;' the pentad and the characteristics of the number five. In arithmology or number mysticism the number five partially pertains to the essence and importance of the decad as being its half and its condensed image, but it is also the
gamos, Aphrodite's number as goddess of the fruitful union, the generatrix of love, and the abstract archetype of reproduction."

- From the 1931 investigation of phi and the pentagram: The Golden Number: Pythagorean Rites and Rhythms in the Development of Western Civilization, written by Matila Ghyka, a book I've just read recently. The above quote is the first of five similar brief references to Aphrodite and the pentagram found throughout the book; three of which combine Aphrodite with the Egyptian goddess Hathor.*

But, while Ghyka acknowledges an ancient connection between Venus/Aphrodite and the pentagram, it seems to rest upon the idea of marriage, a "fruitful" union... conjugal bliss. He calls this incarnation of Aphrodite "Gamelia," that is, "of the wedding." But, Gamelia was a name given to many, if not all the Olympian gods; specifically Hera, wife of Zeus; it wasn't amongst Aphrodite's major titles, and, regarding the pentagram, is too literal a translation of the "love goddess" and is, ultimately, a red herring.

Inset left, above: the contemplative Venus of Arles... (seemingly) holding an apple up to her broken-off, handheld mirror. Inset right,  is an actual bronze mirror (Greek, circa 460 BC) supported by the figure of Aphrodite who is holding a dove, while 2 winged erotes stretch their hands towards a point centered above her head. If you connect the point at the top of the goddess's head with the two doves perched above on the mirror, you'll have an inverted golden triangle. Below, inset left, is another similar bronze mirror from the same period, currently in the Met Museum. This one features dogs chasing hares; the hare is another Aphrodite symbol. Hint: we've seen the dogs and the hares before.**

So, mythologically speaking, marriage was never known to be Aphrodite's strong point... although love, both carnal and celestial, was. The Venus of the pentagram, however, is significantly more complex, and we have to approach her from a higher ground, so to speak. From Theoi we have:

"According to the cosmogonic views of the nature of Aphrodite, she was the personification of the generative powers of nature, and the mother of all living beings. A trace of this notion seems to be contained in the tradition that in the contest of Typhon with the gods, Aphrodite metamorphosed herself into a fish, which animal was considered to possess the greatest generative powers." 

And, there is so much more. One might say, Ghyka was trivializing the Venus/Aphrodite connection. While he inserts the word gamos in his brief comment, he neglects the word heiros (sacred). Nor does he infer in any way that the "wedding" is, in essence, chymical. So, my strongest impression of the book is that the deliberate omissions Ghyka (amusingly) accused other writers of making - including Vitruvius - might've been tip-offs to those of his own. That being said, he probably connects more esoteric "dots" in the pentagram's long journey within the pages of this book than you'll find anywhere else. But, keep in mind, the author may be sticking to traditional "fraternal" codes of secrecy all the while... something also mentioned quite often in his book.

"The mirror also, in turn, symbolizes revelation and truth: the mirror often shows the face, and the eyes, as shown in the painting Venus At Her Mirror or Rokeby Venus or Venus-Aphrodite by Diego Rodriguez de Silva y Velazquez, in which the goddess gazes into the mirror with only her face revealed. The eyes, in turn, are the paths to truth: they are the “window to the soul”, or, ever-more interestingly, the “mirror of the soul.” Aphrodite, in gazing into the mirror, is therefore not merely enjoying the sight of her own beauty, but is acknowledging the truth of all that resides within her – for, as Aphrodite Ourania, she is that which keeps together the entire cosmos and continues the survival of all."

- A quote found here, introducing us to to the most important facet of Aphrodite: Aphrodite Ourania or Urania, the celestial Aphrodite, and the Venus/Aphrodite who will concern us most in upcoming posts. She shares some of the attributes of Urania, the Greek muse of astronomy and the stars, in some cases mistakenly,  but, in others, possibly an indication of the evolution of Pentagonal Venus and the golden meme.


The diagram above, a sort of gateway into the Venus subset of the golden meme, is already somewhat obsolete regarding the pentagonal journey I unintentionally began taking around Thursday, April 4, 2019, after my vision of the rose pentacle. I wrote at the time that I anticipated something "larger." But, really, I had no idea of the many roads left to travel. Needless to say, Venus/Aphrodite had her foot in the door from Day 1.

I have since learned that the Venus pentagram, in its entirety, is not merely one arrangement of five symbols but, possibly, an arrangement of 10. Moreover, each of first five symbols composing the "mirror" represent metaphorical hallways with numerous doors. In other words, the Venus pentagram remains true to the pentagram's nature - it's a fractal - very possibly as colorfully layered as (what appears to be) a phi-based Julia set, inset right. Additionally, Venus/Aphrodite is not merely 2 goddesses in one... she's worn many hats over the course of her very long career and has several hybrid forms as well, so, it's a very tangled web, indeed. 

In spite of all of the above, the first five symbols I'd originally chosen for the diagram still stand, but, while I had hoped to address each of them briefly here, I've come to realize it isn't presently feasible. There is no "brief" in this discussion. 

As for now, I will include in this post some bits which have already been written regarding several symbols in the diagram.

(Note: I was originally going to conclude this post with one more bit of information via Prince Ghyka which might interest us, and it concerns Albrecht Dürer. This "bit of information" was eventually accompanied by other bits of information to necessitate adding a dedicated Part III to Albrecht Dürer and the Divine Ratio Parts 1 & II. As I said, there is no "brief" in this discussion. Stay tuned.)

(Continued after the jump...)

Wednesday, May 1, 2024

The First of May

Fetal Venus & Her First Dove (sketch #1) - digital - 2024, DS.

I am sorry for the delay with posting. I must have totally revised the intended post several times this past month, but it is still stalled; it may never be published on this blog. Meanwhile, in the real world... well, let's face it, if it's not one existential crisis, it's another... or, perhaps, one on top of the other... a parfait of stress factors.

As I write, however, it is May Day... which means two very different things, depending upon who you are and where you live. For some it is International Worker's Day. For others its a banking holiday. But, for many of us, it's the daytime leg of the ancient Celtic holiday of Beltane/May Day... specifically those of us in the Northern hemisphere. In the Southern hemisphere, Samhain is celebrated, (if  I'm reading this correctly). And, this provides us with an interesting symmetry.

For some reason, things always seem a bit more peculiar around the April/May cusp for me, and I don't think it's an astrological phenomenon. Beltane and the Germanic Walpurgis Night (see St. Walpurga) are very much like Samhain (or Halloween) in that the veil between the living and the dead is at its thinnest; the difference being that the dead who walk on Beltane night are seeking rebirth. Perhaps, they're hoping they'll be conceived on May Day... a very Venus/Aphrodite day, due to the planet's astrological rulership over the sign of Taurus the Bull, and Aphrodite's role as fertility goddess. However, while May Day evokes chaplets of spring flowers and circling around maypoles, Beltane, the nocturnal leg of the holiday, is a fire festival. The following quote was sourced from James Frazier's classic, The Golden Bough:

"The people believed that on that evening and night the witches were abroad and busy casting spells on cattle and stealing cows’ milk. To counteract their machinations, pieces of rowan-tree and woodbine, but especially of rowan-tree, were placed over the doors of the cow-houses, and fires were kindled by every farmer and cottar. Old thatch, straw, furze, or broom was piled in a heap and set on fire a little after sunset. While some of the bystanders kept tossing the blazing mass, others hoisted portions of it on pitchforks or poles and ran hither and thither, holding them as high as they could. Meantime the young people danced round the fire or ran through the smoke shouting, 'Fire! blaze and burn the witches; fire! fire! burn the witches.' In some districts a large round cake of oat or barley meal was rolled through the ashes. When all the fuel was consumed, the people scattered the ashes far and wide, and till the night grew quite dark they continued to run through them, crying, 'Fire! burn the witches'."

Apparently, in the 18th century, the Beltane fires were meant to metaphorically burn witches. Possibly, in previous centuries Beltane fires literally did burn witches. It's not really clear to me. But, it is ironic that it is the Wiccans and "heathens" (of the future) who've essentially revived the bonfire tradition... and ran with it!

However, we've moved past the night into the prettier and more benign May Day. Wiki, in the Beltane entry, describes the traditional May Tree or May Bough as being a: " a small tree or branch - typically hawthorn, rowan, holly or sycamore - decorated with bright flowers, ribbons, painted shells or eggshells from Easter Sunday... The tree would either be decorated where it stood, or branches would be decorated and placed inside or outside the house... the tree would remain up until May 31st. The tree would also be decorated with candles or rushlights."


Enjoy your holiday!

Regarding the two images above - Fetal Venus With Her First Dove (above), and Fetal Dove with the Planet Venus (inset right) - both are sketches I made earlier this year when I possessed a more competent graphics program. I no longer do and am not sure when I will, so, these images will have to stand for the originals; I just wish they were the images I originally had in mind. (Note: Phi shells look as if they were designed for the embryonic; don't you think?)

What I failed to mention in this post, however, is that my "fetal" images had an actual precedent, and my original idea was to post my embryonic Venus images along with the image that inspired them (posted after the jump). It's interesting to juxtapose spiral images of Venus from various periods. And, considering she's been an artistic subject for almost 3000 years, there's a lot of images to cover; I may as well start now!


Sunday, March 31, 2024

Apple Blossoms

Apple Blossoms in Hopewell's Orchard - Photo credit: NPS/A. Kane.

"It is remarkable how closely the history of the Apple-tree is connected with that of man. The geologist tells us that the order of the Rosaceæ, which includes the Apple, also the true Grasses, and the Labiatæ, or Mints, were introduced only a short time previous to the appearance of man on the globe.

...Pliny, adopting the distinction of Theophrastus, says, “Of trees there are some which are altogether wild, some more civilized.” Theophrastus includes the apple among the last; and, indeed, it is in this sense the most civilized of all trees. It is as harmless as a dove, as beautiful as a rose, and as valuable as flocks and herds. It has been longer cultivated than any other, and so is more humanized; and who knows but, like the dog, it will at length be no longer traceable to its wild original? It migrates with man, like the dog and horse and cow; first, perchance, from Greece to Italy, thence to England, thence to America; and our Western emigrant is still marching steadily toward the setting sun with the seeds of the apple in his pocket, or perhaps a few young trees strapped to his load.

...Men could afford then to stick a tree by every wall-side and let it take its chance. I see nobody planting trees to-day in such out-of-the-way places, along the lonely roads and lanes, and at the bottom of dells in the wood. Now that they have grafted trees, and pay a price for them, they collect them into a plat by their houses, and fence them in,—and the end of it all will be that we shall be compelled to look for our apples in a barrel."

- Via the entertaining 1862 essay, Wild Apples, by American naturalist, Henry David Thoreau, available for free on Project Guttenberg. Note his Venus/Aphrodite-related description of the apple tree: "It is as harmless as a dove, as beautiful as a rose..."*  Biologically, the apple and the rose belong to the same family.**

Interestingly, in our eventual (future) back yard, we had a small apple tree which grew close by a wild rose bush with small clusters of blossoms, resembling those of the apple tree. The apple tree eventually withered, but the rosebush retained an apple-like fragrance.

Thoreau concludes his essay with the biblical quote (JOEL, Chapter I, Verse 12): “The vine is dried up, and the fig-tree languisheth; the pomegranate-tree, the palm tree also, and the apple-tree, even all the trees of the field, are withered: because joy is withered away from the sons of men.”

"But the first verdict seemed the worst verdict
When Adam and Eve were expelled from Eden;
Yet when the bitter gates clanged to
The sky beyond was just as blue.

For the next ocean is the first ocean
And the last ocean is the first ocean
And, however often the sun may rise,
A new thing dawns upon our eyes.

For the last blossom is the first blossom
And the first blossom is the best blossom
And when from Eden we take our way
The morning after is the first day."

- An excerpt from the poem Apple Blossom, by Belfast poet, Louis MacNeice. For another poem by MacNeice, see my latest post on PMB, A Lost History Retrieved.


I'm not a terribly nostalgic person, but as the years go by, holidays are increasingly becoming "remembered" things... belonging to experiences in the past-tense; belonging to childhood only; a time of wonder.

My earliest memory of Easter is posing with my little brother in front of an immense apple tree, one of two, in our - or, rather, the landlord's - backyard, while my dad snapped B/W photographs of us decked out in our annual Easter outfits. We both wore coats or jackets and "bonnets"; spring was very chilly in New England in those days. My coat had a lavender and white checkered pattern. My brother's jacket had a black and white pattern known as houndstooth.

Funny, the things you remember... while the rest you'd' rather forget. But, today I will go no further down memory lane beyond the grandeur of those two immense apple trees in the landlord's backyard, the like of which I'll probably never see again. The like of which only a few of us will ever see at all. Privilege is a relative thing.

The apple tree is an icon; it is also a meme and/or an element of a meme. It speaks of the beginning of new things and the continuous cycles of life - a "tree of life" - and the processes of growth. On an apparent (logical) level, living things grow and eventually decay. But, on a mythical (mythos) level, the life force is in perpetual expression; it can be transformed or modified, but never annihilated.***

But, I am no sage. I merely observe.

And, what I presently observe is that it's a rabbit, egg and apple sort of holiday for some of us - even if only in memory. And, yet, at its core (pun intended) it is a celebration of spring and a renewal of vitality.

Coincidentally, it just happens to be a beautiful spring day in NM. I hope it is wherever you are, too.


* "Harmless as a dove" is not an exactly an accurate phrase and, as a naturalist, Thoreau would've known this. He was quite possibly referring to domesticated doves, which we might expect to be more submissive. Wild doves, however, in spite of their elegant appearance, will gladly take out any creature threatening their nests. I once saw a mourning dove throttle a blue jay - a more belligerent bird - holding it by it's neck.

** Cherries are also part of the Rosaceae family... and, like the apple and the rose, are celebrated with parades and festivals in the states and abroad.

*** (Regarding the revision): my original statement here was, for the most part, unintelligible. I must've dropped the thread and then, unsuccessfully, attempted to patch it up. I've since revised the statement, but, I'm afraid the original "thread" was lost. And, it's probably just as well!

Tuesday, March 19, 2024

An Auspicious Day

The Three Golden Eggs of Happiness, G-DS-2024
Alternative title: 3 Sleeping Doves

We seem to be experiencing a grand trine of celebratory days today: the first day of spring, the neo-pagan and Wiccan holiday of Ostara, and the first day of the Persian new year! So, put aside your woe and despair and knock yourself out with the pleasure of impractical things - we've hit the jackpot!

I found three (pretty-much) perfect golden eggs (sourced here)... and, now, you have!

Smile. You'll feel better. It's a known fact. :-)

Sunday, March 17, 2024

The Paisley Pattern & the Golden Meme (Revised 3/18/24)

Phi shells served up on a modern paisley fabric (sourced here).
Geometry: 2024, DS.

"memen. a unit of practice or belief through which a society or culture evolves and that passes from one generation (or even one person) to the next. In this sense, the term — coined in 1976 by British  biologist Richard Dawkins (1941–  ) and derived from the Greek word for “imitation”— is a kind of metaphorical parallel to the term gene."

-  An interesting definition of "meme" found in the online American Psychological Association's Dictionary. As a "metaphorical" gene, in lieu of its lack of actual physicality, we can expect a meme to operate in a manner similar to the gene: subconsciously and/or subliminally. Unlike the gene, it is transferred from one mind to the other... often in the form of a symbol, although not necessarily a visual symbol.

The images posted above, and inset right, and inset left and right below are the result of a new 2-dimensional toy I developed last year and introduced in The Universe in a Phi Shell: phi shells. I can't remember what inspired me to make phi spirals into phi shells, but it was probably pretty elementary. However, once I saw the effect the shells produced on some images, I was astounded. In so many cases the phi shell was like an envelope - die-cut to fit a golden spiral - or a custom-made template... with maybe a little wiggle room.

Sometimes, working with a phi shell leads to repositioning the original spiral. Inset right is a prime example: Boy Blowing Soap Bubbles by 17th century Dutch painter (and Bentvueghel), Karel Dujardin. (See the post for the former spirals.) As concise as this spiral appears, it could be larger and better... but, As we see time and time again, some portion of the necessary image area is missing. What sold me on the orientation of this spiral, however, is the way the boy's right arm - holding the bubble pipe - falls so neatly into the small triangle. It's as if the boy is drawing the spiral in the air.

"Proponents theorize that memes are a viral phenomenon that may evolve by natural selection in a manner analogous to that of biological evolution. Memes do this through the processes of variation, mutation, competition, and inheritance, each of which influences a meme's reproductive success. Memes spread through the behavior that they generate in their hosts. Memes that propagate less prolifically may become extinct, while others may survive, spread, and (for better or for worse) mutate. Memes that replicate most effectively enjoy more success, and some may replicate effectively even when they prove to be detrimental to the welfare of their hosts."

- A cautionary quote from the Wiki entry for "meme," and one that might make a certain amount of sense to those acquainted with what I've come to refer to as the "Golden Meme." I could even go so far as to says this might be reflected in my own experience... including the "detrimental to their hosts" bit, but, I try not to entertain fantastical innuendos that accelerate my level of paranoia.

"When Pacioli (the Renaissance mathematician) wrote in his Divina Proportione: '.. just like God cannot be properly defined, nor can be understood through words, likewise this proportion of ours cannot ever be designated through intelligible numbers, nor can it be expressed through any rational quantity, but always remains occult and secret, and is called irrational by the mathematicians'... he seemed to be equating the golden ratio with a variety of magical formula. That is, through his recognition and understanding of "this proportion of ours" he was attempting a marriage between several disciplines: science, art and mysticism. And, this wasn't unusual for the early Western scientists, many who dabbled in astrology, numerology and alchemy side by side with astronomy, mathematics, botany... and the arts. The scientific field was wide open - the hard lines were still blurred...

The "new scientists" were seeking enlightenment, autonomy, discovery, truth. And, it was into this slowly reawakening world that the pentagonal spiral made its appearance... and not necessarily for the first time. Similar to the sign of the pentagram - by which the followers of Pythagoras "knew each other" - perhaps, the "sign" of the pentagonal spiral enabled the Baroque artists to identify like-minded artists... kindred spirits in a world in which freedom did not, yet, ring."

"Keep in mind that the spiral was also present in Baroque ornamentation... and possibly somewhere in Baroque music, too.  So, were the artists, artisans, and musicians completely aware of its proportions and using it as a measure of perfection and beauty? Or, was the proliferation of golden spirals (at that time) an almost paranormal thing... or a variety of subliminal meme... an unconsciously recognized icon which was possibly a symbolic presentiment on the part of an artistic community who were, without noticing it, heralding an evolutionary phase of a whole society?"

A Baroque flourish (w/ phi shells). G-DS-2024.

- Two quotes via my blogpost: The Gentilischi Spirals and a Series Afterword. One painting appearing in that post is the image reposted (inset right) above the flourish: Orazio Gentileschi's Young Woman with a Violin. It's phi shell is looser than the original spiral I gave it, and resting on her lap like a humongous sea shell gives the image a somewhat comical appeal... but the spiral and triangle intercepts the young woman's bow perfectly.

The flower painting, inset left, above her, however, was the work of Dutch Baroque artist, Willem Van Aelst and can be found in Part III of the Bentvueghel series, where we become acquainted with the curious appearance of the garden snail in flower paintings. And, really, if the Dutch flower painters used the snail as a clue indicating that a larger spiral was deliberately embedded in their paintings, well, all I can say is, what a brilliant idea!

Directly above is a portion of a Baroque ornament recently found (now decorated with phi shells). Not all Baroque flourishes were as golden as this one, but, considering the quantity of spirals appearing on furniture, architecture and textiles, the golden meme was alive and well.


For the past few years, I've spent a lot of quality time pondering the meaning of the appearance of the pentagonal golden ratio in works of art, artifacts and in architecture. I can't say for sure why this is so anymore, but, once upon a time, there seemed to be a point. I had a plan. Nowadays, I have to ask myself, why in the world would anyone actually sit down and spend a great deal of quality time blogging about obscure topics to an evaporating audience while everything in the surrounding area (the world, that is) seems to be either rapidly going down the tubes, or up in flames?

And, this isn't the first time I've asked myself this question!

On the other hand, blogging about obscure topics to an evaporating audience - in view of the tubes and flames - presently encourages my survival. Perhaps, blogging is kind of a therapy after all. Or, perhaps, in an absurd world, doing nonsensical things in response to Armageddon is appropriate.

And, it is in this state of mind I will reveal my latest pentagonal phi epiphany... which just happens to regard a description I've used previously: subliminal meme. The kind of meme that just might slip into an image while the artist was totally unaware of it, as it did in 2012, when I created Metamorphosis Interrupted, reposted (inset left), with a phi shell that fits the unfortunate worm like a chrysalis! (See: A Virtual Can of Worms.) 

Generally speaking, the unconsciously produced spirals in the images referred to in the Worms post were not all that strong. However, the worm's proportions in Metamorphosis is very close to perfect. But, while I was aware that I was being very finicky about the shape of the worm's body while creating it, in reality, at the time of its creation, I had never even heard of the pentagonal golden spiral!
So, is this an indication of phi "on the brain"? Are our brains somehow wired to its aesthetic or does it go deeper than that? Is this why the ratio was referred to as "divine" - in that it is embedded so deeply in our consciousness we assume it is an outer force? Or, when we find it in organic life, is it merely the imprinted diagram of actual physical forces science is already aware of?

Or, taking into consideration the nature of a meme, can you actually "catch it" like the common cold?*

Speaking of brains, I was delighted to find an image of one that seamlessly fits into a phi shell (inset right). Of course, humans tend to have rounded skulls, so it should come as no total surprise. But, then again, the image is an illustration and not an MRI, so, this is an artistic interpretation (and a very nice one) by Hank Grebe.

I didn't notice this till I was adding my geometry "stamp" to the image, but what lies at (a very important) 3-cornered junction of the spiral nearest to the coil - technically, the apex of the smallest golden triangle (and its spiral) - is that notorious pine-cone-shaped brain element, the mysterious pineal gland or "third eye." Oddly enough, it is depicted here as another tiny golden triangle (click for enlargement.)

(Continued below the jump...)

Monday, March 4, 2024

The Universe in a Phi Shell

M74, or the Phantom galaxy... embellished with a phi shell. (G-DS-2024).

'The most merciful thing in the world, I think, is the inability of the human mind to correlate all its contents. We live on a placid island of ignorance in the midst of black seas of infinity, and it was not meant that we should voyage far. The sciences, each straining in its own direction, have hitherto harmed us little; but some day the piecing together of dissociated knowledge will open up such terrifying vistas of reality, and of our frightful position therein, that we shall either go mad from the revelation or flee from the light into the peace and safety of a new dark age."

- H. P. Lovecraft, The Call of Cthulhu, 1926.

"Messier 74, also called NGC 628, is a stunning example of a "grand-design" spiral galaxy that is viewed by Earth observers nearly face-on. Its perfectly symmetrical spiral arms emanate from the central nucleus and are dotted with clusters of young blue stars and glowing pink regions of ionized hydrogen (hydrogen atoms that have lost their electrons). These regions of star formation show an excess of light at ultraviolet wavelengths. Tracing along the spiral arms are winding dust lanes that also begin very near the galaxy's nucleus and follow along the length of the spiral arms.

M74 is located roughly 32 million light-years away in the direction of the constellation Pisces, the Fish... In its entirety, it is estimated that M74 is home to about 100 billion stars, making it slightly smaller than our Milky Way."

- Via this Hubble page, we are introduced to the M74 galaxy discovered by the French astronomer Pierre Méchain in 1780. It is a kind of go-to spiral galaxy for those seeking spiral perfection. Of course, for the golden (pentagonal) spiral hunter, gazing at it's sinuous, twisting, mollusk-like proportions is sheer bliss... as, if there were a God and It had a face, this would be it. (God the mollusk! The Mollusk god. So, burn me at the stake. See if I care.)

"Messier 74 (also known as NGC 628 and Phantom Galaxy) is a large spiral galaxy in the equatorial constellation Pisces. It is about 32 million light-years away from Earth. The galaxy contains two clearly defined spiral arms and is therefore used as an archetypal example of a grand design spiral galaxy. The galaxy's low surface brightness makes it the most difficult Messier object for amateur astronomers to observe. Its relatively large angular (that is, apparent) size and the galaxy's face-on orientation make it an ideal object for professional astronomers who want to study spiral arm structure and spiral density waves. It is estimated that M74 hosts about 100 billion stars."

- This quote was sourced from the Wiki entry for M 74, along with the galaxy image used in this post (inset left). That the Phantom is considered an "archetypal example" of a "grand design spiral galaxy" is kind of interesting... in light of the fact that it is composed of 2 near-perfect pentagonal golden spirals. (And, when discussing massive deep-space objects, "close" is about as good as it gets.)

"This Hubble Space Telescope photo of Messier 74 reminds us that spiral galaxies are some of the most beautiful and photogenic residents of the universe. Nearly 70 percent of the galaxies closest to the Milky Way are spirals. New research finds that spiral arms are self-perpetuating, persistent, and surprisingly long lived."

"How the spiral arms form continues to puzzle scientists. One theory suggests the galaxy arms could be the result of density waves traveling through the outer disk. Encounters between galaxies could cause such waves as the mass of the smaller galaxy could affect the structure of the larger galaxy as the two combine."

- Two quotes from this Space.com article... reminding us of how little is known - and how much there is to know - about our cosmos.


I'm a busy woman these days, and I hate it. All it amounts to is more stress. But, tell that to an artist's muse. Time perplexes muses. They simply don't get it.

So, I currently have five posts sitting on the back burner, an unfinished digital image - my first formal digital image since my 2022 car accident - a ceramic project that MUST go into production this month, and a number of tedious life issues to deal with... and deal with, and deal with. But, then, last night I came across this photograph (inset right) of M74, the Phantom galaxy and, within minutes, I found myself absorbed in playing with my latest graphic toy, a phi shell (rhymes with seashell)... trying to decide which color to use from my ever-growing palette. In the end, only the white shell did the Phantom any justice, and even this had to be almost transparent... to endow it with that certain marine life appeal which, for whatever reason, seemed appropriate.

Anyway, I decided to post 2 images to the blog for the (freakish) enjoyment of those who, like myself, delight in this sort of thing. (Yes, in case you haven't noticed it, we constitute a minority.)

Oh, and incidentally, the "phi shell" - my attempt to give the common mathematical diagram a little extra-dimensional pizazz - will be appearing quite often in the coming days. I confess; I'm hooked on phi shells now, and there's no help for it!

Adieu, till next time... :-)

Via the ESA: a multi-observatory view of the Phantom.

Sunday, February 11, 2024

The Heart Nebula & the Flaming Heart of Venus

The Heart Nebula (detail) - Photo credit: 2022, Ernie-Jacobs. Geometry: 2024, DS.

 "What's that inside the Heart Nebula? First, the large emission nebula dubbed IC 1805 looks, in whole, like a human heart. It's shape perhaps fitting of the Valentine's Day, this heart glows brightly in red light emitted by its most prominent element: hydrogen. The red glow and the larger shape are all created by a small group of stars near the nebula's center. In the heart of the Heart Nebula are young stars from the open star cluster Melotte 15 that are eroding away several picturesque dust pillars with their energetic light and winds. The open cluster of stars contains a few bright stars nearly 50 times the mass of our Sun, many dim stars only a fraction of the mass of our Sun, and an absent microquasar that was expelled millions of years ago. The Heart Nebula is located about 7,500 light years away toward the constellation of the mythological Queen of Aethiopia (Cassiopeia)."

Cardiod animation

- Via this NASA archive page: In the heart of the heart nebula. Basically, the Heart Nebula - in the constellation Cassiopeia, composed of 5 major stars - seems to be an artifact created by gases from the birth processes of new stars, but, well, nothing I've read simplifies it to that degree, so it's merely my guess. In any case,  it's very impressive looking... like looking into the innards of an exploded star (nova) or a bubbling cosmic cauldron. Also, see its companion: the Soul Nebula... and an interesting star, φ Cassiopeiae.

Inset right is a cardioid animation, created by Atomic Shoelace, and sourced from the Wiki entry for Heart Symbol. I've never seen this before, but, looking at it now, it appears that the cardioid and the pentagonal golden spiral have something in common.

(continued after the jump-break...)

Wednesday, January 10, 2024

A Belated Christmas Spiral: The Annunciation by "Juan de Flandes" (Revised 1/19/24)

The Annunciation, 1508, Juan de Flandes. Geometry: 2024, DS.

"Juan de Flandes ("John of Flanders") was a Flemish painter active in Spain from 1496 to 1519. His actual name is unknown, although an inscription Juan Astrat on the back of one work suggests a name such as "Jan van der Straat". Jan Sallaert, who became a master in Ghent in 1480, has also been suggested. He worked in the Early Netherlandish style.

He may have been born around 1460 somewhere in Flanders, Flandes in Spanish, which encompassed modern Belgium, Netherlands, Luxembourg and bordering regions of France. He evidently trained in his home country, most likely in Ghent, as his work shows similarities to that of Joos van Wassenhove, Hugo van der Goes and other Ghent artists. He is only documented after he became an artist at the court of Queen Isabella I of Castile, where he is first mentioned in the accounts in October 1496. He is described as "court painter" by 1498 and continued in the queen's service until her death in 1504..."

- Via Wiki's entry for Juan de Flandes.

"In a list of 23 February 1505, thirty-two of them, including the Last Supper, were acquired by Margaret of Austria, Regent of the Netherlands and sister of Isabella’s son-in-law (in whose collection they were seen by Albrecht Dürer). Dürer noted in his diary: ‘And on Friday Lady Margaret showed me all her beautiful things, and among them I saw about forty small pictures in oils, the like of which for cleanness and excellence I have never seen’."

- Although not applicable to The Annunciation,  the quote was sourced from this page in which a full account of Juan de Flandes' mysterious collection of small paintings is given.

(Update:1/19/2024) I'm adding a new spiral to the mix (inset left). While not the "magic act" posted above it still has 2 things going for it: it terminates on the "holy spirit," and it begins to form a pentagram over Mary. There's a third thing. Gabriel conjures up an important golden relationship with his wand. What is it?

(Correction: Gabriel conjures up (at least) 3 golden relationships with his wand. What are they?)


I had a different post in mind for the first post of the year, but, as it happened I had the image (above) on file as a possible Christmas alternative, but, it needed research and I didn't have the time for it last year.

As it was, the trail grew cold fairly early on in my search for information, and there doesn't seem to be any clear cyberspace description of who the artist actually was, although we do know he was fairly successful, especially in Spain. (See: Juan de Flandes and His Financial Success in Castile.)

But, Juan de Flandes' Annunciation is possibly one of the most unique of all the many paintings in that genre produced in the late Renaissance, another being Botticelli's Cestello Annunciation (1489), the first spiral painting I documented on this blog... and, still, to this day, one of the best spiral paintings.

The spiral in this somewhat odder Annunciation painted by the Flemish painter, however, has one special feature. It's spiral presents us with a little conjuring effect, a magic act. (Why, it even has a dove!)

I've inset the unembellished original so you can witness it for yourself. Click on the image, and when the "slideshow" pops up, click on the spiral version. Jockey back and forth.  Do you see it?

Abracadabra! the angel performs a magic-trick with his staff... as if it was (an overly long) wand. (Presto! Mary is now with child.)

(Now, there's a mixed metaphor...)

Sunday, December 31, 2023

Five Spirals for December - #5 Song of the Morning by Nicholas Roerich

Song of the Morning, 1920, Nicholas Roerich. Nicholas Roerich Museum, NY.
Geometry: 2023, DS.

"The Awareness of beauty will save the world."

- Nicholas Roerich

"The pursuit of refinement and beauty was sacred for Roerich. He believed that although earthly temples and artifacts may perish, the thought that brings them into existence does not die but is part of an eternal stream of consciousness—man’s aspirations nourished by his directed will and by the energy of thought. Finally, he believed that peace on Earth was a prerequisite to planetary survival and the continuing process of spiritual evolution, and he exhorted his fellow man to help achieve that peace by uniting in the common language of Beauty and Knowledge.

...Nicholas Roerich died in Kullu on December 13, 1947. His body was cremated and its ashes buried on a slope facing the mountains he loved and portrayed in many of his nearly seven thousand works.

As he wrote: 'Let us be united—you will ask in what way? You will agree with me: in the easiest way, to create a common and sincere language. Perhaps in Beauty and Knowledge.'"

 - All quoted text above was sourced from Roerich.org. - the first (and most comprehensive) port of call for all things Roerich. But, the story of Russian Symbolist painter, Nicholas Roerich and his wife Helena is unusually extensive. The couple's Neo-Theosophical spiritualism was particularly influential in the States in the earlier half of the 20th century; so influential that there is actually a term for it: Roerichism. It is hard to believe that the Roerichs somehow faded into obscurity in America during the latter half of the century but they did.

(Note: Nicholas Roerich was first introduced on this blog in the Nijinsky post.)

"In December 1923, Roerich and his family arrived in Darjeeling, India in search of a mythical kingdom called “Shambhala”. Not to be found on any map, the Roerichs travelled across 25,000 kilometres of uncharted road to find the Kingdom that the Buddhists, Hindus, Tibetans and local healers so firmly believed in.

According to legend, with the spread of materialism, humanity would deteriorate and the people of Earth would unite under an Evil leader. This leader would attack the Kingdom of Shambhala with  terrible weapons and that’s when he would be defeated, ushering in a new Golden Era of peace and harmony. "

- Via the fascinating article: Explore the Himalayas : Paintings by Nicholas Roerich. Regarding the legend of the "evil world leader"... well, if prophetic, the question might be: which one?

The Hunt, 1937, Nicholas Roerich.

"Through the desolate summits swept raging intermittent gusts of the terrible antarctic wind; whose cadences sometimes held vague suggestions of a wild and half-sentient musical piping, with notes extending over a wide range, and which for some subconscious mnemonic reason seemed to me disquieting and even dimly terrible. Something about the scene reminded me of the strange and disturbing Asian paintings of Nicholas Roerich, and of the still stranger and more disturbing descriptions of the evilly fabled plateau of Leng which occur in the dreaded Necronomicon of the mad Arab Abdul Alhazred."

- A paragraph from At the Mountains of Madness by H.P. Lovecraft. Interestingly, Lovecraft references Roerich's "strange" paintings of the Himalayans several times during the tale. The painting above might be an example of what Lovecraft had in mind.

"Through his spiritual journeys into the Himalayas, Roerich also developed a deep sense of the role that the feminine principle had in the evolvement of humanity. Several of his paintings depict this importance, particularly, The Mother of the World. The Letters of Helena Roerich, written by his wife, explains the importance of this work: “The ‘Mother of the World’ is at the head of the Great Hierarchy of Light of our planet. Read in the Cryptograms of the East the narrative about the Mother of the World, and accept it as the highest reality.”

Helena Roerich further explains the inspiration for the painting, 'The star of the Mother of the World is the planet Venus. In 1924 this planet for a short time came unusually near to the Earth. Its rays were poured on Earth, and this created many new powerful and sacred combinations which will yield great results. Many feminine movements were kindled by these powerful rays.'"

- Via this Theosophical article: Nicholas Roerich: The Treasures Within.

Roerich was very close to his wife Helena. One might say they enjoyed a soul-mate relationship. Both were feminists and it was their belief in the World Mother that brought them into conflict with the Russian Orthodox Church (see: Sophiology) . Inset left is one version of the Mother of the World painted by Roerich in 1937.* The reference to Venus as the Mother's Star is interesting, and in the course of this post we will meet another reference.


There's something very special about the woman in Roerich's Song of the Morning.

To begin with, she's extremely beautiful. She's a brown woman... possibly Mongolian... with a fruit-of-the-earth, nuts and berries kind of beauty... but, she has another outstanding feature...

Monday, December 25, 2023

Five Spirals for December - #4 The Vision of St. Cecelia by Orazio Gentileschi

The Vision of St. Cecelia, Orazio Gentileschi (1620). Geometry: 2023, DS.

In ways, this post is an addendum to my previous (2022) Gentileschi post where the image above was first introduced. At the time, I had just had an automobile accident and my computer was stolen from the trunk of my destroyed vehicle. So, I wasn't able to overlay a spiral.

But, I have that capability back again, and, once tested, The Vision of St. Cecelia proved to be as golden as I suspected; and, actually, a little bit more! Orazio's spiral accomplishes what every good spiral ought to; it behaves like a clockwork.

As you can see from the images above, inset left, and inset below - and to see them best, click on any one of them for a sort of slide show - regardless of the spiral's size or orientation, its basic relationship to the image is not changed; it's proportions are, instead, systematically measured.

The spiral's activity in relationship to the painting, in this case, is determined by the apex (or acute tip) of the triangle and its direction in relation to the angel. The smallest spiral - and the most basic - informs us of the general focus of the design, which, as we might suspect, begins with the angel - Cecilia's "vision" - but  inevitably terminates on the body St. Cecilia.

But, Orazio has gone one step further. His spiral can be rotated (clockwise) and he shows precisely where it ought to go. First, the apex is turned from the angel's waist - its robes tied up in what appears to be an enormous bow - to the end of the white fabric. Note that the triangle's side is now facing Cecilia's pipe organ (inset right). Note also that, after every shift, the spiral still terminates on some portion of Cecilia and/or her clothing. How well this works, of course, relies on the size of triangle - the further the spiral has to turn will require a larger spiral.

Lastly, we carry the spiral to the furthest notch: indicated by the end the palm branch held in the angel's hand. The spiral is now enlarged (see below) and the triangle's side is up against the pipes of the organ... a perfect alignment. Is this significant? Well, yes, because, as it happens, Saint Cecilia is the Patron Saint of Musicians and Music. So, the spiral has made a cryptogram.

Of course, it might help to know Saint Cecilia's official story. But, I'll have to be brief, because I can't quite get it myself.

In Orazio's painting, the man in red facing Cecilia is most likely her formerly pagan husband, Valerian (who converted to Christianity), and the man in the doorway is her brother-in-law. All that's missing from the frame is the Roman soldier who was eventually martyred along with the rest of them. (although I haven't the faintest idea why). In any case, she and her husband must've never consummated their marriage before their untimely deaths as Cecilia died a virgin. She also "sang in her heart to the Lord"... and, along with martyrdom and her virginity is how she became a saint.

BTW, the 2 small wreaths of flowers - in the angel's hand and behind Cecelia, on the pipe organ - are chaplets of roses and lilies.

Another image that appeared in the first Gentileschi post in which I also found a spiral is the painting of Mary Magdalene, created by Orazio's daughter, Artemisia... an artist rediscovered, perhaps, fifty years ago or less. And they are still discovering her! (Image is below the jump.)

Anyway, Artemisia has finally come into her own in the modern world, and, if you have little prior knowledge of her, I suggest you read this older Green Women post...