Wednesday, June 29, 2011

Melencolia I (1514) by Albrecht Dürer

Conjectures and theories about Dürer’s solid: an overview

(See also “A new link between Melencolia I and the golden ratio” and “A Melencolia sequel: tracing Dürer’s point of view”)

The copper engraving “Melencolia I” (1514) by the German Renaissance artist Albrecht Dürer (1471 – 1528) remains one of the most enigmatic works in the history of Art. At first view, the composition seems as a jumble of apparently unrelated objects, some of which cannot even be recognized or named. Behind the first layer of perception, that the eye catches immediately, a closer inspection reveals smaller objects, chaotically scattered or even half hidden.

The picture is dominated by a sitting angel, presumably feminine but muscular, whose glance is fixed somewhere in the distance. With her right hand she holds a seemingly idle pair of compasses, while with her left hand she supports her head, gloomy faced, as if dried up of inspiration. At her feet, and half hidden underneath her long garment, lay several carpenter’s tools: a plane, a pair of pliers, a saw and some nails. On her right, a putto sits on a round millstone or grindstone, occupied with scribbling something on a small board fixed on his knees, hiding his work from the viewer with his left hand. Remarkably, the millstone seems to have an axle socket slightly off – centered. Diametrically of the angel stands a large geometric solid, which we will refer to as “the Dürer solid” in this essay. Only four faces of the solid can be seen from the viewer’s point of view. In front of the solid, a dog is calmly rolled up before a sphere. In the background of all these stands what seems to be part of a building, two walls meeting seemingly perpendicularly and fixed on them appear a balance, a hourglass and a bell. A magic square is inscribed on the wall facing the viewer: a square arrangement of the numbers 1 to 16 such as the sum of each row, column and diagonal is equal to 34. A ladder leans on the wall behind the structure while far in the horizon, through the only part of the composition which is not crammed with objects, providing thus an unobstructed view, some sea or lake can be seen and even the houses of some village on its shores. A luminous celestial object on the night sky, possibly a comet, sheds light over the landscape, which is covered by a bright arc, resembling a lunar rainbow. And against the sky, over the whole composition of objects, earthly and heavenly, hovers an imaginary bat – like creature, having only two clawed front legs and the tail of a lizard. The creature stretches its body and its wings to reveal, as if tattooed on its own skin, the inscription “Melencolia§I”.

The conundrum presented by the engraving has been studied extensively in the past by many researchers, and several different interpretations have been offered. Most of these are attempts to guess Dürer’s own intentions, while some could even be described as wild guesses or long shots. Sometimes the interpretations given are not very different from the interpretations of a rather abstract poem: it seems that none of them is more or less “correct” than the others. Dürer has left for us a riddle whose answer is highly susceptible to subjectivity, without leaving any instructions on how this enigmatic work should be read after all. Only a few features of the picture are rather clear, such as the year 1514 appearing in the base row of the magic square. Dürer signed his work AD1514, giving himself the solution to this minor puzzle (note that AD stands for “Anno Domini” and “Albrecht Dürer” at the same time). The inverted 5 of the magic square is generally agreed that represents the month of the year 1514 in which Duerer’s mother died, an event undoubtedly linked to melancholy. It has been suggested that the magic square itself is borrowed from a talisman of the German occultist Heinrich Cornelius Agrippa von Netteshime (1486 – 1535). Cornelius Agrippa intended the talisman, or “Jupiter’s square”, as a shield from the bad influence of planet Saturn, which could cause melancholy (in medieval times, scholars associated Saturn with melancholy, one of the four humors of ancient medicine). This explanation is at least possible and reasonable, and is supported by the fact that Agrippa had visited Nurnberg, Dürer’s city, in 1510. Little else, if anything, about the picture has been resolved satisfactorily, let alone undoubtedly.

The comet, the bat – like creature, the landscape, the lunar rainbow, the building, the ladder, the scales, the hourglass, the bell, the putto, the millstone, the angel, the tools, the sphere have all been, and still are, subjects of guessing. Even the title of the work, “Melencolia I”, remains largely unresolved, partly due to the misspelled “melancholia” (in latin) and partly due to the mysterious “I”. Of special interest in this essay is the Dürer’s solid, the geometric object dominating the left part of the engraving. The proposed theories or conjectures about the solid can be distinguished in two categories. The first is comprised of theories about the exact geometric nature of the solid. The second collects all theories about the possible message the solid conveys or about what the solid stands for in the general context of “Melencolia I”.

Examining the first category, these are some of the proposed solutions of the riddle:

1. The truncated rhombohedron hypothesis: Most researchers agree that the solid is probably what in Solid Geometry is called a “truncated rhombohedron”. The name of it may seem somewhat repelling, however it is a rather simple solid with six faces, all of which are rhombi. A rhombus is a quadrilateral with all of its sides having the same length (by school Geometry it can be proved that it is also a parallelogram). The rhombohedron is a solid constructed by repeating a rhombus six times and looks like a dice whose angles are not right. A “truncated rhombohedron” is a rhombohedron with its two facing vertices cut off. It must be stressed that the assertion that the Dürer solid is a truncated rhombohedron is simply a conjecture and is not supported by any of Durer’s writings or by any other data. The only reason leading to a general agreement on the truncated rhombohedron hypothesis is that the Dürer’s solid simply looks like a truncated rhombohedron.

2. The truncated rhombohedron – 72° hypothesis: Some researchers have even made a step further, to assume that the angles of each face of the rhombohedron are 72 and 108 degrees (i.e. 1/5 and 3/10 of the full turn). Mathematically, these values are of special importance and link the Dürer’s solid to the notorious number φ, the ancient “golden ratio” or “golden mean”. Indeed, the above angles refer to a special kind of polygon, called a regular convex pentagon, which is a shape with five equal sides and five equal angles (all of them equal to 108 degrees). It can be proved that the chords of such a shape are in golden ratio to its sides.

3. The scaling hypothesis: It has also been argued that a vertical compression of the engraving by a factor sqrt(φ) turns the picture into a square and the solid into a truncated cube, providing thus another link to the golden ratio.

4. The truncated rhombohedron – 80° hypothesis: According to another point of view, resulting from measurements based on quantitative assumptions, the angle of the rhombohedron is approximately 80 degrees, which by mathematical standards is a rather humble number with no miraculous properties as the ones mentioned above.

5. The heptagon hypothesis: Despite the apparent humbleness of the number 80 mentioned above, it has been observed that it is suspiciously close to the number 77,2, which would be the value of an angle of the irregular pentagon produced if five of the seven vertices of a regular convex heptagon were joined.

6. The truncated rhombohedron – circumscribed sphere hypothesis: According to this theory, Dürer observed that while six of his rhombohedron vertices lie on the same sphere, the other two, those at the bottom and the top, stick out of it. In order to make his solid beautiful, the artist cut of the two protruding vertices in a way that the resulting truncated solid is circumscribed in a sphere. The theory is supported by the familiarity of the artist to the so called “plan and elevation” method, i.e. the using of projections of solids on planes, which would have been of great assistance in the process of the truncation.

7. The rectangular slab hypothesis: Apart from the rhombohedron conjecture, the Dürer’s solid has also been viewed as a nearly rectangular slab, or parallelepiped (a matchbox), with two opposite corners cut off.

8. The ambivalence hypothesis: This can be summarized in physicist David Finkelstein’s phrase “I propose that Dürer designed the Octahedron [meaning the solid] to be ambivalent, irresistibly construed as a truncated rhomboid in one orientation, as a truncated slab in another, and as something else from yet another”.

Number 1 of the above hypotheses may be traced to Erwin Panofsky (“The Life and Art of Albrecht Dürer”, 1943), a standard reference on Dürer. Numbers 2 and 6 have been proposed by Peter Schreiber (Historia Mathematica 26, 1999, 369 – 377). Numbers 5 and 7 are found in works by David Finkelstein, who attributes number 7 to personal communication with Dr. Basimah Khulusi, apparently a medical doctor. Number 4 has been proposed by C. MacGillavry (Mac Gillavry C., Nederl. Akad. Wetenschap. Proc. Series B 84, 1981, 287).

A number of explanations have been proposed about some meaning conveyed by the solid, or why Dürer included it in “Melencolia I”, putting it in such a prominent position. Some of these are:

1. The philosopher’s stone hypothesis: According to a point of view, the Dürer’s solid is a symbol, or image, of the medieval philosopher’s stone or the “stone of Saturn” (this is the same stone that, in ancient Greek mythology, Saturn swallowed instead of Jupiter). Such alchemic positions can be found in

2. The star of David hypothesis: David Finkelstein sees a Jewish star (a hexagram) of David on the projection of the solid on the ground. In his opinion the “circumcised rhomboid” is a sign of Hebraicism from Dürer’s part.

3. The human skull hypothesis: Many see a rather deformed human skull in the strange stain on the front face of the Dürer’s solid, with unclear meaning.

4. The four ghosts (or hidden faces) hypothesis: David Finkelstein again noticed in 2004 that “anyone who steps back several paces from a good print and focuses on the shading of the front face of the polyhedron patiently for a minute or so, will soon find or construct a face” which, by some sort of Necker’s cube illusion, could be at the same time either a man’s or a woman’s face. After mentioning that these faces may represent Dürer’s mother and father, Finkelstein denotes that “I am less certain of a third hidden face”. By 2008 Finkelstein had discovered two more faces on the solid which he claims first to have inferred and then found. His overall theory about “Melencolia I” is a medley of subliminal images, gematria, rebuses and anagrams, to arrive, as far as the solid is concerned, to the conclusion that it is a puzzle “unsolvable in principle but appears to solve itself in some views. It declares that the Intellectual World may have a mathematical design but, if so, that design is inaccessible to us”.

I must say that some of the above theories and explanations seem to be, to me at least, too farfetched and perhaps many of them did not cross Dürer’s mind, not even as a distant possibility. It must be taken into account that Dürer was an artist of the Renaissance era and, a genius as he might be and a master of the burin, his capabilities were certainly within the boundaries of human ability. It is rather doubtful whether, apart from some basic principles of perspective, Dürer could have rendered his solid with such precision and mastery to deliberately create such a subtle illusion as the ambivalence hypothesis requires. As pointed out in a previous essay (Albrecht Dürer’s eyer lini), the bell of the upper right corner of the picture is after all inaccurately drawn. The 72 degrees hypothesis is plausible and rather tempting to adopt, however there is nothing specific in the engraving to support it. I have more respect to the 80 degree hypothesis, as it is the result of a calculation with data taken directly from the picture, be it so under certain assumptions. The scaling hypothesis is simply wrong: it takes some imagination to link the dimensions of the picture to number φ, and the solid does not look like a cube under the scaling by a factor sqrt(φ). The circumscribed sphere hypothesis is the only one providing a reasonable explanation of why and how Dürer arrived to his solid. The heptagon hypothesis is supported by a newly found sketch of Dürer, showing an irregular pentagon inscribed in a circle with an apex angle approximately equal to 80 degrees (Weitzel, Hans. A further hypothesis on the polyhedron of A. Dürer, Historia Mathematica 31 ,2004, 11). If this is accepted as a rough sketch of the solid, then MacGivallry’s result is confirmed. The rectangular slab hypothesis is plausible but unsupported.

I have several times ventured to discover for myself the ghosts that allegedly haunt Dürer’s solid and I have indeed found several figures that might be taken for human faces. I have also found other figures that might be taken for the head of a rat, the head of an alien and the head of a lamb. It seems to me that there might be still other such figures that remain undetected, which however are of not very different nature than the face on Mars.

Instead of trying to explore Dürer’s mind, by sometimes wild guesswork, in order to understand what he could possibly have intended to depict with this enigmatic solid of his, I find much more interesting the problem of understanding what he actually did depict. And the only way to achieve this, in my opinion, is to build up solid arguments based on direct measurements and calculations.

Tuesday, June 28, 2011

Denderah Zodiac from the Temple of Hathor

The temple to Hathor at Denderah, Egypt was constructed as a monument and a learning venue for Motherhood. The design of the monument carries the celestial theme throughout its grand architecture. Within each chamber and hall are magnificent and profound works of art which glorify the qualities of benevolent motherhood. In the entrance areas are fully inscribed columns with the Nile Lotus replaced by the head of Hathor wearing her traditional wig over cow ears. The temple is the product of the Hellenistic Period after the conquests of Alexander the Great during the Ptolemaic Dynasties, just prior to, and overlapping the initiation of the Christian era.. The Nile Lotus and the Wig of Hathor are both symbols of the object held in the hand of the constellation Virgo. Modern astronomers recognize the wig as Coma Berenices; the wig of Berenices. However it should be self evident that the stars at the top of the axis of the galactic sphere can be connected to look like a sprig of wheat, an ear of corn, a Red Pandanus, a Nile Lotus, an Indus Lotus, or any fertile flower. The essence of a fertile flower is the theme of the temple to Hathor, and the essence of benevolent motherhood. This essence is drawn directly from the positions of the stars at the top axis of the galactic plane of the Milky Way, perceived from earth.

In the ceiling of the Osiris chamber the architects installed a cosmic story about the deflowering of motherhood. The story was told as a prophecy, and an example teleology; the belief that design and purpose are apparent in nature. It is derived from the Egyptian record of the experiences of life, and their recorded knowledge of the most ancient traditions of the whole world. It is the quintessential document which relates to the conversation between the Egyptian priest and Solon in Plato's dialogue by Critias. For the monument reads like a history book, in which the clock of the heavens defines the space-time frame, and the rhythm of Plato's Dance of the Gods. Besides a history of the ancient world, the monument tells a story about the maturation cycle of human souls in the whole of Western Civilization for an additional 8600 years to come. The solar eclipse of August 1999 demonstrates that the great accuracy of the celestial knowledge of the temple architects included dramatic teleological aspects. Within the presentation of their material science, the spiritual message of the Egyptian priests and priestesses is clearly visible. It is a story about love and reason, arrogance and wisdom, banishment and salvation. Most of all, it is a story about a man child (Horus) and his mother (Hathor), and the dialog of mankind with the Creator.

This is the Greatest Story Ever Told. It has been told around the world. In Polynesian traditions it is the legend of the Land Under the Land, where a disenchanted wife meets a stranger and goes to the Land of Shadows to have her husband's child before returning to the top of the world with peace in her heart, and a Red Pandanus in her hand. In Mesoamerica it is the legend of Hanahpu and his brother Xbalanque. In Mesopotamia it is the legend of Gilgamesh and Enkidu, among many others. In India it is the dance of Shiva. In China and east Asia, it is the dance of the Sun Dragon. The Bible tells the same story with many different players, including Adam and Eve, Isaac and Ishmael, Aaron and Moses, John and Jesus, and the Child of the Madonna. On August 11, 1999, the sun winked as it entered into the Kingdom of Heaven. To let this incredibly important moment pass without recognition of its historical value would be a sacrilege against all that has been sacred since the beginning of the dialogue between man and Creator. For, to be silent is repression. Thus, this story is being told in the hopes that a modern world is not so cosmically deaf that it has lost all value in its ancient sacred meaning.


The reason why so many of the world's sacred myths have a familiar ring to them is the fact that they were all taken from the same patterns in the stars. To say that the stars contain Plato's Lost Island of Atlantis would not be an exaggeration, for the stars contain all the lost legends of our past. The legends are all testaments to the observational brilliance that accumulated at the predawn and early dawn of civilization. They are the foundations of our modern Jungian archetypes. Within these great legends is an unmistakable common belief in the benevolence of Mother Nature. It is the trust and faith that humans gave to Mother Nature that caused her to be deflowered as Western Civilization became a domain of the rule of law. Civilization is not to blame for her being deflowered. It is a process which humanity must pass through. Mother Nature deflowered herself so that the man child could grow to full maturity. The stranger she had come to know was the rationalizing husband and the immature son that she loved so dearly. They had forgotten her value as the vessel of life. In her own words, as spoken through her son, we find the echo of all the great legends.

John 14:1-3
1 Let not your heart be troubled: ye believe in God, believe also in me.
2 In my Father's house are many mansions: if it were not so, I would have told you.
I go to prepare a place for you.
3 And if I go and prepare a place for you, I will come again, and receive you unto myself;
that where I am, there ye may be also.

The deflowering of motherhood is the essential path which all children must follow, sooner or later. For you see, it is not the mother's body that is deflowered, it is the ideal of the goddess she represented when we were young. Our rational mind searches for scientific truth, in the hope that one day we can be independent of the myths we learned within the mothering bonds that we hold in the memories of our youth. When we finally reach the end of our adolescent umbilical cord, it breaks, and we search for the mothering bonds to find out what went wrong. As we turn back, we can become a "pillar of salt" and lose all emotional bondage, or we can shed the tears of our temporal identity and be grateful that the Cosmos was willing to give us mothering when it was needed. After the bond is broken, it can be restored again by the simple grace of gratitude. Then we become wise and are capable of passing on to those who follow us, the stories we once believed to be the material and maternal facts of life. At the end of our mother's cord we learn that she had prepared us to break the bondage of the rational umbilical cord, which was our dream of her divine nature. A few years wandering in the wilderness is all that any rational person needs to recognize that we do not live in a world dominated by the rational ideas of men, we live in a world bound to the breast of Mother Nature. We cannot break that bondage by death. We can only stress and distort it by denial during life.

Hidden behind the logic of this microscopic maturation cycle of a single human soul is the macroscopic maturation cycle of civilizations. Like individuals, civilizations strive to break the umbilical bondage. And like individuals, health is restored when a civilization recognizes that Mother Nature knows only Tough Love. She gives us a scepter of flint as proof that we are capable of self inspiration when we walk in the pits of our emotions. She gives us a flail as proof that we can overcome physical stress. She gives us a crook as proof that we can lead others to the Promised Land. And last, she gives us a door through which we can pass a material record of our lives on to her children who come after us. In Egypt the door was Rostau, where Osiris ascended to Orion, and the Great Pyramids promised his return. It is through that door that the present essay will lead. The metaphor of the way of the door has been fabricated many times and in many ways. The ceiling monument in the chapel to Osiris is a detail of Mother Nature's passages through the cosmic door into the mystery of the heavens.|

Read More Here:
Denderah Zodiac
S. Battaglia, G. Miglietta and Opera Laboratori Fiorentini

Full scale reproduction of the Zodiac from the Temple of Hathor in Denderah, the first depiction of the heavens in which use is made of stereographic projection. The outermost circle shows 36 decans, each scanning a 10-day interval; the first decan is level with Sirius, the star that heralds the beginning of the year and the arrival of the Nile flood. The second circle shows the constellations in the southern hemisphere. The third circle shows the constellations in the northern hemisphere. The figures of the Zodiac are aligned along the eccentric circle of the ecliptic. The axis joining Cancer to Capricorn (the solstices) points in a north-south direction. The temple's transverse axis is perfectly oriented towards the heliacal rising of Sirius, which rose at 18o east-southeast. Sirius, portrayed as a squatting cow, is correctly shown at the point of maximum culmination on the celestial meridian of the southern hemisphere.

Found Here:

During the Napoleonic campaign in Egypt, Vivant Denon drew the circular zodiac, the more widely known one, and the rectangular zodiacs. In 1802, after the Napoleonic expedition, Denon published engravings of the temple ceiling in his Voyage dans la Basse et la Haute Egypte.[4] These elicited a controversy as to the age of the zodiac representation, ranging from tens of thousands to a thousand years to a few hundred, and whether the zodiac was a planisphere or an astrological chart.[5] Louis Charles Antoine Desaix, a member of the expedition, decided to remove the relief to France and so, in 1820, the antiquities dealer Sébastien-Louis Saulnier commissioned Jean Baptiste Leloraine, a master mason, to remove the circular zodiac with saws, jacks, and scissors constructed for the job. The zodiac ceiling was moved in 1821 to Restoration Paris and, by 1822, was installed by Louis XVIII in the Royal Library. In 1964, the zodiac moved from the Bibliothèque Nationale to the Louvre.

The controversy around the zodiac, called the "Dendera Affair", involved people of the likes of Joseph Fourier (who estimated that the age was 2500 BC[6]), Thomas Young, Jean-François Champollion, and Jean-Baptiste Biot.[7] Johann Karl Burckhardt and Jean-Baptiste Coraboeuf held, after analysis of the zodiac, that the ancient Egyptians understood the precession of the equinoxes. Champollion, among others, believed that it was a religious zodiac. Champollion deciphered the names of Tiberius, Claudius, Nero and Domitian on the ceiling of Dendera's temple, and placed the zodiac in the era of Roman rule over Egypt.[8] Baron Georges Cuvier placed the date 123 AD to 147 AD.[9] His discussion of the dating question is an interesting summary of the reasoning as he understood it in the 1820s.

Over the last 10 years, I have been presented with a star-code science called cosmogenesis. From 2000 July 17th to 2007 July 17th a pattern of psycho-physical reference points to astronomical events were presented over and over again to engage and synchronize my attention to the placement the Egyptian’s called Heaven’s Gate: The Cancer-Leo Boundary. The code cosmogenesis uses to construct meaning is based on an epistemology designed by the Egyptian Cosmologists. For the sky-priests who closed the Egyptian Empire’s Old World Order, they designed an ingenious calendar wheel at the Temple to Hathor was called the Denderah Zodiac. I made it my concern to follow the on-line study of this remarkable cosmology with the help of Rush E. Allen an engineer and astro-archaeologist whose mentorship provided me the firing-pins for walking the path of the True Priest.

To present the revelation of the Little Lion, we will walk through the basics of Egyptian Cosmology. The modus operandi for their vision of reality is not unlike that of the modern sages like Einstein, Podolsky and Rosen who argued that a quantum state corresponds not to reality but to the information we have about a subsystem of the universe. The quantum state for The Egyptian sky-priests who authored the science accepted two fundamental bodies of knowledge. One was called the Horus of the Two Horizons and the other was Coming Forth By Day, or the study of the First Fire Ceremony. Between these two pillars of science did the Egyptian cosmologists attempt to construct based on information gathering how the planets, milky way and stars were set to the house of creation that made room for one very unique ally in creation: man and his perception of self within a living system. The D.Z. concretized all these aspects with one important axiom. To acquire the experience of Truth one must suffer the burden of cosmic responsibility once he has put his ‘feelings straight’.

A scientific paper was presented during the time of Seti I to define the ability to understand the science of predicting outcome. The Hunefer Papyrus was the Occam’s Razor of the 17th Dynasty.

Found Here:

The Hidden Geometry of the Zodiac in the Temple of Hathor, Denderah
1. The eight Horus figures holding the circle of the heavens define a pair of crossed axes by the lines defined by knees and elbows. 2. A hexagram may be drawn in by joining the points where the Isis figures hands meet on the circumference of the circle. 3. A pair of horizontal lines are drawn as shown by connecting the "shoulders" of the Horus figures, then point where their headresses terminate. These lines occur at the correct location to each form one side of the regular pentagram which can be inscribed in the circle. Thus two pentagrams are formed; one upright and one inverted. 4. With the remaining lines of the pentagrams drawn in, confirming proof that these constructions represent the intent of the designers is provided by the placement of the straight lines which connect the two fishes of Pisces. Note how one of these two lines not only forms a small section of the side of the pentagram itself, but occurs at the point where this line is intersected by the horizontal diameter of the circle.

Related Pages:

Hidden Geometry in Art:
Pierro Della Francesca:
Fra Angelico
Fra Luca Pacioli
Denderah Zodiac

Geometrical Analysis of the East Meon Crop Circle

Etheric technology of ancient Egypt: The Temple of Hathor at Denderah

Original Article Found Here: