Gnomonicity Part One: the Study of Relations Between Luminous and Reflective Bodies.

Gnomonicity is the study of relations between luminous and reflective bodies. Any “risen” body, shape or form, that permits the differentiation between light and shadow, be it naturally occurring or man made, has the potential to be used as gnomic indicator of significant information which is projected as immaterial phenomena such as the Sun’s shadow.

The term “gnomonicity” has been recently coined by Midhat J. Gazale (1999 “Gnomon: From Pharaohs to Fractals”,  Princeton University Press. ) in reference to the progression of the discipline of geometry, made possible by the creation of sophisticated computer programs, which permits the construction of an imaginary field or “cube” of space, within which both simple and  complex higher-dimensional “surfaces” or domains may be constructed. Supporting the study of a four dimensional “trace through space over time”, which constitutes the imaging of a moving reflective body, illuminated by one or more sources of light, as can the study of a simple or complex object which may be topologically transformed or rotated, its internal and external spaces mapped and observed, or even “traveled” through, has provided an exceptional progression of tools used by both Artists and Scientists, effectively unifying the disciplines of Art and Science, theory and practice. Modelling both real and imaginary shapes and forms, in digital space, has merged the science that provides the analogue model of real shapes and forms whose functions are defined as geometrically dynamic systems, with the art of creating virtually real shapes and forms and virtually real systemic functions. Distinguishing between the real and the imaginary, demands as it always has, the description of the domain and all elements with inform the parameters of the domain, with regards to proximity between the observer and the observed, the source and nature of light which illuminates the subject matter, and the source of data sets which are determined by graphic images of real phenomena which includes significant indications of both immaterial and material information, derived as “signatures” from both naturally occurring or man made projections that are either proven to be real through the process of natural deduction, or remain hypothetical or imagined proven to be merely based on inferences or speculations of the real, and hence cannot be proven to be real, through the process of natural deduction.

The process of natural deduction will lead to either the foundations which inform the building of the cube of space itself, and hence will be informed by the foundations of the electromagnetic spectrum and its analogue model, or will lead to the geometrical modelling of the cube of space itself, commonly known as the “algebraic matrix”, or the “intuitive domain” of Klein Geometry, or as higher dimensional immersions in “‘Hilbert Space” (4 space where motion and time is objectified as the fourth dimensions, inherent in the action of scribing a generalization of a cycle, or pattern of motion or kinetic action, in space over time, which constitutes the process of gathering raw data and hence plotting “points” within Cartesian Co-ordinate space of both 2 and 3 dimensions, where the “axis” time is objectified according to the “arrow of time” which permits the modelling of 4 and higher dimensional “traces” or “signatures” of the relation between light and motion, of reflective and luminous bodies, and their interaction with specific domains such as the “camera effect” of light as it passes through a singular point or “hole” or “aperture”; the “trace” of the image which of light as it falls into a hemisphere or bowl, that can be manipulated by the external sources of motion and so on; the “trace” of any moving reflective or luminous object in either an illuminated space or a space devoid of any source of light other than the object itself; and so on.

Both Science (theoretical and applied) and Art (theoretical and applied) demands the critical awareness of all spatial and temporal parameters which define the context of the product, and identifies both the “scientist” and the “artist” as either an active participant involved in the generation of information, or as a passive observer of the transformational process that is naturally occurring and self revealing as information generated in and through space over time, hence engaged in the act of making images that either directly or indirectly “capture” a series of events according to the technological apparatus employed for the purpose of the exercise. The scientist and artist is either directly involved with the evolution of ideas and images, patterns, cycles and sequences, scripts and narratives or expressions that are inspired by the intention to reveal to the self or to others, through the process of induction of others, be it a student or audience, of facts deduced, impressions inferred, or emotional responses triggered by the audience participating either passively or actively, directly or indirectly in the process of induction itself.

The contemporary work of the animator who creates imaginary objects in virtual space with or without a likeness to the “real”, and the scientific study of actual objects whose surface and internal complexity is now able to be  “captured” by exceptional imaging technologies which transcend the limits of the common camera, the common telescope and microscope, or technologies of transportation, designed with the singular purpose of traversing the tyranny of distance in space and time, and the tyranny of ignorance that is approached through the process and means of communication, have all been advanced to such a degree that was in the past quite possibly unimaginable to the common person, yet predictable to those engaged in the so called “cutting edge” of both Science and Art.

The progression of the science of computation, advanced historically as the project of  “ars combinatorics” either inspired by speculations of ancient symbolic codes, or as coherent “systems” of symbolic communications, unrelated to ancient cannons leading to various ciphers which demanded the “key” to unlocking the information communicated, led to the consolidation of a new “Principia Mathematica” given by Bertrand Russell and Alfred North Whitehead, which focused on the coherent aximatisation of set theory and propositional logic in the early Twentieth Century have permitted both the scientists and the artist to transcend the common parameters of 3 dimensional Space and Time, by providing a virtual “cube of space”, within which all simple and complex objects may be constructed from the “ground up” or reconstructed from data sets gathered as “moments” of frozen space-time, which can be reanimated in “real-time” being 24 frames per second. The “cube” of virtual or imaginary space within which the object is immersed, constitutes either one or all of the 8 “quadrants” of Cartesian coordinate space, hence the “barycentric center” of the “subject matter” mapped according to Cartesian Space, is postulated both as an imaginary “point”, “hole” or “void”: (0,0,0) from which spacetime is imagined as emergent in all directions: “>|<“, or as a “positive” integral point, located in the positive domain emergent from the imagined “void” (0,0,0) in three perpendicular (90 degree limit) dimensions of positive space (x,y,z):  “|<” where the fourth parametric of time “t” defines the independent real variable of time, and the object is generalised as an integral whole, and point, whose motion over time is able to be mapped according to the positive domain of “real space”. It is for this reason that in Logic, the “truth value”: “T” is written as “1” in contrast with “0”, which as “_|_” indicates a Falsehood. The rotation of an simple or topologically complex object immersed in three dimensional space, and the mapping of an objects motions through space over time, a mapping of the surface of a complex object or the motion of an object on the surface of complex object is realized as an immersion of an independent object that is either imagined or real, in the positive quadrant of “real space” constructed in virtual space.

It is clear that the unification of both the imaginary and real, leads to potential problems in interpretation of any given data, unless a student is inducted into the intricacies of geometry and the logic of the symbolic indications which signify the elements, in a rich overlay of information, which demands contextualisation. Just as the nomenclature of terms in any given dictionary of terms, common to a specific linguistic codification, offers numerous definitions for the same term, which are determined by the context of the word itself, and the object or phenomena signified, or intention of the communication, the dictionary of terms provided for a specific discipline, reveals the same complex overlay of related information, indicated by the symbol, which in the case of the universal language of mathematics betrays a similar complexity, and demand for identification of the domain and its elements.  For example, such things as the transposition of a matrix: “T”; the distinction between real variables in time “T”, an independent variable in parametric equations that are often non angular: “T”.in contrast with the angular parameter indicated by the use of the “character “theta”;  the relation between physical “units” “T” (abreviation of “terra-” used in the System International); and with the inverse of the “T”-  the notation of a falsehood “_|_”, and the indication of the orthogonal, perpendicular, or singular: “_|_”. (E. J. Borowski and J.M Borwein.1989., “Collins Reference Dictionary of Mathematics”.)

It is also plain that the specialized use of terms within a specific discipline, may or may not be clear with regards to the general use of the same term, within the nomenclature of a dictionary of terms of a nations linguistic codification. For example, the term “gnomon”, is defined in the dictionary of English Terms, as 1. an early astronomical instrument, consisting of a vertical shaft, column or the like, for the determining of the altitude of the Sun, or the latitude of a position by measuring the length of its shadow cast at noon; 2. the raised part of a sun dial that casts the shadow, a style, and the final definition specific to Euclidean Geometry, given as 3. the part of a parallelogram that remains after a similar parallelogram has been taken away from one of its corners. This term is from Latin, referencing “the pin of the sundial”, and from Greek, having the meaning “one who knows” or “inspector”. The apparent obscurity of the third definition, is only made plain when one returns to the discipline of studying the domain which remains outside the sphere of human influence, indeed that which informs the foundations of astronomy, as the study of the light and motion, conducted in the broad light of day, and from this, the bridge between that which appears to be entirely obscure, and that which is made plain as intelligible information, which transcends mere appearances, and informs the inherent complexity of theoretical science which grants the apprehension of limits and constants, as determined by terrestrial motions, which then inform the construction of applicable scripts derived from analogue models.

It is the awareness of the bridge between the obscure, and hence that which remains implicit, and the explicit, that is essential for the clarification of issues raised in science today, which regards to the foundations of science and technology, and the means through which science is determined as either hypothetical or theoretical.

It is at this point that the above mentioned reference to the work of Midhat J. Gazale demands attention, for his generalization of the foundation of mathematics, definitively rejects the relation between the term gnomon as it is used in Euclid’s definitions given in the “Elements” or definition “3”, and the first and second definition, pertaining to the foundations of astronomy. Gazale speculates upon the Egyptian Obelisk, stressing spuriously that was never used as a gnomic indicator, but rather, it was used to record the imperial conquests of the Pharoah, specifically in the New Kingdom period. There is no mention of the fact that Thutmoses III removed Obelisks from Heliopolis, to Hermopolis, placing them in prominent positions at the entrance of significant buildings, and used the faces of the obelisks to record his conquests.

The Narmer Palette is one of the earliest examples of Ancient Egyptian graphics. It dates to the pre-dynastic era, (circa 3,100BC) and was found in Hierakonpolis. Marked by Red circles, evidence of the “setchat” flower appears on both sides of this tablet.

Gazale also speculates on the use of the structure called the “Setchat” without providing images of this device, inferring that this structure was developed as a tool for the use of recording angles of the sun’s shadow, in the New Kingdom, when this structure is graphically portrayed in the earliest depictions, and in various ways, which infer the clustering of numerous circles about a common center, the most common figure indicating 6 circles clustered about a common central point and circle. As more rigorous research reveals, the object known as the Setchat, informs an alternative method that supports the construction of a cube cage, isometric lattice, and orthogonal lattice, to the simpler configuration of two equal circles, whose measure of their radius is equal to the distance between the centers, commonly known as the vesica piscus, and featured as the structural foundation of Euclid’s first Proposition, of his “Elements”. The setchat, be it configured as 6 circles or 8 or more circles clustered about common central point, informs a model which can support a complex matrix construction, in a manner that provides more control over the extremities of the sets of the lines ruled.

Left: The Merkhet; Centre: Isis is depicted with the setchat above her head. Right: an icon of the Setchat.

As a clustering of 6 circles, the setchat is a flower like pattern, made from the intersection of the circles scribed, is commonly located above the head of ISIS, and informs that which is commonly referenced as “the seed” of the “flower of life”, a structure which informed the work of Leonardo De Vinci, as his folios reveal, and indeed remaining outside the public domain, due to this specific page being excluded from the “complete reproductions” of De Vincis folios, until its appearance online in 2006. After some decades of “new age” speculation regarding the importance of this shape for the foundations of geometry and mathematics, it appears that the common perspective cultivated in the public domain, prevails in support of mathematical idealism rather than mathematical realism.

We can only speculate that the failure to comprehend that which informed ancient systems of governance known as Nomarchy, is a failure due to the impoverished interpretations of pictographic material, made without a detailed knowledge of geometrical information, and without a coherent understanding of Egyptian Hieroglyphics, and indeed indicates a failure, conscious or not, to comprehend that which once informed the foundations of natural and common law, to which the rulers themselves were bound or yoked, as the givers rather than makers of laws, and mediators over justice, who moderated the natural rise of corruption in the administrative body, hence acted as representatives of the “completed fabric of the garment of god”, which is the linguistic codification of Ancient Egyptian Hieroglyphics,  in defense of the common person who remained by their own choice, in a world which cherished the nobility of education and learning, disenfranchised from the system of knowledge that informed and was informed by the linguistic codification of the elements of both concrete and immaterial significants.

It is not necessary to make any appeals to any culturally given device, product of geometry, or system of determinations specific to a specific time, or a specific nation of peoples, or a religious creed or ideology. What follows is a simple generalization of the facts that are self revealing, self generative, and inform the translation of the primary signatures of light and motion, to the prototypical analogue model which informs the foundations of the theoretical underpinnings of the electromagnetic spectrum, and the transposition of information from the orthographic projection of the sun’s shadow on the ground plane, to the source of light, the source of motion and the source of mind, leading to the apprehension of the distinction between the naturally occurring “knot which cannot be undone” and the process of natural deduction which effectively does in fact undo the “knot which cannot be undone”, and is considered by Plato, to be the cause of the greater evil, if indeed that which is undone in the process of reaching the conclusions which are counter intuitive to the appearances of the apparent motions of the Sun, as the action of the Artificer to which Plato gives the facts in a process of reverse engineering the model, is itself undone. (Timaeus) It is the process of natural deduction, that permits the scientist to transcend the intuition gained from appearances, which provides conclusive facts regarding the Earth’s complex diurnal motions, and reason for the Earth considered anthropomorphically to have masculine qualities, and two running speeds (Narmer Pallette Bottom Register) where the Earth’s axial rotation informs the flourishing of life on earth, and where the degree of axial tilt informs the apparent eccentricities in the Sun’s apparent path and location on the dawn horizon which shifts from the day of the winter to the summer solstice: “^^^^^^”, which constitutes the significant influence of the axial tilt, and the reading of its precise angle without which it is quite reasonable to suggest, the tree of knowledge itself would never have flourished as a profound unity of complementary attributes, perspectives, and reciprocal relationships which remains concealed by the imposition of an impoverished exploitation of dialectics, and crude binary logic. As it is the latter which informs the spurious notions of absolute “good” and absolute “evil”,  which are characteristically qualified with gender distinctions that impose a conditional cultural entrapment of fatal paradoxes which  perverts the integrity of science – theoretical and applied, the integrity of law and justice, as well as perverting the integrity of Theology and Mathematics. As each of these disciplines are dependent on a coherent understanding of geometry and its foundational formative influences determined by  the acknowledgement of limits and constants, drawn and born in the signatures of light and motion itself, which remain outside the sphere of human influence, and inside the sphere of human experience, it is this that  underpins Plato’s intention, given late in life, after the rise of tyrannical abuses of power form his own familial relatives, to arm a majority of peoples, with the very elements necessary for a proper induction into that which is intelligible, and transcends the merely visible, as the only possible insurance of the integrity of democratic forms of governance, and the integrity of science and of all of the arts, beyond the impoverishment of these by anarchists, autocrats, despots and oligarchies, whose power is raised by and through the cultivation of ignorance in the public domain.