Crystallography: Part One- Orthographic and Fischer Projections.
The study of crystallography, has informed the application of the isometric matrix to the analysis of shapes and forms that are naturally occurring. As the isometric matrix is built from the ruling of sets of lines, each set informing new perspectives and inferring higher dimensional structures, all immersed on the two dimensional plane, the self generative modelling inherent in the design, constitutes an orthographic projection of information, derived from the singular point of the gnomon, which unifies the three primary sources of light, motion and the human mind.
As the human species is endowed with bicameral vision, we naturally observe external reality with two vanishing points, which direct the vision of each eye to points on the circle of the horizon. Drawing an image with the element of perspective, is performed with the location of these vanishing points, ruled according to the lines of sight, with regard to the horizon. the resulting image, appears to distort the topological information of the surface, in a manner that is true to the visual cortex.
With an increase in the focal length and distance of the “camera” (single lens and bifocal eyes) to infinity with a perspective drawing the results constitute an orthographic projection presenting a means whereby a three dimensional object is able to be represented as an immersion in a two dimensional plane. The view direction is orthogonal to the projection plane resulting in every plane of the scene, appearing as an affine transformation on the viewing surface. It is further divided into multiview orthographic projections and axonometric projections. An orthographic projection corresponds to a perspective projection, with a hypothetical point of view. (see http://en.wikipedia.org/wiki/Orthographic_projection)
In “Of Analemmas, Mean Time and the Analemmatic Sundial” 1994, Frederick Sawyer states that “Vitruvius alluded to the theory of the analemma as a graphical procedure equivalent to what is known today as an orthographic projection.” The term “orthographic” is derived from the Greek: orthos meaning “straight” and graphē meaning “drawing”, and began to be used in the early 17th Century. The term “analemma”, is defined as “a scale shaped like the figure “8”, showing the declination of the Sun and the equation of time for each day of the year.” From the Latin, the term is defined as the “pedestal of the sundial, and sundial.” From Greek, the term analemma is defined as “support”. (Webster’s Encyclopedic Unabridged Dictionary of the English Language.)
The building of the isometric matrix, from the topological translation of the analemma, as twin circles, whose radii is equal to the measure between their centers, constitutes an orthographic projection, and affine transformation of the viewing surface.
From A.H. Corwin and M.M. Bursey’s textbook published in 1966, and titled “Elements of Organic Chemistry as Revealed By the Scientific Method” we are introduced to the problem of illustrating representations of 3 dimensional Tetrahedra, and other 3 dimensional objects considered to cause too much confusion, on account of the failure to reveal all vertices of a shape as the generalization of the solid as an orthographic projection, appears to flatten the figure, and in the case of representations of the cube, one of the 4 “diameter lines” enclosed by the vertices of the cube remains implicit, at the apparent “center” of the 2 dimensional hexagon, which constitutes the corresponding figure that in 3 dimensions informs the cube. A more convenient method of representing the three dimensional figure in two dimensions, was provided by Emil Fischer, whose method is known as “Fischer Projections”. While this method of representation shifts the model of the solid for the purpose of revealing information that appears concealed in the orthogonal projection, the resulting dissociation from the orthogonal projection has led to the diminishing of the original scientific method determined from the progressive building of the isometric matrix from the twin circles that translate the figure of the analemma. Indeed, throughout the history of modern science, the rendering of the process of ruling lines from the twin circles, remains implicit, as if the analogue model of the primary signatures which inform the relation between the Sun and the Earth, the orbital cycle, informed by the axial tilt, and the Human mind, remains implicit, as if the “eye of god” structure which informs the “genera” of the orthogonal representation of embedded solids, that constitutes the isometric matrix, is merely a template or support for illustrations used as representations of the elements of material chemistry.
The model of a crystal structure, given by Rene Hauy in 1803, was reprinted in the Journal of Ultrastructure Research in 1958, and again in 1966 in the above mentioned textbook, titled “Elements of Organic Chemistry as Revealed By the Scientific Method” in reference to the exceptional correspondence between the early models, and the micrographs captured in the twentieth century. It is clear from Hauy’s model, that the foundational inspiration is clearly inferred by the ruling of lines which inform the building of the isometric matrix.
As the science of crystallography progressed, through the following decades, models of the “fringe patterns” returned to the orthographic projection, and while the superstructure remains implicit as if a template, the rigors of theoretical physics and its applications to natural science is now able to be revealed explicitly.
In the 1970’s Scientific America published “readings selected and introduced by Victor A. Bloomfield and Rodney E. Harrington. In the edition titled “Biophysical Chemistry”, significant readings concerning basic biomolecular structure are provided, and of specific interest, M.F. Perutz provided a reading titled “The Hemoglobin Molecule“, originally published in 1964. In it he illustrated three dimensional fringes, which have been redrawn here, using the isometric matrix as the superstructure. He states “these fringes are needed to build up an image of protein molecules. For this purpose, many different x-ray diffraction images are prepared and symmetrically related pairs of spots are indexed in three dimensions: h, k and l and
h, k and l. Each pair of spots yields a three-dimensional fringe like those shown here. Fringes from thousands of spots must be superposed in proper phase to build up the image of the molecule.”