Like architecture, geometry constructs dimensional relationships and associations; in the process, it can also build a network of impressions. For example, anything can be mapped through a zero-dimensional point, as if forced through a tiny hole in space not unlike a black hole singularity, to reemerge on the “other side” in a different zone (the coordinate plane), unchanged but turned. Metaphorically, the subject now has a new viewpoint. The transformation, called point reflection, actually rotates the pre-image around the origin by an amount of exactly 180 degrees. We can consider that network inception arises with this timeless point in zero-dimensional space, which is actually just a location. The property of a singular point is an identity because of its isolated nature, an unconnected vertex that is equivalent to a self-aware state – “I am” and “there exists” characterize and define it. Mathematically, this property is known as reflexive, A = A. Thus, the main attribute of this network property is that of perception; an analogy from Nature is a seed, the potential for growth as a nascent vitality. With unconnected points as individual phenomena, the intellect is within them, but there is no context, no relationship among them, as Abbott described in Flatland:
“That point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself his One and All, being really Nothing.”
Using this taxonomy as a general structure, the next dimension results when two vertices are connected in space, forming a geometric segment, analogous to a stem rising from the soil. A geometric transformation using a line is known as a reflection, which maps an object across the line of symmetry, as if flipping it over a fence, i.e., if A = B, then B = A. Symmetrical objects are prevalent in Nature: most animals have bilateral symmetry where a line of reflection can be drawn through their bodies, producing two matching halves. A line represents not only a bridge, serving as the connecting edge between two or more vertices, but also polarity since the vertices may be conceptually opposed; yet even then, a stronger bond is established between the formerly isolated points as the structure increases. The property of a line is therefore duality, and its attribute is relation. Thus, selected affinities in art, science, and Nature which have dichotomy characteristics will be represented: is/is not, light/dark, good/evil, odd/even, smooth/lumpy, plus/minus, etc.
If the progression of dimensions is like that of subatomic particles, having rapidly formed during an inflationary period, then the advent of one-dimensional structures necessitates the introduction of a temporal unit. Time now exists along the infinitely narrow band – it just goes back and forth between the two nodes. However, there is still no surface to the object; this happens later in geometric evolution when another vertex is added noncollinearly, creating the simplest polygon, a triangle. It is easy to see this possibility only from a viewpoint standing outside the given dimension: we have achieved a transition to another dimensional level that likewise requires new sources and perceptions.
Two-dimensional structures are flat areas with no thickness, whose main network property in this poem is complexity. As the system has increasingly grown by adding vertices and connecting edges, the form is now essentially an infinite sheet of surface, extending in all planar directions. Once a third vertex was added and connected outside the line’s boundary, a triangle was formed (it has the minimum number of vertices and edges to make a planar figure) – a more intricate structure from Nature is a leaf. There was also simultaneously created a closed pathway or circuit that is directional, and thus also temporal, where the first and last vertices are the same, like connecting dots. It is similar to both the transitive property, if A = B, and B = C, then A = C, and the logical syllogism which is also a directed process, forming a conclusion given two or more premises. Examples of two-dimensional affinities are: direction, orientation, transit, periodicity, resonance, pattern, cycle, triangulation, as well as polygon associations such as stability, equilibrium, mosaic, tessellation, etc. Thus, within the planar figure, there is a continuous function of relay among the nodal elements.
As a further development, more than one transformational event could occur in planar space and higher dimensions. A composition is a procedure in which two or more transformations take place, one following the other, wherein the image of the first transformation becomes the pre-image of the next one, allowing the transfer of identities. It has been proven that every isometry (a transformation that preserves distance between points) can be expressed as the composition of at most three line reflections. For example, a translation is the composition of two reflections across parallel lines and a rotation is the composition of two reflections across intersecting lines.
While any object’s identity is aligned with the zero-dimensional analog, volume is the three-dimensional space that it occupies, i.e., the projection of Nature as a set of impressions. Area (two-dimensions) is its surface, and perimeter (one-dimension) is the surrounding linear enclosure. Two planes form a line when they intersect, and multiple intersecting planes connect to form a volume in space. Intersecting edges within planes in turn create new vertices and the groups of formerly separate relations and circuits become incorporated into a composite mass. This sum total is a wholly formed entity with no spatial limitations, as a tree might grow. The property of a volume is therefore found in the concept of unity, and the attribute of this network property is that of summation. Examples of three-dimensional affinities include cohesion, integration, recognition, and equanimity. Unlike other transformations, a dilation is not an isometry as the pre-image can enlarge or diminish the image, according to a scale factor. A constant of k > 1 will increase the distance between all vertices in the structure, and if 0 < k < 1, then the image will collapse. If k = -1 the image will vanish through the origin, turning 180 degrees as it does so, and reemerge unchanged in size and in another location:
Since the world that we inhabit is three-dimensional, then the foundation of our universe requires the integration of all previous dimensions, including time, without which recollection, progress, movement, and growth would not be possible. Motion of objects can only be measured relative to each other and stationary sources, such as a fixed distance, or the background of stars (which itself is in motion). If a tree represents volume, then a grove becomes a symbol of concurrence: space and time operate together as matter and memory to define all things, places, and events; this lends a network property of ability. The four-dimensional structure is called the worldline, a representation of the spacetime continuum. It allows moment and eternity, presence and absence – in short, our ability to exist in a world that is best suited to our means, knowing of no other.
Because the speed of light is constant, Einstein predicted that in a moving frame of reference objects approaching near-light speeds would undergo transformations in the form of time dilation, length contraction, and a relativistic increase in mass. Since all forms of electromagnetic radiation already travel at this light speed, the Lorentz Transformations predict the impossibility of spaceships attaining light speeds, but they do not account for objects that always travel faster than light, such as the alleged tachyon particle, whose lower speed limit is that of light, or the neutrino, also predicted to travel faster than light. Undoubtedly, such possible encounters would result in a rethinking of the laws of cause and effect: could simultaneous events be separated into dimensions greater than four? Since present theory postulates that the Universe began as a singularity, then this point would also have no time-value, existing indefinitely as the location of potential space. It would then seem reasonable to suggest that coincident to the post-inception inflationary period geometric vertices as dimensions, including time, were sequentially created in conjunction with subatomic particles – it would also grant the possibility that any additional whole or fractional dimensions beyond spacetime were simultaneous to this event as well, but presently remain unknown.
Since the zero-dimension is comprised of one vertex, the first dimension requires a minimum of two vertices, and the second dimension needs only three vertices, a pattern can be seen to develop where each dimensional state is equal to the required minimum number of vertices minus one. There is also a specific symmetry aligned with the numbers in a section of Pascal’s triangle, as detailed by mathematician George Pólya:
“The analogy between segment, triangle, tetrahedron has many aspects. A segment is contained in a straight line, a triangle in a plane, a tetrahedron in space. Straight line-segments are the simplest one-dimensional bounded figures, triangles the simplest polygons, tetrahedrons the simplest polyhedrons. The segment has 2 zero-dimensional bounding elements and its interior is one-dimensional. The triangle has 3 zero-dimensional and 3 one-dimensional bounding elements and its interior is two-dimensional. The tetrahedron has 4 zero-dimensional, 6 one-dimensional, and 4 two-dimensional bounding elements, and its interior is three-dimensional.”
3 3 1
4 6 4 1
By extension, we can enlarge this pattern to include a time vertex, originating with the third dimension but in a separate row, along with further dimensions comprising multiple temporal and spatial vertices, all aligned with the binomial expansion of Blaise Pascal. To maintain the symmetrical pattern, such additional units of space and time become fractional amounts; and as hypothetical transparent vertices they could geometrically describe multi-spatial and polytemporal features existing undetected among the known dimensions. Only through the viewpoint of a meta-observer, standing outside the familiar worldline, can sources and events in these interstices or quasi-dimensions be directly witnessed:
3 3 1
4 6 4 1
5 10(3/5) 10(2/5) 5(1/5) 1
6 15(4/6) 20(3/6) 15(2/6) 6(1/6) 1
If these other dimensions exist, then they contain authentic impressions, which we only partially apprehend, referenced by each archetype and source-entity – it is the domain of irreducible phenomena. And since it is defined as beyond our experiences (a priori), we can call the potential field a transcendental state. Occasionally this world is revealed through projections and glimpses of non-ordinary reality: dreams, apparitions and phantasms, delusions, mirages, hallucinations, premonitions, precognition and déjà vu feelings, ecstatic visions and exultations, revelations, epiphanies, miracles, and states of grace. Because these experiences are more generally “felt” than understood, such as sensing a “presence,” traditionally it has been easy to ascribe them to the realm of another dimension. These experiences are ubiquitous and resonate worldwide in religion, art, and philosophy, and also include their more ancient forms of spiritualism, animism, shamanism, and magic; for that reason, if no other, they must be also be an aspect Nature by their predominance among humanity. Returning to the pre-science era of natural philosophy, linked with cultural studies of ancient myths as Joseph Campbell, Frazier, and others have done, there are often found unusual explanations, representations, and the means for expressing these eccentric or supernatural phenomena as actual events. Those experiences should be compared to contemporary models by referring their similarities to a common historical baseline in the idealization, as Husserl suggested, which is valid for geometry and all other spiritual structures. In terms of writing the analytic poem, the baseline is the intersection of common associations in Nature, and the idealization refers to the dimension of source-entities – affinities should also be drawn from it to complete the network.
To freely speculate, this transcendental state is a density of impressions, not unlike Jung’s idea of an inherited collective unconscious, but here in this context, an alive and active source. This realm, then, is the reservoir of universal intellect (νους) and the medium for transmigration (μετεμψύχωσις) – the relic of past lives, the vessel of present thought, and the fabric of future possibilities:
“The sluggish cream wound curdling spirals through her tea. Better remind her of the word: metempsychosis. An example would be better. An example.
The Bath of the Nymph over the bed. Given away with the Easter number of Photo Bits: Splendid masterpiece in art colours. Tea before you put milk in. Not unlike her with her hair down: slimmer. Three and six I gave for the frame. She said it would look nice over the bed. Naked nymphs: Greece: and for instance all the people that lived then.
He turned the pages back.
–Metempsychosis, he said, is what the ancient Greeks called it. They used to believe you could be changed into an animal or a tree, for instance. What they called nymphs, for example.”
Through the framework of geometry, achieving that next step in this taxonomy requires only an additional vertex to gain another dimension, now called v+. The vertex unit for this multi-dimensional network is a membrane – a hypergeometry that is an extension of all prior networks and affinities, consisting of myriad cognitive nodes folded and packed into an edgeless density. The v+ network is made by grafting variables of cause, potential, and alternative reality. It is a compression of multiple worlds no longer containing any unique identities, the membrane wholly emerging as a new singularity beyond all previously known experiences. Although it is formally detached from our own world, all features of dimensions zero through four are present with it, just as a flat panel is part of a box, an edge is part of the box’s panels, and its corners define the edge’s length. A popular depiction of an object found in this dimension is the hypercube, described by astronomer Carl Sagan and others as a cube-within-a-cube, where all edges are of equal length and all angles between the edges are 90 degrees. The hypercube model (or tesseract, from L’Engle’s, A Wrinkle in Time) is a three-dimensional representation, i.e., merely the projected shadow of the authentic one that is beyond our capability to perceive:
Leonard Euler not only formalized graph theory, but he is also known for his comprehensive assessment of two-dimensional polygons through a formula that today still bears his name, Euler’s Formula: V – E + F = 1. Simply stated, the number of vertices minus the number of edges, plus the number of faces for any polygon or map, will always equal one (all zero dimensional units, minus one dimensional units, plus two dimensional units, is one).
And, for the next dimensional level, polyhedra, the formula is: V – E + F = 2. Can we assume that the fourth spatial dimension will likewise have the same pattern for the hypercube, V – E + F = 3 ? More generally, our modified version of Euler’s Formula aligns all units for each dimensional state, 0 through v+ :
V – E + F = S + T – Mx + 1, where S and T represent space and time values and Mx is a manifold unit.
If it is possible, then, to depict such higher dimensional levels architecturally, even knowing that they are false projections and poorly represented in our world, we can also assume that their poetic metaphors are not only valid, but perhaps more substantial and powerful as they are inherent to ourselves, arising from mind and spirit. With these tools, a comprehensive map could be made from diverse sources in geometry, art, philosophy, and science enabling connections among all dimensional levels – including transcendental states. The analytic poem aligns those selected affinities, initially subjective to each observer, but ultimately aspiring to convey universal impressions, as a comprehensive statement of Nature.
I am: In the Bible, book of Exodus (3:14): “I am that I am” (אֶהְיֶה אֲשֶׁר אֶהְיֶה) was the response given to Moses when he asked God His name. From the Hebrew letters, without vowels, is the word YHWH (יהוה), pronounced as “Yahweh.”
Edwin Abbott Abbott, “Flatland: A Romance of Many Dimensions,” p. 93-94. Princeton University Press, Princeton, 1991.
worldline image: 320px-World_line2.svg.pngen.wikipedia.org
George Polya, “How to Solve It. A New Aspect of Mathematical Method.” Princeton University Press, 1945, p.44
James Joyce, “Ulysses.” Vintage Books, New York, 1990, page 64
density: “More is unknown than is known. We know how much dark energy there is because it affects the Universe’s expansion…It turns out that roughly 70% of the Universe is dark energy. Dark matter makes up about 25%. The rest – everything on Earth, everything ever observed with all of our instruments, all normal matter – adds up to less than 5% of the Universe.”
NASA, Astrophysics press release: http://science.nasa.gov/astrophysics/focus-areas/what-is-dark-energy/
hypercube image: 190px‑Hypercube.svg.png en.wikipedia.org
see Carl Sagan, “Cosmos – The Edge of Forever –The 4th Dimension,” 1980.