Relationship between Algorithm and Geometry

Mathematics is the order, structure, logic, relationship and quantification of the language of numbers and symbols.

Geometry is a branch of mathematics - an architecture - relating to points, lines, angles, surfaces, and solids.

How does geometry operate on the properties of its architecture (points, lines, angles, surfaces, and solids)? Operations on geometric objects such as:

• Boolean
• Tessellation
• Merging
• Extending
• Trimming
• Triangulating
• Tweening
• Twinning
• Tracing
• Filling
• Shadowing
• Resizing
• Extruding
• Curve fitting
• Interpolating
• Extrapolating
• Splining

Geometry operates (applies functions) on objects through a rule set – an algorithm.

How does one determine the characteristics (properties of its architecture) of a member of a geometric set? Determination of member characteristics such as:

• Resultant numeric properties (size, location, etc.)
• Comparative characteristics (max, min, difference, ratio, arrangement, etc.)
• Set membership (shape, # of vertices, angle, symmetry, state, topology, any other)

Geometry computes object properties through a rule set – an algorithm.

RELATIONSHIP: Algorithm is the program which operates within the Geometry's architecture.

Algorithm and Geometry

Just until recently, the notion of space remained Cartesian. The determinant shift from high modernism to today’s notion of architecture is the replacement of the background with fixed ordinates and coordinates by a new notion of space and matter acting as a single entity. This turning point has modified architectural thoughts and designs and redefined architecture as a material fact.
The Cartesian paradigm has lost its influence on current architectural philosophy and the existence of its universal space is redefined as a material field of omnipresent difference. The universe, now a matter field, replaces the unchanging essences of a fixed field with interactive entities. Architecture incorporates speeds rather than movements and therefore becomes “as much matter and structure as it is atmosphere and effects.” (Atlas of Novel Tectonics, Resier + Umemoto, p.23)
Architecture involves assemblages of multiple models, surfaces, and materials, and therefore not continuous anymore but rather composed of several organizational models operating at different scales. Architecture must perform with a series of techniques in order to provide the right balance between material and force.
The notion of hierarchy in the modernist projects is currently used in an innovative way such that the order is not just constrained to the scale or to what lies above or below it. In fact, instead of processing in a top-down order from the general scheme to the specific detail, the new concept of hierarchy allows the particular to influence the universal and reciprocally. Indeed, inventive architectural organizations and effects emerge out of entities or wholes, which cannot be reduced to their parts, and become readable not as elements to a whole but as whole-whole relationships.
Typology is significant in the material practice and allows for a wide variety of architectural organization. When selecting a specific typology, a correlation between a rough typology and a practical or structural criteria is possible. Typology is therefore not just used as a mean of classification at the end of a process but also used in its rough state as a device during the design process. Typology is less a “codification” than it is the source for a method of controlled material expressions.
Geometry used to be thought by the modernist architect as an abstract regulator of the materials of construction and is now perceived as a notion that unites matter and material behaviors. In contrast to the prior concept of geometry as a regulator of the irrational or accidental state of matter, the latest theory “must be understood not as a supercession of measuring but as the interplay between intensive and extensive differences.” (Atlas of Novel Tectonics, Resier + Umemoto, p.74)


David Deutsch explains that quantum mechanics should not as a predictive tool but as an justification for how the world works. If we are to take quantum theory at its true value, we must come to the conclusion that our universe is “one of many in an ensemble of parallel universes” that physicists frequently call the “multiverse.” Deutsch believes that the photons in the two-slit experiment are prevented from falling on some parts of the film because they are being obstructed by invisible ''shadow'' photons from a parallel universe.
The “multiverse” version of quantum theory is one of the most developed theories od Deutsch along with theory of computation: the idea, developed by the mathematicians Alan Turing, Alonzo Church and others, that all material procedures can be simulated on a computer. Also essential is the theory of evolution and an “epistemology” (theory of knowledge) that takes science not as a human build but as an ever-improving diagram of the world.

Algorithms and geometry

Algorithms and geometry share an intimate relationship. Algorithms may operate on geometry (real or virtual) and algorithms may operate on other algorithms. They almost always come down to operating on geometry. Geometry on the other hand is created by algorithm. Or was that the other way around? It is the old chicken and egg dilemma; did whatever material contained in whatever blew up in the Big Bang (arguably the originator of all geometry) have an algorithm or a set of conditions as to when it was to explode? Or was there one miraculous appearance of original geometry and then all other geometry (i.e. everything) behaves according to an algorithm? And if this is the case what governs this algorithm who got to write it?

Geometry and algorithm cannot be separated it appears because without rules nothing would work. (Like the anvil that you try to drop on the roadrunner but it was just a second too late, without rules the anvil might have dropped faster than it should have and then it would have killed him and then we wouldn’t have any more coyote vs. roadrunner cartoons.)

Also without rules there would be no interesting opportunities or exciting leaps new level to the rules is discovered.

Researching Lost and Perceptible/Imperceptible Time

The quantization of matter is at the root of the perceptible dichotomy of the discrete and continuous. In physics quantization; the root name of quantum physics, “asserts that all measurable quantities are quantized (or discrete as opposed to continuous)” (Deutsch, 54) Quantization therefore allows for the most important aspect of quantum theory, that of interference, alluding to the Multiverse. In perception psychology this plays out in the difference between what we perceive the world to be and its physical characteristics. For instance the most obvious of these would be the fact that we have a perception that the world is flat despite the fact that we know our planet is round. Psychologically, this concept was first taken on by the Gestalt Psychologists Christian von Ehrenfels, Max Wertheimer, Wolfgang Kohler and Kurt Koffka. They looked to understand why, “elements come together to form wholes, objects that are qualitatively different from the sum of their parts, and can not be understood in terms of their parts.” (Levitin, 74) In this sense they endeavored to explain how discrete parts can become continuous.

But physical and mathematical research has resulted in theories which are averse to such unification. Since Max Planck kicked off the quantum age by postulating that atoms could “absorb or emit energy only in discrete amounts,” physicists have struggled to answer how one escapes continuous motion to merely explain how you get from particle A to particle B. Modern quantum theory describes a world where discrete values are assigned to observable quantities, but despite this, motion and change are continuous (Deutsch, 1). Upset by the narrow focus in which most particle physicists propose a ‘theory of everything’ and, “because the fabric of reality does not consist of only reductionist ingredients such as space, time and subatomic particles, but also for example, of life, thought, and computation,” David Deutsch has argued for a Theory of Everything with a wider scope (Deutsch 30). Epistemologically, considering Reality as a malleable fabric allows for the overlay of quantum theories, computational theories and cognitive science; all of which have roots based in the disjunction between discrete and continuous.

In the first book of his epic A la Recherche du Temps Perdu: A Cote de Chez Swann, Marcel Proust explains the discrepancy between the world in which we live, the world which exists and the role in which memory plays in a quantization of time and continuity of memory. He describes 2 different walks his family would take from their house in Combray: the Geurmantes way and the Meseglise way. Both distinctly different in the flora and fauna one would encounter as well as the time each walk would take and the repercussions which would ripple into the evening; determining whether or at which time he might receive his bedtime kiss from his mother. For Proust, these routes, “linked with many of the little incidents of the life which, of all the various lives we lead concurrently, is the most episodic, the most full of vicissitudes; the life of the mind.” (Proust 258) As specific ephemeral qualities would be realized within his perceptibility so too they would be realized as physical veracities. These interweave to create an intricate network of time and space which was no truer without the knowledge of its physical traits as it was with them; the network creating a conception which depended on the existence of all possibilities:

All these memories, superimposed upon one another, now formed a single mass, but had not so far coalesced that I could not discern between them – between my oldest, my instinctive memories, and those others, inspired more recently by a taste or “perfume”, and finally those which were actually the memories of another person from whom I had acquired them at second hand – if not real fissures, real geological faults, at least that veining, that variegation of coloring, which in certain rocks, in certain blocks of marble, points to differences of origin, age, and formation.

Because the majority of information we receive cognitively is incomplete, perception becomes a process of inference and the relegation of probabilities. When we receive insufficient information our cognitive processes provide it themselves, and are typically very good at it (Levitin, 99). This was developed as an evolutionary trait, but it now has become manifest within a deeper observation in epistemological lineage and our understanding of the universe. Cognitive science allows for a link between our measurable or quantized realities, and those which are continuous. Memory and perception begin to broach the link where singularities combine to form a whole greater than the addition of their individual quantities, but which are dualistically perceptible as quantifiably individual and intrinsically continuous within a complete whole.

Discrete or Continuous?

The human brain processes incoming stimuli though its system of neurons. Once activated, a neuron speeds an electric pulse along its axon. When the pulse reaches the axon’s endpoint, it triggers a chemical release to bridge the synapse gap; thereby activating the next neuron in line. Although this constitutes a processing system of discrete events, our consciousness perceives it as continuous.

Both the definition and implications of continuous include “uninterrupted”, “extent” and “undivided”, yet it is deemed to be sub-dividable; a sequence consisting of moments or events that are attached. In other words, the continuous is reducible to a series of the discrete, albeit functioning or perceived as analog.

The machine which interprets the discrete as continuous is the human brain with the qualia (phenomenal qualities) of its consciousness.

One could conclude that what we experience as continuous is either a fine series of the discrete or a continuity that is algorithmically processed by the brain in discrete steps. As such, we cannot discern the truly continuous. Having said that does not minimize its importance as we live in a continuous world. It only serves to emphasize that it is our analog experiential relationship with the surroundings that most significantly determines the true quality of human life; it should be provided by our architecture. It is true that a discrete moment, discrete image or singularity – the digital input – can provide a powerful sensory experience permanently impregnated and forever recalled; but the discrete is a monumental moment. It does not represent the continuum of quotidian. It is the discrete view of architecture from afar, not the continuous experience of architecture up close.

life and the discrete

Let's assume that the physical, material dimension has no discrete elements. This is a major assumption. If no thing that exists had no end unto itself. Molecule becomes atom becomes nucleus, then quarks, gluons and so on --- forever. Can this be assumed ?

Is there an absolute zero ? Kelvin thinks so. We haven't gotten there yet, and anyhow is it possible to mechanically, or electronically, transmit anything where absolutely nothing can happen ? I don't know. But there is no disagreeing with Kelvin on absolute zero. It's mathematically proven -- that must mean it's right, no?

Look, math is the ideal. It needs discrete values, or else it's just a mess. We take from math so readily, perhaps because in a, albeit troped, Darwinistic model of estimated self-interest, the idea of the division of resources -- of the group versus the individual -- or vice versa, is so fundamental to survival.

So the idea of the discrete preceeds our rational understanding of what discrete is. There is one of something. Now there are two of something. RNA, for instance, works on this principle.

The RNA does not know what discrete is -- it merely knows how to act upon its occurring. And math allows us to document as a form of a statement its occurance.

Through the wonderful manipulation of energy all around, life has been able to evolve without cognitively knowing the difference between one and two, or nothingness and existence. Yes, mate selection, the instinct to fight or flee, a horde of aphids eating not one, but two lettuces on a given summers day -- all of these things do not require the comprehension of the discrete -- they just act on its existence. And with math we have been able to discuss concretely about the matter.

Literature as Diagram

I just wanted to add a passage of Faulkner's to expand on Deleuze's description of Faulkner as "the great literary illuminator" and perhaps you'll notice the irony in that statement. This is from The Sound and the Fury, June Second 1910 pp. 214-217. The syntax is all taken directly from the book. What is important is the overlap of time; Faulkner breaks the book into 4 sections, each comprising one day, this particular one day is described over 100+ pages. The passage is mainly in reference to the eldest son of the Compson family; Quentin, as he commits suicide in a bathtub while at Harvard. While the exactitudes of this situation aren't understood until much later in the book, certain information is divulged which elucidates earlier references in the chapter (day) and earlier chapters. The key to the writing is Faulkner's movement of language, time, syntax, point of view and senses to describe a single situation. A single situation so complex that it can not even be understood directly after its occurrence, not until years later:

I turned out the light and went into my bedroom, out of the gasoline but I could still smell it. I stood at the window the curtains moved slow out of the darkness touching my face like some breathing asleep, breathing slow into the darkness again, leaving the touch. After they had gone upstairs Mother laid back in her chair, the camphor handkerchief to her mouth. Father hadn't moved he still sat beside her holding her hand the billowing hammering away like no place for it in silence When I was little there was a picture in one of our books, a dark place into which a single weak ray of light came slanting upon two faces lifted out of the shadow. You know what I do if I were King? she never was a queen or a fairy she was always a king or a giant or a general I'd break that place open and drag them out and I'd whip them good It was torn out, jagged out. I was glad. I'd have to turn back to it until the dungeon was Mother herself she and Father upwards into weak light holding hands and us lost somewhere below even them without even a ray of light. Then the honeysuckle got into it. As soon as I turned off the light and tried to go to sleep it would begin to come into the room in waves building and building up until I would have to pant to get any air at all out of it until I will have to get up and feel my way like when I was a little boy hands can see touching in the mind shaping unseeing door Door now nothing hands can see My nose could see gasoline, the vest on the table, the door. The corridor was still empty of all the feet in sad generations seeking water. yet the eyes unseeing clentched like teeth not disbelieving doubting even the absence of pain shin ankle knee the long invisible flowing of the stair-railing where a misstep in the darkness filled with sleeping Mother Father Caddy Jason Maury door I am not afraid only Mother Father Caddy Jason Maury getting so far ahead sleeping I will sleep fast when I Door door Door It was empty too, the pipes, the porcelain, the stained quiet walls, the throne of contemplation. I had forgotten the glass, but I could hands can see cooling fingers invisible swan-throat where less than Moses rod the glass touch tentative not to drumming lean cool throat drumming cooling the metal the glass full overfull cooling the glass the fingers flushing sleep leaving the taste of dampened sleep in the long silence of the throat I returned up the corridor, waking the lost feet in whispering battalions in the silence, into the gasoline, the watch telling its furious lie on the dark table. A quarter hour yet. And then I'll not be. The peacefullest words. Peaceful words. Non fui. Sum. Fui. Non sum. Somewhere I heard bells once. Mississippi or Massachusetts. I was. I am not. Massachusetts or Mississippi. Shreve has a bottle in his trunk. Aren't you even going to open it Mr and Mrs Jason Richmond Compson announce the Three times. Days. Aren't you going to even open it marriage of their daughter Candace that liquor teaches you to confuse the means with the end. I am. Drink. I was not. Let us sell Benjy's pasture so that Quentin may go to Harvard and I may knock my bones together and together. I will be dead in. Was it one year Caddy said. Shreve has a bottle in his trunk. Sir I will not need Shreeve's I have sold Benjy's pasture and I can be dead in Harvard Caddy said in the caverns and the grottoes of the sea tumbling peacefully to the wavering tides because Harvard is such a fine sound forty acres is no high price for a fine sound. A fine dead sound we will swap Benjy's pasture for a fine dead sound. It will last him a long time because he cannot hear it unless he can smell it as soon as he came in the door he began to cry I thought all the time it was just one of the those town squirts that Father was always teasing her about until. I didn't notice him any more than any other stranger drummer or what thought they were army shirts until all of a sudden I knew he wasn't thinking of me at all as a potential source of harm, but was thinking of her when he looked at me was looking at me through her like through a piece of coloured glass why must you meddle with me dont you know it wont do any good I thought you'd have left that for Mother and Jason.