abstraction and transformation: #designprocess

Abstraction is essential for making architecture. Architecture involves many scales, utilities, materials, situations. An idea needs to be carried from iteration to iteration,, from model to drawing, from spatial organization to detail, from one material to another. This idea is the abstract order that permeates and holds the whole.

Abstraction is the “formation of an idea, as of the qualities or properties of a thing, by mental separation from particular instances or material objects.”


affine transformations: #geometry #transformation

We are going to use transformations… (Affine transformations) A function which maps geometry from one context to another, or one orientation to another,… a simple mapping from one context to the next. (Move, rotate a certain number of degrees, etc.)

affine transformations have the property of preserving parallel line relationships, but not length or angles. Translation (move), rotation, scale, and shear are affine transformations.

In geometry, an affine transformation, affine map[1] or an affinity (from the Latin, affinis, “connected with”) is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence. Every linear transformation is affine, but not every affine transformation is linear.


algorithm: #geometry #transformation #codification a series of steps, a procedure. A set of rules that precisely define a sequence of operations.

Algorithmic Process: Composed of three fundamental parts: Inputs, Processing and Output

Input (ex. Values, points) into Processing (ex. Addition, drawing a line) into Output (ex. Results, line and line length)

to square :size
repeat 4 [forward :size right 90]

to flower :counter
if :counter > 100 [stop]
rotate Z :counter * 0.1
square | sin (:counter) |
flower :counter + 1
flower 0

An algorithm is a very specific set of instructions for carrying out a procedure that generally includes an instruction to stop. It may be long or very short. The word is almost certainly a Latin corruption of Al-Khwarizm, the name of a Persian mathematician who wrote a treatise on calculation in 825 AD. The idea of an algorithm or programme as a formal procedure predates the existence of electronic computers, but to use a computer always means to activate an algorithmic procedure that transforms input to output. In architecture, this means formalizing the design process as a set of procedures and specific instructions. These may be opaque to the designer through the interface of a CAD program, in which a library of generic geometrical objects, relationships, attributes as potential inputs, and their visual representation as outputs is already encoded. They are more visible to the designer who programmes or scripts to control the software inputs and customize the outputs.

Recipe for Spiraling
1. Pick an angle (d).
2. Imagine a circle. Plot one point on this circle at d degrees from the origin.*
3. Plot another point at d degrees from the last point on a concentric circle that is slightly bigger than the circle before it.
4. Repeat step 3.

Recipe for Packing
1. Create a shape* of a random size.
2. Pick a random point.
3. a) If the shape is inside another shape, or overlaps another shape, throw it away and go back to step 1.
b) If not, place it. Go to step 1.

Recipe for Weaving
1. Start drawing a sin curve: a line that goes around a circle at a steady rate, spread out over time.*
2. Loop the curve by adding a term–a mathematical function, like cos()–that speeds up and slows down the line as it goes around the circle.
3. Add more terms to create more loops, overlaps, and squiggles.
4. Mirror the curve for a denser, interlocking figure.

Recipe for Blending


allographic vs. autographic arts:

In his extensive discussion of the question of notation, Goodman distinguishes broadly between two type of art forms. He calls autographic those arts, like painting and sculpture, which depend for their authenticity upon direct contact of the author. In music, dance and theater, the concept of authenticity is described differently. These arts, where the work exists in many copies, and can be produced without the direct intervention of the author, he calls allographic. Allographic arts are those capable of being reproduced at a distance from the author be means of notation. In Goodman’s account, despite different circumstances of performance, changes in interpretation or instrumentation, every performance of a particular musical composition counts as an authentic instance of that work. The guarantee of that authenticity is not the presence of the original author, but the internal structure of the work as set down in the score. The use of notations is the defining characteristic of the allographic arts. Aloographic arts do not imitate or reproduce something already existing, they produce new realities, imagined by means of abstract systems of notation.


ambiguity: #shape grammars

Ambiguity is intrinsic in what there is to see and has many uses. You’re encouraged to play around freely with whatever you see in an open-ended way. Order, predictable results, and steady plans aren’t the goals.


analysis: the breaking down into parts of what already exists


aperture: #space


architecture#space #thebody architecture is a cultural production

Architecture organizes and brings together a series of independent conditions; a series of thresholds.

Architecture is an art of form and material. It is also a social product that participates in economic, political, cultural contexts. There is a language of architecture lodged in the space of the drawing that utilizes geometry and the same language of architecture can have agency in the world. Materials, technologies, and systems of architecture are tools through which we are able to design buildings and environments. This studio will introduce design approaches to architecture students by way of rule based systems that will gradually lead to questions of material production and architecture as mode of cultural production.


boundary: #space


calculating: #shape grammars counting what it contains.


cellular automata: #codification    Points with labels or weights located in a grid, usually square, that has one or more dimensions. Neighboring relations are specified in rules that are applied under translations. The rules work in parallel to change every point at the same time, with the suum of the individual outcomes as the overall result. The best-known rules of this kind are the ones for John Conway’s game of life. In words, it goes like this:

Survival:    If an occupied cell has two or three neighbors, it survives.

Death:        If an occupied cell has four or more neighbors, it dies from overcrowding.

If an occupied cell has four or more neighbors, it dies from isolation.

Birth:         If an unoccupied cell has exactly three neighbors, it becomes occupied.

Stasis:       If an unoccupied cell has less than three neighbors or four or more, it stays unoccupied.


closed shapes: #geometry  creates a closed boundary


codification: the act, process or result of arranging in a systematic form or code. Towards formalization of syntactical elements through simple rule based procedures.


compound transformation: #geometry #transformation     accumulated basic transformations.

define translation order.


constituents: #shape grammars a part of something, a component made to calculate with


constraints: #parameters


concept: defined in the second semester as the meeting of program of the body, site with formal languages. Can be defined as the synthesis of disparate ideas.


coordinates #geometry:

A Cartesian coordinate system is given by three mutually perpendicular oriented axes called the x,y, and z-axis.

The thee axes are labeled x,y.z and pass through a common point 0 called the origin. On each coordinate axes we use the same unit length.

local vs. global vs. polar coordinates: #geometry

We have been using global (world absolute) coordinate system. For geometric design, it is often desirable to employ local (user defined, auxiliary, relative) coordinate systems to simplify modeling tasks.

In addition to planar Cartesian coordinates, there is another useful way of defining planar coordinates. Polar coordinates of a point p measure the radial distance r of the point p to the origin o, and the angle 0< _ 360 to the polar axis. The polar axis is usually chosen as the positive ray of the x-axis. Whereas planar Cartesian coordinates measure the distance to two orthogonal axes, polar coordinates measure the radial distance r to the origin and the angle between the ray op and the polar axis.


descriptions: #shape grammars  representations like lists, strings, graphs, networks, trees, schemas, sets, structures


designing: does not conform to preconceived standards of hierarchy and order but relies on fundamental elements and rules, which can be made explicit. Designing can also be seen as an evaluation of known and unknown relationships that yield opportunities for discoveries. You are designing when you are looking at the work after it was produced; hence discovery.

ontology of design: (the necessary elements that need to exist when we design something and the ways they relate to each other. The three main parts are

1. elements- the abstract geometric things you use when you design

2. rules- the procedures that change the geometry of their designs

3. material- pen, paper, sticks, chipboard, etc. (at 1:1 scale and in representational ways)

A ‘design approach’ in this context is defined as the operations that act on geometry executed in a material. You will use these methods differently over the two semesters. in 101 you create a genealogy or a body of work, which you evaluate in order to curate a selection. Discoveries and inventions lie in recognizing the formal value of a design and then deciding to continue to work on it. In 102 you use similar discoveries and contextualize them in terms of site and program and challenge preconceived relationships between these architectural constructs.


diagram: Diagrams belong to space and organization. A diagram can be a notation but a notation can not be a diagram.

-“A diagram is a graphic assemblage that specifies relationships between activity and form, organizing the structure and distribution of functions. As such, diagrams are architecture’s best means to engage the complexity of the real.”

-“The diagram is understood as a visual tool designated to convey as much information in 5 minutes as would require a whole day to imprint on the memory.”

-“A diagram is not a thing in itself but a description of potential relationships among elements, a map of possible worlds.”

-“Diagrams are not regulating devices but simply instructions for action or possible formal configurations.”

-“Architecture’s most powerful tool to think about organization. “Its variables include both formal and programmatic configurations: space and event, force and resistance, density, distribution, and direction. Diagrams are highly schematic and graphically reductive, but they are not simply pictorial. Diagrams are syntactic and not semantic, more concerned with structure than with meaning. In an immediate apprehension of the relations between the parts, while the process of reading a notational schema is more extended, unfolding in time, like reading a text or a musical score.””

-“Diagrams function through matter/matter relationships, not matter/context relationships. They turn away from the questions of meaning and interpretation and reassert function as a legitimate problem, without the dogmas of functionalism. The shift from translation to transposition does not so much function to shut down meaning,, as to collapse the process of interpretation. Internal relationships are transposed, moved from part-to-part from graphic to the material, by means of operations that are always partial, arbitrary, and incomplete. The impersonal character of these transpositions shifts attention away from the ambiguous personal poetics of translation, and its associations with the weighty instructions of literature, language, and hermeneutics.”


dilation#geometry #transformation define center of dilation, scale decrease factor


dimensions: #geometry  the number of coordinates needed to specify a point on the object. (point/symbols dimension I=0, lines, planes, and solids dimension I>0). For example, a rectangle is two-dimensional, while a cube is three-dimensional.


divisions: #math #grid


drawing:drawing is an explorative tool. It can ask a particular question and reveal an idea. It is an instrument of thought.

abstraction through projection. Drawing clarifies; it enables us to see what we cannot yet see. It serves to register what is gradually known. Orthographic projections (plans, sections, and elevations) are powerful analytic tools. They slice through the (conceptual) object, making visible the measured attributes of the design.

Orthographic projections can also be understood as cuts through light where the station point or viewpoint is infinitely far from the object. The picture plane cuts through and registers the rays of light that reflect off the object and are perpendicular to the picture plane. Because the viewpoint is at infinity, the projecting rays are parallel. Because the projecting rays are parallel, measured information is kept intact. In other words, certain measurement of the corresponding parts of the actual object. in this sense, orthographic projections present an objective point of view as oppose to a subjective point of view. The measured relations from the direction of the viewpoint are kept intact. The principles of this method of drawing address the abstract order of what is being depicted. With these drawings, once can examine the geometric properties and mentally inhabit the order of a thing. From these drawings, one can read accurate, quantifiable, and specific dimensions, location, and relations needed to build.

Orthographic projections simultaneously offer a noumenal picture and a view which is infinitely far from the object. These drawings are pictures of what is inside our mind, and  at the same time, infinitely away from our body. perhaps this is why they serve abstraction so well and the sensation that one is traveling through from outside to inside to being the object.

architectural drawing: rules for future actions that has not yet happened… potentials for events.


embedding: #shape grammars identity among constituents. Its relation. The ability to seize fresh aspects in concrete things. There’s always a chance for something new. Rules are defined when shapes are combined in pairs… and rules apply via embedding. Seeing alternative ways. To show how the rules work, I first have to give the embedding relation, and then tell what options where are to satisfy it. The constituents in the rule may be transformed as a whole arrangement-they can be moved around, reflected, or scaled-to determine the right correspondence with constituents in the shape. The shape has a finite set of constituents.

The trick to a good rule is to find a suitable embedding relation, and then show how M can be embedded in S, and how together with P, this changes S. This is a dynamic process with plenty of ambiguity.

Embedding and transformations together make rules work. This is a neat relationship that lets me see what I do and change it. Visual reasoning—using your eyes to decide what to do next. Counting is habit worth having.

The rule applies to a shape when all of its constituents (points) are also in the shape. Moreover, the constituents in the rule may be transformed as a whole arrangement—they can be moved around, reflected, or scaled—to determine the right correspondence with constituents in the shape. The shape has a finite set of constituents. Its 24 line segments each one defined separately by its endpoints—first are the 4 sides of the large square and their halves, then the two diagonals of the square and their halves and finally the horizontal and the vertical and their halves.

You can extract 3 types of triangles by defining different amounts of lines to extract… three lines, four lines, six lines. 2 lines, 4 halves, etc. And my rules find all of these figures wherever they are in the shape. There are five squares, four rectangles, six bow-ties, and two crosses of each type.

There may be too many options since this rule is too open… you can then define constituents by their endpoints and start to define rules in there. I can record horizontals (H), vertical (V), and diagonals (D) in three different lists.


expansion: #geometry define center of expansion, scale increase factor


folding:#geometry #transformation   fold line is a material event… potential embedded in the material itself. The lines indicating the axis of creases or folding. Valley or mountain fold… guides indicate the action to be performed.

ex. folds, pleats, creases

paper folding workflow: through arrays of grids of points generated through attractors and points on surfaces

how do you generate your deign motif and then how do you determine if it is a mountain or valley fold?

symmetry and chains, transformations and continuity, different types of symmetry on a 2d plane, translation, rotation, glide reflection, reflection, boundary is important to the tiling



force is just any influence causing change in speed, change in direction, or change in shape.


Function: #geometry #transformation #codification A function is a relation that uniquely associated member of one set to members of another set. You can used a predefined one or create one yourself.

ex. predefined spiral function




generative: #geometry #transformation #codification




gradient: #blending #interpolating an increase or decrease in the magnitude of a property (e.g., temperature, pressure, or concentration) observed in passing from one point or moment to another. Transition between two states.


grammar: #shape grammar


graphs: #geometry #codification #categorizing #sorting #exploring 


grid #geometry: A grid usually refers to two or more infinite sets of evenly-spaced parallel lines at particular angles to each other in a plane, or the intersections of such lines. There are many types of grids: orthogonal, hexagonal, triangular, diagonal grid, circular. refer to Islamic patterns.

concepts of grid:

grid as generated from a pattern set of points

grid as regular tessellation of a component (tiled) (look at Islamic Patterns)

grid as generated from a set of points moving in two axis

grid as a rotation of a pattern tile and extending of lines infinitely from it



how a grid is structured as a 2d system

the key is working with points

Points are represented by an ordered set of numbers called coordinates, most likely Cartesian in nature

Coordinate System (XY Plane)

Point in World Space (X,Y,Z)

Origin point…. base point

local vs. world coordinate system

how do you generate a grid?

Creating a 2d array of planar points: base point, row/column directions, row spacing, column spacing, row count, column count, sequence, interval, division


Describing a Grid

-divisions of a curve

-modulation of the grids

-utilities performed on grid

-ways of paneling in 2d and 3d


read Krauss reading on “Grids”




horizontal movement: #space #thebody



a thing made by combining two different elements; a mixture.

solving between two states

8737399_orig TYPOLOGIES_90551d4a407454fd_a.ud_Koerner_2x8_610x392_560bb058d8a3ca2b854c82b223dd5bb7b aliceIMG_5180Density_Openess_Reviseted_005_rocker lange



index: #geometry #codification #categorizing #sorting #exploring 



indexes “are not abstract signs but physical artifacts.”  Moment of physical contact.  animal tracks, handwriting, medical symptoms, “fingerprints, fossils, the tracks of gulls in the sand, the trajectories of electrons in a cloud chamber or the images on the shroud of Turin can be classified as indexical by virtue of the spatial connection of the object in question to the chemical surface of the photographs negative.”  “A sign which refers to the object that it denotes by virtue of being really affected by that object.

“Index is doubly marked… by the definiteness of physical contact and by the uncertainty of interpretation.”  “Index initiates a narrative process.  Narrative of detective story.  Reconstruction of cause from effects.”  Medical symptoms typically do not reveal their cause… (indirectness= secondness… Charles Sanders Pierce).

“In architecture, the index does not point to a moment of physical contact between the designer and the fabric of the building, but it points instead to a set of virtual movements (cuts, displacement, grid shifts, shears, inversions, rotations, or folds) registered through the abstract codes of representation. ” “…a building is to function as an index of the drawing, it will be through the further mediation of complex social and technical operations involving large numbers of people, capital investment, building codes and techniques of construction, regulatory agencies, the properties of materials, and the limits of available technologies. Even more radical operations of process (invoking chance, complex geometries or computer simulations), are forces at a certain moment to arrest movement and capture a particular instant of the process.”

cross-reference with traces


inside/outside: #space


Intervals: #math #grid  a pause, a break in activity


line #geometry: dimension one yet can be embedded in higher dimensional spaces

constituents such as edge line, polyline, start point, endpoint, midpoint, vector, sub -lines, intersections, etc.

concepts of line:

line as the relation of one point to another,
line as the movement of one point through space,
line as the transposition or mapping of one point to another location in space,
line as the transformation of an ideal set of points into a real one – or bi si versa.

how do you draw a line?

ex. define point and movement along a vector and distance, define points from set a and points from set b and connect, etc.


line annotation #symbols #annotation: pens weights and lines the hierarchy we give to a line to represent its meaning, defined by you!


lists#geometry #codification #categorizing #sorting #exploring  In order to work with multiple things you need to work with lists.

Lists are a collections of things sorted by an index

Each thing is stored in a particular location in the List denoted by its Index. In an order that is managed by the Index File…. a collection of points is understood in a numbered index next to the item… the entirety of the index and items is called a List. The list is taken from an output which shows the points that resulted in order to solve that geometry.


lindenmayer systems#geometry #codification #lindenmayersystem #aggregation

A Lindenmayer System is a formal grammar that was initially conceived as a theory of plant growth. L-Systems can describe complex forms of plants with relatively few simple rules.

L-System consists of two parts: a generative and an interpretive process. The main concept of the generative process is string rewriting, in which the letters that comprise an initial string are replaced in parallel by other letters according to pre-defined rules. The replaced letters form a new generation of string which is then subjected to the same replacement rules. This string rewriting process is usually repeated for several generations.

In the second part of the L-System, the letters of one or multiple generations of string are interpreted. This project explores the interpretation of string as geometric forms. Several methods of visualization are considered, among them mapping a string and turtle graphic interpretations.

Generative Process


-Iterations: 6

-Initial String: a

-Replacement Rules:

1) a -> aba

2) b->ac

Replacement Process:

0) a

1) aba

2) abaacaba

3) abaacabaabacabaacaba

Interpretative Process


-String: abaacabaabacabaacaba

-Interpretation rules:

a= go forward

b= turn right

c=turn left

Visual Interpretation:


definition by Michael Hansmeyer


LSystem is a parallel string rewriting system. A string rewriting system consists of an initial string, called the seed, and a set of rules for specifying how the symbols in a string are rewritten as (replaced by) strings. Let’s have a look at a simple LSystem:

seed: A


A = AB

B = BA

The LSystem starts with the seed ‘A’ and iteratively rewrites that string using the production rules. On each iteration a new word is derived.

n is the derivation length = the number of iterations

n=0: A

n=1: AB

n=2: ABBA




n=0:         A           start (axiom/initiator)
            / \
n=1:       A   B         the initial single A spawned into AB by rule (A → AB), rule (B → A) couldn't be applied
          /|    \
n=2:     A B     A       former string AB with all rules applied, A spawned into AB again, former B turned into A
        /| |     |\
n=3:   A B A     A B     note all A's producing a copy of themselves in the first place, then a B, which turns ...
      /| | |\    |\ \
n=4: A B A A B   A B A   ... into an A one generation later, starting to spawn/repeat/recurse then

77px-Cantor4.svg      729px-Cantor_set_in_seven_iterations.svg










local distortions: #geometry #transformation


looping: #geometry #transformation

A shape produced by a curve that bends around and crosses itself

A length of thread, rope, or similar material, doubled or crossing itself, typically used as a fastening or handle



open relationships, new relations between sets of information


define martix: #geometry #grid  where pattern and material are married

rectangular array of numbers. A matrix dimension is m-by-n where m: number of rows    n: number of columns

Transformation matrices are responsible for moving, rotating, projecting, and scaling objects. Matrices are also used for transformations between coordinate systems, for example from the 3-D world coordinate to the 2-D screen coordinate system. We can define transformation as a function that takes a point (or vector) and maps that point into another point (or vector).


morphing: the smooth transformation of one entity into another, typically achieved through the sampling of a base and a target space.


networks: #geometry #codification #categorizing #sorting #exploring 


notation: Notations belong to time. Notations differ from diagrams.

-ANTICIPATION: “Notations always describe a work that is yet to be realized.”

-INVISIBLE: “Notations go beyond the visual to engage the invisible aspects of architecture.”

-TIME: “Notations include time as a variable.”
-COLLECTIVE: “Notations presume a social context, and shared conventions of interpretation.” “Notations operate according to given codes and shared conventions of interpretation.”

-DIGITAL DIAGRAMS: “Notations work digitally.”

-The many hands of construction is what the notational language is for.

-reading is extended, unfolding in time, like reading a musical score… as appose to a diagram which can be read instantaneously.

note: “all notations are diagrammatic, but not all diagrams are notational. Those aspects of space and organization contained in a notational score, such as the position of a dancer’s body in space, would be diagrammatic while the overall score could still be considered notational. Time and change over time might be implied in a particular diagram, but the precise description of time-based phenomena is a property of notational systems. This is consistent with the fact that notational systems operate according to shared conventions of interpretation, while diagrams are by definition open to multiple interpretations. Notations are, strictly speaking, digital, while the diagram retains some analog properties. For Goodman, this is the determining characteristic of notation: that each score designates a unique work, allowing little little latitude for change.


open shapes: #geometry  creates an opening within a boundary


operations: #geometry #transformation #codification an active process; a discharge of a function


parameters: #geometry #transformation #codification A Parameter, in the most general sense, is a factor that helps to define the overall limits and performance of a system.

Parametric Model (a model wherein the parts of a design relate and change in a coordinated way as defined by the various Parameters and Dependencies stated by you.)

What are the HOLDS (not transform) vs. what is allowed to TRANSFORM

ex. Height width depth are the parameters

position: Its location in space (x/y/z coordinates)

orientation: The directions it is facing (3 unit vectors representing the up-right, right vector, and front-vector)

Distance: How far to move each time (LE, scalar for front-unit vector)

Thickness: How thick of a tube to draw (RA, radius for the tube)

Rotation: The incremental degree with which to turn around each axis

Working “Parametrically”

-need to understand How Data Flows

-need to Divide a Model into Manageable Parts

-require to Think Abstractly… instead of concretely or tactility

-using mathematical knowledge and Think Mathematically… numerical values, expressions, functions, etc.

-Think Algorithmically…. very fluid….


performative models:  models with behaviors and properties, dynamic, and can be described beyond their dimension but their relationships


plane #geometry

A plane is a two-dimensional doubly ruled surface spanned by two linearly independent vectors. The generalization of the plane to higher dimensions is called a hyperplane. The angle between twointersecting planes is known as the dihedral angle.


point #geometry: dimension zero

examples: origin point, endpoints, midpoint, division points

-points are one of the most basic geometrical elements

-points typically serve as the underlay for generating more complex geometric types

-points can also be easily generated from, thus dependent upon, other more complex geometric types.
-points reside within a specific coordinate system (ie. “Space”).

-typically this is the “World” or “XYZ” Cartesian Coordinate System, but there are other “Spaces”…

Points are represented by an ordered set of numbers called coordinates, most likely Cartesian in nature

Coordinate System (XY Plane)

Point in World Space (X,Y,Z)

Origin point…. base point

local vs. world coordinate system


point annotation #symbols #annotation


procedures: #geometry #transformation #codification a series of actions conducted in a certain order or manner.


process: Ideas reappear as familiar, in new forms, or transform as one goes from model to drawing to sketch to construction or in solving a structural problem or writing program. One needs to be on high alert for these conceptual reappearances. recognition is the key to these confirmations of the creative process. And as Charles Olson said, “One perception must immediately and directly lead to a further perception.”

The creative process is a ground for discovery. Recognizing the capacity for a discovery to be useful in someway brings about invention.


problem making: #process


problem solving: #process


processing: #geometry #transformation #codification set of single or sequential operations that are defined that will occur… simple or layered/complex rule.


projection: #geometry #transformation  define projection plane to be projected and plane to be projected onto

the projection point of a given 3D point P(x,y,z) on the world xy-plane
equals Pxy(x,y,0). Similarly, a projection to xz-plane of point P is Pxz(x,0,z). When
projecting to yz-plane, Pxz = (0,y,z). Those are called orthogonal projections

A projection is the transformation of points and lines in one plane onto another plane by connecting corresponding points on the two planes with parallel lines. This can be visualized as shining a (point) light source (located at infinity) through a translucent sheet of paper and making an image of whatever is drawn on it on a second sheet of paper. The branch of geometry dealing with the properties and invariants of geometric figures under projection is called projective geometry.


psuedo code: #geometry #transformation #codification (design intent) a plain language description, not bound by syntax, of what you want the script to do and how it should do it. Writing Psuedocode is the first step to writing actual code.

Line script example

In: length

Out: A box

  1. get the length from the user
  2. Create an array for each Point
  3. Use these arrays to create the Lines

In: 4 point

Out: A box with a cross through the diagonals

  1. get the 4 points from the user
  2. create the arrays of points for each polyline
  3. use these arrays to create the polyline border
  4. draw the diagonals


reasoning: the ability to deal with novelty


recursion: #geometry #transformation  recursive rules: repeating until a quality emerges…. you may start out with a function such as scaling and rotating and discover a spiral.

One of the  most important concepts in complexity is recursion, a method of defining functions in which the function being defined is applied within its own definition. Within a procedure, therefore, one of the steps is to run the whole procedure again; another way of saying the function becomes the input of the next iteration. The Fibonacci number sequence is well known mathematical example of recusion, where n = (n-1) + (n-2), or the current term is the sum of the two previous terms in the sequence, each of which is the sum of two before it.

The movie INCEPTION also uses a recursive function in the plot device diagram of the dream within a dream sequence within a dream within a dream sequence. The architect is responsible for creating the dreamscapes. As an architect you responsible for knowing the systems that you work with and to create the best solutions (e.g. Never Ending Staircase) to new problems that need to be solved (Inception). Being able to see the world around you as objects with state and behavior is a useful skill for both understanding Object Oriented Design/Programming.

Examples of the Recursive Function from Pamphlet Architecture 27: Tooling

recursion image

By recalling its source shape recursively, cracking generates a geometry of self-similarity.

recursion image 003


In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:

0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\;

By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.
LINDENMAYER SYSTEMS #lindenmayersystem

See linedenmayer systems


Examples of Subdividing Space functions:
Constructing divisions in the successively smaller units of the subdivided starting shape… creating finesse within the system.
repetition and modularity, the characteristics of tiling that provides such inspiration is the issue of adjacency.


Rules for tiling:
1. take a set of points
2. construct a bisector between one point an all the others.(the bisector is halfway between two points at an angle perpendicular to a line that would connect the two points.)
3. The voronoi cell is bounded by the intersection of these bisectors.
4. repeat for each point in the set.

At its simplest, a Voronoi diagram is the division of a space into contiguous neighboring cells. The cells relate to a set of points (Voronoi sites) in that space. Each point has an associated cell consisting of all the points closer to that site than to any other.The Voronoi diagram, named after mathematician Georgy Voronoi (1868 – 1908), is also known as a Dirichlet tessellation

Each of these experiments involves the creation of a Vornoi tiling from a point set. They result in cellular patterns where each cell contains all of the space that is closer to its points than to any other point. They form a collection of shapes that can look like squares, honeycombs, crystals, or boulders–the nesting of other orders within a patterned, minimal enclosure.

subdivision examplevoronoi diagram



  1. Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).
  2. Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole – because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpinski’s triangle.)
  3. Repeat step 2 with each of the smaller triangles (image 3 and so on).

Note that this infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpinski triangle. Michael Barnsley used an image of a fish to illustrate this in his paper “V-variable fractals and superfractals.”[4]



reflection: #geometry define mirror line or plane


rotation (pitch, roll, yaw)#geometry #transformation define center of rotation and rotation angle

Take a point on x,y plane P(x,y) and rotate it by angle(b).


rules: #geometry #transformation #codification

Rules are used in an open-ended process to decide when and how to change shapes with all of their ambiguity.

Using a pre-defined algorithm (ex. Spiral by Expression) vs. creating your own

One of a set of explicit or understood regulations or principles governing conduct within a particular activity or sphere.


scale shift: #space


schemas: #geometry #codification #categorizing #sorting #exploring 


semantics: the meaning of a word, phrase, sentence, or text. You will discover the semantics after you make a drawing or model.


Sequence#math #grid  A sequence is an ordered set of mathematical objects.


sequences: #geometry #codification #categorizing #sorting #exploring 


series: #geometry #codification #categorizing #sorting #exploring   sequence of numbers within a list.

Create a Sequence of numbers spaced according to a space value to a Step value


sets: #geometry #codification #categorizing #sorting #exploring 


shear: #geometry #transformation  define invariant line or shear factor



(it is related to time)

Simulation simply means to inform a virtual system, which, during the processing of that information, takes on an actual structure that is a registering of [its inputs]. In formation

Lars Spuybroek “The Structure of Vagueness”


stretch (1 way): #geometry #transformation  define invariant line and scale factor


stretch (2 way): #geometry #transformation  define invariant lines and scale factors
P’ = ScaleFactor(S) * P


strings: #geometry #codification #categorizing #sorting #exploring 


structures: #geometry #codification #categorizing #sorting #exploring 


syntax: #shape grammars how something can be broken down into parts and defined how they relate to each other. A way of investigating spatial complexes in an attempt to identify its particular structure that resides at the level of the entire configuration. To define shapes in terms of their lowest level constituents or alternatively atoms, buts, cells, components, features, primitives, simples, symbols, units, etc.








tectonics: syntax, or tectonics, is the lawful relation between words and things. Syntax/tectonics is the articulate meeting. The arrangement and accumulation of these meetings are the governing principles of the whole. The relation of joint to joint, space to space, inside to outside.


threshold: #space


tiling: #codification #recursion #grids regular geometric relationships between component parts in a coherent progression. Component parts repeat.

Recursion as tiling process.





“Traces as indexical signs”

“Traces as absence and erasure”

“Trace as regulating geometry”

“Trace as the ephemeral marks that its abstract instruments of design and projection leave on the built work.”

crosses-reference with indexes



transforming: #geometry #transformation

a qualitative change

We can define transformation as a function that takes a point (or vector) and maps that point into another point (or vector).

(mathematics) a function that changes the position or direction of the axes of a coordinate system

a rule describing the conversion of one syntactic structure into another related syntactic structure

(genetics) modification of a cell or bacterium by the uptake and incorporation of exogenous DNA

(transform) subject to a mathematical transformation

In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term “transformation” refers to a function from X to itself which preserves this structure.

the act of transforming or the state of being transformed; a marked change in appearance or character, especially one for the better; the replacement of the variables in an algebraic expression by their values in terms of another set of variables; a mapping of one space onto another or onto …

Note that when transformations are specified with respect to a coordinate system, it is important to specify whether the rotation takes place on the coordinate system, with space and objects embedded in it being viewed as fixed (a so-called alias transformation), or on the space itself relative to a fixed coordinate system (a so-called alibi transformation).


translation/moving#geometry #transformation define displacement vector.

Moving a point from a starting position by certain vector is calculated as follows:
P’ = P + V


trees: #geometry #codification #categorizing #sorting #exploring   not a list since it is not just a single number but a path


vectors #geometry:

Vectors are abstract data types that describe Direction and Magnitude.

Geometrical vectors are defined by three distance values (delta-x, delta-y, delta-z)

Anchored at a base and moving at a direction…. the force by which we will manipulate our particle. A numerical value {dx,dy,dz} will be assigned to it to say the start point and the end point… if pointing down this would be gravity. The force by which we will manipulate the particle.

define direction and magnitude


vertical movement: #space #thebody


void: the presence of absence. example


volume #geometry


void: #geometry the absence of presence. example




diagram vs. notation


inductive vs. deductive readings


reflective vs. generative writings… before and after


evaluation towards invention