The mathematical models that Eisenman is rejecting include the linear, static and deterministic models that dominate traditional architecture theory. He states that static conditions in traditional arch theory of figure/ground are not able to explain true complexity in the urban condition. He believes that a new reading, from another frame of reference, of figure/ground is needed to expose immanent and unseen conditions that allow for new urban structures.

Time of experience has changed, and a new time, conditioned by the discontinuity and autonomy of the sound bite, prevails. Eisenman states that architecture can't be bound by static conditions any longer, and the current situation requires that architecture deals with the Event. He suggests that the fold, which breaks the Cartesian order of space and allows for a temporal multiplicity of conditions, has the potential to reinterpret urbanism and allow for new events to unfold from the immanent conditions of existing urbanisim.

Eisenman is rejecting the mathematical models of Cartesian space and the "two static conditions of the object: figure and ground." He rejects them because they do not "explain the totality of urbanism."

He is replacing them with Thom's Catastrophe Theory because, in looking at the form of the fold, Cartesian order of space is broken up with a "possible new relationship between vertical and horizontal" in which form is continuous.

1: What are the mathematical model that he is rejecting?
I guess when he argues against figure/ground architectural thinking he could be talking about Cartesian coordinate system--however, Eisenman only mentions Cartesian coordinate system as only a description of some Leibniz' thinking as 'turning its back to Cartesian rationalism' because he suggested 'the smallest element in the labyrinth of the continuous is not the point but the fold.' Then goes on to talk about the fold as articulating a different kind of relationship between vertical and horizontal, or between figure and ground--something as revolutionary as breaking up Cartesian orders of space.

What is it about that continuous transition between ground and figure or vertical and horizontal that is so impossible in Cartesian space?

2: What is he replacing?
I suppose the answer you're looking for is topology since its been what we've been talking about. But in my limited understanding of topology, I get the impression that topology more about the relationship or connectivity of things rather than the elasticity of space as shown in the folded or bent Cartesian coordinate system that Eisenman obviously has interest in.

3: and why?
Searching in the space between figure and ground (what he calls fold) I think he hopes to find a bridge between Architecture and building--(The ideas associated with building and the manifestation of building). As an idea, its not a bad way to approach the problem but I still don't quite follow the how or the execution of these ideas within any processes yields the non-traditional, or non extruded architecture he envisions.

An extruded plan on a folded coordinate system is still topologically an extruded plan.

Eisenman is rejecting the model of Cartesian space and figure/ground contextualism. He is replacing it with the model of catastrophe theory. Catastrophe theory, characterized by sudden shifts in behavior arising from small changes in circumstances, can be used to give another reading or "frame" to the notion of figure/ground. Eisenman states how the idea of figure/ground is static, and dismisses the architectural "event". This concept of the fold allows for celebration of architectural events, can be used to re-frame context, and can allow new conditions to form that may have been repressed or ignored.

Eisenman rejects the model of Cartesian space due to the manner in which it discreetly divides space in a regimented uniform way, since that might in turn also imply that the matter represented therein should also maintain the same uniform consistency within its own fabric and construction. Space is framed within the Cartesian plane, and thereby susceptible to false contextualization. He rejects this model because it can not account for continual variation. He calls this concept the 'fold', as coined by Deleuze, in order to depict the sense of variation and inconsistency of density that can occur in space, whether that be material space, such as terrain, temporal space, such as the human experience of physical space, or some combination of the two, such as the urban fabric.

Eisenman rejects Cartesian rationalism and its notion of the point being the smallest element, and the Modernist interpretation of figure/ground, the linear ground slab as "the neutral datum, projecting its autonomy into the future."
"In the idea of the fold, form is seen not only as continuous but also articulating a possible new relationship between figure and ground, thereby breaking up the existing Cartesian order of space."
"continual variation of matter... characterized through the agency of the fold." "first defined culturally in the baroque"

Eisenman replaces traditional architectural theory with the notion of the fold as a "third condition", somewhere in between plan and section, figure and ground.

"the idea of the fold gives the traditional idea of an edge a dimension," opposed to an abrupt line the fold offers a volumetric dimension which Eisenman argues can be used as a re-framing of sorts, and applies it to social organizations and the urban environment.

Peter Eisenman is rejecting the traditional Cartesian rational model. He wants to replace this model with catastrophe theory that is more appropriate for the present day. Eisenman believes that architecture historically has been focusing on only space and place, but ignoring the idea of events. Architecture today needs to see the hidden possibilities in objects. Studying them gives new interpretations of existing organizations.

Eisenman rejects the rigid cartesian math model and figure/ground contextualisim. He states that there is a need for the possibility of reading figure/ground from another frame of reference in order to revel conditions that may have been repressed in the urban fabric. Eisenman supports the position of Leibniz as a non-cartesianist and adheres to Thom's and Deleuze's ideas of the fold. The concept of the fold allows the possibility of refocusing or reframing what is pre-existing in any site which denotes the fold as the aforementioned location of repressed conditions. Eisenman believes this is important because architecture can no longer be bound by conditions of space and place. Figure/ground contextualism ignored the idea of event which can be addressed through the catastrophe theory. The fold helps achieve this by revealing possible relationships between figure and ground, thus breaking up existing cartesian order.

Eisenmann suggests that architecture must not only deal with static conditions of space and place or figure/ground contextualism and that architecture must rather deal with the problem of event and time. He rejects the idea of the Cartesian coordinate system such as in the straight extrusion from a plan and replaces it with the idea of the fold as presented in René Thoms catastrophe theory. The idea of the fold affects both plan and section thus breaking up the Cartesian order of space and, as Eisenmann argues, since the fold is both figure and ground it can be used as a way of projecting new social organizations into an existing urban environment.

In folding in time, Eisenman rejects the mathematical model of the Cartesian rationalism. He explains that we have entered a new era of mediated environments and experiences in which past architectural expressions such as brick and mortar, structure and foundation, are no longer relevant. He replaces these ideas with Deleuze's and Thome's theory of "the fold" in which new relationships between vertical and horizontal, or figure and ground create a continuous space breaking up the Cartesian order of space and experience as it has been previously defined.

Eisenman, like Leibniz, is "turning his back on Cartesian rationalism," and instead focusing on objectile events as described by Gilles Deleuze and furthered by René Thom's catastrophe theory. In order to address historically ambiguous site or programmatic conditions in more complex and exhaustive ways, Eisenman argues that the catastrophic fold can represent a "non-dialectical third condition." This categorical differentiation suggests any architectural resolution is a distinct event, "already in place in the structure" of the site, context or condition, and thereby creates an architectural solution "of the past and the future," not merely a zeitgeist or static object.

1: What are the mathematical model that he is rejecting?
Geometry, Cartesian order of model, figure/ground contextualism

2: What is he replacing?
Topology, the event based interactive way of capturing space such as the catastrophe theory

3: and why?
Because architecture can no longer be bound by the static conditions of space and place. Traditional architectural theory which focuses on figure/ground contextualism has largely ignored the idea of the event. Now it is time to face with the problem of the event which the catastrophe theory can deal with.

Peter Eisenman states, " Today, a rock concert might be considered an architectural event..." The rock concert becomes an environment with lights, sounds, and movements. He tries to inform us that concerts are more than just music. Other factors like the lights and people create the complete experience. The same idea is applied to Figure and ground. These are two separate entities with one sitting on another. The fold is a volumetric edge which defines the space by framing the figure, rather than just a straight horizon line.

1) A syllogism is a logical argument. In two statements, call 'premises', terms are related in categorical terms. In the classic example;

All animals are mortal.
and; All humans are animals.
then; All humans are mortal.

"animal" is a categorie of "mortal". Also, "human" is a categorie of "animal". Thus, the conclusion of the syllogism is that "human" is a categorie of "mortal".

2) On page 3, Barr talks about a line being like a string, where it can be manipulated into many shapes without individual points that make up the line changing in order. Going further, the string could even concieved as an elastic band, where the distance between the points can possibly change.

Every arrangement of that string or elastic band has its own "geometry", but its "topology" is unchanging, provided the string isn't cut or attached back onto itself.

3) In many stages of design development, architecture is looked at in purely topological terms. For example, a program diagram may show the different building spaces using vague (bubble) shapes and lines that represent physical connections. This allows the designer to begin to plan the spaces without introducing specific geometry.

A syllogism is a logical argument and is thereby exactly what Aristotle is using to describe and organize his system of categories. Just as a syllogism is an equation of truths, the categories are simply equations for understanding and quantifying composition or structure.


The terms topology and geometry explored by Barr on pages 2 and 3 describe the permanence associated with topology versus the contingencies associated with topology. Furthermore, the strict parameters of geometry are identified as characteristically opposed to the understanding of topology.


Social and emotional affects and effects of architecture are not always direct results of physical geometry. Although light, air, orientation, acoustics and other environmental influences are often quantified using computational software, the attitudes and persuasions of a designer are difficult to describe or predict in pure geometry.

Syllogisms seem to work by relating things first in a more general way and then in a more specific way. The relationship of the general and specific supports the argument by either agreeing or denying the relationship.

Probably everything Barr says can probably be construed to somehow relate to geometry but before going to far into it, I found the discussion starting on page 5 about how something can still be alike even when its been taken apart and reassembled so long as the order is still the same. Not sure why it bothers me, perhaps its because I think of it in terms of geometric operations of stretching and for for cutting and reattaching there isn't such a neat geometric operation.

Again, probably everything can eventually be construed to relate geometry to architecture (at least the parts of it we like) but for the sake of discussion I think one area where it doesn't quite relate is when architects design an effect or environmental change without necessarily designing how it can happen.
If for example: if an architect may only be concerned with the temperature of the room insomuch that it is habitable by people, they wouldn't care about the geometric arrangement of the room, or mechanical systems (geometrically designed to perform) necessary to produce a specific temperature, just that the effect is achieved.

1. Syllogism is a kind of system which can be used to prove logics. Things have to be categorized when you use syllogism.

2. On pages 2~3, Barr mentioned about main geometric characteristics of the things which don’t change by distortion or stretching.

3. Geometry is not used in the concept, program diagram, adjacency diagram, style, and color.

1. How does syllogism relate to the categories?
Syllogism depends on Aristotle's categories by using systematic logic to determine relationships of things.
2. Where in Barr's text do you find something different than geometry?
Barr begins to describe topology as "a state of mind-and its own goal" and "space as we usually understand it is left far behind" once topology is understood. p2.
3. Where do we not use geometry in architecture?
Perhaps in virtual space when generating algorithms, but even so, the experience of these things are happening in a space dependent on geometry.

1) The syllogism is a manner in which common attributes of categories and sub-categories are parsed to form logical relationships in human language through the use of the transitive property for the purpose of discerning a greater fundamental understanding of an objects function, use or nature.

2) Just as Aristotle used the syllogism to create broader and broader categories within a membership by reducing its description to as few common characteristics and discarding unlike elements, topology seeks to find a more fundamental insight to the spatial relations between elements by reducing their description and relinquishing both measurable quantity and orientation that are so crucial to the representation of Euclidean geometry within Cartesian space. For example, on page 4 of the reading on topology, Barr states that a "line does not have to remain straight" in topology. In a strict geometric interpretation, where a line is defined by three points within a straight relationship, once this alignment is broken, the element would now have to be called a curve or suffer division into two independent 'straight' lines. In the topological sense, the definition of a line is reduced to "the quality of being continuously connected along itself."

3) The aesthetic appreciation of the visceral experience of architecture is not done by algorithm. Nor the budget.

1: How does the syllogism relate to the category?
Syllogism is a rule and system. Categories are produced by syllogism which is all the rules.

2: Topology where in Barr's text do you find something different than geometry?
(Page.2) ...the study of continuity: beginning with the continuity of space, or shapes, it generalize, and then by analogy leads into other kinds of continuity-and space as we usually understand it is left far behind.

3: where do we NOT use geometry in architecture?
Time, perception, space, change, continuity, distortion, stretching

1. Syllogism seems to be the logical argument which one is chosen over the other. Aristotle uses syllogism, when characterizing "man" in an appropriate way. We categorize man as a specific ethnicity or the action he has done, instead of just being a type of species.

2. Barr states on page 2 that typology started as a kind of geometry...one might say it is a state of mind - and is its own goal. If typology is perceived as a state of mind would it be considered "real"?

3. Architecture is not just the design of geometric forms, but also the study of programmatic layouts. In that case we do not use geometry in architecture when we research and formulate a strategy for program.

1. In order for syllogisms to affirm or deny, the subject must be absolute...they depend entirely upon categories to make distinct the relationship between the two terms.

2. One area of where Barr highlights a difference is on pgs 2-3. Here he points out that a topologist is interested in the "most permanent" properties of a thing. The properties that geometry is interested in are relative...contingent rather than permanent qualities. Ex: A straight line is must lies evenly with the points on itself.

3. The concept phase, or the entire area of architecture that happens before form-making can be carried out without the use of geometry.

1. How does the Syllogism relate to categories?
In last week’s class, we learned that one of the functions of categorization is to isolate a way of thinking that is systematic. Syllogisms are very systematic in the way that they use deductive reasoning, a tactful approach using logical consequences.
2. Where in Barr's text do you find something different than geometry?
Geometry is described as the study of figures in a space of a given number of dimensions and of a given type. On page 2, Barr says that in topology, “space as we usually know it is left far behind.” Wouldn’t the removal of space disqualify something as geometry?
3. Where do we not use geometry in architecture?
I’m going to have to agree with Andri and say that we are always using geometry in the design side of architecture.

1. Syllogism relate to categorism
Categories are based on distinct identities. I believe that catergories are aided by syllogism which reveals (intentionally and unintentionally) an array of finite characteristics and function.

2. Barr
What about the entire underlying notion that is based on viewing geometric shapes topologically according to their geometric definition. For example, on page 12 there is a pentagon and a topological pentagon. The only thing that makes these similar is the fact that they each have 5 verticies and 5 edges. Whether or not the lines are straight is irrelevant. That is not to be considered by the viewer. It's a differnt state of mind in a way.

3. No geometry in architecture
Other than the obvious negotiating and writing of contracts one could argue that the assembly of material schedules does not use geometry. Yes the physical materials relate to geometry but simply specifying a particular brand and assigning materials to their respective locations does not require geometry.

1. How does the Syllogism relate to categories?

Aristotle defines syllogism as "a discourse in which, certain things having been supposed, something different from the things' supposed results of necessity because these things are so." Meaning if all animals are mortal and all humans are animals then all humans are mortal.

2. Where in Barr's text do you find something different than geometry?

Since topology is so difficult to define, it can have a different meaning than just being a kind of geometry, Barr states that "one might almost say it is a state of mind" and that it has it's own goal.

3. Where do we not use geometry in architecture?

In design we are always using geometry in one way or another. The only place I can think of where we are not using geometry is when we are arguing with clients about issues such as time schedules and budget, although those arguments are evidently about geometrical shapes of some sort.

1. How does the Syllogism relate to categories?

My interpretation/understanding of Aristotle's definition of syllogism relates to the ideas of categories by its correlation to the idea of definable characteristics that in turn make it understood as specific functions. Similar to how Allen Turing created computation as a way to rationalize a specific set of commands to perform a range of functions, Aristotle's syllogism uses categorizes as a way to identify certain characteristics which can then form a more complex range of meanings.

2. Where in Barr's text do you find something different than geometry?

Throughout Barr's text it seems that the subject of geometry is a constant however on pg. 4 in which he explains the transformation of topology creating different functional uses of a given object, it becomes apparent that topological manipulations have the ability to alter what we recognize as a certain object(in this case a taurus to a coffee mug) while maintaining its fundamental organization.

3. Where do we not use geometry in architecture?

Perhaps when we focus on the use of the algorithm to perform a specific sequence of tasks, we are relying on the intelligence and looped system output of the equation to generate form rather than relying on particular geometric systems for their building potential.