Delueze: Smooth and Striated

From the section of "Smooth & Striated" in which he discuss mathematics in the context of multiplicities:

"...it might be thought that the Number would belong exclusively to the other multiplicities, that it would accord them the scientific status non-metric multiplicities lack ... the number is the correlate of the metric: magnatudes can striated space only by reference to the numbers, and conversely, numbers are used to express increasingly complex relations between magnatudes, thus giving rise to ideal spaces reinforcing the situation and making it coextensive with all of matter. There is therefor a correlation within metric multiplicities between geometry and arithmetic, geometry and algebra, which is constitutive of major science..."

Although he goes on to provide exceptions, he explains, in relatively clear terms, how numbers can define magnitudes which define space.

In the first model (the technological model), Deleuze uses fabric as an example.
"...it is constituted by two kinds of parallel elemnets...there are vertical and horizontal elements, and the two intertwine, intersect perpendicularly...one is fixed the other mobile, passing above and beneath the fixed."

Based on the assimilation of fabrics, this math model allows us to determine which is smooth or striated. Felt, however, as the "anti fabric" "...has neither top nor bottom nor center, it does not assign fixed and mobile elements but rather distributes a continuous variation." There is no defined assimilation or intertwining like other various fabrics. Felt is like a continuous flow of meshed fibers, different, but continuous. "It implies no separation of threads..." It's a continuous variation revealing the principles of topology.

In the technological model, Deleuze discusses fabric, felt and quilts, using quilts or patchwork as an example of a mathematical model: "An amorphous collection of juxtaposed pieces that can be joined together in an infinite number of ways." (P. 476)

Deleuze argues that the patchwork is literally a Riemannian space, which is essentially a space made up of small domains. Explaining further, thanks to wikipedia: a Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length of curves, surface area, and volume. From those some other global quantities can be derived by integrating local contributions.

Deleuze then uses the the smooth space of patchwork to demonstrate that the smooth does not mean that it is homogenous but rather "an amorphous, non-formal space prefiguring op(tical) art"

The Smooth and the Striated

“Magnitudes can striate space only by reference to numbers, and conversely, numbers are used to express increasingly complex relations between magnitudes, thus giving rise to ideal spaces reinforcing the striation and making it coextensive with all of matter. There is therefore a correlation within metric multiplicities between geometry and arithmetic.” (Page 484)

Numbers can be used to striate smooth space. The multiplicities of smooth space are difficult to measure or calculate without using numbers or units. However, dividing a thing into a smooth space limits the understanding of the characteristics of the smooth space.

Reading Schedule

Hello everyone, for the week following spring break this is our reading list
Please make sure that you are caught up on the reading.
I am giving you a short assignment in order to wrap things up in the first section on mathematical models and the tradition of Euclid, Descartes, the rupture with digital design and calculus based and topology based techniques.
Two pages, singles spaced: discuss the use of Cartesian and non-Cartesian models of mathematics in any of the contemporary architects (Lynn, UN Studio, FOA, etc.) and identtify how the transformation of models signals a new ontology for architecture.

Cecil Balmond, Informal (Munich; New York, 2002) 347-365
Ben van Berkel, Basically (for now) Three Topics, in C. Davidson ed., Anyhow (MIT Press: New Cambridge Mass., 1998) 88-93.
Alejandro Zaera Polo, FOA, Phylogenesis, passages.
Gilles Delueze, Fold: Leibniz and the baroque (Minneapolis, 1993)
Greg Lynn, “Blob Tectonics, Or why Tectonics is Square and Topology is Groovy, in Folds, Bodies, and Blobs (Originally published in Any Magazine, no. 14.)

DeLeuze uses the concept of fabric and the weave as a way to explain the organizational properties of the smooth and striated. Deleuze states that "A fabric presents in principle a certain number of characteristics that permit us to define it as a striated space" He then identifies these different qualities as being: the vertical and horizontal, fixed and mobile, the delimited, and having a top or bottom.
These princples give us a mathematical model in which we can asses weather the particular fabric is either smooth or striated by nature of its organization. Further, Deleuze states that felt uses a different set of organizational principles, and is in a sense an anti-fabric. "In which there is no top or bottom, and is infinite in scale and complexity." This continuous entanglement of felts microfibers creates a smooth space that is not bound by princlples of sepratation and intertwining(as in other fabrics), but instead is based on constant variation of connectivity and topology.

Deleuze uses a Mathematical model in his description of the sea. Through his distinction of striated and smooth space Deleuze identifies the ocean as a "smooth space par excellence." He notes the behavior of striated space is that "lines or trajectories tend to be [subordinate] to points," whereas the opposite is true in smooth space. This identification however disproves the properties in which smooth space can be determined since in his 'Maritime Model' Deleuze points out that "the striation of the sea was a result of navigation on the open water."

Using the ocean as an example of the geometric and topological mathematical models seen in smooth and striated space, Delueze physically realizes the intermingling relationship between the two systems.

mathematical model

Deleuze uses the mathematical model of three dimensional space to describe the manner in which the musical concepts of harmony and melody relate to each other. He states, "Returning to the opposition [between the smooth and the striated], the striated is that which intertwines fixed and variable elements, produces an order and succession of distinct forms, and organizes horizontal melodic lines and vertical harmonic planes." By equating harmony to a greater level of geometric complexity, he is drawing the distinction between the two elements such that he concludes harmony is the determination of melody plus some other elusive multiplicity. He is somehow trying to suggest that there is a greater level of conceptual depth within the construction of harmony than there is in melody. I chose to highlight this use of a mathematical model because I believe his analogy fails.

Harmony and melody are bound and ruled by the same lines and differ only in the mechanics in which they portray the same underlying complexity. Harmony portrays the (traidic) chord structure of a piece through the use of unison, melody through the use of sequence and arpeggiation. In melody interstitial tones are inserted not only to create stepwise motion along the scale, which may lead one to incorrectly conclude that melody and harmony are fundamentally different, but also as way to create delays or rhythmic way stations that allow the composer to hold off from using a tone which would establish the next chord of the harmonic progression before the chord infrastructure dictates. On average throughout a piece, these interstitial tones will appear more often on the off beat of the rhythmic meter, where as the three tones that make up a chord (or base triad of a seventh chord in jazz) will more often occupy the primary positions of the rhythmic structure. These primary tones make up the very same notes of the backing harmonies played by whatever music ensemble backs up the lead melody.

I would more view the concepts of tonal and rhythmic orders within a piece of music to be the multiplicities that act in different, often contradictory, ways which are the parameters whose intensities a composer will modulate to create the fundamental nature of their creation.

The Smooth And The Striated

"Fractals are aggregates whose number of dimensions is fractional rather than whole, or else whole but with continuous variation in direction." (Page 486)

The line with two end points and a center point starts out smooth and straight. By continuously adding triangles to the old and each new mid point, striated surfaces form. Eventually the striations become so dense, the surface eventually becomes smooth again.