# Choicelessness

## Higher and Higher

Posted in architecture, Uncategorized by johnsnavely on October 6, 2007

I am in bed with the flu today. I’m sick like a dog. It blows.

Yesterday my crit went pretty well. Mostly thanks to the input from my friends Pete, Stephen, and Bryan. One of the results is that I have a new precedent (other than Monticello) and she’s gorgeous:

Such nice hands!

And of course her star turn in this movie, which is what my project is secretly based on. I had a conversation recently with someone, I don’t remember who exactly, who insisted that Ghostbusters II was better than the original. Something so contrarian must be you, right T? The argument (which I think was very quickly abandoned) was that the Statue of Liberty scene was “really uplifting”. Hopefully, my good friend Mason (who’s coming to visit, yay!) will rattle off the countless lines of dialog from the original, all spoken entirely by Rick Moranis. Although in defense of GB II, “(Your Love Keeps Lifting Me) Higher and Higher” is a great song and deserves a cover. Maybe by someone like Cat Power.

Last but not least, I’ve put up a “new” blog for all the Rhinoscripting stuff I work on. It can be found here. And unlike the Invivia blog, it will actually be updated with real content. A workshop at MIT is planned for mid-January. Stay tuned!

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### 10 Responses

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1. son1 said, on October 7, 2007 at 5:06 pm

“Something so contrarian must be you, right T?”

Oh, hell no. Ghostbusters II?? Such a corruption of their original vision. I was disappointed with it, even as a child.

2. johnsnavely said, on October 7, 2007 at 5:17 pm

Damnit. I’m trying to remember who said it.

I suspect it was someone trying to make a joke, but, missing the sarcasm, I was all over them like Marion Barry on a gram.

-1 for me.

3. Mlle. LeRenard said, on October 9, 2007 at 5:48 pm

Notes (from our conversation) on Metaphor in architecture (rather than simile/surrealist juxtaposition a la Rem K)

Example of possible programmatic innovation: Rem’s Seattle public library, called hyperrealism. (As Vidler proposes these come from a data-driven computational generation of program.) But Rem emphasizes an intense disjunction that is more akin to surrealism than the term he prefers, hyperrealism. The projects juxtapose various elements of program in unexpected collage … one piece is next to the other … but one element is not actually the other.

John proposes rather than a relation of collage/ juxtaposition/simile, a relation of metaphor between unexpected programmatic elements. A metaphor is a transfer of terms from one discourse to another (Aristotle, Poetics). The metaphor simultaneously says is and is-not (Paul Ricouer, The Rule of Metaphor). Just as the duck-rabbit or crone-girl simultaneously is an old woman and is not, the out-line in poché simultaneously is a movie theater and is not (as it is a 99 cent store). Richard (The Kind of England) is a lion (i.e. has, experientially the many qualities of a lion)… and yet is precisely not literally a lion. A set of terms from one discourse (about lions) is applied to another set of terms (those about King Richard). The influence, while not equal, is mutual… lions become kingly, and kings become lionly. The vehicle affects the tenor and vice versa (I.A. Richards).

Metaphor is not a mere rhetorical device suitable for the realm of poetry and art, but a tool of science. Science posits models that assert metaphorical relation to the world; an atom both is and is not actually Niel’s Bohr’s model (Max Black). Once established precisely, the metaphor-model allows new creative thinking to be done in science. The new transformations and extrapolations based on the model have specific, not arbitrary relationship to the world (M Black); just as Juliet is the sun has a specific rather than arbitrary relation to Romeo’s experience of a young woman in his life.

Metaphorical transformation between two content elements (Juliet, the sun) occurs in a way much like isomorphic transformation in topology (Ricouer, and Henly). The imagination works by isotopy, to use a term borrowed from Ricoeur and Henly and ultimately from mathematical topology (c.f. Ricouer, Study 8). In topology, a mathematician performs transformation on one surface – say, a torus (a donut-shape) – and another shape – say, a coffee cup; though these shapes appear very different, in topological terms, they have similar underlying structures. The mathematician can work on one shape according to the same set of rules and transformational patterns as she can work on the other shape. The same thing happens in isotopic or symbolic reflection on ideas presented in metaphors.

Uhhh…I think that’s what we said 😉

4. son1 said, on October 21, 2007 at 2:49 pm

“The imagination works by isotopy, to use a term borrowed from Ricoeur and Henly and ultimately from mathematical topology (c.f. Ricouer, Study 8). In topology, a mathematician performs transformation on one surface – say, a torus (a donut-shape) – and another shape – say, a coffee cup; though these shapes appear very different, in topological terms, they have similar underlying structures.”

You probably want to use a term like homeomorphism, or maybe homotopy, instead of ‘isotopy’ — which is, properly speaking, a kind of homotopy.

The coffee-cup and the donut actually appear to be the same in topological terms, even though visually they may appear different. In topology, you’d treat them both as surfaces. Because their surfaces are homeomorphic to one another, they are indistinguishable in topological terms.

This is, in a sense, what basic topology is: the study of properties which are preserved under homeomorphism.

In the case of the donut and the coffee-cup, one of the ‘properties that is preserved’ is the genus of their surfaces — the number of ‘handles’.

5. johnsnavely said, on October 21, 2007 at 3:21 pm

So then what’s isotopy? My understanding (from the wikipedia article) is that isotopy is a set of homeomorphisms in which each discrete transformation gives rise to another homeomorphism. This means that all the surfaces (functions) created while transforming a donut into a cup are also homotopic. Is this right?

When would you have two things which are homotopic but not isotopic? Maybe the trefoil to torus? Or something similar?

6. johnsnavely said, on October 21, 2007 at 3:32 pm

Nevermind, found my answer here.

“passing through itself”, ahem. Which is why I guess isotopy is so important in knot theory..

So, taking what I just learned and applying it to that awesome sphere vid you sent me:

A sided sphere is homotopic to an inside-out sphere but not isotopic. Is that right?

7. son1 said, on October 21, 2007 at 3:51 pm

Well, I think it’s a little complicated, although the Wikipedia page on Homotopy gives a good example of two things that are homotopic but not isotopic:

However, the map on the interval [−1,1] in R defined by f(x) = −x is not isotopic to the identity [map].

And then it explains why:

any homotopy from f to the identity [map] would have to exchange the endpoints, which would mean that they would have to ‘pass through’ each other. Moreover, f has changed the orientation of the interval, hence it cannot be isotopic to the identity [map].

Homotopy is one of these ‘higher-order’ math concepts, that’s hard to get your mind around when you first think about it. The ‘maps’ that they’re talking about here are what you and I would call ‘functions.’ Let’s call $f(x) = -x$ the ‘inverse’ function; it takes in numbers in the range [-1,1] and outputs numbers in [-1,1]. Similarly, we could talk about ‘the identity function’, a function g which doesn’t change its argument, $g(x) = x$.

Computer scientists, and some mathematicians, often use a kind of descriptive notation for functions. If $f$ ‘takes in’ a value in the set [-1,1], and ‘outputs’ a value in the set [-1,1], then we say that [-1,1] is both the domain and the range of the function, and we write $f:[-1,1] \rightarrow [-1,1]$. Another way of saying this is that $f$ is a member of the set named $[-1,1] \rightarrow [-1,1]$. Obviously, $g: [-1,1] \rightarrow [-1,1]$, too.

A homotopy is a map of maps — it ‘takes in a map’, and for each such map that it takes in, it outputs another map. Furthermore, a homotopy has a ‘dial’ attached to it — think of it as a knob, which you can turn from 0 to 1. So you give a homotopy a map, and you set the dial, and ‘out comes’ another function. (We require that, if you set the dial to 0, then the homotopy always outputs the same function you put in — it becomes ‘the identity function on maps’).

Mathematically, we could write a homotopy $h$ like this: $h : ([-1,1] \rightarrow [-1,1]) \times [0,1] \rightarrow ([-1,1] \rightarrow [-1,1])$. Basically, $h$ takes a function and a number from [0,1] in, and outputs another function out. A function from functions+numbers to functions.

Topology gives us a notion of ‘smoothness’ for these kinds of higher-order functions — so we require our homotopy to be ‘smooth’ in this sense (which I won’t go into here).

If we have a homotopy $h$, such that $h(f,0) = f$ (as we required above) and $h(f,1) = g$, then we say that “h is a homotopy from f to g.”

Okay, that’s the math. Now, the applications.

You might ask, “how is this concept of homotopy applicable to things like coffee-cups and donuts? They aren’t functions, they’re surfaces, and homotopies are defined between functions!”

But actually, coffee-cups and donuts are functions, if you think of them correctly. They are “embeddings”, of a locally-2D surface (the torus) into a 3D space (our world, say). An embedding is a function; it maps every point on the torus to a corresponding point on the coffee-cup or donut.

So now we could think of two new functions, $C : Torus \rightarrow CoffeeCup$ and $D : Torus \rightarrow Donut$. Fair enough? And when we say that “the coffee-cup and the donut are equivalent in a topological sense,” what we’re really saying is that there exists a homotopy $E : (Torus \rightarrow CoffeeCup) \times [0,1] \rightarrow (Torus \rightarrow Donut)$, such that $E(C,0) = C$ and $E(C,1) = D$.

(Actually, the relationship is a little bit stronger — in this case, the coffee-cup and the donut are isotopic, I think, because as you “turn” the dial on $E$, you get successive embeddings of the torus that are themselves ‘smooth’ in a particular sense. But there are homotopies that aren’t isotopies. For example, if your homotopy “flipped” the coffee-cup inside-out as it transformed it. The point I was trying to make is, “isotopy” is a stronger concept than “homotopy,” and homotopy is usually what people think of, when they think of topological transformations.)

8. son1 said, on October 21, 2007 at 3:52 pm

When we meet up today, let me edit that comment, and I can fix the parsing errors.

9. son1 said, on October 21, 2007 at 3:53 pm

“A sided sphere is homotopic to an inside-out sphere but not isotopic. Is that right?”

That would be one example, yes.

10. johnsnavely said, on October 21, 2007 at 4:28 pm

Well, it’s all beginning to make a bit more sense. I’ve got the coffee cup “function” thing.

Like we were talking about before, I work in two and three dimensions more or less exclusively. Being creative in 3d is pretty hard (for me anyway; I need more brain exercise in that area), which is why architects often map-down 1 less dimension to plans and sections.

Maybe I should try working in 4 dimensions and then try to map down afterwards.

Anyway, isotopy seems to be much more useful to “real world” applications. I need to show you that tool that I made for Goulthorpe’s studio (runs a little slow in a browser) again. Maybe you could help me with a java/processing rewrite? I had trouble with the opengl canvas in processing when I tried to port over from flex 2.