Email this link to a friend HOME : ARTICLES
Printer Friendly Version Articles: a079
Date Published: 3/2/2000
Arthur and Marilouise Kroker, Editors

The Small

Steven Whittaker

First time I saw the self-sumppumping ampersand surface that hyperbolic topologists call a Klein Bottle I was queasy in the pineal. Gland of mind.

I got over it.

A surface fisting itself. I represent Warfarin Unlimited Home Augments. Our Klein Condo project wants your order. The final soft solution wants another second-to-last liquescence. Design your unit now. Secure your suite parameters before it's too late. Construction has already broken ground, broken with (3D) ground. Let me invite you to the site. Smack in the heart of the city.

Yes there's precedent. Kleins turn up under every rock lately. The topology of the Klein Bottle has been enlisted to describe a host of incommensurable domains. The Klein has served as a figure for virtually everything, except figurality itself (leave that to me). The Klein Bottle may look like a visual parable of autoreferentiality, but it is also a very plastic sign. Like the ideally elastic plastic or hyperbolic rubber of which any topological surface is made, the Klein Bottle is all give. Before I show you a home, here are 13 ways of looking at a Klein Bottle.

A physics (of electron spin)1. A logic prop (for the class of classes which are not members of themselves)2. A metaphysic (of insideless Nature)3. Another metaphysic (of "(w)holeness", an auto-outing interiority, or intimate objectivity)4. An aesthetic/an epistemology/a critical sociality (regarding complex surface as exacerbated immanence. "Otherness is a twist in becoming.")5. A pathology6. A phenomenology (of browsing)7. A profile of political intransigence8. A scifi technology (container for antimatter, or a time travel portal)9. An "embedology" (models for embedding programs to instantiate "nonlinear temporal series" to accomodate "the twistiness of being"). And a signifier of time as horizoning dimension10a&b. A theology (this one in dire need of apologies to Occam)11. A cosmology (ourobourus in rewind, disgorging itself)12. A barometer, a hat, and a purse13.

Home is already a Klein. At Warfarin Unlimited we're just making it literally so.

And architects have been playing with hyperbolic topology since before Gehry got a sense of algorithm. There's the Tsimtsum Expo done in that topologic wedgie known as Boy's Surface. The architects had Cristo cover it with ten acres of untenable tea cozy after he'd finished draping an impossible cerecloth over the gutted and refurbished Reichstag. Like the Klein the Expo's a topologically necessary, physically impossible surface.

Impossible, yet we've got Kleins in every first world city, near enough, and even in parts of the developing world. Klein Condos, K Condos, Kondos, commanding horizons wherever horizons are found. Dissolving horizons. And no this isn't cookie cutter architecture. Every Klein is an original.

Every Klein Condo is an original without copy. We make it seem so, anyway. At Warfarin Unlimited we franchise the original without copy. Customerize.

The Caritas Condo of Calgary is not the Koan Do of Kyoto is not the Etruscan Venus of London is not the Small Wall of L.A. But they are all Kleins.

(Koan Do: the silence of one surface containing itself.)

Berlin built the decentralized centre of the Bundesbank in mobius form. We've been going one further. Beside a Klein, the Bundesmobius seems as much edge as surface. We know, we built Checkpoint Condo, a discretely disorienting boomer barracks just by the bank in the fully immersed form of a self-sumped, selfsubsuming Klein.

We at Warfarin Unlimited build Kleins. How?

By sleight of edge we join a mobius to its siamese, itself to itself, edge to one edge.

Mondo Condo. Order now or offer void.

We mate mobius with mobius. Most of us know the mobius. Escher's famous infinity treadmill for worker ants. One mobius seems to have 2 sides, 2 edges, looked at locally. But take in the thing totally, and you find one surface, one edge. Here I am local on the surface of my mobius, and look, it seems to have an other side. But pan out. Now there is no other side. What seemed the other surface is just this surface, one half-twist along itself. Dual in situ, a mobius is, taken tota simul, one surface bound by one edge, one.

Our Klein's the same but different. Like I said it's a mobius glued along its single edge to the single edge of another mobius. Or if you're a minimalist in hyperbolic surfaces you make the edge of one mobius meet itself. Same thing. Make those two edges, or your one edge, meet, and you've got a surface that passes through itself without a single tear. The single edge that contained the single surface disappears itself in disappearing its equal other. The Klein has no edge. All surface, no edge. Makes Jack a dull boy.

No no. The Klein is a continual fascination. Fascinans without tremendum, depth-field 'on the outs' with depth.

Another way to the Klein is you pull a cylinder's end around through its own distending midriff to meet its nether from within, as it were. The ends cancel to yield the continuity of the one surface.

So what, you say, a sphere's also a surface with no edge, and what pedigree the sphere. Aquinas' and Pascal's God, for instance, sphere Who's centre is everywhere and Who's surface is nowhere.

Yes. But thinking in terms of surface we can't ignore that an infinite sphere bounds an infinite cavity that can't issue outside. Of course, everywhere inside that infinity is God Central so it's a cozy cavity. The surface known as the Klein, on the other hand, is all issuance. Nowhere centre, everywhere surface. No cavity, no void contained.

If infinite sphericity is God's provenance, the Klein surface is something else.

God's colostomy bag.

Devoid even of void. But this is a good thing. It makes your Klein the appropriate place for forgetting you've lost your sense of loss.

Another thing about a Klein. This surface can be immersed, but not embedded, in 3-space. This is of the essence. The key to the Klein. Embed means represent without compromise. Immerse means compromises. But we've thrown all that out the window. We promise to embed every Klein before the gong sounds the end of The Long Now or before your mutuals peter, whichever comes first.

In the meantime our Kleins will seem the limit of seeming itself, embedded. Until we embed the seeming, you'll at least seem embedded.

The world your vestibule. To enter your home, when your home is a Klein, you must be governed by seeming. The very grid of the city will do a coriolis towards the Klein to accomodate your immersion in it. Watch from your panonymous terrace the streetlevel unorientable coriolis of all coordinates for the sake of the Klein. The elsewhere in which your Klein unit seems embedded must do a fatal convolve on your condo, or we'll buy it back and house you, well, elsewhere.

The surface of your future Klein, your privy unit, will be one ecstatic dysentery of all elsewhere, or I'm no agent.

Wherever we have built a Klein, the structure dominates the horizon. Imposing in its repose. Nothing puts the Klein in relief. The Klein puts the Klein in relief.

But what of a southern exposure? you ask. Southern exposure to what? North-South disparity? In truth the Klein faces, on every side, its own wake of unorientability. It dissolutes its surround. It makes dissolute into a transitive verb. Your Klein will command your horizon. Your horizon.

And a westering facade? Well, this is the thing about a Klein. No Klein is orientable. Every Klein is all wester, no west. All figure, no ground (except its tacit 4th dimension). A Klein is insideless, topless, backless, and your ideal habitus all the same.

Every Klein alludes to infinity, but has a sort of finality. The zero perimeter and lack of cavity seem to incarnate the infinity of self-containment.

Hegel's good infinity, on a bad day.

It's only one reflexive thought from self-containing to self-causing. Welcome to the Causa Sui Condo. Finality's coccoon. But more of this later. I regress.

Felix Klein, whose name means Happy Small, found or made the first Klein bottle surface about a century ago. He by the way married Hegel's granddaughter Anne.

I like Klein Condo, I like K-Condo, and I even like Kondo. These can be generic and cover every self-intersecting hearth we at Warfarin Unlimited have ever surfaced. But I really like The Small. In the future I'd like to see Small in the name of all of our structures, no matter how grand.

We've franchised Smalls all over the developed and the developing world, and we leave it up to local architects to materialize the basic idea. Some do better than others. Incarnation is a talent, when the always already disembodied extension of your hyperbolic surface is all you've got to work with.

We work with two basic representations of the Klein14. There's the lazy-8 Klein you get by coupling a mobius to itself or to another mobius. Think herniated power-upped infinity sign, overdetermined frisbee, dimensional torque-convertor. The Caritas Klein of Calgary is one of these, ranching the horizon the way it does. The Caritas, like all lazy-8s, is a compelling construction with no visible means of entry. I like the lazy-8, the infinity of repose it seems to lord over its landscape. But given my druthers I like the other figuring of the Klein surface, the amped ampersand, tubular self-eversion you get by pulling one rim of a cylindrical surface around to meet its opposite, but performing a 180 degree twist on the way, making rim rimjob rim. Without that crucial twist you'd get a torus, donut of the topos world. That twist is of the essence. Who wants to live in a donut?

To accomplish that topologic titty twist we must ghost the edge of our surface through itself, so that we can, in bringing our edges to meet, disappear both. This also means that unlike the donut surface, we have no inside or outside proper. Just one continuous conduit of invaginating self-eversion.

Done well, there's nothing elegant the way an ampersand type Small is. Ampersand. &. Now I think of it, this sort of Small's more like an amped ampersand amputee. It doesn't have the unmeet extremities. All of any Small's extremities are meet.

I like Sydney's billowing white Lucid Dreamtime Condo, a.k.a. The Boomerangst. Or Toronto has one in virtual plaid, antiplaid really when you factor in its unorientability.

My least favourite is our rival franchiser Frank Rued-Left's Akron Anachron Condo, the only Small I've seen that collapses under its own design pretensions. It finds itself between a 3-space Euclidean rock and a hyperbolic hard place. Wrought entirely of a two-way mirror, meant to allow residents to see out and those outside to see only themselves, the structure fails miserably to attain unilateral transparence, as anyone who knows the Klein surface could have foreseen. The mirror that would gate the window doesn't work with the Small. As the Small surface becomes its own other side, the two-way mirror makes a sort of frigid beast-of-2-fronts. Wanting a one way window the Anachron instead achieves the doubled opacity of a mirror giving itself a cloacal kiss.

Probably my favourite Small is London's wharf transform, a wry version of artist Donna Cox's romboy homotopy Etruscan Venus form of this sort of Klein.

The Cox Venus kneels, omphalic with its own ostriched head, before the city, its steep lap a sheen of, well, the idea of plastic. The Venus has been realized in scotchguard. She is scotchguarded scotchguard. Against a skillfully concealed skein that arcs to cross, without breaching, its own surface, veil after see-thru veil of this dirticidal resin has been draped, until all the singular transparencies form a cumulative opacity, or not quite opaque as it does let in light. The lobby of The Venus immerses you in a sallow glow, the wall a liver allowing light but no form.

Only a blanched suffuse is admitted, elsewhere's leachate. The architects have played with the shadow and wizened glow of passing others who flicker mild optic noise through the framing walls. Hung on those walls, liquid crystal windows reconstitute the outside like eidetic Tang, in a constant, constantly otherwise, facetiation of .....whatever.

Constantly otherwise, yes, these images playing over the lobby, the immersed unembedded commons of the hyperbolic Venus, but not, in Emmanuel Levinas' sense of the infinite, truly hyperbolic, alterity, Otherwise Than Being. These images, rather, are figural puree.

The architects of the Cox Venus have converted the noise of faces into the signal of profiles. "The infinite faces of the living", in Robin Blaser's phrase, are here totalled as silhouettes.

It works for me.

We've lately even been selling units that cater to a cadre of Klein Condo theorists cum connoisseurs, ktheorists they call themselves, and nearly all insist on a suite where the everting continuance of surface self-intersects. Try selling a ktheorist a unit along the surface's handle. No go. She knows where the action is, and will settle for nothing less than a balcony leaning over the self-extrusion.

We've even been developing a Small that is nearly all self-intersection, with the rest of its surface minimized. This way we can cram more units along the surface's singularity.

Fancy a unit in The Hyperbolic Small, the one with the spread-eagle singularity? Trump's New York travesty, The Big Small, may house more residential units than any other Small in the world, but its self-intersection is a gaudy, 3-space Euclidean, stigmata compared to ours. Our Hyperbolic Small promises to be virtually all singularity, all soft stigmata, all friction-free self-crossing. Live at the critical, the crucial, zone.

Reserve your basement loft now. Be immersed in the insideless surface. Know to the hilt of hyperbolic gnosis your self-intersection. Rupture without aperture.

Imagine the ideal anonymity of a zero perimeter privacy.

Imagine yourself a finale in perpetuity, invaginating threshold, your own soteriological singularity.

Think the lethal aletheia, the lossless, lessless clarity, the mind in its unitard, you.

Listen. Most ktheorists we've polled would give their high teeth and hind legs to live along the critical zone of the topos. All the cynic volupts of volition vie for a Room with an Eversion. And our new improved Small is hyperbole raised to the power of hyperbole. We're making hyperbolic the one facet of an already hyperbolic surface, because we've got some takers.

What would your self-intersecting home in such a Small be like? Think the ganglia-engorged lips or hyperbolic hands of a textbook homunculus. We'll stretch the fascinans of the surface of The Small, the extruding singularity of its self-intersection, at the same time we shrink the rest. Would you like to touch my homunculus?

You'll be immersed in the highest concentration of selfcritical percipience ever. You won't be able to get out of the right or left side of the bed for trying. Everything you enter, every room or thought you approach lefthanded, you will leave right. The opposite is also true. There was a 60's scifi novel with a great title: The Infinitive of Go. At home in your Hyperbolic Small, the infinitive of your go will be the infinitive of your come. You will lean forward in an everted recumbence in your ampersand or lazy-8 Hyperbolic Small, and you will grin, though critically, and a bit rictal, in the split infinitive of spasm.

Waking up is hard to do. Commit to your Small and history will no longer be the nightmare from which you must awake, but the lucid dream you may tweak.

Your self-reflexivity will be ipso facto otherous, as your most dilating yearnings will be a sphincter clenching a small morphing thing.

Topoi R Us. So make me your agent.

I think I've reached the end of my trope.


1. See "Intrinsic Spin of an Electron: A Study of Time and its Components" p2. <>.
2. See "Logical Types". <>.
3. "To me, nature is a Klein Bottle ... with no inside holding its contents." <>.
4. Steven M. Rosen in "Wholeness as the body of Paradox", for whom the Klein figures that which must be engaged by a sort of proprioceptive agape:
The necessary hole in our Kleinian text calls neither for a continuation of symbolic mediation by itself, nor for the meditative immediacy of "pure experience"; rather it invites what we may call medi(t)ation: the mutual permeation of externally mediated activity and internal self-realization - of reflectivity and the prereflective. Thus our proprioception would not be a pure self-thinking but a thinking of other that flows right through the hole in this Kleinian other and back into itself.[....] Put in alchemical terms, the inside-out Kleinian vessel would be bene clausum (hermetically sealed): not only entirely open to its own prereflective source but - because of this openness - also entirely closed, complete unto itself. p23. <>.
5. Brian Massumi in "Introduction to Deleuze, Guattari and the Philosophy of Expression" invokes the Klein and other single-sided surfaces as adequate figures for the process of becoming, as he introduces essays exploring "a contagious alternative to depth for the postmodern aesthetic: the intensification of the surface." see <>
6. Jacques Lacan, "Structure and Reality" p5. see <>.
7. "Market Philosophy/Assumptions" at <>. "The net is based on categories, boxes within boxes that, fourth-dimensionally, do Klein Bottle type movements to move the browsing mind from one idea to another."
8. See Bruce Sterling's "Premonition of Spasm or Why I Read Arthur Kroker": Kroker is "so far beyond left that even to map his position in the political spectrum would require some kind of non-Euclidean hyperspatial Klein bottle." <>.
9. The time portal Klein apparently appears in "Star, Bright", a story by Mark Clifton, referred to at <>. I can't remember where I read of the antimatter container Klein.
10a. In "Delay Vectors as Perceptual Chunks: Understanding Nonlinear Time Series Analysis as Self-Organizing Cognition", Ben Goertzel sees an analogue of program embedding procedures in the ways topological surfaces such as the Klein, accomodate self-overlaps with additional dimensions. <>.
10b. "Eric", the author of "The 4th Direction and Klein Bottle Theory" suggests the Klein's "fourth dimension could serve as the 'time' direction for all dimensions". <>.
11. The cover of Daniel Shepard's futurist theology "The Image of God" sports a Klein floating over an idyll. <>. The text is in this vein:
The universe surrounded by God is like a bottle that, to someone outside the bottle, appears to have no outside. To God, the bottle appears to have no inside. God, however, having created the Klein bottle and having been in and out of these bottles many times, knows differently. God knows exactly why It created the bottle and also knows the emptiness It feels from having used the bottle. The empty feeling God senses is the same empty feeling we feel when we lose a loved one, when we lose a limb. God senses a loss because It is missing a portion of Itself. It is missing the portions of Itself It has sent into seemingly infinite numbers of Klein bottles, the many Klein bottles It has created within Itself and dispersed throughout Itself.
12. My notion.
13. Barometer: Clifford Stoll's site at <>.
Hat: "Knit yourself a Klein Bottle" by "Mary in Manchester, England" at <>.
Purse: "The Klein Bottle and Variants", by Nathaniel Hellerstein at <>. The "Mobius Bag" is actually a 3-D mockup of a Klein. I haven't found anything yet using the Klein Bottle as a model for an economy.
14. Nathaniel Hellerstein mentions both the lazy-eight and ampersand Kleins, as well as two other topological equivalents, at <>.

Steven Whittaker is a writer living on Vancouver Island.
© CTheory. All Rights Reserved