I recently acquired a copy of my old friend Clifford Pickover’s new tome, The Math Book , a really attractive and reasonably priced volume with 250 full page color illustrations, each illustration accompanied by a single-page description.

In Pickover’s words, “My goal in writing The Math Book is to provide a wide audience with a brief guide to important mathematical ideas and thinkers, with entries short enough to digest in a few minutes.”

[Breather Pseudosphere, Copyright (C) 2006 by Paul Nylander. See a larger image on Paul’s site.]

One of my favorite images in The Math Book is Paul Nylander’s rendition of the so-called Breather Pseudosphere. The idea behind this surface is that it has a constant curvature of -1, as opposed to a sphere, which might have a constant curvature of +1, and also as opposed to a plane, which has a constant curvature of 0. You’re supposed to ignore the ribs, and you need to accept that the surface intersects itself along a circle, which is clearer in the image below, by Xah Lee. You can rotate this image on Xah’s site.

[Segment of a Breather Pseudosphere, Copyright 2006 by Xah Lee.]

From the arcane math references that I’ve consulted—see for instance the Wikipedia “breather” page—I gather that the breather pseudosphere can in fact “breathe” in the sense that, by diddling a certain parameter, someone (not me anymore) could create a sequence of images of it and then assemble these into a video in which this negatively curved object will pulsate like some omnivorous space squid from Dimension Z. If any of you ultra-math-and-CS maniacs out there has access to such a video—or feels the urge to create one—share the link with us via a comment on this post!

[A traditional Beltrami pseudosphere, Copyright (C) 2006 by Richard Palais and the the 3DXM consortium.]

As Pickover’s book explains, the notion of pseudospheres was invented in 1868 by the mathematician Eugenio Beltrami, who formulated the more familiar “double trumpet” model, as shown above, created by the 3DXM Consortium . (3DXM is a graphics program, now called 3D-XplorMath.)

You can find further images of various kinds of pseudospheres (these images by Xah Lee, Luc Bernard and other members of the 3DXM consortium) on the Gallery of Pseudospherical surfaces at the Virtual Math Museum. This page includes an essay “About Pseudospherical Surfaces,” which explains (amid much gnarly math) that, at least when depicted in our normal 3D space, any surface of constant curvature -1 will include “singularities” in the forms of self-intersections or cuspy lines where the surface has a crease in it—like those ribs in the breather pseudosphere or like the edge where Beltrami’s two trumpets meet. But you can smoothly embed pseudospheres into 4D space, I believe.

As I was discussing in an earlier post, another way to create a negatively curved space is to start with a disk of some ductile material, and the keep stretching the disk all over, but without overly stretching the outer edge. The inside of the disk acquires more area than one would find in a regular flat disk. The extra room is there because the interior is now a negatively curved surface. And I think this surface is something like a pseudosphere. (A different approach is to stretch the outer edge of the disk to infinite length, and this is a different model of negatively curved space called the Poincaré plane.)

[Copyright (C) 2009 by Vonda N. McIntyre . White hyperbolic anemone with red veins.]

Some knitters and beaders have crafted physical objects like the edge-stretched pseudosphere, some of which appear among a wild Crochet Reef show in 2009. Among the goodies on display was a bead construction of a pseudospherical sea anenome by no less a personage than the fantasy and SF writer Vonda N. McIntyre —see the MathCrafts section of her home page.

When I was snorkeling near Palau a few years ago, I noticed that many naturally occurring soft corals are indeed negatively curved surfaces.

It’s worth remembering, by the way, that if you’re living in a very small house, it might be nice to have negatively curved space inside…

It occurs to me that the yuels and the jivas in Jim and the Flims going to be better characters if they can talk—but that their speech should be strange.

[A jiva and a yuel.]

I think of the Unipuskers in Frek and the Elixir, who only talk in imperative sentences, that was rich, a great gimmick for expressing bossiness. And the devilish-looking Wackles in Spaceland, they talked in a kind of Beat poetry.

Here’s a simple and powerful idea: the yuels speak in strings of verbs, and the jivas speak in strings of nouns! The whole thing is to give the aliens’ speech a different texture.

It expresses a nice distinction between them. The yuels are more Zen-like, in action, flowing, shapeshifting. And the jivas are more capitalist, hoarding, acquiring.

[A yuel dinosaur and a jiva sun.]

Originally I had the jivas talking in a somewhat dull orotund society-lady Mrs. Earbore kind of way. Like a standard Hollywood SF-movie alien with a deep voice and *ugh* a British accent—and no use of contractions. It’s so stupid that superintelligent aliens in Hollywood can never master the trick of saying, like, “we’re” instead of “we are.”

It’s gonna be more fun, mysterious, poetic, and alien if my jivas speak in strings of nouns.

Nouns seem easier to string together than verbs, by the way.

I’m still getting the hang of it. These are alien modes of thought! So I need to practice.

Suppose that a jiva and a yuel are going to rent a cottage at the beach with their families. Maybe they’d say, respectively, something like this:

Jiva: “Ocean house family days.”
Yuel: “Surge soak sleep eat nurture.”

Feel free to send in (as comments) some further examples of noun and/or verb speak.

Yesterday I was out a favorite spot on Four Mile Beach, north of Santa Cruz. That forest fire they had inland has pretty much died down. I did a little routine that I’ve often done while working on a new novel in recent years—I draw a logarthmic spiral in the sand and I letter the slogan, “EADEM MUTATA RESURGO”, which is Latin for “The Same, Yet Changed, I Re-arise.”

The mathematician Jakob (also called James) Bernoulli (1655-1705) has this inscribed at the very bottom of his tombstone in Basel along with a picture of a spiral (the engraver mistakenly depicted a regular Archimediean spiral instead of a wildly growing logarithmic spiral, like the kind you see on seashells).

I started this particular routine while writing Frek and the Elixir, and in fact the slogan is mentioned in that novel. If you feel like it, you can search my blog for “resurgo” to see the other spots where I’ve mentioned it.

I did some good work on my novel-in-progress Jim and the Flims today. Here’s a fun passage I wrote today, about a sinister alien plant. [Note that what I’m printing here is only the first draft, and I will in fact be revising it repeatedly on my way to finishing the novel.]

In the morning, Durkle woke me with a nudge of his foot. I heard a babble of voices nearby. Sitting up and looking around, I saw something like an umbrella projecting from the ground a short way off.

“It’s an offer fan,” said Durkle. “Did I tell you about them? A mobile plant—see those writhing roots at its base? They can walk, a little bit. It must have teeped us here. Mostly they live in the swamp, a few miles off. Isn’t it cool? I’ve heard you can get anything you want from an offer fan—if you’re quick enough. Watch me.”

All sorts of desirable objects were dancing beneath the offer fan’s greenish umbrella. It seemed that the offer fan could read my mind, for as I stared, it produced some items that I myself might want right about now: a cup of tea, fried eggs on rye, a map of Flimsy, a bag of pot with rolling papers, and a cantaloupe—all marching in giddy circles with the others across the roots beneath the thick umbrella, everything seemingly quite real. Perhaps the odd, alien plant had perfected a kind of direct matter control?

It was of course obvious to me that I shouldn’t try grabbing for the goodies, but Durkle was a willing fish for the bait. Having grown up in a land without computers or television, the boy was naive. Manfully he approached the offer fan, skipping to the left and the right as if he hoped to outmaneuver it.

The fan’s umbrella made continual slight adjustments in its position—the broad cap sat alertly atop a flexible stalk the thickness of a leg. I noticed that the underside of the umbrella was spongy and damp, as on certain unwholesome mushrooms. The thing’s roots were twitching and obviously eager to act. I was guessing that the fan’s kill technique would involve poison spray as well as strangling.

Durkle seemed oblivious of the risk—his eyes were fixed upon a scuffed-up board identical to Flam’s, a tasseled orange racing cap, a little chessboard, a shiny sword, and a fleshy glob that was forming itself into the body of—a naked woman, but with rounded off arms and legs and a smooth bulb for a head.

“Stop right there, Durkle!” cried Ginnie, sitting up beside me.

“I know I can beat this thing,” said Durkle, glancing back at her. “You want me to get you something too, Ginnie? Make her something, fan! I dare you.”

Sensitive to our group’s dynamics, the fan extended its offers to include a steaming mug of coffee and a very fashionable pair of sunglasses in wide tortoise-shell frames. Durkle crouched, preparing for his final dash.

I couldn’t let this continue. I ran forward and grabbed him around the waist.

“Senile loser!” he hollered. “You’re just jealous that I’m young and fast! Ginnie wants me, not you!”

Maybe I was old, but I had a jiva inside me. Durkle wasn’t going to break free of my grip. But he did manage to knock me off balance. The two of us fell to the practically into the shadow of the offer fan’s umbrella—a very bad place to be.

Fast as a whip, the thing had its roots around our wrists and ankles. And now, sure enough, an evil-smelling mist began wafting down from the cap. Most of the offers had disappeared now that the fan was getting down its real business. It’s central stalk tilted over, maneuvering the umbrella so that it could flop down right on top of us. I felt drowsy, and the spray was stinging my skin. As well as being a soporific, the spray was a digestive fluid.

Suddenly the umbrella tumbled off to one side. Ginnie’s jiva had cut the stalk! The offer fan let out a telepathic scream that filled my mind with red and yellow jaggies. Ginnie was circling around us, lashing at the carnivorous plant with her jiva tendrils and now—slow as always to defend me—my own jiva, Mijjy, began attacking the offer plant as well.

Durkle had managed to free one of his wrists and he’d gotten hold of that little sword the plant had made as bait—this desirable item had remained on offer to the end. It was indeed a real and solid blade. The boy slashed away at the roots, freeing our hands and ankles. He crawled a few feet away and tugged me after him. Slowly the fan’s frenzied alarm waves within my head died down—and the mist drifted away. I could think again. In a belated coup de grace, Mijjy set the remains of the offer fan on fire.

“Got any more good deals for us?” I asked Durkle.

The personable Tamara Vining of Seattle has posted a 50 minute video of my Clarion-sponsored appearance at the University of Washington Bookstore. First I read from the first chapter of Hylozoic for half an hour, and then I do some Q & A.

Thanks, Tamara!

This week I’m cranking up work on my novel Jim and the Flims again. I want my characters Jim and Durkle to tunnel down through purgatory beneath Flimsy and to drop out of a floor hatch in the lowest level of purgatory, and to find themselves falling from the Flimsy sky, that is, from the dome of living water above Flimsy. It’s dramatic. But how do I explain it?

I’d really like to lay this issue to rest, as I’ve been hung up on it for way too long, and I feel like it’s blocking me from proceeding on the book. This said, it could be that there’s something about the issue that’s core to my psyche and my state of resonance with the world of Jim and the Flims.

(Old Explanation: Bent Purgatory) In my July 19, 2009, post “Towards a Topology of the Afterworld,” I entertained the notion of having purgatory somehow bend around past the disk of Flimsy to be on top of it as well as under it, as shown in the reprinted illustration above (my guys enter purgatory at level 1, and exit it at level 3). But this is kludgy, and asymmetrical, and it poses the problem of why the snail-tunnel happens to go through from Flimsy straight to Earth, intersecting the living water, yes, but not intersecting purgatory. Also I want to bring in the notion of having an infinitely expansive center to Flimsy.

In other words, I need a tighter idea. Chip? Dale?

(New Explanation: Living Water Flow) Maybe I suppose that there’s a powerful and rapid flow of the living water, and that when Jim and Durkle tunnel down out of the lowest level of purgatory, they’re rapidly swept up into the sky, and they jump down from there. This is a fairly attractive notion, as it’s pretty easy to understand. And we might suppose that the infinitely distant hole in the center of Flimsy has a recycling fountain-like quality, creating a more or less toroidal flow through the central Helaven, with living water emanating from it, streaming around Flimsy and eventually dropping back into the Helaven from the sky. Okay, fine, let’s go with that. Here’s a picture, followed by some remarks on what I see in the picture.

Note the border snail on the right, sticking through to make a tunnel. Note also that I have an “Endless Sea” around the central Helaven, this can be like the shallow waters of Pyramid Lake.

A fine rain of souls is falling from the sky above Helaven. The souls are absorbed into the light, and new water and souls flow outward below.

More observations: Purgatory is a bunch of corridors and boxes, somewhat randomly accreted, like the Winchester Mystery House in San Jose, like an airport terminal forever under construction, or like a repeatedly renovated mall.

Jim and Durkle go through the Monster Pit, through Purgatory, and are swept with the living water in the sky, and they dive back down from there.

The tapering of the landmasses towards the center indicates that the center is, in effect, infinitely far away.

And there might very well be a great leviathian in that great gulf of living water at the bottom. Suppose it’s a gnarly tentacle-laden Supreme Jiva, a puppet-master who in fact controls all the sun-sized jivas in Flimsy—and I may as well suppose that some mid-level jivas are controlling the jivas in Weena, Jim, and so on. (Turns out that thing I was calling the Supreme Jiva a few posts ago was only the Easternmost Jiva.)

Groovy.