More Inside Story

:: flatworms by M.C. Escher ::
(geometry of nature)

Some of you have maybe joined our book circle around King of Infinite Space, the well-told story of "the man who saved geometry", Donald Coxeter.

As I was discussing in Pointful Gossip (gossip with a point), Donald had an uneasy relationship with Bucky Fuller, on the one hand okaying the latter's choice to dedicate Synergetics to the master, on the other hand wanting to keep Bucky in his place, as a great man maybe, but not a great mathematician.

Dr. Fuller was a troublemaker for the settled disciplines. He got a lot of notoriety for those domes, but even here, the game was to discredit him if at all possible, deflate the guy's brief for having "the right stuff" (paraphrase).

Bucky was subversive of academic authority because his philosophy ran counter to the conventional wisdom of his day, and within earshot of students. Bucky was a dangerous maverick whose influence needed to be checked, or at least channeled.

In terms of channeling, an original discovery of Bucky's to which Coxeter was willing to give his stamp of approval, was 10 * F * F + 2: the number of balls per each layer of a growing cuboctahedron of balls i.e. a key access point to the CCP and/or face-centered cubic lattice (FCC), what Bucky himself called the "isotropic vector matrix" (with a focus on edges, more like Zome's).

This formula could be proved using high school level math. Geometry could afford a small accolade to the maverick polymath at this juncture.

As Siobahn Roberts tells it:

When a reporter from LIFE magazine called in 1970, Coxeter gave Fuller a somewhat backhanded -- but then accidentally glowing compliment.... Coxeter sent back a letter saying that one equation would be 'a remarkable discovery, justifying Bucky's evident pride,' if only it weren't too good to be true. The next day, Coxeter called: 'On further reflection, I see that it is true.'

This very same formula was turning out to have applications in virology, as it explained aspects of the viral capsid, the hard shell containing the RNA coil, relating its protein capsomere counts to the icosahedral numbers.

X-Ray diffraction had newly revealed the geodesic dome pattern in nature's micro-architecture. This was years prior to the discovery of buckminsterfullernene, the naturally occurring carbon cages with similar properties (i.e. five fold rotational symmetry).

However, Fuller himself had a bigger discovery in mind, when he hooked up with Coxeter as a leading light, and that was in connection with his modular dissection work in a related sub-branch of geometry.

Fuller goes into meticulous detail in 950.12, with related Figure, noting how page 71 of Coxeter's Regular Polytopes gives three tetrahedral space-fillers, one being a component of the other two and hence the more primitive.

Fuller emphasizes in Synergetics that this more primitive space-filler is his own discovery, the MITE, whereas Coxeter is merely describing and renaming it, while at the same time omitting all mention of the A & B modules (two cellular automata, both left and right handed). The MITE consists of two As and one B and has a volume of 1/8 in the prefrequency concentric hierarchy.

:: MITEs Cube with
pg. 71 of Regular Polytopes ::

Donald lived long enough to see this priority claim in print, and apparently published no objection to this characterization, which came without accusation of plagiarism or anything so undiplomatic. Fuller was playing some masterful chess here, dedicating the work to this great geometer, a mentor, while upping the ante with some chips of his own.

If 10 * F * F + 2 were a lucky break, then this MITE would seal the deal: Fuller was no charlatan and keeping him safely at bay, pigeon-holed as some renegade architect, would no longer be feasible. Geometers would need to acknowledge him for more than "just domes".

As it happened, academia continued to draw a line in the sand, by refusing to adopt any of Fuller's nomenclature, which, in effect, would be to accredit his discoveries, make them mainstream.

Scientific American had succeeded in cutting 10 * F * F + 2 out of the virus story, by finding an alternative telling, wiring through Michael Goldberg's studies in Japan. The Scientific American later made up for this somewhat, by publishing about Yashushi Kajikawa's synergetics-inspired modular dissections in the five-fold realm (likewise David Koski's focus), but only in its Japanese edition.

As of 2009, the intelligent reading lay public is still safely clueless, not connecting the dots between this Medal of Freedom winning radome architect, so admired by U.S. Marines and Russian intelligence services (and proponent of the global grid), and high caliber geometry breakthroughs outside of dome design, in a realm closer to the cellular automata.

When A New Kind of Science was published, obvious associations were not made, and to this day the MITE is missing from Wolfram's MathWorld on the web, while on Wikipedia, Synergetics is confused with some Springer-Verlag publication by the same title [note: we've done subsequent disambiguation work].

"Obfuscating Fuller" continues to be the name of the game in some circles, some thirty years later. Fortunately for students though, we have this Coffee Shops Network, other venues, for sharing this esoterica. Plus not everyone in academia has been trying to keep the lid on this story.

Arthur Loeb, the MIT crystallographer, dared to speak openly about A&B modules, came forward with the brilliant Amy Edmondson as his protege and author of A Fuller Explanation, seeking validation through Birkhäuser (a respected academic publisher, and subsidiary of Springer-Verlag). This book drew heavy fire from some corners, but helped a lot of die-hard buckaneers better understand their great pirate "phantom captain" (insider joke).

Likewise Ed Applewhite, chief collaborator on Synergetics and former "Deputy Inspector General and Chief of the Inspection Staff" for the CIA, groomed a few proteges, me included, for more diplomatic cockfighting of this kind, plus he amassed lots of "collateral" (aka "ammo") some of which he sent off to the special collection at Stanford.

His chief concern: that academia would get away with bleeping over Fuller's work "with impunity". I'd say there's no chance of that now. These stories are just too good to keep under wraps, plus young scholars need to get on with their studies, can't afford to just dilly-dally. It pays to know your history and your heritage.

Tygerology

Disney version of
The Jungle Book tiger Shere Khan,
here depicted with Kaa

William Blake's famous poem

Tony the Tiger
said "grrrreat!" a lot,
subliminal code for
"eat mass quantities of
Kellogg's Frosted Flakes"

Esso from S.O. i.e. Standard Oil.
The "in your tank" tiger of the 1960s
was resurrected when the company
changed in name only to Exxon
i.e. "same stripes"

The Disney version of
A. A. Milne's Tigger
happy go lucky
friend of Pooh, Eeyore et al

Princeton Tiger, one of several

Number Lines

A "number line" need not be "infinite" in order to be useful, nor must it be "perfectly straight". A good example: the equator, or line around the edge of a clock, with notches for hours and minutes.

On a 24 hour dial face clock, you could keep the solar disk at high noon and show which time zone was "on top" as the world kept turning, with midnight directly opposite, at the antipode.

A number line that goes all the way around the world is locally straight enough to serve as a ruler, or we might look at actual rulers as short segments taken from this great circle line.

In "arrowhead geometry" we point four number lines out from a common origin at (0,0,0,0), use 4-tuples for vector tip addressing. Michelle, Jason and I were comparing notes on whether to say "toople" or "tuhple" after the last PPUG meeting (and how about Ubuntu?: Michelle says "oobuntoo" whereas I say "ooboontoo").

We're not "anti-Euclid" just because our school inclines towards Karl Menger's "claymation station" approach to points, lines and planes. They're all "lumps" as otherwise they wouldn't reflect rays the way they do, and this is a ray tracing geometry that we're learning here, and so want our axioms and definitions to fit the application.

It's an innocuous enough move, but the dogmatists will start barking at this juncture, as "dimensionless points" are an article of faith for them, a perceived foundation for hyperdimensional polytopes and all the rest of their edifice.

I tend to make reassuring remarks at this point, reminding viewers that it's not either/or, plus in gnu math we have APL and J, hungry consumers of "hyperdimensional arrays" i.e. there's no reason to let "extended Euclideanism" fall by the wayside, even as we pioneer in different namespaces.

What we're doing is mostly backward compatible, is what I'm endeavoring to put across, even if individual notations, such as Python's, have their quiet breaks with the past.

When it comes to simple non-rationals (irrationals, incommensurables) such as 2nd and 3rd roots of simple integers, phi and pi, the natural place to begin is with simple diagrams constructible with string and a scribing surface ("a sandbox" or "white board").

Our CSO Glenn Stockton does some good teaching in this domain, including drawings in perspective. From here, it's but a small step to constructing Polyhedra, with toyz like Zome, vZome and StrangeAttractors.

Our exploratory constructivism has a soft spot for such geometric constructions, embraces Euclid like a favorite teddy bear. We don't forget about Euclid's Method for the GCD either, like using Guido's Pythonic implementation thereof.

Another Creature

:: another Koski original
using vZome by Vorthmann ::

I'm at Rams Head (McMenamins), feeling fortunate to be getting some of the latest and greatest.

I'm at bat for Silicon Forest today, making a case for keeping students in the 21st century, not allowing as much backsliding. Mostly I'm working with math teachers, as they're the furthest behind, the most in need of remedial attention.

Fortunately, the new curriculum is as colorful and entertaining as it is effective in getting the concepts across. We're not shy about saying our stuff is the right stuff.

Twas my privilege to provide computer services to Common Man Studios yesterday, the outfit behind Jimmy Lott. Given I'm marketing Portland as "the Nashville of Open Source", it makes sense that I'd be criss-crossing the music, not-music boundary, obscuring the difference in some ways.

Kirby Urner
4Dsolutions.net

---

Excerpting from PolyList:

There is a total of 6 different rhombs that make up the 132 sided shape; 12+12+12+24+24+48=132.

There will be a total of 220 hexhedra that make up the form which I will follow up on.

I do want to apologize for the ugliness of this beast, and hope to put it back in its pen as soon as possible.

David Koski
Minneapolis MN

From PolyList

by D. Koski using vZome

Well, I was encouraged by Scott Vothmann to try again on the great rhombicuboctahedron.

I didn't exactly succeed but did generate this 132 sided rhombic form. It has 48 squares and 24 (60,120) rhombi and the three other rhombi I have not identified yet. 12 of the fattest one, 24 each of the ones that make the eight pointed stars.

I figured the form out by using the Parallel Projection Process. Vorthmann's vZome has a root 2 system which has 24 brown strut positions or 12 different axes.

So, 12*11 will be the amount of rhombic faces of which there are five different types. Therefore, 12/1*11/2*10/3 = 220 total hexahedral cells.

Yes, those are flat hexagons and there are 4 zero volume hexahedra that correspond to them.

Amounts of different hexahedra will soon follow.

BTW, The late Russell Towle refers to the Archimedean Truncated Cuboctahedron (s/b Great Rhombicuboctahedron?) on this page:
http://home.inreach.com/rtowle/Zonohedra/zonohedra.html scroll down to middle of page

He calls it A Notorious Zonohedron, does anyone know why?

Is this shape cataloged anywhere, I would like to find more out about it.

Thanks
David Koski
Minneapolis MN

Followup to PolyList: I made a mistake. There are not 48 squares, but they are rhombi of course. David Koski

Coffee Table Discussion

Click the title (above) for the preprint of this paper: Inference to the Best Decision by Patricia Smith Churchland, Philosophy Department UCSD, La Jolla CA 92093.

Below, a summary quote for further discussion.

My main point in this context, however, is that naturalism in ethics should no longer be hobbled by the dictum that you cannot infer an ought from an is. Fine; you cannot deduce an ought from an is. What you can do, however, is come to a decision about what you ought to do without relying on any normative rules or maxims. That is what humans, and undoubtedly other animals, in fact do. From this perspective, many new questions in ethics arise. These questions present philosophers with a unique opportunity to collaborate with scientists on matters of great social importance.

The "house rules" for debate are up to your coordinator -- remember to leverage asynchronous media if your network is distributed across time zones. IRC isn't always the best way.

Feel free to send blog links if you'd like others to be aware of your discussion.

My thanks to Nat Bobbitt for starting this circle around Portland.

Commercial Space

CSN is looking for ways to share screen space with vendors.

Come forward with commercials, but also games, like the ones at your web sites.

We're looking to give our customers a way to build identity around buying your products, yes, but also committing some of the proceeds to causes of their own choosing.

We're even proposing to reward heroic efforts, as some of these games may be quite puzzling.

Finding solutions helps your friends and family.

In the News

OK, so late 1980s. Cutting and pasting from alt.education.

ARCHIVE COPY

From The Oregonian (local newspaper), Portland section. Don't have the exact date handy (1987 or so, given it says I'm 29). Came with a picture of me holding one of those Fuller Projection postcards up next to my head.

=======================

Young Portlander entertains dream of lighting up world

By Peter Carlin
Correspondent, The Oregonian

Kirby Urner has a plan.

It involves geodesic domes, a new world map, an international power grid, world peace and the teachings of R. Buckminster Fuller.

And Portland might play a big part in all this, Urner contends.

One of Fuller's most distinct images is his Dymaxion Projection map, a super-accurate projection of the earth's land masses and oceans that the Medal of Freedom winner copyrighted in order to guarantee that it would never show national boundaries.

While living in Jersey City, N.J. during the early 1980s, Urner attempted to interest some corporate donors in constructing a huge, wall-sized projection of the Dymaxion map out of millions of tiny light bulbs, much like the scoreboard screens at football stadiums.

Urner said the map project met with some interest, but he found investors unwilling to gamble their funds on the future of a ravaged urban center such as Jersey City.

"Portland, on the other hand, has a much brighter future," Urner said. "If OMSI goes ahead with their plans to develop a new site on the Willamette, I expect many Fuller-related artifacts to become part of that project." He noted that there were Fuller displays at the Louvre in Paris, at Disney's Epcot center in Florida, and at Expo in Vancouver, British Columbia.

The map, Urner said, "would be a great way to educate people about the Earth, as well as providing businesses with a unique method of advertisement. Companies could get access to the map on a time-share basis to show off their assets. ...I suspect airlines would love to use it to show their routes around the globe."

Meanwhile, Urner, a 29-year-old computer consultant with the Center for Urban Education, is laying some of the groundwork for acceptance of Fuller's ideas. He teaches computer skills to the underprivileged and is making plans to expand the center's educational influence through the use of video technology.

"I don't expect people to buy into all this immediately," Urner admitted. "But, as Buckminster Fuller showed, we have access to all the tools we need to transform the world into a more prosperous, peaceful place. We just need to pick them up."

Urner spent some of his childhood years in Portland, before his father's career as an urban planner carried his family to a series of far-flung places around the world.

Back in the United States, Urner studied at Princeton University and then spent some time teaching high school in New Jersey before moving back to Portland in 1983. He has been working with computers at the center since 1984.

Fuller, who died in 1982, was a modern philosopher, mathematician, architect, theorist and writer who could count President Reagan, Henry Ford II, former CIA chief Stansfield Turner and Soviet premier Leonid Brezhnev among his more influential admirers.

Fuller's revolutionary mathematics, explained in his books, "Synergetics Vols I and II," are the basis for his concepts of design and education. Products of these theories include architectural designs such as geodesic domes and Fuller's international energy grid, a proposal that would link the power supplies of nations around the globe.

Urner said he hoped that Fuller's approach to science and math would open up peoples' perceptions of the world's future.

"Frustration with science and math is the main roadblock between the world today and the world we want to live in," Urner maintains.

"True awareness of the globe and the solutions to problems like hunger and war -- which Fuller says are under our noses -- will present themselves much more clearly if we acquaint ourselves more with the basic principles of math and science.

"Media extravaganzas like scoreboard-sized maps and video programs are important, because they are dramatic methods of education, and education is the key to the future."

----------------------------------------------------

Kirby Urner "ALL realities are 'virtual'" -- KU

Outreach to Japan

:: csn esoterica ::

I'm glad to see Hillary making headway in Asia. The Port of Portland is definitely facing in that direction, looking North along the armpit of North America (Alaska), out across to Tokyo Bay, Kyoto, Hiroshima and those. Not forgetting about Korea or my Korean sweetie in high school, later college (she went to Bryn Mawr, had a blast).

I'm basically an out of the closet Asian, as you know, which doesn't mean I can't follow football, or read the ingredients on a Snickers bar. I know what spam is, thanks to Wawa and friends. Nor am I biased against non-Asians -- those would be a lot of my best friends, we grew up together.

When Don and I ventured into Backspace between Livio talks, CSO yakking outside on cell, he asked if they'd all stand up and clap now that Mr. Esozone had arrived. I said it wasn't like that, but we do wish to support Japanese business executives wishing back office, high bandwidth access to companies in other time zones, closer to home.

Portland is a great place to visit, Todai a fine restaurant, but there's no place like Kansas at the end of the day (however that translates). We played some chess, didn't finish the game, made it to Powell's Technical for 2 of 2.

Whether Backspace is yet equipped at that level is a micromanagement concern, not CSN business. Like, I carry Visa, but that doesn't mean I get to sweep Visa's floors.

The USA's rail system is a natural wonder, a source of tourism revenues if the xenophobia ever abates.

USAers are great about providing medical assistance overseas, but are loathe to accept therapy in state ever since the Bhagwan Chapter, or at least in Oregon that backfired (talking about Rajneesh Puram, a long story, and no I wasn't there -- was retiring in Florida at the time, had a gig with McGraw-Hill before that, didn't return to Portland until 1985, from Bangladesh).

Ranchers might accept foreign exchange students wanting to learn about horses, but not if they have to paint themselves blue and pray to elephant heads in their closets, not that I'm being disparaging, having a soft spot for elephants myself (Packy and I are about the same age). But there's only so much change people can handle. Future shock in measured doses: is that too much to ask? Just asking.

Anyway, you need that wifi to work on your Amtrak trains, and how about a DVD in every chair like on Airbus, with documentaries about the scenery, Google Earth, not just mindless lies about hackers or snakes on a plane -- topics Hollywood knows next to nothing about. Or make up new kinds of media car? Like a bizmo, but part of a circus train, like the caboose used to be? Surely we haven't exhausted the science of train car design. The USA used to have a reputation to protect in that area i.e. dome cars and like that. Any pride still?

Speaking of Japan, this new product from SONY is creating a stir, netting a few chuckles in business meeting (um, sorry, my bad).

:: CSN esoterica, 2008:
soundtrack by Jimmy Lott ::

Giving Thanks

> 'Necessary' is hyperbole too. It is very important
> however.
>
> Now that those are out of the way, would you please
> indicate how you see middle and high school math
> would be enhanced by the use of graphing calculators?
>
> Richard

My feeling about calculators, graphing or no, is similar to Haim's towards his Education Mafia (never met the beast). They're available, use them if you must, but this is 2009, and we actually do have rather well defined notions of how to reboot the engineering curriculum at least, even if maths proves terminally moribund, and that involves running low cost FOSS on commodity hardware, not talking about calculators, sorry.

I am talking about group theory, elementary version, don't care if you get to Sylow's or those, Euclid's Algorithm, Euler's Theorem for Polyhedra, totatives, totient, polyhedral numbers, and so on (nothing I haven't posted about a hundred times on this list) -- this is about opening doors, not going all the way down the hall to the far end. Choose a hallway in college maybe, but get a sense of your options. If there's no good engineering taught in your school, how will you know?

In CS, we're pretty happy with Python, so there's not much objection from further up the pipeline. Kids are into this appengine business, looking for entrepreneurial angles, ways to differentiate outside of Facebook and Myspace.

Does that mean we have all the kinks worked out? Per my letter to the school superintendent, there's still a lot of discovery needed.** We have a lot of budgeted inservice and a lot of talent in the private sector, and a community service requirement in some companies.

Mentor Graphics has been stellar, in supporting our sciences, geek subculture (ISEPP.org etc.). The late Doug Strain's ESI a role model too. Bruce Adams of Applied Materials... great group out here, proud Pacific Rim capitals (Seattle another hot spot for innovation, thx to Gates Foundation probably, other initiatives).

I only know a fraction of what's going on, but that's enough to keep me focused. Lots of data centers in our present and future, especially if you count the casinos (and why wouldn't you?).

Speaking of Microsoft, many thanks to Paul Allen, and also to Joyce Cresswell, our director, without whom very little of the present Saturday Academy scenario would have been possible.

Kirby

Notes:
http://en.wikipedia.org/wiki/Sylow_theorems (cool)
http://www.youtube.com/watch?v=6WmJNswqfKA (promo)
https://wanderers.pbwiki.com/PyconPromo (** from above)

Click title for Math Forum version

Pointful Gossip

I'm saying this gossip isn't pointless, needs to get out there, more in the "thousand dots of light" genre, in need of interconnections.

My own little life became a little more complicated when the University of Buffalo decided to kill Geodesic, a long-running list frequented by crackpots, wackos and other die-hards, including myself on occasion. Joe Moore is a serious scholar. There's actually a lot of good sharing on that list. The university has made this material all the more interesting and valuable for having pulled the plug after all these years. Now we have a better story than ever.

The Wild West Internet is like that; we were warned...

My ISP Internet Arena slunk away in the night that time, no warning, sites gone. I'd fallen in with them when Teleport shriveled. Then when Drexel inherited the Math Forum from Swarthmore College, all kinds of URLs fell by the wayside, as the indexing scheme was completely revamped.

For writers like me, like ducks to water with hypertext, massive link loss just goes with the territory. But that's not to say I'm always jumping up and down with joy when my "geometry of thinking" gets interfered with in this way.

Think of a spider, going home mad, after a long day at the office, when some kid comes along and... But hey, there's always archive.org for a lot of what's gone.

To the matter at hand:

Journalist Siobhan Roberts does some excellent storytelling in her King of Infinite Space, a book about H.S.M. Coxeter, the great 20th century geometer.

Here's finally a place to start up with the Bucky stuff, making those links, telling the story of our own lifetimes. Fuller dedicated Synergetics to Coxeter, with permission, as a part of a long history of communications.

One of the juicy bits (here's the gossip):

M.C. Escher's son George bumped up against the Fuller machine, otherwise known as the DoD, as a newly naturalized citizen of Canada, was incensed to discover the DEW-line radomes could be patented to the extent they had been, given their natural occurrence in nature, and as mathematical concepts.

"These should be open source!" he was thinking (paraphrase). Coxeter felt the same way. Who is this interloper, this non-mathematician who thinks he's hot stuff, warping the legal structure so badly as to give the whole idea of "intellectual property" a bad name? Arthur Loeb agreed: we need to watch this Bucky guy, he's got charlatanical aspects, could mean trouble for us.

This was long before any talk of a dedication, having dinner with Fuller and his wife in Carbondale.

Fuller's reputation as an architect was internationally recognized, so it wasn't like there wasn't a ready pigeon-hole for the guy. What was maddening (crazy-making) about Fuller is he wouldn't stay in his hole, not that he didn't have one (he had several -- wore many hats over the years).

Dr. Loeb, a respected MIT crystallographer and teacher for Amy Edmondson, author of A Fuller Explanation, later became friendlier towards Fuller, to the point of writing a frontispiece.

Loeb's sincere wish that Synergetics not be construed as occult esoterica or dark ages metaphysics from hell, is more mutedly expressed in this intro, as for the most part there's a lot of good stuff here that crystallographers might use to pass on their discipline, i.e. reading Synergetics at a place like Harvard couldn't be all bad, even if the occasional nut case felt moved to burn it as a "Witches' Bible" in a public square or someplace, Boston being what it is (if you know your history).

But getting academia all riled and foaming at the mouth was part of Fuller's grand strategy for world domination, as we in the open source world are wont to say (the FOSS community has always expressed that as a goal, somewhat in self spoof (shades of Subgenius)).

Fuller was out to make a point, about "what one individual could do" amassing credit to himself in heaping helpings ("ungodly amounts"), knowing full well the patents would expire eventually, but wanting to tout himself as "not a corporation or government" i.e. "just some guy" (or "an average human being" as he liked to put it). He aimed to champion "everyman" -- call it a literary endeavor, The Adventures of Guinea Pig B (good title for a children's book eh?).

Like he really was quite distinct from the military-industrial complex, had that "maverick" label right from the get go, and with good reason (because he was one)). People will call him a "cold warrior" and that may be so, but you have to read the sections on cryogenics in Synergetics to really know what this "cold war" thing is all about (hint: no outward weapons need apply if the strategy is working).

Towards the end of his life Dr. Fuller cashed in on his never having sold out to some "boss corporation" and wrote these "tell all" epics that were all over the map in terms of displaying his loyalties. Here was stuff the Russians could use for ammo, not just the Americans, although that Medal of Freedom from Ronald Reagan really helped me on the USA side of the fence (I've had rather limited time in the former Soviet Union, went through there en route from Kabul that time, exiting to Helsinki I think it was, then on to Findhorn in Scotland, an intentional community of some esoteric nature).

Part of what he wrote in Grunch of Giants was subversive of intellectual property rights, which you might consider hypocritical given how he'd lived, but his whole point was it's we humans who do the real work in this city (Spaceship Earth) whereas our "big institution" Grunchies are but mythical animals, puppets, not really real in the same way that a human being is real, e.g. as a source of real, human intelligence, kindness, compassion and so on.

It's a philosophical point I suppose one could say. In the Wittgensteinian namespace we'd underline that he showed (pregnant pause) what he meant, didn't just say it. He was a "walk your own talk" kind of engineer, "ate his own dog food" (geek expression), lived in a dome (sometimes). He also aimed for a responsible degree of transparency given the level of intervention he was performing, kept his life an open book. Blogs hadn't been invented yet, but if they had been, he'd've likely had one, maybe several.

Fuller's writings have a very definite flavor compared to most flavors of corporatese, which, like lumps of clay, tend to mean anything you want them to mean (very slippery), in case the political winds shift. For example Operating Manual for Spacehip Earth is an easy read and works well in Chinese, because it's so comic book clear, so full of pirates and so on.

Fuller's writings are not all equally difficult, any more so than Nietzsche's are.

That being said, the magnum opus Synergetics, originally published by Macmillan in two volumes, is a lot of obviously difficult reading, especially if you mistake it for a physics or a geometry textbook, whereas it's clearly labeled as philosophy ("geometry of thinking").

This is not a work one tackles just for the heck of it. 99% of humanity is otherwise employed. Universities mostly said "pass", meaning they kicked the can down the road for a couple generations, leaving career diplomats to fend for themselves for the most part.

Diplomats had to say something intelligible because of the networking Fuller engaged in. Fuller knew a lot of heads of state. He'd circuited the world many times well before the term "jet set" had yet been coined. His good friend Ed, aka Sonny, was a spook, and very out of the closet about it.

My advice: maybe read Love's Body by Normon O. Brown to get in the mood, followed by Wittgenstein's Philosophical Investigations. Fuller's Synergetics (as distinct from Haken's) gets very literal and specific in many places, but on the other hand Fuller has high tolerance for "polymorphism" as we say in hermeneutics.

Brown's aphoristic tirade against literalism (shades of Nietzsche) is a good antidote if you're not used to "reading for meaning" in the humanities sense, whereas LW's PI reminds us about "spin" (Fuller was a strong "spin doctor" within his invented namespace).

Synergetics is a circuit diagram yes, but not for anything you might literally build, any more than you could find a recipe for Monads (a Leibniz idea) in Make: Magazine.

That being said, there's lots you might make after studying this magnum opus, with spheres and domes just the beginning. Read a lot of H.S.M. Coxeter too, and L. Euler, and W. Gibbs... other greats Fuller admires, features prominently in his pages. Plato, Democritus, Kepler... Boltzmann.

Science writers tend to eschew metaphorical thinking except when they acknowledge writing "a popularization" of science, whereas laypeople aren't supposed to need access to the "real deal" i.e. "the hard stuff" (very mathy, very Springer-Verlag and considered non-metaphorical because "not literary" because "formal and logical" (strut puff)).

Synergetics isn't a popularization of anything, is the hard stuff point blank, but is more prose than squiggles, more accessible to the humanities trained. Consider it a contribution to American Transcendentalism, if you want a pigeon hole. Esoteric, yes, but it's not Dianetics.

When I corresponded with that Most Beautiful Molecule journalist, he thought I was admitting something, throwing in some towel, when I claimed Synergetics was a work in the humanities (like Poe's Eureka! in some ways).

To his way of thinking, that was like admitting defeat i.e. "the hard stuff" is by definition on the other side of the C.P. Snow chasm, and if Synergetics were to be taken at all seriously in future, it'd need to stake and defend turf on that far side, stop sounding so much like Neoplatonism (not in vogue).

But then what if Synergetics were itself a bridge across said C.P. Snow chasm? What if there were no more chasm? That'd be more like the old days, when Philosophy straddled a more wholistic vista. It all gets rather zen-like at this point: no difference, no sameness, non-duality etc. (kind of est -- a philosophical work, true, but also a performance, a kind of engineering, or reality TV).

Synergetics isn't about sweeping away all that came before (a paranoid fantasy). It's about wiring stuff up, with plenty of continuity, lots of smooth gear shifting. It's backward compatible (e.g. there's the Synergetics Constant), but it's also forward compatible, and that matters too.

Fuller's later writings hearken back to the days before "corporate personhood" when "artificial persons" were more clearly perceived as such. Portland's Thom Hartmann has done some excellent scholarship in this area.

There's another juicy bit in Siobhan's narrative (quoting from his diary) where Coxeter bolts from a Bucky talk, 1500 students or so present, because he doesn't like the frequency mix, which sounds way overinflated, not like anything a conservative academic could ever sponsor or endorse.

This was Bucky the "rock star" doing his exhortationary shoptalk, pretty much all self-invented, and quite intelligently designed (built to work, code worth keeping).

It wasn't purely mathematics or geometry though, this stuff he was saying, these stories he was telling, and whereas western cultures give their pulpit preachers a blank check for fiery fulmination, the social contract is they should in return leave science to the scientists and math to the mathematicians -- a contract many scientists feel is now broken, given all those militant creationists and questioning Pentacostalists ("Sarah Palin types"), other brands of home schooler, many highly unorthodox.

In Coxeter's view (at the time of his bolting), Fuller was simply trespassing too deeply, apparently giving himself credentials he didn't deserve and therefore gaining access to the very same inner sanctum (student body) he would be professionally involved with. If everyone became some "raving 4D-ist" as a result of Fuller's great pirate talk, how would that affect the tenor of his classes? One cannot blame a good professor for feeling like Fuller might be stirring up trouble, as that's precisely what he was doing, and students loved it.

Civil Unions

I'm mostly an onlooker when it comes to the practice of law, not having the requisite credentials to come before a judge in anyone's defense except maybe my own, but I've often wondered if a group of senior partners sufficiently skilled in navigating state codes, could safeguard assets, assign powers of attorney and so on, such that hospitals would have plenty of guidance regarding whom to consult in case of life and death decisions. Dependents would receive willed assets and so on. Transitions might occur relatively smoothly, with a minimum of fuss and bother.

Whether any of these partners were married or not, to each other or to people off camera, would not have to matter, as the contracts would be implemented as business relationships, not domestic partnership agreements.

Perhaps my theory breaks down if minors are involved and the partnership wants to devise some kind of kibbutz-like shared responsibility of a care giving nature. Even if the roles are crystal clear within the "compound", judges have limited patience for any but a few cultural templates and overly creative lawyers tend to give the profession a bad name, or so I would surmise. On the other hand, the cultural template in question might be well established and some thousands of years old.

Thinking of my own case, my first legal agreements with my future spouse were of a business-like nature and based on internally devised codes. I wasn't at that time aware of Ed Applewhite's daughter's work in the area of civil contracting among unmarried individuals, but that's the ballpark we were in, not yet having certified anything with the state regarding our marital status. Had we both been of the same gender, certifying with the state as "married" wouldn't have been an option in any case in 1990, although we could have registered our template with the Quakers (and later did so).

The difference between a "together law firm" bound by clear and definite agreements, and amorphous relationships of an unspecified nature, is when the latter come under duress, because of illness or financial hardship, the surrounding society has no clear "operating manual" and can't easily sort out the disposition of assets.

Encouraging clear and sharable codes is what the "open source" movement is all about in engineering circles, where the word "template" also applies. Given Oregonians have a lot of liberal receptivity to FOSS, it wouldn't surprise me to see "together engineering firms" taking a similar path in future.

About Constructivism

First submitted to Math Forum:

Most of us associate the word "constructivism" with Piaget, who studied how children develop and internalize a model of reality in stages. Not every concept is equally accessible in each age range. There's a progression, and a sense of prerequisites, i.e. a child needs layer A to build layer B and so on. In other words, children need to "construct" their own understanding of the world in stages and this is a dynamic activity involving manipulation of the environment, getting feedback (reality checks) from others, including one's peers (learning as a social activity).

Constructivism was in some ways a response to a more Prussian vision of education as a well oiled machine in which students sat in rows and columns and quietly absorbed (received) the teachings coming from the one authority in the room. Students were discouraged from communicating with one another or teaching their peers. Teamwork and group activities were strictly verboten.

At some layer in cultural archeology, questioning that system comes across as "rebellious" whereas in others it's more common sense that children learn through play, discovery, activity, as do adults i.e. the authoritarian model isn't even remembered, so there's nothing to rebel against. Almost all education in the USA would be considered "constructivist" by some of these older European standards.

What's important to remember is constructivism is not a synonym for "undisciplined" and the play is not without structure or "scaffolding". Rather, the model constructivist is a research mathematician exploring new areas, developing new mathematics. This implies focus, concentration, devotion to the work, not simply recitation of known material or rote memorization of facts i.e. there's inventiveness involved, hence the word "constructivism".

Constructivism might have been called inventivism (not really a word), with students encouraged to develop those habits of mind they will need in order to contribute new thinking, not simply pipeline what's not original with them -- although there's nothing wrong with pipelining others' materials, teaching it to peers (that's also part of the constructivist model: teaching others, not just leaving that role to "the teacher").

Where constructivism was perhaps most successful was in the reform of science education. The experimental sciences are so obviously based in hands-on activities with actual equipment, involve discovery, inducing the rules or patterns in nature from especially designed tests. A key figure in bringing constructivist models to science teaching was Robert Karplus. I've been working in cahoots with one of his main students, Dr. Bob Fuller, University of Nebraska, on various aspects of the physics curriculum, and so have developed some appreciation for his outlook and approach.

Another key figure, in mathematics this time, was Caleb Gattegno. Gattegno took those Cuisenaire rods, colorful rectilinear blocks of various pre-defined lengths, developed by the Belgian mathematician for whom they are named, and built a constructivist algebra curriculum around them. His approach was to label the blocks with letters and introduce the four arithmetic operations using these letters, with students in discovery mode, deducing relationships. This forms a nucleus of the AlgebraFirst curriculum, with proponents at Stanford and the UK.

Since the PC and open source revolutions starting in the 1980s, constructivism has received another boost, as working with computers is clearly an exploratory or investigative activity, a kind of play, although again we're talking about developing self discipline for a later career.

Seymour Papert rebranded his thinking, based around the Logo language, as "constructionism", yet the constructivist legacy was clear. His focus on including machine executable math notations (as Kenneth Iverson called computer languages) fed into a movement that has been gathering momentum ever since, although not always within the USA, or at least unevenly therein.

The twin goals of self study and peer teaching have been increasingly achieved with the aid of the Internet, wherein many subcultures have formed into "teaching communities" each intent on perpetuating a critical mass of skilled users of whatever tools -- the process by which civilizations persist from one generation to the next. A lot of these tools are mathematical in nature, so the next step may be to move more concertedly beyond calculators and more into executable math notations, as is already happening in some schools. This would be consistent with the USA administration's goal of providing a world class education and bridging the so-called "digital divide".

Mathematics as a controller for technology is an old theme, tracing back to Leibniz and Pascal. What has changed is the ubiquitous nature of this technology, as well as our level of mastery and control i.e. electronic computers are no longer regarded as "too esoteric" (or "too expensive") to share with children. Indeed, a youthful demographic is trailblazing the new curriculum, following in the footsteps of some of the aforementioned elders or ancestors.

Kirby

Endnotes: Re Karplus; Re Gattegno; Re Iverson

A Glass Bead Game

We're huddled in the Pauling House with Mario Livio. It's time to play with "glass beads" (metaphorically), the smoothly worn stones of our scientific subculture.

I've just given a "lightning talk", or more a "thunder talk" (somewhat longer), whipping through my slides for Chicago, soliciting feedback, about 4D vs. 4D vs. 4D in particular.

Mario's comments: check out Birkhoff on Beauty; also maybe cast Stephen Hawking as a bridge figure between Einstein.4D and Coxeter.4D as his focus is making time "less special" in terms of its mathematical treatment. Mario is an astrophysicist, so I took this as good authoritative feedback. He also accepted my use of Akbar font, but suggested I vary the color in some slides. Buzz caught a typo (thanks guy).

Julian is here, as well as George Weismann. David Feinstein just walked in (Shomar out in his car, per usual). Lynne Taylor, Allen Taylor... there's a "gang's all here" feeling, although we're missing Jon Bunce (we phoned him last night), Bob McGown. Barbara is in California etc.

We have a young grad student, Valarie, a math teacher and grad student at PSU, whom Terry invited. She encourages more computer use (e.g. Maple), and reports that PCC is better equipped than PSU in this regard.

We discussed calculus, Julian hoping we don't lose the wisdom there, even as we shift the emphasis to more discrete math. Julian is eyeing a $9K differential equations solver as a possible tool for designing new sculptures.

Mario was born in Romania, with Romanian his first language, then grew up in Israel. In a more technologically well endowed "other tomorrow", we might have a live link, to Lionel in Jerusalem etc.

Now Terry is reciting his own ABCs, a complicated spiel to be sure. Buzz interrupts with his "just in time reality" meme. Mario: "nothing" (in the sense of "cosmic zero") simply reflects conservation and symmetry laws, keeps our bookkeeping tidy. From nothing, we tunnel to something.

We're kicking around a lot of "global variables" here, like "universe", absent any strongly shared namespace. This gives our meeting a Tower of Babel flavor. Mario is professionally adept at speaking about dark energy, dark matter, string theory and so on, which is helping Wanderers stay current. His use of the verb "tunneling" is interesting.

Regarding Fuller's "non-unitarily conceptual Universe" it's obvious enough we each have limited bandwidth i.e Universe never squeezes through anyone's thinking as "one thing" any more than a whole dictionary does (Fuller's analogy). We're linear creatures in that sense, chaining thoughts into "trains".

Mario clarifies that not all mathematical thinking is centered around axioms and theorems. He thinks inventiveness occurs around initial concept formation, but then the consequences, the ramifications, the realizations that stem from these concepts, is a process of discovery. Regarding "discovery", in a Wittgensteinian investigation, we'd look for grammatically related concepts like "surprise", "unanticipated", even "synergetic".

Glenn is now giving his presentation, with Allen Taylor taking over on camera. He's summarizing information about "gnomon studies" per Hamlet's Mill and other sources, doing some simple derivations of phi, root-2, root-3, root-5, with a piece of string on the white board. "You need these lengths to build the Platonic polyhedra" he says.

Glenn takes his story back thousands of years before the Greeks, guesses Ea and Enlil, the two tropics, might trace to common ancestors in Africa. A stick in the ground (a gnomon), casting a shadow, is the original computer, and the basis for all trigonometry. Each feather on the winged solar disk, popular with ancient Egyptians, describes a successive gnomon-cast shadow, a measure of big wheels turning in our solar system. Good mnemonics.

We adjourned to Tanh Thao. Mario has remained affable and diplomatic throughout, fielding pitches from every angle. He well deserves a quiet and relaxing afternoon on a beautiful spring day here in Portland.

Contemplating Mathematics

Mario Livio is contagiously passionate about his subject, a good thing.

PSU students in 53 Cramer were clearly into his rap, including the doubled slides (adjacent duplicates). During Q&A they wanted to hear more about this "intelligent jelly fish" that wasn't into natural numbers or discrete math, yet was nevertheless mathematical in its outlook -- a nice philosophical chew toy.

Mario's questions are about (a) why math proves relevant even when it's practitioners may be proudly dismissive of "the real world" and not looking for applications and (b) whether math is discovered or invented.

To the first question, he feels we're likely constituted in such a way that we can't help but reason effectively (we've been conditioned by long experience, inheriting from predecessors).

For every other-worldly geek there's another equally driven to find those practical applications. These possibly not-mathematicians act as enzymes or catalysts by connecting the dots, maybe tying knot studies -- originally proposed for atomic modeling (not so useful) -- to computing the energy required for DNA transformations (a current application).

Fuller's 10 * f * f + 2 would be another example, in connecting icosahedral numbers to the structure of the virus.

Mario's funniest joke, perhaps unintentional, is when he recapped Descartes' "cogito ergo sum" argument, then said, appreciatively, "an unbelievable guy!" (meaning "awesome brilliant" but you appreciate the irony maybe).

He focuses a lot of the "lore" of that discipline (western civ math), which fact I hope to bounce off tomorrow during our confab.

I've been invited to riff off his topics and will do so by introducing my "lore axis" then diving head first in to the 4D vs. 4D vs. 4D thread I've been developing.

He might find that fun, plus I'm guinea pigging my Chicago talk, hoping for some useful feedback.

Mario's storytelling takes us from Plato etc. to the "three worlds" of Penrose, to Bertrand Russell and his letter to Frege, pointing out a paradox. That letter wasn't really about the barber who cuts his own hair (when off duty? -- so not a barber then?), but about the set of all sets that don't contain themselves, or something similar.

Godel's Incompleteness Theorem gets touched on, reminding us of how axioms don't need to be "true" are more just "rules of the game" e.g. the rules of chess aren't "true" just give us a fun pass time (Don and I actually played most of a game of chess at Backspace, passing time until Dr. Livio's next after-dinner appearance).

Indeed, we probably shouldn't go around calling axioms "true" because that confuses them with proved theorems. Axioms are "assumed" and/or "presumed" and/or "postulated" (called "postulates" sometimes) i.e. we're allowed to make them up out of whole cloth, no proof required.

I was happy to take in this talk (and the slides) two times within the space of a few hours, the 2nd time at Powell's Technical. Terry, the consummate roadie, lugged the equipment in his van.

Once again, the audience was large and attentive and asked a lot of good questions.

Mario was generous with his time, enthusiastic with his answers, and signed quite a few books.

In Terry's van later, returning Mario to The Heathman, we yakked about the sometimes stressful aspects of jetting here and there on speaking tours, how one needs to fly through hubs in curious ways.

Mario is bothered by the turbulence out of Denver, lost a pilot friend to that sometimes unforgiving weather system (in a twin engine plane).

I didn't ask any questions out loud, was mostly thinking how the "three worlds" of Sir Roger might be reduced to two, i.e. the physical and the psychological.

I see no need for Plato to have a third special world all his own, where all the perfect stuff lives. His is just another psyche and we each have one of those (slave boys too), a private sky or private Idaho.

I have an easier time dispensing with "perfect circles" (Platonic) because of my Princeton training in Wittgenstein. Synergetics came later, with a different idea of perfection (as in "not merely imaginary").

Platonists (also called "Realists" -- versus "Nominalists" an opposing camp) depend on a mental model of naming, familiar to Python programmers. In this conventional view, names point to objects and these must be Platonic if the math is truly pure (a bias).

But what if "pointing" isn't really what's going on so much? Rorty calls Wittgenstein a "nonrepresentationalist" -- another way of saying he outgrew the Augustinian model, in turn Neoplatonist, ergo Platonist, and based on names pointing.

Instead of a Platonist, I could say I'm a Play-Doh-nist, an allusion to Claymation Station i.e. our "geometry of lumps".

The question of why math keeps syncing with nature might point us back to the fact that we we both actively invent and passively discover our own psychological namespace, mathematical to the extent that it's precise, self-consistent, and well-ordered (logical), if it is (depends on who's, how much homework, housekeeping).

While at Powell's, I snapped some pictures of really cool books, including ones about innovative dwellings and XRL (extremely remote living).

Glenn showed me a thick book called Space Structures that (a) omitted Dr. Fuller from the index (par for the course) yet (b) mentions geodesic domes and credits Fuller and his licensees for doing most of the work in that area. The U.S. Marine projects get mentioned.

Meditation on Math

Mario Livio will be in town for the next couple of days, making appearances at PSU, Powell's Books, and the Linus Pauling House.

His newest book has sparked some discussion on the Wanderers list.

From Mario's chapter one (Mystery):

A few years ago, I was giving a talk at Cornell University. One of my PowerPoint slides read: "Is God a mathematician?" As soon as that slide appeared, I heard a student in the front row gasp: "Oh God, I hope not!"

I contributed this essay this morning...

---

Re: is God a mathematician (Cornell student: "oh God I hope not"), I'm thinking of Keith Devlin's book about non-human creatures and their significant processing powers in physical interaction with their environment

Some say "that's instinct" which means they don't have to think about it -- like humans don't have to think about thinking or using language (we "just do it" -- with practice).

So where is the line between language and not-language, between mathematics and not-mathematics?

With university trained faculties (a double meaning) we're conditioned to compartmentalize and see "math" (M) over here, "not math" (~M) over there, with walls between language and not-language (L|~L), thinking and not-thinking (T|~T).

That's a view I'm inclined to counter (see summary below).

When you get really into it, a theology can seem precise, rigorous and logical, no accident that many mathematicians were fervent in some faith or other.

That sense of seeking something beyond, but as a corporate process (with others) gives rise to debate, revisited threads, a set of concepts that interact, get fine tuned.

Adherents feel comforted by the "rigor" of their dogmas and enjoy reciting, reviewing, reinforcing their creeds.

I am not alone in thinking it makes sense to see "God" as yet one more operational signifier, at work (or not), within language games, like a "zero".

We also have uses for "time", "nature", "balance"... these are difficult words too, in the sense that it takes a lifetime to appreciate 'em (and then some).

I tend to like polytheisms when it comes to "balancing energies" e.g. the village elders and younger learners, future leaders, sit around the campfire and debate what to do next.

If we're in some routine period when the environment seems unchanging (rhythmic but constant), aren't being invaded or attacked, then those background patterns get taken for granted.

But humans themselves tend to evolve quickly (using memes more than genes) and that brings about change, including to the environment. Change becomes the accepted backdrop then.

In any case "pleasing the gods" is a psychological exercise in keeping the tribe viable.

A lot of western writers (Nietzsche, Freud, Norman O. Brown) will pick up Apollonian and Dionysian as their polarity, but is one polarity sufficient? Multiple antipodes suggests a sphere, a polyhedron, working to stay centered.

All of which is to say: I think natural processes, including that of humans doing their best to anticipate the future, adapt, keep an even keel, are mathematical processes, whether or not one uses the ideas of axioms and theorems.

Furthermore, thinking about God, deities of any kind, ancestors, projecting psychological qualities into various creatures and balancing them, is a kind of information processing, and therefore mathematical in nature.

Finally, I think we all experience not being fully in control, i.e. are part of a larger process in which we find ourselves that "has a life of its own" (Ouija board, invisible hand...) and there's this real phenomenon of prayers being answered, dreams coming true, thoughts proving reliable -- as well as the contrary.

Our sense that "God is a mathematician" derives from this sense of integrating into a larger picture that's also "thinking" (i.e. "doing the math") in some way. We find ourselves rewarded for thinking logically. We feel God must be too, or Nature, or... (something more than "just me" or "just we").

There's this prevalent stereotype of "religion" (which fits in some cases) as being primarily about believing something impossible / incredible, and making one's "faith" a test of one's willingness to sacrifice one's better judgment and/or powers of reason, on the altar of some cosmic fairy tale.

However this is just one of many Hollywood movie situations.

Having a belief system (ideology, paradigm) that's not adapted, not serving, is a cause of suffering for sure, so any strong tradition (science included) is going to have processes for surviving implosion, rebirth and reconstruction of the psyche. No one is immune from having to let go of old certainties from time to time. What once seemed reliable, no longer does.

Let's think of the American Dream as a shared fantasy, and with the added advantage of being secular in some way, even though there's some eye on a pyramid. Per those National Treasure movies, it's secular more in the sense of being "all traditions" (lots of heritage), not any one of them lording it over the others, no "official religion" backing an emperor, pope or king (for so long the pattern, democracy not trusted, thought of, or tried).

No "official science" either, lots of competing paradigms.

Still that need for balance. Plus the environment is changing. Plus we're aware it's a small planet. Our mathematics needs to be pretty strong, is my feeling. That heritage will serve us, is my hope. I say "our" as I'm including myself as one of the American dreamers here.

Per Hegel, there's a relationship between Logic and History we need to be thinking about. I've written a longish essay here, probably few have the patience. I'm doing the math, reasoning to a point where we see the future of the USA, lots of components and subsystems, as a mathematical challenge.

What I'm balancing against (countering), is this university-inspired notion that "mathematicians" are just this select group in the Ivory Tower, proud of how their discipline "has no meaning" in terms of reality.

Making the "real world" be something distant, exotic, is a literary conceit of many academic cultures, but that's for anthropologists to study, has nothing really to do with the ongoing computation.

I'm not saying "their" work (including some of "mine") is unimportant or irrelevant, merely that I don't choose to see "mathematics" as some esoteric process that takes place exclusively in some rarefied academic environment.

I know how to talk as if that were the case (i.e. I'm as "schooled" as the next guy), but I don't really believe it.

So, on with the math! In God we trust (or in Apollo or whatever).

Kirby

Launch!

:: a fine grind production ::

In late 2008, I started branding some works as Fine Grind Productions, named for a local coffee shop, then owned and operating by artist in residence Jody Ahn, with assistance from her capable staff.

FGP's first productions were a slideshow about Py3K (Python 3000, a backward incompatible leap into the __future__ by the Python computer language), and a backdrop projection for my debut as a "Dymaxion Clown" at Portland Center Stage (IEEE lecture, election night, 2008).

The Py3K slides presaged my focus on coffee shops as an outlet for exclusive high definition content, including interactive games with an eye towards supporting worthy causes.

Our customers themselves are "worthy causes" and our charitable giving games help customers build their profiles, chronicle exploits, even while supplying vendors and potential sponsors with market-relevant analytics.

Unlike my others, this blog will exploit AdSense and Adaware technologies.