Saul-Paul Sirag's New Math Book

Saul-Paul Sirag in Eugene, Oregon plus some Japanese numerals

SAUL-PAUL SIRAG'S NEW MATH BOOK

Back in the 1950s Reader's Digest ran a monthly feature "The Most Unforgettable Character I've Ever Met". In my long life I've met many such characters but Saul-Paul Sirag must be near the top of the list.

Saul-Paul was born in Borneo of Dutch-American parents who were fundamentalist Baptist missionaries to the Dayak headhunters. When Japanese troops occupied Indonesia in 1942, they imprisoned Saul-Paul, his older brother Mark and their mother Sylvia in the Banju Biru concentration camp in Java. His father, William, was interned in a separate camp where he died before the end of the war. Each morning the occupants of the camp had to line up in front of their cell blocks to count off in Japanese. I like to think that this prison-camp countdown was Saul-Paul's first experience of an exotic form of mathematics.

Back in the States, Saul-Paul was educated in various ultra-conservative Baptist schools, most notably Christ's Home in Warminster, PA and Prairie Bible Institute in Three Hills, Alberta, Canada. During high school in Canada both he and his younger brother David showed exceptional interest and ability in physics and math. In 1960 Saul-Paul graduated from PBI and soon ended up in Berkeley (Berkeley in the 1960s!) where he continued his science education but also befriended a number of artists and musicians, notably Don Buchla and Charles MacDermed.

Saul-Paul Sirag plus a Hair poster

Energized by 1960s Berkeley, Saul-Paul moved to New York City where he reviewed books, wrote and acted in plays at La MaMa in the East Village, participated in an Andy Warhol Fashion show and, by his own account, inspired a Broadway hit: "Jerry Ragni, who wrote Hair, told me that he got the idea while watching me dance to the Grateful Dead in the East Village." Ah, Saul-Paul Sirag: woolly-headed icon of the Age of Aquarius.

By the early 1970s Saul-Paul was back in Berkeley where he resided at Arthur Young's Institute for the Study of Consciousness (ISC) on Benvenue Street, a kind of boarding house and meeting hall for people interested in the mystery of human awareness. I first met Saul-Paul in 1973 at ISC and was impressed by his uniqueness. Everyone was into Tarot cards at the time but Saul-Paul had created his own private deck, made up of images (including porn) that had significance only to Saul-Paul himself. Saul-Paul was privately schooling himself in math and physics and was writing a popular science column called "The New Alchemy" which was syndicated in college newspapers. ISC was a nexus for people interested in ordinary and altered states. And through Saul-Paul and his friends at ISC I was able to meet many smart people who helped me write my own book on consciousness.

Saul-Paul's room at ISC was stacked floor to ceiling with unusual books. One of my favorite finds in the Sirag scriptorum was a German-language edition of John von Neumann's classic Mathematical Foundations of Quantum Mechanics which announces behind the title page: "Copyright Vested in the U.S. Alien Property Custodian 1943, Pursuant to Law." I don't know exactly what this means but it conjures up the image of a famous German physics book seized by the Allies as "spoils of war." ISC was only a few blocks from the UC Berkeley campus. No day spent with S-P was ever complete without a jolly tour of campus bookstores in search of exotic codices. This was before the Internet, when bookstores in Berkeley were as common as pizza shops.

Saul-Paul was a founding member of the Consciousness Theory Group (CTG) which held meetings first at ISC, then later, across the Bay in San Francisco when Saul-Paul moved into Henry Dakin's Washington Research Institute (WRI) where both consciousness and the Soviet Union were central topics of concern. While at WRI, Saul-Paul served as president for a few years of the Parapsychology Research Group (PRG) in which Russell Targ (of SRI Remote Viewing fame) was a prominent participant. While at WRI, Saul-Paul devised an ambitious hyperspace model of consciousness which, unfortunately, only Saul-Paul was able to understand.

A description of S-P's hyperspace model appears in Jeffrey Mishlove's authoritative tome The Roots of Consciousness. Saul-Paul penned the Afterword for Robert Anton Wilson's Cosmic Trigger and published at least two papers on obscure physics phenomena in Nature, Earth's most prestigious science journal. Saul-Paul has also produced an irreverent and unpublished manuscript about his school days at Prairie Bible Institute entitled Jumped by Jesus.

Saul-Paul Sirag at Esalen Institute (1980 - 1988)

 During the 1980s, Mike Murphy invited Saul-Paul and me to host invitational seminars on physics and consciousness at Esalen Institute in Big Sur, California. The major focus of these seminars was Bell's theorem which proves that if certain quantum experiments are correct, then reality must be "non-local". Bell's theorem is unique in that it is a statement not about experiment nor theory but about "reality itself". Moreover, BT is no mere conjecture concerning the nature of reality but a rigorous mathematical proof.

Many of the participants in the Esalen Seminars on the Nature of Reality were members of the Fundamental Fysiks Group (FFG) founded in Berkeley by Elizabeth Rauscher and George Weissman as recounted in David Kaiser's recent book How the Hippies Saved Physics.

Saul-Paul was also present at the creation of Sirag Numbers and Siragian Triangles, two small mathematical curiosities with no known real-world application.

Saul-Paul Sirag's New Math Book

Earlier this year (2016) Saul-Paul published a new math book ADEX Theory: How the ADE Coxeter Graphs Unify Mathematics and Physics.

Saul-Paul Sirag and the five Coxeter graphs

ADEX Theory is basically a taxonomy of mathematical objects similar to the classification of life forms on Earth, into fundamental categories such as kingdoms, phylums, classes, genuses and species. All animals, both alive today and in the fossil record, can be organized into only 16 phylums with names such as Protozoa, Anthropoda, Mollusca and Ctenophora. Humans belong to the phylum Chordata along with cats, dogs, lizards, birds, sharks and sting rays. "Possessing a backbone" is the passport to membership in phylum Chordata.

Compared to mathematical objects, life on Earth is easy to classify because life has a common origin and all life forms are members of a single tangled family tree. A mathematical object, on the other hand, consists of any structure whatsoever that the human mind can invent and find interesting. Categorize the human imagination? Impossible.

Saul-Paul describes in ADEX Theory the successful classification of a large variety of seemingly unrelated mathematical objects using only five "mathematical phylums" called Coxeter graphs (after Harold Scott MacDonald Coxeter). Here I will confess from the outset that the task of reviewing this book is well above my pay scale. Only a person with considerable mathematical sophistication can truly appreciate ADEX Theory.

What amazes me, as a mathematical pedestrian, is 1. the enormous variety of mathematical objects that can be brought together into a few communal families by this simple classification scheme and 2. the peculiar nature of the five "mathematical phylums" that are able to tame this giant population of unfettered figments of the human imagination.

Get this: the mathematical phylums, called Coxeter graphs, are symbols for something called Reflection Groups which are analogous to a set of mirrors in multidimensional spaces. Each Coxeter graph stands for, if I understand things correctly, a particular KALEIDOSCOPE in hyperspace. So, instead of a backbone, each member of the same mathematical phylum possesses, in some sense, the same brand of kaleidoscope.

And here is a partial list of the variety of mathematical objects which Saul-Paul shows how to classify using just five mathematical phylums (five Coxeter graphs):

1. The Coxeter reflection groups, or Weyl groups

(the objects these diagrams were originally designed to represent)

2. The Thom-Arnold "simple catastrophes" of dynamical systems

(the "butterfly catastrophe" lives in phylum A5)

3. Digital error-correcting codes

(Hamming-8 code lives in phylum E8)

4. Knot and Braid theory

(E6, E7, E8 phylums important for classifying knots)

5. Maximal quantum entanglement of 3 qubits

(related to the E7 phylum)

6. Roger Penrose's "twistor theory"

(twistors live in every ADE phylum)

And that's just some stuff I can almost begin to understand. Saul-Paul goes on to show how ADE phylums can help organize Lie groups, Klein-DuVal singularities, McKay correspondence groups plus Kac-Moody and Heisenberg algebras.

Moving on to more practical topics, the guy that inspired Hair uses ADE to clarify and unite topics in string theory, black hole physics, the holographic principle plus his own rendering of the possible ADE underpinning of our current Standard Model of elementary particle physics.

Oh, and this theory can also help us understand something called "quivers".

This book should appeal to specialists in any of the above-named fields who will appreciate Saul-Paul's passion for these esoteric topics and his fine attention to detail. In a rare literary aside, Saul-Paul shares this: "In his preface to Regular Complex Polytopes, Coxeter wrote: 'Its relationship to my earlier Regular Polytopes resembles that of Through the Looking Glass to Alice's Adventures in Wonderland,' This book which Coxeter says he 'constructed ... like a Bruckner symphony' is the most beautiful mathematical book which I possess (among several hundred)."

And you there reading these words now. Which is your most beautiful book?

The author of ADEX Theory lives in Eugene, Oregon with his wife Mary-Minn.

Saul-Paul Sirag at the Eugene, Oregon Ken Kesey statue
plus the first nine Sirag Numbers

Harry Houdini Metaphase Breakout Challenge

Saul-Paul Sirag at the Ken Kesey Memorial in Eugene, Oregon

In the history of American counterculture, 2015 is a year for celebrating anniversaries: a few days ago I attended a celebration in Santa Cruz of the 50th anniversary of the First Acid Test initiated by author Ken Kesey, Ken Babbs and the Merry Pranksters to spread the gospel of Lysergic Acid Diethylamide to the unstoned masses. To honor this important historical event, the city of Santa Cruz is constructing a memorial bus stop near the site of Ken Babbs's farm (long since replaced by fancy condominiums). At the celebration in Bookshop Santa Cruz, I met Ken Babbs and talked about his role as an editor of the Kesey-inspired magazine Spit in the Ocean, one of the few printed artifacts to emerge from that era's participants -- whose insights and epiphanies, except perhaps for The San Francisco Oracle, Tim Leary's numerous publications, Tom Wolfe's picturesque report, Be Not Content and a bunch of poetry, were mostly non-verbal.

Spit in the Ocean # 3, edited by Dr Timothy Leary, was dedicated to the subject of Jailbreak and featured a story by Saul-Paul Sirag and Ken Babbs in which I was directly involved. In the early '70s, while working at a Smith-Corona research facility in Palo Alto CA, I invented and constructed the Metaphase Typewriter, which transforms quantum uncertain processes into pseudo-English text. I was hoping that spirits (whether living, dead or inconceivably other) might possess this machine and turn its quantum-random text into intelligible communication.

My friends and I operated the Metaphase Typewriter in many "high-energy" psychic environments,  producing a few remarkable synchronicities, but no sustained clear-text message "from the other side". MIT professor David Kaiser describes the MT in his best-selling How the Hippies Saved Physics and Queensland, Australia artist, Lynden Stone replicated the metaphase device using 21th-century tech and generated real-time quantum text messages for Philadelphia museum-goers. One of our most memorable Metaphase Typewriter experiments was held in the basement of a medical research center in San Francisco on the hundredth anniversary of magician Harry Houdini's birth. Saul-Paul wrote a mostly accurate account of that event for SITO to which Ken Babbs added a fanciful ending.

HARRY HOUDINI CENTENNIAL BREAKOUT CHALLENGE

A Heisenberg-uncertain typewriter has been set up at an undisclosed Northern California research center. Its sensitive inner quantum mechanism appears to be free enough from every known physical law to permit takeover as a communication terminal by sufficiently skillful discarnate entity. Metaphase Typewriter is a presumptive open mike to the Void. Should you decide to accept this challenge, HARRY HOUDINI, and successfully impress your intentions upon the stream of random anagrams endlessly flowing from the teleprinter, you will be warmly welcomed by our little band and most justly ranked among the great masters of escape.

 

 This is it.

That's the thought that went through my mind when the metaphase typewriter began spewing out letters.

We've made contact.

With who or what wasn't clear, especially since only the first line made any sense to me. But let me give you the setting.

Though a series of fortuitous events I found myself to be part of a newly forming research group whose avowed aim was to develop a physics of consciousness. I had a rather polyglot background in theology, philosophy, mathematics and physics -- in such order. That the world is stranger than I had been led to believe I learned for myself during numerous psychedelic trips. Lately I had been investigating psychic phenomena and related fringe activity, such as unidentified flying objects. I was trying to fit my findings into a pattern, but they just wouldn't jell. There was too much to look at all at once. Gradually it dawned on me that until I could understand the physics of ordinary, everyday consciousness, other unexplained phenomena just couldn't make sense, About then I ran into Manny Hilbert, a nuclear physicist working for a typewriter company in Palo Alto. Out of his private musings and researches the metaphase typewriter was born.

Actually the metaphase typewriter isn't a typewriter at all. It's a teletype machine hooked up to a small computer and the computer is hooked up to the output of a geiger counter, recording beta decay events in a small sample of a radioactive element. Its function is to provide a clear channel of communication for disembodied conscious entities -- spirits, if you like.

For Manny, the metaphase typewriter began as a joke, a tongue-in-cheek way of challenging the far-fetched but intriguing theory of Harris Walker that consciousness functions as a set of hidden variables in a quantum mechanical system. It ended for me on a dreamlike note, a no-audienee display of human consciousness roaming beyond science's wildest expectations.

On the hundredth anniversary of Harry Houdini's birthday, Manny gathered a small group of consciousness researchers around his newly set up metaphase typewriter in the borrowed, cramped computer room of a little-used section of a large San Francisco research complex. Hours had been spent altering the hardware of the borrowed Nova computer and using Manny's program which would translate time intervals between beta decay events (electrons streaming out of the nucleus of Thallium 204) into second-order English language statistics and thence into rapid-fire teletype print-out. The more likely the length of time between electron emissions from the Thallium, the more likely would be the English letter combinations printed by the teletype -- one recorded emission would produce one printed letter. A rather suspect communication channel you might think. But then you haven't encountered a mind as strange as Manny Hilbert's, or been entertained by his poetic notion that "Quantum mechanics says the Universe is a randomly strobed four-dimensional digital display."

The word "display" suggests that there is something behind the computer putting on the display. For Manny, it is the realm of mind, or spirit, or sub-quantum level -- take your pick. The metaphase typewriter was to be a literary doorway to this realm, a non-protoplasmic spirit medium. It was not his fiendish sense of irony that led Manny to pick Houdini's 100th birthday as the day to open the quantum-mechanical channel to the spirit world. Rather it was the fact that although Houdini had made himself famous for unmasking spirit mediums, he had issued a promise to his friends before he died. He would return, if at all possible, and communicate!

If anyone could escape from the quantum sub-levels, Houdini, the great escape artist could do so. To reinforce this notion, Manny had prepared pictures of Houdini in his many attitudes of constraint: in a coffin being lowered into into San Francisco Bay; upside down in a water tank on stage (it was after such a stunt performed successfully while suffering from an internal rupture that led to Houdini's premature death in 1926) [unfortunately, no photos remain of that tragic escapade]. And now with these pictures staring down on the metaphase typewriter, Houdini was being issued a new challenge.

Speak to us from beyond the grave. Though you have no vocal cords, no body, we have offered you a channel in which to encode your spirit.

It was a preposterous proposition. But what if the world really is a randomly strobed 4-D digital display with consciousness as a wild card -- what then?

 "What a crazy idea!" said the janitor who had come in to clean up, when he found out what was going on.

"But don't you understand? Manny thinks the underlying substratum of the physical world is linguistic in nature."

The janitor pondered. "Why English?" he said.

"Any language will work, just so somebody can read it. On a deep level each of us here in this room are part of the sub-quantum realm, according to Walker's theory, and this implies English."

"You mean it might be we ourselves influencing the beta decay?" he said, a light dawning.

"Sure, if English comes through, that could be a way to tell."

He stood thoughtfully, finger tapping pursed lips, I waited for a minute, but when he still didn't answer, I shrugged and turned away.

It was time for Manny to test the hook-up. He gingerly pushed the manual advance button. Nothing happened. There could be a thousand things wrong -- it was a borrowed computer.

Fortunately the janitor, Olaf Johnson, turned out to be a electronic hardware freak and a magician with computers. There's more and more science geniuises like him refusing to work for big business and big government and supporting themselves with physical labor instead. The janitor and Manny conferred and soon lights were flashing again on the Nova computer's front panel.

"It's working but the paper isn't feeding," Manny exclaimed. Line after line of output was being spat out on the same space. I was if somebody were trying to get out and couldn't make it, couldn't push open the door. Manny looked over and saw what was wrong: the paper was in crooked. He quickly pulled it straight and the transmission began. But wait a minute, the first lines were dingbat strange, letters printed on top of letters, lines skewed this way and that. But in the middle of the chaos, framed by itself in the jumble of characters sat one clear phrase: anininfinitime.

"Maybe it's trying to say that at this rate it'll take an infinite amount of time to get a decent message through," I said.

"Ja," Manny replied dejectedly. "It's telling us about the monkeys at the typewriter, that at this rate it'll take an infinite amount of monkeys typing forever -- and they'll never come up with Shakespeare or the Bible."

By this time pages and pages of gobbledegook -- bearing no resemblance to Shakespeare and company, rather a coded WWII message needing to be broken --had come through. "Aw shit," Manny said, throwing his arms up in the air and heading for the beer and snacks the women had provided upstairs.

"Anininfinitime." The word-phrase repeated itself in my mind over and over as pages and pages of nonsense streamed from the metaphase teletype machine, until only Houdini's promise remained unfulfilled unless -- yes, that was it. That's what he was trying to say. Would the metaphase typewriter work? Would we be able to identify Houdini for sure. Yes. In an infinite time. We were given all the time we need, even infinite time, What an assurance to go ahead!

I bounded out of the computer room to tell Manny the good news but I couldn't find him anywhere. I considered writing off the whole event as a joke, writing it up as a spoof to send to Martin Gardner for the amusement of his Scientific American fans, but then I noticed the janitor in the corner watching the growing pile of printout in thoughtful contemplation.

"It's too slow," he said.

"What?" I answered, amused.

"That's why you're not getting anything that makes sense," Olaf said. "The channel you're dealing with is a high-energy stream conducting information faster than your machine can pass it on. You're only getting one phrase out of a hundred,"

"Well," I said. "If you've got anything better to add, cough it up. Everyone else has given up. And I'm about to join them." I took one last frustrated look at the metaphase typewriter relentlessly spitting out its gibberish and prepared to join the party.

"Cygnon-17, " Olaf muttered quietly.

"What," I yelled, whirling around. "Are you mad? Possession of Cygnon 17 is a mandatory twenty years in the slammer. Besides, there hasn't been any Cygnon 17 around for years."

"I've got some," Olaf said, no hint of superiority behind his friendly smile. He opened his hand and held it out for me to see, a glowing pink capsule radiating a light, shiny glimmering aura: Cyg-17 without a doubt. I hadn't seen one in fifteen years, not since its mind-altering experiments had been ruthlessly banned. Cyg-17! I had only taken it once but that was enough to convince me its powers were greater than LSD, MP-14, STP or any of the other hallucinogens scientists, sociologists and self-proclaimed astronauts of inner space were ingesting. When Cyg-17 took you on a trip you literally went there. It was the same for everyone who took the drug. Everyone who came back insisted they'd been in a different place, a world in which our straight-laced laws of physics were loosened and lost in a welter of rubber-like, opened-junctured rules, a world in which there were no ends or beginnings, instead open holes through which unending dramas flew.

I suddenly wished Manny hadn't given up and gone elsewhere. His goofy metaphase experiment was taking an unexpected turn.

"I'll drop the capsule and plug into the computer with earphones bypassing the metaphase teletype..." Olaf was mumbling to himself.

"Are you crazy?" I said. "That's a twelve-hour commitment. Outcome totally unpredictable."

"It'll take that long to absorb the computer language well enough to translate into my own. I know the risks. I'm prepared to play that card."

I knew Cyg-17's powers. The contact high off the person taking the drug was fantastic. "All right," I said. "We'll try it. I'll volunteer as ground control."

I unhooked the teletype machine and plugged in the earphones, watching Olaf swallow the capsule out of the corner of my eye. He puts on the headphones.

I take out a notepad and pen and sip a cup of cold coffee. "Tell me everything you see or experience."

I grabbed my pen and began writing feverishly, keeping pace with Olaf's steady stream of words.

"...a metallic taste in my mouth, tips of my fingers tingling, my stomach revolting (he turns and calmly retches into the wastebasket), vision narrowing and concentrating to a single point of light at the end of a long tunnel I'm rushing down faster and faster, penetrating inward, elastic boundaries stretching, breaking, membranes opening, holes enlarging, portals passing, free-space beckoning, I'm through, loose on the DNA, time-energy stream, hurtling past centuries and planets, traveling instantly everywhere the life code has run and will ever have been..."

Olaf was taking me there too, but I hung on to the Grail: "Houdini, man, " I stammered through clenched teeth.

"Next stop, London," Olaf announced in his best tram car conductor's manner. "All out for Swyndomme on Marne, curses if this ain't the damnest fix I ever got myself into," his voice part cockney, part blarney.

"Who are you, man?" I asked, sweat popping off my forehead, droplets spattering the pages of my notebook.

"Who? Who, you say but were I to clue you there would be no who...who indeed, sirrah? Ask yourself who was the greater, me or that French blowhard who claimed I learned all my escapes from him. Whereas I and the world I have proven it to know it was the other way around. The world, alas..."

His words faltered.

"It's you then, Houdini?" I said, giggling with happy inspiration, then immediately grasping the seriousness of the situation, I tried to center myself in ordinary reality. His words came to me from a long echoing distance. Forcibly, with bells clanging, I yanked myself up into a sitting position and continued transcribing.

"...and once the act was booked, I had to go through with it. The great Houdini never goes back on his word. I made good the escape but the strain was too much. Who would have guessed the irony? To escape from death so many times and then, when death came, to fail the simplest task of all. A turned head, a glimpse of passing skirt, the outbound train was gone and I was left at the station, stranded, unable to get away."

"I've paced and paced and thought and thought ever since. It should be a simple matter to leave on my own. My mind, a cold crystal, has analyzed every conceivable possibility: to escape Earth's gravitational pull, the atmospheric blanket, Van Allen's radiation zone, and the final Thyandrocal membrane. They must be breached in a single maneuver else I exhaust my energy in successive struggles. You see, I must escape Earth. First the acceleration, then slide edgewise like a knife, then spread cellular tissues wide enough for atomic particles to pass through, but the last, the Thyandrocal layer, the escape eludes me, telescreen blur of spaatering, craxkled, alarffed, gzggle to dark madkavlod marganalkindlomejj..."

Olaf or Houdini or whoever he was lurched to his feet and was weaving towards the teletype. The alien creature inside Olaf's body forced the janitor's bony arms and fingers into the teletype machine. Smoke curled out of the electrical cord. The keys leapt to life. The human apparition, once known as Olaf stood eyes closed, earphones clamped on his head, hands on the machine. The keys clattered in a thundering din.

I rushed to the teletype. Words were pouring onto the page. My eyes devoured the impressions.

AHH, MUCH BETTER, THE DISPLAY PATTERN HAS OPENED UP AND THE THYANDROCAL MEMBRANE IS NOTHING STICKIER THAN BLACKBERRY JAM. NOW IF I CAN JUST SOLVE THE PROBLEM OF THE PHYSICAL STRUCTURE I MUST ASSUME IN ORDER TO SLIP BETWEEN THE SHEETS -- THAT'S IT! HYPERBOLICAL PARABALOID FACTOR NEGATIVE INVERSE RADIUS AND I HAVE IT. MY GREATEST ESCAPE EVER.

The teletype scorched and crackled. Sparks flew from the keys. The metaphase sputtered and quit. Olaf staggered and fell into a chair. He covered his eyes with his hands and moaned: "I helped Houdini go though his last cell door. Harry's finally escaped the chains of Earth and his spirit now sours unfettered throughout the Universe. But tie me kangaroo down, boys. Olaf's only two hours into this twelve-hour Cygnon-17 chemical odyssey!"

Spit in the Ocean #3 and Ken Babbs at Bookshop Santa Cruz

Siragian Triangles

The square of the hypotenuse of a Siragian Triangle equals J(J+1)

Pythagoras's Rule for the calculation of the lengths of the sides of a right triangle is familiar to everyone. So well-known is this piece of elemental math that it has been repeatedly suggested that humans might construct giant replicas of the Pythagorean Theorem on Earth's surface as a means of signaling to aliens. As a warning perhaps that humans had progressed to the level of high-school geometry -- and regressed to a high-school level of adolescent and worse behavior toward our fellow beings with whom we share the Earth.

Motivated in part by analogy with the quantum mechanics of angular momentum and partly by mere curiosity, I introduced a modification to Pythagoras's Triangle which I named after my friend Saul-Paul Sirag. The Siragian Triangle differs from the Pythagorean Triangle only by the way in which you describe its hypotenuse. Phythagoras uses the symbol Z. Sirag uses the symbol J where Z and J are related by the expression:

Z^2 = J(J+1)                         (EQ 1)

The Sirag gambit does not alter the geometry of the right triangle but is merely a conventional change of variables similar to the choice to use Centigrade rather than Fahrenheit degrees to measure temperature.

One of the questions that mathematics have asked about the Pythagorean Triangle is this: How many right triangles have sides whose lengths are integers? The resulting 3 numbers for the integral lengths of two sides and the hypotenuse are called a Pythagorean Triple and can be symbolized PT(X, Y, Z).

The most familiar Pythagorean Triple is the famous 3-4-5 right triangle, but lots of integral-sided Pythogorean Triangles exist.

Similarly if one looks for integral solutions to the Siragian Triangle, one is led to the notion of a Siragian Triple ST(X, Y, J). Shortly after introducing the Sirag naming convention, I calculated a few Sirag Triples on my pocket calculator and committed a little elementary arithmetic in the privacy of my home.

Saul-Paul was immediately inspired to take the question further. "This problem reminds me of my favorite book," he said. His favorite book tells you a lot about Saul-Paul. It's Albert K. Beiler's Recreations in the Theory of Numbers: The Queen of Mathematics Entertains.

(Apparently that branch of math called Number Theory is widely recognized as "the Queen of Mathematics.")


With our meager little Sirag Triple problem in hand, opening the cover of Beiler's book was like walking through a door from a dingy alleyway into the crystal palace of the Queen of Mathematics. Our precise question was not answered in Beiler's book but we did find many closely related math problems that had caught the attention of Western mathematicians as well as Chinese, Arabs, Indians and Greeks. Saul-Paul's favorite book does indeed contain the keys to solving the problem of Sirag Triples, but for the mathematically adventurous, it holds many more treasures besides.

The Square Numbers S (N) = N^2

I learned in Beiler, for instance, about the concept of Square Numbers, which are not numbers that lack hipness, but numbers that can be formed into a square. The Square Number S(8) = 64, for example, represents the number of squares on a chess board.

The Triangle Numbers T(N) = N(N+!)/2

 Also in The Queen of Mathematics Entertains I encountered the concept of Triangular Numbers which, as their name suggests, can be arranged in a triangular array. The most familiar example of a triangular number is T(4) = 10, which pictures the way that pins are set up in a bowling alley.

But what do these cute numerical squares and triangles have to do with the Sirag Triples? Quite a lot, as it turns out. For our work in Her Majesty's service, the Queen of Mathematics Herself bestowed on Saul-Paul and me a most marvelous gift. A gift that shines. A gift that will last forever.

What we expected: Two long strings of numbers.

What we got instead: A mathematical miracle.

To understand the nature of our gift, consider the special case of the isosceles Siragian Triangle, a right triangle that possesses two equal sides, which I will call "X". The Sirag Triple then takes the isosceles form ST(X, X, J). The numerical values in this triple answer this question: What two pairs of integers X and J will satisfy (EQ 2)?

 X^2 + X^2 = J(J+1)           (EQ 2)

By trial and error, and using a pocket calculator, one can compute the first three solutions to the Isosceles Sirag Triple problem. These three solutions are ST(1, 1, 1), ST(6, 6, 8) and ST(35, 35, 49). You can check for yourself that each of these three pairs of numbers satisfies (EQ 2), hence is belonging with full benefits to the illustrious Sirag Triple Society. Here is a larger table of Isosceles Sirag Triples that cost a day and a half of my time using pocket calculator, graph paper and lots of strong coffee.

The first nine Isosceles Sirag Triples

To the casual glance, these two columns of numbers appear completely arbitrary except perhaps for the fact that they both alternate between "even" and "odd" as you progress down each column. However the apparent patternlessness of X and J conceals a remarkably simple relationship, which is the gist of Her Majesty's Mathematical Gift. Her Gift is precisely this: If we let the numbers X in the left column label the Square Numbers S(X), and the numbers J in the right column label the Triangular Numbers T(J), then anytime X and J satisfy the formula (EQ 3), then X, X and J form an Isosceles Sirag Triple.

S(X) = T(J)              (EQ 3)

In plain language, the problem of finding an Isosceles Sirag Triple is exactly the same as finding an integer that is both a Square and a Triangular Number. Dismissing the integer "1", which is a trivial case, the lowest square triangular number is 36, which is the square of 6 but also can form a triangle with base 8. For this case (EQ 3) reduces to S(6) = T(8) = 36. As you can see from the green diagram, numbers that are both Triangular and Square are exceedingly rare. Searching through the first 2 trillion numbers, I was only able to find 9 such magic numbers that satisfy this condition. (Of course, I couldn't actually test all 2 trillion digits to construct the green diagram, but used tricks to get close to the right number. Then I conducted a local search on a few likely suspects to nab the culprit.)

Given the scarcity of square triangular numbers (and their largeness, which rapidly outstrips the capacity of a pocket calculator), how does one actually find and capture more of these rare beasts? The Queen of Mathematics Entertains suggests many complicated strategies for carrying out this Quest. But there is no method shorter and more efficient than that devised recently by Armando Guarnaschelli, an amateur mathematician who lives in Argentina.

The Isosceles Sirag Triple ST(X, X, J) is not strictly speaking a triple since two of its members are alike. These numbers are in effect a Sirag Couple. Saul-Paul, playing around with Siragian Triangles,
has discovered an infinity of such triangles with unequal sides. But all such non-isosceles Siragian Triangles that Saul-Paul has so far derived conform to the type ST(X, J, J); that is, one of the triangle's sides always has the same length as the "half hypotenuse" J. Generalizing from this experience, Saul-Paul conjectures that there are no true Sirag Triples. The only Sirag Triples that exist in nature are in effect Sirag Couples which come in two varieties, namely ST(X, X, J) and ST(X, J, J). Saul-Paul's conjecture may well be true. But it could be suddenly demolished by a single counter example.

I would like to thank Saul-Paul Sirag for inspiring this effort and for putting together most of the pieces. (And thanks also for more chances to learn to spell the word "isosceles"). And deep gratitude to Her High Radiance, the Queen of Mathematics, for deigning to bless our work with such an unexpected and breathtakingly elegant outcome.

Saul-Paul Sirag in front of the Ken Kesey statue in Eugene, Oregon

Sirag Triples

Physical gyroscope and iPad with a digital gyroscope inside.

Nature seems to like to spin. Rotating systems are everywhere, from the Sun, the planets, the moons and Our Galaxy itself, to Ferris wheels, bicycle wheels, hard drives and CDs. The classical physics
that describes rotating systems such as gyroscopes and the Earth is particularly elegant. Classical rotation is certainly admirable but the quantum mechanical description of rotation is one of the most beautiful cultural achievements of mankind comparable to a grand cathedral or symphony.

The first quantum discovery concerning rotation is that spin is quantized. Spin only exists in nature as integral multiples of Planck's constant h. (Systems with integral values of spin are called Bosons; a second class of systems exists called Fermions consisting only of spins with 1/2-integral values.) For the sake of simplicity I will limit this discussion to Bosons.

An object can be spinning with its spin axis pointing in any of three directions. The direction of the axis defines the "direction of spin" Physicists call these 3 axes X, Y, and Z. In accord with the picture above of a spinning iPad, I will also call these three axes Red, Green and Blue (RGB). What it means for a spin to be quantized is that if we measure the spin of an object along the Red direction, the only values we will ever get are 0, 1, 2, 3, 4 ... units of Planck's constant. Spin is digitized!

Any rotating system can be conveniently labeled by its TOTAL SPIN "J" where J is related to the spin values (X), (Y) and (Z) along the X, Y and Z axes by the formula:

(X)^2 + (Y)^2 + (Z)^2 = J(J+1)        (EQ 1)

The sum of the squares of the values of the spins in all three directions is equal to J(J+1). The reason why the sum of the three squares is equal to J(J+1) instead of J^2 is a quantum thing which I take on faith but which I have never been able to understand.

Besides quantization, the next most important quantum discovery is a fundamental linitation on what you can measure and what you cannot -- an extension of the Heisenberg uncertainty principle to rotating systems. For a spin system, what you can measure is the total spin J, plus one of the three Spin components. If you choose to measure Red Spin, then Green and Blue Spin become uncertain.

If you choose to measure Green Spin, then Red and Blue Spin are fuzzy. This Heisenberg restriction is sometimes explained by saying that a Red measurement "uncontrollably disturbs" the Green and Blue Spins. But for me a better way of thinking about this restriction is that quantum systems with one definite spin direction actually exist in nature. But quantum systems with two definite spin directions do not. To ask what is the value of the Green Spin after I have measured the Red Spin is like asking how long is the fourth side of a triangle. Four-sided triangles do not exist; neither do rotating quantum systems that possess two or more well-defined spin directions.

A classical system possesses all of its properties in a well-defined manner, whether we observe these properties or not. For a quantum system only some of its properties are well defined (sometimes called "good quantum numbers"); the remaining properties dwell in a peculiar sort of quantum limbo similar to the fate of Dr Schrödinger's famous cat.

A general spin system possesses two good quantum numbers: its total spin J and the value of one of its spin components, Red Spin, for instance. Thus a typical quantum system may have a (total) spin of 3 Planck units and a Red Spin value of +2. For a spin-3 system, the only possible values that Red Spin is allowed to take are -3, -2, -1, 0, +1, +2, +3. The plus and minus signs indicate whether the spin is in a clockwise or counterclockwise direction.

Summing up this quantum spin lesson. Quantum systems have two good quantum numbers: Total Spin "J" and Red Spin "X" (where "Red" could be any one of the three basic directions.)

Recess: I learned recently that every new iPad has three digital gyroscopes inside it, that measure the gadget's rotation in the X, Y and Z direction. These gyroscope are not made of rotating chunks of metal but are ingenious computer chips called MEMS devices (short for Microelectromechanical Systems). How this digital gyroscope actually works is described here in great detail. Despite its name, the digital gyroscope is not digital in the quantum sense, rather it's a clever implementation of classical mechanics.

Return to class: Quantum mechanics says that deep down every spinning system is a digital gyroscope with certain natural restrictions on which of its properties you can measure.

Although a general quantum gyroscope possesses only two good quantum numbers "J" and "X", there may exist certain very simple systems that (accidentally) possess more than two good observables.

For instance a system with J = 0, possesses no spin at all. Its total spin is zero. And its spin in all three directions is zero. That's 4 good quantum numbers instead of two for this simple system.

Next up is a system with J = 1. If we measure Red Spin and get 0, we know that the Green and Blue spins must be some fuzzy mixture of +1 and/or -1. That is, we know the absolute value of these unmeasurable spins but do not know their sign. This amount of knowledge is equivalent to saying that when we know that Red Spin = 0, we also know that the SQUARE of the Green Spin = 1. And the SQUARE of the Blue Spin =1. So the spin-1 system (because it's so simple) possesses FOUR good quantum numbers, the number J = 1 and the SQUARES of the spins in all three directions.
 

These three square have the value 1, 1, 0 and satisfy the fundamental spin addition formula (EQ 1). Namely, for a spin-1 system (EQ 1) reads: 1 + 1 + 0 = 2.

The spin-2 system or higher spins do not seem simple enough to possess any more good quantum numbers than the standard two-to-a-customer measurement restrictions that apply to all sufficiently complicated rotating systems. So regular physics stops here. And sci-fi physics begins.

Both the spin-0 system and the spin-1 system permit the SQUARES of the spins in all three directions to be good quantum numbers. Suppose there existed a new kind of quantum physics (call it LEGO PHYSICS) for which the squares of all spins were good quantum numbers. How far could LEGO physics be pushed before it ran into trouble?

How do we test LEGO physics? We pick a total spin value, say J = 3, assume the squares of all the spin components are good quantum numbers. For J = 3, these squares can only take the values 0, 1, 4 and 9. The test of spin-3 LEGO physics is whether some combination of these 4 numbers can be found which will satisfy the fundamental spin addition formula (EQ 1). For a Spin-3 system, this is possible: 4+ 4 + 4 = 12 does the trick. 

The question "How far can LEGO physics be pushed?" leads to what I have called the "Sirag Numbers" in honor of Saul-Paul Sirag, a brilliant mathematical physicist whose birthday is August 31. A Sirag Number is defined as any integer J, for which the LEGO conjecture fails. That is, if the sum of three squares that obey the rules of quantum mechanics cannot be made to sum to J(J+1), then that integer J is a Sirag Number. The story of the Sirag Numbers can be picked up here.

Since Saul-Paul's birthday is approaching, I was thinking about giving him another math present.

Let's use up one of the good quantum numbers and set Spin (Z) = 0. Then the fundamental spin equation reduces to:

(X)^2 + (Y)^2 = J(J+1)       (EQ 2)

We apply the LEGO conjecture which allows both (X)^2 and (Y)^2 to be good quantum numbers and ask the question: For what values of X, Y and J, do integral solutions exist? 

Although this question arises (as did the Sirag Numbers) in the context of VERY DUBIOUS PHYSICS, it is a well-posed mathematical question which may lead to interesting results.

The question of the existence of integer solutions to (EQ 2) closely resembles the question of the existence of PYTHAGOREAN TRIPLES (X, Y, Z) which satisfy the simple formula:

X^2 + Y^2 = Z^2      (EQ 3)

This expression (EQ 3) is the Pythagorean expression for the sum of the squares of the lengths of the sides of a right triangle. A Pythogorean triple is a set of three numbers for which the sides of a right triangle are whole numbers, such as the classic 3-4-5 right triangle.

The question of what integers satisfy the LEGO model formula is almost identical to the question of the existence of Pythagorean triples. Indeed the "LEGO formula" (EQ 3), loosely derived from quantum gyroscopes, might be construed as the "quantum version" of the Pythagorean Theorem.

In honor of Saul-Paul's upcoming birthday, I would like to designate any 3-tuple of integers (X, Y, J) that satisfies the "LEGO formula" (EQ 2) a SIRAG TRIPLE. Simple examples of Sirag triples include (0, 0, 0), (1, 1, 1) and (9, 3, 9). Just as there now exists a formula for generating all Sirag Numbers, there must also exist a formula that generates all Sirag Triples, but I don't yet know what it might be.

Happy Birthday, Saul-Paul.

Sauk-Paul Sirag lecturing at Esalen Institute, Big Sur.

Sirag Numbers Enter the Canon

Saul-Paul Sirag lecturing at Esalen Institute

The Sirag Numbers, first arising in the context of "accidental commutation" in the quantum theory of angular momentum (and dedicated to Saul-Paul on his 72nd birthday), were first defined on this blog and partially "tamed" by myself and Saul-Paul Sirag.

Recently these numbers have attracted the attention of the larger mathematical community (thank you, Antti Karttunem) and are now listed in the Online Encyclopedia of Integer Sequences as Integer Sequence A196224. The Sirag Numbers were "completely tamed" by Jack Brennan, Ray Chandler, Charles P. Greathouse and Harvey P. Dale who devised formulas which generate all Sirag numbers. Happy birthday, Saul-Paul! Now that this sequence has been officially recognized, the one thing lacking is to discover an application for Sirag Numbers in the real world.

Also in the news this week, sci-fi author Rudy Rucker paid me a visit at the Quantum Tantric Ashram where we discussed philosophy and literature in my redwood forest retreat. Over tea and stollen, Rudy presented me with copies of his book of paintings Better Worlds and his latest novel, set in Santa Cruz CA, Jim and the Flims which chronicles a throroughly Ruckeresque vision of the afterlife. Hint: the Rucker afterlife is not even close to what you've been taught in school; it more resembles some seriously twisted DMT trip.

Also this weekend Beverly Rubik, a colleague from the Consciousness Theory Group and the Esalen Seminars on the Nature of Reality, invited me to Berkeley to give a talk at her Institute for Frontier Science.

Dr Beverly Rubik, PhD

I took Beverly's invitation as an opportunity to express my latest conceptions about quantum tantra and to read from my work-in-progress Harlot Nature. The talk was well-attended by a responsive crowd which included Dana Ullman, Jim Johnson and Allison Kennedy, the co-founder of the innovative Berkeley-based psychocyberdelic magazine Mondo 2000.

Nick performing at Institute for Frontier Science

After the talk Beverly treated Allison and I to supper at P. F. Chang's Chinese Bistro in Emeryville. Walking to Chang's in the dark our group was surprised by what appeared to be a hidden portal to another reality but when entered turned out to be a time tunnel memorializing the landscape's early inhabitants. Inside the portal, a parade of black stone slabs, each etched with text and pictures, told the story of Emeryville's former human habitation, brief by geological standards. A beautiful and informative piece of public art--the city of Emeryville's Shellmound Memorial.  

Entrance to Emeryville Shellmound Memorial

The Sirag Numbers

Saul-Paul Sirag, a "hippie that saved physics"

Yesterday, I issued a challenge to my physicist friends to solve a problem that, as Jeffrey Bub informed me, does not even exist. My challenge was based on the fact that for quantum spins 1/2 and 1, the squares of the spin components J(x), J(y), J(z) commute although the spin components themselves do not. I erroneous believed that this result was TRUE FOR ALL SPINS--what I called the "Commutation Conjecture" and asked for either a proof of this conjecture or a refutation. Within a few hours Jeffrey Bub (physicist at University of Maryland) replied that this conjecture was false and that no one in the field had ever believed otherwise. Casey Blood (physicist formerly at Rutgers) also sent a refutation a few minutes later. This contest is now closed.

My so-called "challenge" was a mistake from the start, but it led to an interesting discovery--the "Sirag Numbers"--named after my colleague Saul-Paul Sirag now living in Eugene, Oregon. Conversations with Saul-Paul concerning the quantum spin operators spurred my own refutation of the "Commutation Conjecture" that invokes a brand-new set of numbers.

If we assume that the Commutation Conjecture is true--that for every spin J, the operators J(x)^2, J(y)^2, J(z)^2 commute, then these spin squares are simultaneously observable. And furthermore these three observables must add up to J(J+1).

Now define the Sirag number J* such that the spin squared components of J* no matter how selected cannot be made to sum to J(J+1). If a Sirag number exists then the Commutation Conjecture is refuted because it leads to a contradiction.

Proof that "3" is a Sirag Number (ie, that its spin components squared cannot add up to 3(3+1)

The spin components of a spin-3 system are 3, 2, 1, 0, -1, -2, -3.

The squares of these components are 9, 4, 1 and 0.

It is impossible (by inspection) for 3 of these numbers to sum to 12.

Thus 3 is a Sirag number and the Commutation Conjecture is refuted.

Several questions immediately present themselves:

1. Is the number of Sirag Numbers finite or infinite?
2. Does an algorithm exist for calculating all J*?
3. Have these numbers been discovered before?
4. Is "137" a Sirag Number?

A little research shows that "3/2" is the smallest Sirag Number and that "3", "7/2" and "12" also belong in the class of Sirag Numbers. Today the largest known Sirag Number is "12" but that record is not likely to stand for long.

[Breaking news: the Sirag Numbers have been redefined to exclude 1/2 integer values; "3" has been shown NOT to be a Sirag Number; Mark Buchanan, president of Optical Alchemy, using an ad hoc
iPad app, has declared the lowest THREE Sirag Numbers to be 12, 15 and 19.

The Sirag Number J* is now officially defined by this equation:

X^2 + Y^2 + Z^2 = J*(J* + 1)

A integer J* is a Sirag Number, if no triplet (X, Y, Z) of integers exists that satisfies this constraint.

Saul-Paul's birthday is August 31 and Nick's birthday gift to him is this set of numbers.]

HAPPY BIRTHDAY, SAUL-PAUL!

[[ Sirag Number cognescenti will not fail to appreciate that the Buchanan Program, through some lucky circumstance, has revealed 1. J*(1) = 12, the first EVEN Sirag Number. 2. J*(2) = 15, the first ODD Sirag Number and 3. J*(3) = 19, the first PRIME Sirag Number. The BIG QUESTION now is: What is the value of J*(4), the fourth Sirag Number? ]]

[[[ Late Breaking News!!! On Saul-Paul's birthday, Mark Buchanan calculated ALL SIRAG NUMBERS between 0 and 1000. A partial list: 12, 15, 19, 44, 51, 63, 76, 83, 108, 112, 115, 140, 143, 147. Sad to say, Saul-Paul's favorite number 137 is not on this list, but the numbers 371 and 911 do make an appearance. ]]]

Direct from Mark Buchanan's smoking iPad: the first 92 Sirag Numbers:

12 15 19 44 51 63 76 83
108 112 115 140 143 147 172 179
204 211 236 240 243 255 268 271 275
300 307 332 339 364 368 371 396 399
403 428 435 448 460 467 492 496 499
524 527 531 556 563 575 588 595
620 624 627 652 655 659 684 691
716 723 748 752 755 780 783 787
812 819 844 851 876 880 883
908 911 915 940 947 960 972 979