Jabir's Observatory

Jabir consulting the planisphere in his Tree House Observatory

A few years ago I built a high platform secured to three redwoods close to my house and have been sleeping up there on warm summer nights. Most of the sky is covered by trees but there is a 15 degree slot in the southern horizon which is open to the sky. Since living outside gives me a continual view of the night sky I decided to turn my tree house into an observatory in the spirit of the medieval Muslims who accurately mapped the stars and gave them Arabic names, most of which persist today.

Sometime in the 1980s I caught wind of the Moorish Orthodox Church through the writings of Hakim Bey and in 1986, in his apartment near the Nicholas Roerich Museum I was initiated into the MOC by Hakim Bey himself and later received in the mail a scroll declaring me an "adept of the seventh chamber". Orthodox Moors are supposed to take Muslim names and since at that time I knew hardly anything about Arabic culture I asked Bey for help. "Well you could do worse than Jabir, the Eighth-Century Arab alchemist," he said. So Jabir it was. To which for euphony and reverence I added 'abd al-Khaliq.

Jabir's Observatory is equipped with two 7 x 50 binoculars, one 10 x 42 monocular, a planisphere , a bunch of astronomy books (including Stephen J. O'Meara's  Observing the Night Sky with Binoculars and A Field Guide to Stars and Planets. Plus a red head lamp to read these books without destroying night vision.

The planisphere is the modern descendant of the Arab astrolabe which was a star-based mechanical computer, the iPad of the Islamic Golden Age. One treatise on the astrolabe (by the medieval star scientist who discovered the Coathanger) lists 1000 uses (1000 astrolabe apps), including telling time, locating the constellations, and finding the direction of Mecca. My planisphere only shows which stars are visible at my latitude and where they are located for any date and time of night.

Recently I've acquired an iPad whose Sky Guide app can mirror the heavens in any location. Hold up the iPad and the stars on the screen are in the same position as the stars in the sky. You can even set the iPad on the floor and look through the Earth at Southern Hemisphere objects such as the Large and Small Magellanic Clouds. Sky Guide is a planisphere on steroids.

Even though I only possess a viewing slot of 15 degrees, each month brings a new zodiacal sign into my observatory window. Late October's hottest attraction is the appearance of Orion and his hunting dogs high in the predawn sky.

In addition to relearning the constellations, I am trying to collect Messier objects which are 110 fuzzy objects in the sky which French astronomer Charles Messier listed as nuisances in his search for comets. Today no one remembers any of Messier's comets but Messier's non-comets are some of the most remarkable and well-studied objects in the night sky. M1, for instance, is the Crab Nebula, the remnant of a supernova explosion recorded by Chinese astronomers in 1054 AD. M45 is the Pleiades.

Bell's Theorem on xkcd

Bell's Theorem (Randall Munroe at xkcd)

Is Jack a Tacit Muslim?

Jack Sarfatti, celebrity physicist

IS JACK A TACIT MUSLIM?

When al-Sarfatt at night emerges
Into Herb Caen's Baghdad by the Bay
Seeking fortune, frivolity and fame
The stars that shine upon his aging brow:
Most bear medieval Arabic names.

Lacking algorithms, his Mac and iPhone
Would simply fail to compute.
And Jack frequently resorts to algebra
Despite its well-known Arab roots.

Like Omar Khayyam before him,
Jack depends on Arab alcohol
To soothe his cares away.
And we know Jack faces Mecca
Several times a day.

Do you spurn our Roman numerals, Jack
Because you hate the West?
"I love my Muslim numbers, Nick
Arabic numerals is the best."

Church of the Origin

Williamina Fleming's Triangular Wisp

CHURCH OF THE ORIGIN
(Inspired by L'Origine du monde
by Gustave Courbet)

Come worship at the Origin, chum.
From here all Christs, all Buddhas come.

Then lick my navel
Kiss my nipples,

Lift up my hair
And kiss my nape.

Then kiss my lipsticked lips
But no tongue please.

I never do tongue
On the first date.

Automating Quantum Weirdness

Islamic geometric pattern: an example of classical entanglement

AUTOMATING QUANTUM WEIRDNESS

A bowl of spaghetti or an Islamic repetitive pattern is an example of classical entanglement -- things get mixed up. And pulling one strand of spaghetti moves another strand far away. But quantum entanglement is different and difficult to express in ordinary language. When a number of particles get entangled in the quantum way, none of the particles possesses definite attributes. Only the entanglement as a whole possesses stand-alone values. Furthermore, whenever a measurement is performed on any one of the entangled particles, the state of each one of the other particles changes instantly, no matter how far it may be from its neighbors. And quantum mechanics is so delicately arranged that these instant collective changes cannot be used to signal faster-than-light. One way of expressing the entanglement situation is that Nature can communicate faster-than-light (and does so all the time). But humans cannot use entanglement to send signals because they cannot break Nature's "strong encryption" that governs the occurrence of each individual quantum jump.

Quantum entanglement is an unprecedentedly original way of getting things done in the world. Physicist Erwin Schrödinger called it "not ONE, but THE WAY, in which quantum mechanics differs most from classical expectations about how the world works."

Even at this early stage in our understanding of this phenomenon, quantum entanglement has found practical use in quantum computing, quantum cryptography and quantum teleportation as well as in many subtle new forms of optical imaging.

For the development of new quantum devices, human engineers are particularly handicapped because quantum mechanics follows a non-human logic that defies human intuition.

Enter MELVIN the robot.

Like his inventors (Mario Krenn and his colleagues at the University of Vienna), MELVIN thinks only classically. But he is able to design and test hundreds of possible thought experiments carried out with any number of quantum entangled particles. MELVIN was built to simulate the entanglement of photons. So he has at his disposal photon-entangling crystals, beam splitters, mirrors, wave plates, polarizers, holograms and perfectly efficient photon detectors. But quantum behavior is so generic, that any new results that MELVIN might discover for photons can almost certainly be exploited in other quantum systems such as electron spins, cold Bosons and superconducting junctions.

Schematic of MELVIN, the quantum engineer.

The first entanglement experiments (called EPR, after Einstein, Podolsky and Rosen whose pathbreaking 1935 paper first focused attention on the phenomenon) considered only TWO entangled particles. Work on two-particle entangled systems has been very fruitful, leading in 1964 to Bell's Non-locality Theorem, as well as the discovery of many new phenomena, notably quantum teleportation.

Entanglement was extended to THREE particles by Greenberger, Horne and Zeilinger (GHZ) who were able (with 3 particles) to construct a particularly elegant form of Bell's Theorem.

One task of MELVIN is to search for new phenomena beyond simple EPR and GHZ experiments by expanding quantum entanglement into the realm of greater numbers of particles and into higher entanglement dimensions. "Dimensions" refers here to particular particle attributes that participate in the entanglement. MELVIN considers three different dimensions of photon entanglement: #1. Path entanglement, #2. Polarization entanglement and #3. Orbital angular momentum entanglement. Since a particular photon can be both path-entangled, polarization entangled and OAM entangled at the same time to one or more other particles, the number of different allowed kinds of entanglements rapidly becomes astronomical.

But MELVIN is up to the task. He rapidly constructs numerous virtual experiments which are tested against certain criteria set by the experimenters. Most tests fail. But those that succeed become new building blocks that increase the odds that more of the tests will succeed. MELVIN is a kind of Darwinian machine: only the fittest experiments survive. While MELVIN the robot tirelessly produces hopeful candidates, the task of the human experimenters is two-fold: #1. to devise good tests for determining the "fitness" of a proposed experiment and #2. to use human logic to simplify the fit experiments and make them as efficient as possible.

MELVIN has already produced some unusual, never-seen-before types of quantum entanglement: partial entanglements, nested entanglements, cyclic entanglements and many more. The beauty of this robot designer is that not only does MELVIN produce exotic forms of entanglement, but he also outputs an exact plan of the hardware that will produce these new entanglements in the lab.

Which leads to a third important function of the human experimenter: #3 to discover new uses, either theoretical. practical or both for the flood of new "quantum lifeforms" brought to life in the lab by this imaginative robot/human collaboration.

A typical experiment designed by the MELVIN/human team

The MELVIN project is just one example of the extraordinary fruitfulness of the notion of quantum entanglement. As they say in show business: "Entanglement's got legs." It's going places.

Which leads to another Shri Nick Predicts.

Shri Nick predicts that either this year (2015) or the next, the Nobel Prize in physics will be awarded to John Clauser, Alain Aspect and Anton Zeilinger for their pioneer work in quantum entanglement. Such an award will also serve to honor the work of John Stewart Bell whose untimely death denied him this prize.

Remember, you heard it here first.

Esalen Institute

 

Orbital angular momentum of light

ESALEN INSTITUTE
(For Michael Murphy's 85th birthday)

For more than fifty years
We have been preparing the optimum site
For the Large Hadron Collider of the Spirit.
We are accepting proposals:
Come show your face.

We offer you the Pacific ocean
We offer you the Big Sur coast.
We offer you the legendary hot springs
We offer you an aware and intelligent staff
Each with a PhD
In one or more of
Aldous Huxley's nonverbal arts.

But be forewarned
We are experienced:

We have witnessed every scam,
Every ego-tripping, self-deluded vision,
bullshit, inauthenticity.

If you've got something big, hot and alive.
If you've been luckily electrified
By some new aspect of Deep Reality
O fortunate, O blessed one
Then bring it here.
That's what we live for.

We yearn for you
To arrive and turn us on.
We'd love to be the womb
Inside of which
Your vision blossoms.

Esalen bath house: outdoor hot tub.

JJCCTT Device for FTL Signaling

Omer and Nick's JJCCTT Device for FTL Signaling

JJCCTT Device for FTL Signaling

Bell's Theorem proves that quantum reality must be non-local.

Belfast-born physicist John Stewart Bell based his important proof about reality on the EPR Device (named after Einstein, Podolsky and Rosen) which uses TWO ENTANGLED PHOTONS whose wave function ψ can be written:

ψ (EPR) = 1/√2 ( |HH> + |VV> )          (EQ1)

Bell showed (and John Clauser subsequently measured) that this quantum wave function's statistical predictions exceed any result that any merely local reality is able to muster. Thus quantum reality is non-local.

Later Greenberger, Horne and Zeilinger used THREE ENTANGLED PHOTONS described by the GHZ wave function:

ψ (GHZ) = 1/√2 ( |HHH> + |VVV> )        (EQ2)

to prove a "Bell's Theorem without Inequalities". GHZ showed from EQ 2 that a local reality predicts a certain result will never happen, while quantum mechanics says that this same result must always happen. Since quantum mechanics gives the correct prediction, one measurement suffices to prove that quantum reality is non-local.

In recognition of GHZ's concise proof, the quantum state described by EQ 2 is usually called a "GHZ state".

Recently, Omer Dickstein, a physics student at Jerusalem College of Technology, proposed to exploit the non-locality exhibited by FOUR ENTANGLED PHOTONS for faster-than-light signaling. His wave function just adds one more photon to the GHZ state:

ψ (GHZ + 1) = 1/√2 ( |HHHH> + |VVVV> )        (EQ3)

As a place holder I have tentatively called EQ 3 the "GHZ plus one" state. How this state will eventually be designated will depend on what we learn from temporarily construing it as the core ingredient of an FTL signaling machine -- the so-called JJCCTT Device where JJCCTT stands for "Joint Jerusalem-California Collaboration on Transluminal Telecommunication".

The state |HHHH> represents the simultaneous emission of FOUR Horizontally-polarized photons, two of which are measured by Alice, and two of which are measured by Bob, each of whom possess two detectors that register "H" when accepting a Horizontal photon and "V" when accepting a Vertical photon.

The state |VVVV> represents the simultaneous emission of FOUR Vertically-polarized photons, two of which go to Alice, and two of which go to Bob.

One nice thing about the JJCCTT situation is that, unlike the EPR situation where Alice and Bob possess only one detector each (hence can obtain only ONE BIT of information), in the JJCCTT situation both particles possess two detectors (hence each can potentially obtain TWO BITS of digital information. The JJCCTT Device represents, information-wise, a more broad-band channel than its EPR competitor.

One might naively imagine that EQ 3 represents a situation in which the 4-photon source emits EITHER a pulse of 4 H-photons OR a pulse of 4 V-photons but that is not the case at all. In this peculiar process (called "polarization entanglement") illegal for all except quantum systems, the source emits BOTH a quadruple of H-photons and a quadruple of V-photons AT THE SAME TIME.

You might think of EQ 3 as describing a kind of "four-photon Schrödinger Cat state". When unlooked-at, "this cat" is in a superposition of both a 4H-cat and 4V-cat. (4H-cat has four Hazel (yellowish-brown) feet and 4V-cat has four Vermilion (yellowish-red) feet). Each foot simultaneously is both colors. That's when not looked at. But whenever it's looked at, whoever looks will always see this cat with each foot having the same color -- either four Hazel-colored feet or four Vermilion-colored feet -- no matter how far apart the cat's feet are. Bob and Alice might be 100 light-years apart and this feet-coloring process will happen exactly the same way.

According to quantum mechanics, when this ambiguous pulse of 4-light (quantum cat) encounters a detector, the detector "flips a coin" and randomly decides which one of these two possible polarizations it will "make real". Will it be the all-H-state or the all-V-state? It is important to understand that quantum theory tells us that this choice of what polarization will be recorded is made at the detector, not at the source.

Once this decision is made by one detector (we can never really identify which one), the other 3 detectors follow suit, so that each detector, no matter how far it might be separated from all the others, immediately comes to the same conclusion concerning which possible polarization state (H or V) it will also make real.

It is easy to see how this apparent instantaneous conspiracy between far-distant detectors to always record the same polarization, when up until the moment of choice both polarizations were actively possible, might embolden some physicists to attempt to exploit this system to send signals faster than light.

The configuration pictured above won't work as an FTL channel between Alice (the traditional sender) and Bob (the inevitable recipient), because EQ 3 allows only two elemental events to happen, either HHHH or VVVV, none of which are under the control of either Alice or Bob.

But here's how Omer from JCT in Jerusalem plans to change all that. We note that in the START STATE (pictured below), only two things ever happen to Alice and Bob. They either both receive 4 H-photons. Or both receive 4 V-photons. Bob always registers HH or VV in his two polarization detectors. And so does Alice. Never anything else.

START STATE: Only two events can happen, either HHHH or VVVV.

In particular Bob never observes a "cross term" such as HV or VH, where one of his two detectors counts a H-photon and the other counts a V-photon.

If there were something Alice could do with her photons that would produce "cross terms" in Bob's results, then Alice would be able to send a message to Bob at superluminal speed.

The essence of the JJCCTT Project is to examine all the things that Alice can do to her 2 photons, while looking for effects that Alice's actions might have on Bob's 2-photon cross terms.

Here's a hint about what Alice might do to induce cross-terms into Bob's detectors.

Alice might decide, for instance, to "make real" states of Right and Left Circularly polarized light rather than H and V polarized light as in the START STATE. Alice can easily configure her two detectors (using a phase plate and a beam combiner) to register R and L light rather than H and V light.

If Alice's R-and-L-making action causes any amount of R and L light to appear at either one of Bob's detectors, this will have drastic consequences. Because R light incident on an H/V detector (the kind Bob has deployed) will always produce a random mixture of H and V counts -- that will certainly lead to cross terms in Bob's data. Likewise L light incident on Bob's detectors will inevitably produce cross terms via the same procedure.

The 2-photon EPR situation offers some hope that this could happen, via a process that Irwin Schrödinger dubbed "steering". Starting with EQ 1, which represents a perfect correlation of H and V photons between Alice and Bob, Alice can transform her detectors from the Plane-Polarized basis H/V to the Circularly-Polarized Basis R/L, where R and L are Right- and Left-circularly polarized photons. This transformation turns EQ 1 into:

ψ (EPR) = 1/√2 ( |RL> + |LR> )          (EQ4)

One interpretation of EQ 4 is that by Alice's choice to measure R/L polarization rather than H/V polarization, she was able to "steer" Bob's distant photons from a mixture of the H/V eigenstates into a mixture of the R/L eigenstates.

Can the same "steering mechanism" that works for the EPR state work its magic on the GHZ +1 state? If Alice can steer even the tiniest fraction of Bob's H/V photons into a R state and/or a L state, then transluminal signaling will be accomplished via the instant appearance of cross terms {of the form HV or VH) in Bob's two HV detectors.

Any physicist familiar with purported FTL signaling schemes will reflexively credit such a vaguely plausible argument as no more than a hopeful conjecture. And will suspend judgement until seeing some actual calculations. Omer at Jerusalem Center for Technology and Nick at Quantum Tantra Ashram in California are currently calculating the 16-term quantum correlation matrix that encodes the full behavior of the JJCCTT device for whatever detector choices Alice can make to try to signal Bob. These are very elementary calculations. But it is easy to make mistakes.

For the record: It was Omer who suggested that the GHZ + 1 system might be a promising candidate for FTL signaling. And it was Omer who proposed that Alice-controllable cross-terms in Bob's HV detectors might function as an FTL signal. And it was Nick who suggested that Alice might use Schrödinger "steering" to remotely create cross-terms in Bob's HV detectors. And Nick did the graphics.

RESULTS: Omer calculated the correlation matrix for the case where Alice chooses to measure R and L photons rather than H and V photons:

Now eight events can happen, but none produces Bob's HV or VH cross terms.

Next Omer considered rotating both Alice's detectors by 45 degrees so that Alice registers Diagonal (D) and Slant (S) polarized photons instead of H and V.

Again eight events can happen, but none produces Bob's HV or VH cross terms.

Neither of these two efforts on Alice's part succeeds in producing cross terms in Bob's detectors. And indeed a more general calculation that allows Alice to effect any possible combination of rotation and phase change in her detectors gives the same result. Nothing that Alice can do will produce an FTL signal in Bob's detector.

So the JJCCTT proposal fails as an FTL signaling device.

"Science is great, but it’s low-yield. Most experiments fail. That doesn’t mean the challenge isn’t worth it, but we can’t expect every dollar to turn a positive result. Most of the things you try don’t work out — that’s just the nature of the process. Rather than merely avoiding failure, we need to court truth." -- Ferric Fang, microbiologist

Citing the FTL signaling Impossibility proofs of Philippe Eberhard and many others, it would be easy to have anticipated our negative result, These well-known impossibility proofs state, in essence, that 1. YES, quantum Theory is non-local (by inspection); 2. YES, quantum Reality is non-local (proved by John Bell) but; 3. NO, the quantum Facts are as local as can be.

Despite the FTL nature of the Theory that represents the World, despite the FTL nature of the Reality which underlies the World, the World Herself displays not a speck of evidence for any FTL connections.

I wish to thank Omer at JCT for proposing this project and I appreciate the fun we had doing these calculations. But now, as in so many other encounters with quantum reality, we end up where we started, back home again at Physics for Beginners.

Omer and Nick: two collaborators separated by 10 time zones.

Palm of my Hand

PALM OF MY HAND

Do you remember the night
I told you
That your underbrush
Was as familiar to me
As the palm of my hand?

But could I really sketch from memory
My hand's Head, Heart and Life Lines?
And how many little creases 
Run across my left hand's Mount of Venus?

Like an alien language
Like the back side of the Moon
Like your underbrush
And quantum reality:
Each as much a mystery
As the palm of my hand.

Roaring Twenties Topless Duet

Betsy on stage (circa 1970)

Today is the 13th anniversary of my wife Betsy's death.

I first met Betsy in San Francisco when she was living on Stanyan Street, studying and teaching modern dance at Ann Halprin's Dancer's Workshop on Divisadero St. While searching for something else in my files this morning I ran across some excerpts from Betsy's Journals concerning those days (during the reign of Herb Caen), when Betsy performed "The Topless Duet".

10/28/82

Lying in bed this morning,  I thought about Maude Meehan's suggestion to write about my topless dancing career. Some opening lines drifted into my head.

Eight of us shared the job, four men and four women. We were part of a dance class taught by AA Leath. He was the guy who scored the gig. We performed in pairs and on alternating nights, the Topless Man-Woman Duet at the Roaring Twenties in San Francisco's North Beach. On the marquee on Kearney St we were listed underneath the Topless Girl in the Swing. She was the feature. Every night, at 10, 11, midnight and 1 AM she climbed aboard a pink-cushioned swing hung 2 stories high in a wide stairwell and arched her back for all to enjoy.

"Come on down tonight," AA said, "See what you think of it." Twenty-five bucks a shot, 4 performances a nite. That amounted to about 30 minutes of dancing for $100. Never in a whole decade's career had I ever dreamed of getting paid like that. It was tempting but terrifying. Nevertheless, at 9:30 that night I found myself bound for North Beach in my bouncy little Citroen.

A big place, the Roaring Twenties, with lots of red plush decor. A long bar off to one side, tables all around the swing's staircase and up to the stage. On stage, a young and sprightly rock band. the manager points me to the dressing rooms upstairs. There's two of them, one for the girl-on-the-swing, the stripper and the topless waitresses and the other for the duet contingent. I open the door to find AA stretching and muttering in his usual fashion and with a southern drawl about getting in touch with some psychological phenomenon. Susan, his partner for the evening, was busy wrapping up in the several yards of chiffon which, atop a G-string, comprised her costume.

It was the Winter before the Summer of Love, 1967, and lots of us were driven by the idea that the Beautiful People could make a difference wherever we found ourselves. AA had that in mind for the clientele of the Roaring Twenties. "We'll show them that they can love their bodies". Sounded intriguing, but looking at AA in his G-string there in the dressing room didn't have me convinced.

I went downstairs, found Jim and Nancy sitting at the bar. They were also in AA's class and considering topless employment. the Girl on the Swing was waving goodbye to three tables of men from this week's convention. We ordered 7-Ups and waited, eager but figgety. the topless waitresses moved about, adorned in high heels, tights, miniskirts and pert aprons. Above the waist they were wearing a sheath of numbness.

The band leader announces The Duet and moves his musicians into a lively number. AA and Susan approach the stage through the audience, from opposite sides of the club, each clothed in chiffon. once on stage there is a lot of self-conscious shedding of chiffon. Reduced to G-strings and each other, the dance of seduction begins. I sit entranced, wondering what it would be like to be in Susan's place, all-but-naked, on stage.  It is dark out here in the audience and the stage lighting is soft enough to be kindly. They seem protected by each other and the movement, which shifts from modern-dancey to Fillmoresque and never leaves the realm of self-consciousness. I look at the tables of convention men and a collection of couples watching. They are looking but they don't seem to SEE. AA and Susan skirt each other, do-se-do, retreat and occasionally touch...

More on Roaring Twenties piece. What I remember about what it was like to perform the duet:

i remember that the band, whose name I can't now recall, who eventually was successful enuf to cut a record, played a piece with the lyrics "GI Joe, come back home. You'll have the best old time you've ever known," Real upbeat.

I remember going to buy my G-string downtown at the dance wear store. What a fuss over absolutely nothing! Well, practically nothing.

The commute to North Beach was by cable car.

Children picking up our bones

Will never know that these were once

As quick as foxes on the hill. 

                ---Wallace Stevens

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