## Category: Mathematics

If four sides of a dreidel and eight candles are proving too simple for you, here is a way to expand the Hanukkah holiday—mathematically and mystically.

The eight-candle menorah is binary, that is, a candle is either lit (1) or unlit (0).

With eight binary places, the menorah is a type of hexadecimal, a code central to digital processing. Each of the eight places in a hexadecimal is occupied by a digit or a letter.

If you assign each of the eight candles, left to right, either a 0 or a 1, you can convert each from a hexadecimal to a numeric value:

First night of Hanukkah = 00000001 = 1
Second night = 00000011 = 17
Third night = 00000111 = 273
Fourth night = 00001111 = 4,369
Fifth night = 00011111 = 69,905
Sixth night = 00111111 = 1,118,481
Seventh night = 01111111 = 17,895,697
Eighth night = 11111111 = 286,331,153

Is this of any use? Some suggestions:

1. A secret code to identify each day of Hanukkah with a number. When greeting someone on the second day of Hanukkah, you might say “Happy 17”. Please be sure to explain the system behind your greeting, lest it is thought you are experiencing a psychological break or are high (assuming you are not).

2. Gematria is a system that assigns numbers to each Hebrew letter in a word, and then calculates a value for each word, which value is then associated with other words of the same value. You can look online to find the gematria associations for each of the above values. In addition to gematria, there are countless systems that assign values to letters and meaning to numbers.

For the larger numbers, you may not find an associated meaning. But you can factor the larger numbers to find smaller associated meanings. So, for example, the eighth night value of 286,331,153 factors to 17 × 257 × 65537. The smaller numbers such as 17 and 257 are widely discussed.

Chag urim sameach! (Happy Festival of Lights). And if I don’t see you, Happy 273!

### Indra’s Net

The Glowing Limit. This illustration follows the mantra of Indra’s Pearls ad infinitum (at least in so far as a computer will allow). The glowing yellow lacework manifests entirely of its own accord out of our initial arrangement of just five touching red circles.

From Indra’s Pearls: The Vision of Felix Klein by David Mumford, Caroline Series and David Wright:

The ancient Buddhist dream of Indra’s Net

In the heaven of the great god Indra is said to be a vast and shimmering net, finer than a spider’s web, stretching to the outermost reaches of space. Strung at the each intersection of its diaphanous threads is a reflecting pearl. Since the net is infinite in extent, the pearls are infinite in number. In the glistening surface of each pearl are reflected all the other pearls, even those in the furthest corners of the heavens. In each reflection, again are reflected all the infinitely many other pearls, so that by this process, reflections of reflections continue without end.

***

Towards the end of the century, Felix Klein, one of the great mathematicians his age and the hero of our book, presented in a famous lecture at Erlangen University a unified conception of geometry which incorporated both Bolyai’s brave new world and Möbius’ relationships into a wider conception of symmetry than had ever been formulated before. Further work showed that his symmetries could be used to understand many of the special functions which had proved so powerful in unravelling the physical properties of the world (see Chapter 12 for an example). He was led to the discovery of symmetrical patterns in which more and more distortions cause shrinking so rapid that an infinite number of tiles can be fitted into an enclosed finite area, clustering together as they shrink down to infinite depth.

It was a remarkable synthesis, in which ideas from the most diverse areas of mathematics revealed startling connections. Moreover the work had other ramifications which were not to be understood for almost another century. Klein’s books (written with his former student Robert Fricke) contain many beautiful illustrations, all laboriously calculated and drafted by hand. These pictures set the highest standard, occasionally still illustrating mathematical articles even today. However many of the objects they imagined were so intricate that Klein could only say:

The question is … what will be the position of the limiting points. There is no difficulty in answering these questions by purely logical reasoning; but the imagination seems to fail utterly when we try to form a mental image of the result.

The wider ramifications of Klein’s ideas did not become apparent until two vital new and intimately linked developments occurred in the 1970’s. The first was the growing power and accessibility of high speed computers and computer graphics. The second was the dawning realization that chaotic phenomena, observed previously in isolated situations (such as theories of planetary motion and some electronic circuits), were ubiquitous, and moreover provided better models for many physical phenomena than the classical special functions. Now one of the hallmarks of chaotic phenomena is that structures which are seen in the large repeat themselves indefinitely on smaller and smaller scales. This is called self-similarity. Many schools of mathematics came together in working out this new vision but, arguably, the computer was the sine qua non of the advance, making possible as it did computations on a previously inconceivable scale. For those who knew Klein’s theory, the possibility of using modern computer graphics to actually see his ‘utterly unimaginable’ tilings was irresistible….

Klein’s tilings were now seen to have intimate connections with modern ideas about self-similar scaling behaviour, ideas which had their origin in statistical mechanics, phase transitions and the study of turbulence. There, the self-similarity involved random perturbations, but in Klein’s work, one finds self-similarity obeying precise and simple laws.

Strangely, this exact self-similarity evokes another link, this time with the ancient metaphor of Indra’s net which pervades the Avatamsaka or Hua-yen Sutra, called in English the Flower Garland Scripture, one of the most rich and elaborate texts of East Asian Buddhism. We are indirectly indebted to Michael Berry for making this connection: it was in one his papers about chaos that we first found the reference from the Sutra to Indra’s pearls. Just as in our frontispiece, the pearls in the net reflect each other, the reflections themselves containing not merely the other pearls but also the reflections of the other pearls. In fact the entire universe is to be found not only in each pearl, but also in each reflection in each pearl, and so ad infinitum.

As we investigated further, we found that Klein’s entire mathematical set up of the same structures being repeated infinitely within each other at ever diminishing scales finds a remarkable parallel in the philosophy and imagery of the Sutra. As F. Cook says in his book Hua-yen: The Jewel Net of Indra:

The Hua-yen school has been fond of this mirage, mentioned many times in its literature, because it symbolises a cosmos in which there is an infinitely repeated interrelationship among all the members of the cosmos. This relationship is said to be one of simultaneous mutual identity and mutual intercausality.

### Dish of Dice

Dish of Dice

“I am going to build a church someday. It will have a holy of holies and a holy of holy of holies, and in that ultimate box will be a random number table.”
Gregory Bateson

Different dice
On the altar
Four six eight sides
Ten and twenty
Sleeping in the dish
Awake and rolling
Prophets with a message
Plan and prepare
To laugh cry and play
The numbers rise up
See their beauty and wisdom
Listen to
The last lesson you need

We had the best education. We went to school every day. I only took the regular course. Reeling and Writhing to begin with. Then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.

Read Alice’s Adventures in Wonderland now. Again if it’s been a while, and definitely now if for the first time.

Lewis Carroll (born Charles Dodgson, 1832-1898) was famously creative as a mathematician and logician. He wove puzzles and tortured logic all through his book Alice’s Adventures in Wonderland.

Puzzles and tortured logic seem likely to be a major component of America in 2018, as they were in 2017.

The leadership and the citizens of Wonderland are variously tyrannical, illogical, stupid, or just plain bizarre. Alice literally does not fit in. While she is only a child, she has more sense than everyone she meets combined.

If I had a news network like CNN, I’d interrupt the futile attempts to understand and explain what’s going on by having different news anchors read aloud one chapter from Alice in Wonderland every day. It would actually be more constructive—and more fun—than just listening to their trying to making sense of the nonsensical.

If Trump’s tweets were taken from Alice in Wonderland, would we know the difference? Would he?

Some Trump/Alice tweets:

We must have a trial. Really this morning I have nothing to do. With no jury or judge I’ll be Judge. I’ll be jury. I’ll try the whole cause and condemn you to death.

We’re all mad here. I’m mad. You’re mad. A dog growls when it’s angry and wags its tail when it’s pleased. Now I growl when I’m pleased and wag my tail when I’m angry. Therefore I’m mad.

Be what you would seem to be. Never imagine yourself not to be otherwise than what it might appear to others that what you were or might have been was not otherwise than what you had been would have appeared to them to be otherwise.

You have no right to think. Just about as much right as pigs have to fly. I give you fair warning either you or your head must be off. Take your choice!

We had the best education. We went to school every day. I only took the regular course. Reeling and Writhing to begin with. Then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.

### Is Zero A Number?

O zero
You fool me all the time
Pretending to be something
When you’re nothing

### Days of Awesome: Day 1 (Rosh Hashanah)

I brought them out of the land of Egypt and I led them into the wilderness. I gave them My laws and taught them My rules, by the pursuit of which a man shall live. Moreover, I gave them My sabbaths to serve as a sign between Me and them, that they might know that it is I the Lord who sanctify them.
Ezekiel 20:10-12 (New Jewish Publication Society translation)

Note from The Jewish Study Bible:

The Sabbath is the foundational sign of the covenant (Exod. 20.8–11; 31.12–17). Scholars have suggested that the Sabbath became particularly significant in the exile, as holy time replaced the vacuum of holy space (the Temple); this might explain why the Sabbath plays such a significant role here. As in Exod. 31.13, 17 (from the Priestly tradition), it is viewed as a sign, namely a symbol acknowledging God as Creator.

Here we are confronted with the phenomenon at the heart of this holiday. At the heart of every holiday. At the heart of religion and reality itself. We are concerned with space. We are concerned with being. We are concerned with time too. But we may not be properly concerned, in a balanced way that accounts for time, space and being.

We can rule space, or at least pretend to. If you visit New York or other great cities, you see how people have shaped space to their liking and purposes. But where in New York or elsewhere have even the richest and most powerful ultimately shaped time? We can mark time, but do we understand? To help us understand, time is set aside. It may be by God, it may be by our society or community, it may be by and for those close to us.

The Sabbath each week, and the Days of Awe each year, are set aside to be different than the other days of the week or of the year. Different in fact than any other days of eternity. In part to remind us of present eternity.

For more, see The Sabbath by Abraham Joshua Heschel and The Time-Being by Zen Master Dogen, which can be found in Enlightenment Unfolds.

This is the first post in a very small project/experiment in random wisdom I call The Days of Awesome. In addition to the standard and traditional forms of worship and contemplation associated with the Jewish High Holy Days (also known as Days of Awe), each day of the holiday I will be studying a randomly selected chapter of the Tanakh (also known as the Jewish Bible or the Old Testament), which has 39 books containing a total of 929 chapters.

Among other things, this is inspired by the I Ching and by social theorist and philosopher Gregory Bateson, who is quoted as saying “I am going to build a church someday. It will have a holy of holies and a holy of holy of holies, and in that ultimate box will be a random number table.”

### Independence Day and STEM Democracy

Is the increasing hegemony of STEM education dangerous to the future of American democracy?

In Science and the Founding Fathers: Science in the Political Thought of Thomas Jefferson, Benjamin Franklin, John Adams, and James Madison, Professor I. Bernard Cohen might see it otherwise. As one of the most eminent historians of science, he makes the case that the familiarity of some Founding Fathers with science inspired the new nation, and that the shape of the new democracy was directly based on scientific principles.

One review notes about Professor Cohen’s theory:

The Declaration of Independence, which he [Jefferson] wrote, reverberates with echoes of Newtonian science, as when he invokes “self-evident” truths or “laws of nature.” Benjamin Franklin, far from being a mere tinkerer or inventor, pioneered the science of electricity. Franklin also developed a demographic theory that North America would become a population center of the British world; this led to the policy according to which the British annexed Canada rather than Guadeloupe as the spoils in the war against the French (1754-63). John Adams, who studied astronomy and physics at Harvard, was a founder of the American Academy of Arts and Sciences in Boston. And James Madison, a devoted amateur scientist, drew on scientific metaphors and analogies in his Federalist articles.

Maybe. But in fact, most of those in Philadelphia for the Continental Congress from which the Declaration of Independence emerged were not scientists or even science fans. And even those whose philosophy was shaped in part by science enjoyed a much broader education, one that gave complete dimension to their thinking, what we now call liberal arts. So that while the intriguing questions that Professor Cohen raises are significant, so is the parallel question: If the Continental Congress had been mostly or entirely filled with 18th century scientists, just what kind of Declaration would have been produced, and more broadly, what kind of nation would we be?

Nowhere can the nexus of Big Science and Big Political Philosophy be better seen than in Richard Rhodes’ magnificent book The Making of the Atomic Bomb. It is sort of a fun house mirror of what Cohen claims for the American founding. Rather than world-changing political thinkers with a scientific bent, we have equally historic scientists with a worldly and philosophical bent. They had been educated in the early 20th century, many in Europe, and the standard for education then and there was broad learning beyond the laboratory. In the end, their science was driven by the realities of World War II and Hitler, but that did not stop them from philosophical ponderings and quandaries about the work they were doing and its ultimate impact.

So, yes, it may be that science did help give us what by all measures is a remarkably robust and resilient democracy, starting with the rousing rhetoric of the Declaration of Independence. And we should educate scientists, to make progress and to advance the liberty, peace, and security we want. But we should also have many other thinkers, scientists or otherwise, who are capable of leading and having enlightening debates about exactly what we do need and want, and about the means we choose to get there, and about where it might lead. We do need scientists, technologists, engineers, and mathematicians. But it is never enough, not nearly enough, at least not in this democracy.

Every picture supposedly tells a story. Actually, every picture is a story.

Beading is a glorious craft. In the hands of a talented artist, the results can be beautiful and enlightening.

But like all art, it can be a messy business. In the case of beads, this can mean tiny items underfoot, and with bits of wire, pretty painful ones. Particularly where barefoot is the custom.

A quick post-beading cleanup led to quite a collection of such detritus, like shells on a beach. Tossed in a white bowl, they looked like something. And so the photo above.

If you are a fan of randomness—and we should all be—you will see in this totally spontaneous display any number of things. Gregory Bateson said, “I am going to build a church someday. It will have a holy of holies and a holy of holy of holies, and in that ultimate box will be a random number table.” Exactly.

Here is a beader in her natural habitat, the largest bead store in New York. It is filled with beads mostly from China which, as in most things, is able to provide whatever we want or need in seemingly infinite supply. So it is all together: geopolitics, economics, ancient tradition, minerals, pottery, glass, color, art, craft, and, of course, beauty. Note, however, that in this emporium, the beauty of the beader outshines all of the beads.

### Notes on Interstellar

1

Christopher Nolan’s movie Interstellar is more interesting than it is imperfect. See it if you like space movies, sci-fi movies, intellectually curious movies, spectacular movies, etc.

It is filled with wonders. It is like the car trunk stuffed with luggage for a vacation, so much colorful and significant luggage creatively crammed in that when you open it on arrival you say: Wow, I wonder how we ever got all that stuff in there?

No spoilers here, but a couple of things.

Look for all the tiny (and not so tiny) echoes of space and sci-fi movies past. Star Wars, Close Encounters, etc., but most of all 2001. Why not? Right now, “they” are probably having a good 5th dimensional laugh watching Stanley Kubrick’s proto-human apes tossing that bone.

Interstellar has the most subtly cool robots ever. TARS doesn’t sing like HAL, but he has moves like Jagger and is great with the snappy patter.

2

The movie is much about cosmology—the origin and nature of existence. Cosmology is the domain of all kinds of people, including religionists and philosophers. But in greater part, we have handed over many of those considerations, as in this movie, to theoretical physicists—Einstein, Hawking, etc. I am a big fan of cosmology.

It is not a spoiler to mention that plenty of people, including some in this movie, believe that the Apollo 11 moon landing was faked. Which raises this way-out-there question: What if the moon landing was real but all the cosmological theoretical physics is faked? Going back before Einstein, theoretical physics spends much of its time (as we understand it) looking for physical proof of those theories. What if all the theory is so utterly astounding and enlightening that when the evidence failed to support it, all the scientists engaged in the study conspired to make it seem as if those theories are supported?

Faking the moon landing mission has never been put entirely to rest because, in fact, only three people experienced it first-hand. Everyone else was second-hand or more distanced from the actuality. But the basic elements of it are well within our understanding: astronauts, rocket, spaceship, lunar lander, moon, television pictures. The cosmological speculation and supporting discoveries are so far beyond anything that most of us can fully—or slightly—grasp that we could easily be fooled into taking it for “reality.”

By the way, for those wondering about the earnestness of all that, be assured that I am just playing. Or am I?

3

We don’t have to be space pilots to experience cosmology, or be theoretical physicists or movie directors to think about it. Cosmology is ordinary. Interstellar and other movies and thousands of works of art and literature point to this. Everybody is a cosmologist, like it or not.

Cosmology is an excellent topic that does not necessarily require specialized knowledge. You may not know a worm hole from a black hole. But you already know a ton about time, space, being, and gravity. You just have to know how to know and that you know.

This is from an essay almost 800 years old. No more or less spectacular than Interstellar, it is no more or less a non-theoretical description:

Do not think that time merely flies away. Do not see flying away as the only function of time. If time merely flies away, you would be separated from time. The reason you do not clearly understand the time being is that you think of time only as passing.

In essence, all things in the entire world are linked with one another as moments. Because all moments are the time being, they are your time being….

You may suppose that time is only passing away, and not understand that time never arrives. Although understanding itself is time, understanding does not depend on its own arrival.

People only see time’s coming and going, and do not thoroughly understand that the time being abides in each moment. Then, when can they penetrate the barrier? Even if people recognized the time being in each moment, who could give expression to this recognition? Even if they could give expression to this recognition for a long time, who could stop looking for the realization of the original face? According to an ordinary person’s view of the time being, even enlightenment and nirvana as the time being would be merely aspects of coming and going….

Mountains are time. Oceans are time. If they were not time, there would be no mountains or oceans. Do not think that mountains and oceans here and now are not time. If time is annihilated, mountains and oceans are annihilated. As time is not annihilated, mountains and oceans are not annihilated.

Dogen
The Time Being (1240)
Treasury of the True Dharma Eye

### Islamic State: Using Arithmetic to Solve Complex Equations

We are not playing three-dimensional chess in the Middle East—partly because all of us will go crazy if we hear that clichéd term one more time.

Instead, we are using arithmetic to solve very complex equations.

The Clay Mathematics Institute offers the famous Millennium Prizes, \$1,000,000 each for solving their current list of unsolved mathematical problems.

Here is description of the Riemann Hypothesis (a manuscript by Riemann of the Zeta function is pictured above):

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications.

The distribution of such prime numbers among all natural numbers does not follow any regular pattern. However, the German mathematician G.F.B. Riemann (1826 – 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function

ζ(s) = 1 + 1/2s + 1/3s + 1/4s + …

called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation

ζ(s) = 0

lie on a certain vertical straight line.

This has been checked for the first 10,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

Right now, in the early days of the campaign against the Islamic State, we are using arithmetic that goes something like this:

1 (U.S.) + x (number of participating nations with wildly different involvement and interests) – IS = conditional victory

The truth is much closer to complex mathematics, as complex as any we may have ever seen on the world stage. There are probably behind-the-scenes discussions that are more subtle, but here in the public we are somehow not supposed to bother our heads about that. The question of why we publicly don’t deal with it this way may be because our leaders can’t handle the truth or because they believe citizen/voters can’t handle the truth or, because of politics and wanting to be seen as doing something, a little of both.

Solving the problem is worth much more than a million dollars. But solving it will take more than simple addition, subtraction, multiplication, and division. There was a time when the world was like that, susceptible to those simple solutions. But those days and that world are gone. Our leaders don’t have to be able to attempt a solution to the Riemann Hypothesis. But they do have to recognize when grade school, old school strategies—when simple arithmetic—will no longer work.