Bob Schwartz

Tag: mathematics

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


To Understand America 2018, Read Alice’s Adventures in Wonderland

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.
Alice’s Adventures in Wonderland

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.

Islamic State: Using Arithmetic to Solve Complex Equations

Riemann - Zeta Function

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.