Besides containing math that helped to advance work on the Riemann Hypothesis, Turing’s paper “Some Calculations of the Riemann Zeta Function” also contains a great snapshot of what it was like to program the first generation of electronic computers. I’ll start with the widely-quoted introduction to the paper, which I’ve annotated a bit:

“In June 1950 the Manchester University Mark I Electronic Computer was used to do some calculations concerned with the distribution of the zeros of the Riemann zeta-function. It was intended in fact to determine whether there are any zeros not on the critical line in certain particular intervals. The calculations had been planned some time in advance, but had in fact had to be carried out in great haste. If it had not been for the fact that the computer remained in serviceable condition for an unusually long period from 3 p.m. one afternoon to 8 a.m. the following morning it is probably that the calculations would never have been done at all. As it was, the interval 2π63² < t < 2π64² was investigated during that period, and very little more was accomplished.

The calculations were done in an optimistic hope that a zero would be found off the critical line, and the calculations were directed more towards finding such zeros than proving that none existed. The procedure was such that if it had been accurately followed, and if the machine made no errors in the period, then one could be sure that there were no zeros off the critical line in the interval in question. In practice only a few of the results were checked by repeating the calculation, so that the machine might well have made an error.

If more time had been available it was intended to do some more calculations in an altogether different spirit. There is no reason in principle why computation should not be carried through with the rigour usual in mathematical analysis. If definite rules are laid down as to how the computation is to be done one can predict bounds for the errors throughout. [Numerical methods, with or without an automatic computer, should be carefully thought out.] When the computations are done by hand there are serious practical difficulties about this. The computer [he is now talking about a human - later he talks about an "automatic computer"] will probably have his own ideas as to how certain steps should be done. When certain steps may be omitted without serious loss of accuracy he will wish to do so. Furthermore he will probably not see the point of the ‘rigorous’ computation and will probably say “if you want more certainty about the accuracy why not just take more figures?’ an argument difficult to counter. However, if the calculations are being done by an automatic computer one can feel sure that this kind of indiscipline does not occur. Even with the automatic computer this rigour can be rather tiresome to achieve, but in connexion with such a subject as the analytical theory of numbers, where rigour is the essence, it seems worth while. Unfortunately, although the details were all worked out, practically nothing was done on these lines. The interval 1414 < t < 1608 was investigated and checked, but unfortunately at this point the machine broke down and no further work was done. Furthermore this interval was subsequently found to have been run with a wrong error value, and the most that can consequently be asserted with certainty is that the zeros lie on the critical line up to t = 1540, Titchmarsh having investigated as far as 1468.” [Buzzkilling bugs introduced by humans were just a common then as they are now!]

Later on, in the section entitled “The Computations,” Turing briefly describes the “Essentials of the Manchester Computer”:

“It is not intended to give any detailed account of the Manchester Computer here, but a few facts must be mentioned if the strategy of the computation is to be understood. The storage of the machine is of two kinds, known as ‘electronic’ and ‘magnetic’ storage. The electronic storage consisted of four ‘pages’ each of thirty-two lines of forty binary digits. The magnetic storage consisted of a certain number of tracks each of two pages of similar capacity. Only about eight of these tracks were available for the zeta-funtion calculations. It was possible at any time to transfer one or both pages of a track to the electronic storage by an appropriate instruction. [A strategy that remains common today for dealing with slow memory.] This operation takes about 60 ms. (milliseconds). Transfers to the magnetic store from the electronic were also possible, but were in fact only used for preparatory loading of the magetic store. The course of the calculations is controlled by instructions each of twenty binary digits. These are normally magnetically stored, but must be transferred to the electronic store before they can be obeyed. In the initial state of the machine (with the magnetic store loaded) the electronic store is filled with zeros. A zero instruction, however, has a definite meaning, and in fact results in a transfer of instructions to the electronic store, thus initiating the calculation. Most instructions, such as transfer of ‘lines’ of forty digits, take 1.8 ms., but transfers to or from the magnetic store take longer, as has been mentioned, and multiplications take a time depening on the number of digits 1 in the multiplier, ranging from 3.6 ms. for a power of two to 39 ms. for 2⁴⁰ – 1.

The results of the calculations are punched out on teleprint tape. This is a slow process in comparison with the calculations, taking about 150 ms. per character. The content of a tape may afterwards automatically be printed out with a typewriter if desired. The significance of what is printed out is determined by the ‘programmer’. [A word so new that it still needs quotes around it!] In the present case the output consisted mainly of numbers in the scale of 32 using the code

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
/ E @ A : S I U ¼ D R  J  N  F  C  K  T  Z  L  W  H  Y  P  Q  O  B  G  ?  M  X  V  £

and using the most significant digit on the right. [Yes, you read that correctly. You are required to read your numbers not only in base 32, but also backwards.] More conventionally the scale of 10 can be used, but this would require the storage of a conversion routine, and the writer was entirely content to see the results in the scale of 32, with which he is sufficiently familiar.”

And what would these results look like? Here is an eminently readable “typical entry”:

ZETAFASTG/F@Q¼B£YNK@:ZSZ"XVMX///SA/////¼0TNR@O//

“This entry has to be divided into sequences of eight characters. In this case they are:

1. ZETAFAST. This occurs at the beginning of each entry. Its purpose is mainly to identify the document as referring to this zeta-function routine.
2. G/F@Q¼B£. This is a number useful in checking results and called the ‘cumulant’. It appears in the scale of 32, with the most significant digit on the right. This is the standard method of representing numbers on documents connected with the Manchester Computer (a decimal method can also be used if desired).
3. YNK@:ZSZ. This is also in the scale of 32 and gives the residue of 2⁴⁰κ₁(τ) modulo 2⁴⁰. Since Z is the symbol for 17 it will be seen that κ₁(τ) is near to ½ mod 1.
4. “XVMX///. This gives the value of 2¹⁷τ; in this case τ is about 239.24.
5. SA/////¼. This was always included in the record due to a minor difficulty in the programming. It did not seem worth while to take the trouble to eliminate it. [Some things never change.]
6. OTNR@O//. This is the value of 2³⁰Z(τ) modulo 2⁴⁰. In this case Z(τ) is about 0.75.”

Finally, here is Turing’s table of what was in memory when this program was run:

The table in the blueprint contains the ratios he would need for mechanical linkage, although as Bill Casselman points out, the table actually contains several calculation errors which would have eventually caused some problems. [My personal favorite is the column for ratios in log base 8.]

From Turing’s application to the Royal Society: “It is proposed to make calculations of the Riemann zeta-function on the critical line for 1,450 < t < 6,000 with a view to discovering whether all the zeros of the function in this range of t lie on the critical line. An investigation for 0 < t < 1,464 has already been made by Titchmarsh. The most laborious part of such calculations consists in the evaluation of certain trigonometrical sums

In the present calculation it is intended to evaluate these sums approximately in most cases by the use of apparatus somewhat similar to what is used for tide prediction. When this method does not give sufficient accuracy it will be necessary to revert to the straightforward calculation of the trigonometric sums, but this should be only rarely necessary. I am hoping that the use of the tide-predicting machine will reduce the amount of such calculation necessary in a ratio of 50:1 or better. It will not be feasible to use already existing tide predictors because the frequencies occurring in the tide problems are entirely different from those occurring in the zeta function problem. I shall be working in collaboration with D. C. MacPhail, a research student who is an engineer. We propose to do most of the machineshop work ourselves, and are therefore applying only for the cost of materials, and some preliminary computation.”

Although this physical machine was never finished, due to the arrival of World War II, Turing continued to putter with the Riemann Hypothesis throughout his career via the zeta function, eventually becoming the first person to use an electronic computer to calculate zeroes, and thereby extending the upper limit for known zeroes to t < 1540. [Minuscule by today's standards, but not bad for work done with paper tape in raw base 32 on a machine with a little over 25,000 bits of memory!] He also devised what is now called “Turing’s method” for easier computational analysis of the function. These exploits are detailed in his papers “A method for the calculation of the zeta-function” and “Some calculations of the Riemann zeta-function,” which are both widely referenced in contemporary math papers.

“Turing took a particular delight in problems, large or small, that enabled him to combine mathematical theory with experiments he could carry out, in whole or part, with his own hands. He was ready to tackle anything which combined these two interests. His comical but brilliantly apt analogies with which he explained his ideas made him a delightful companion.” — DR ALAN TURING An Appreciation, Manchester Guardian 11 June 1954

“For those who knew him here [at Sherborne] the memory is of an even-tempered, lovable character with an impish sense of humour and a modesty proof against all achievement. You would not take him for a Wrangler, the youngest Fellow of King’s and the youngest F.R.S. [Fellow of the Royal Society], or as a Marathon runner, or that behind a negligé appearance he was intensely practical. Rather you recollected him as one who buttered his porridge, brewed scientific concoctions in his study, suspended a weighted string from the staircase wall and set it swinging before Chapel to demonstrate the rotation of the Earth by its change of direction by noon, produced proofs of the postulates of Euclid, or brought bottles of imprisoned flies to study their “decadence” by inbreeding. On holidays in Cornwall or Sark he was a lively companion even to the extent of mixed bathing at midnight. During the war he was engaged in breaking down enemy codes, and had under him a regiment of girls, supervised to his amusement by a dragon of a female. His work was hush-hush, not to be divulged even to his mother. For it he was awarded the O.B.E. He also adopted a yound Jewish refugee and saw him through his education. Besides long distance running, his hobbies were gardening and chess; and occasionally realistic water-colour painting.

In all his preoccupation with logic, mathematics, and science he never lost the common touch; in a short life he accomplished much, and to the roll of great names in the history of his particular studies added his own.” — The Sherbornian, Summer Term 1954

Dip the apple in the brew.
Let the Sleeping Death seep through.

Why did this resonate with Turing? What did it mean to him? Why such a fascination with it? Temptation? Evil? An ironic eternity waiting for True Love’s Kiss? We’ll never know. As Andrew Hodges points out in his outstanding biography, “Alan Turing himself would have been fascinated by the difficulty of drawing a line between accident and suicide, a line defined only by a concept of free will.”

We forget that Snow White and the Seven Dwarfs was a huge sensation among adults when it premiered, and was taken surprisingly seriously. (An incomplete, but not completely off-the-mark, modern comparison might be to James Cameron’s Avatar.) Attending the first feature-length cartoon released for an adult audience would have been a tremendous escape in pre-war England; Turing attended in Cambridge in 1938, soon after he had gone to work at Bletchley Park. The film was a popular topic of conversation in England, both because of the media blitz and merchandising that accompanied the film and because of the considerable debate about the impact of the horror aspects of the film on children. The demise of the Queen is still as violent a scene as it ever was.

Turing was known to repeat the witch’s couplet with glee. In 1937, before the movie was released, he intimated in a letter that he had devised a way to end his own life using an apple and some electrical wires, should he need to. When the time came, he apparently followed his plan, echoing the fairy tale. Here is an abridged account of the coroner’s inquest of 1957 from the Manchester Guardian:

“A verdict that Alan Mathison Turing (41), of Hollymead, Adlington Road, Wilmslow, committed suicide by taking poison while the balance of his mind was disturbed was returned at the inquest in Wilmslow last night. …

Police-Sergeant Cottrell said he saw Dr Turing lying in bed with the clothes pulled up towards his chest. There was a white frothy liquid about the mouth with a faint smell of bitter almonds. On a table at the side of the bed was half an apple from which several bites had been taken.

The witness said that in another room he found a cooking pan with a double container which was connected to electric wires. The contents of the pan were bubbling and there was a strong smell of bitter almonds.

Dr C.A.K. Bird, pathologist, said death was caused by asphyxia due to cyanide poisoning. A man of Dr Turing’s knowledge could not have swallowed it without knowing what would happen, and Dr Bird did not think it could have been accidental. He thought he apple was used to take away some of the taste. …” — Manchester Guardian 11 June 1957

The binary system of computation is particularly well-suited to electronic computers since a complete binary term of a binary number may be expressed in terms of either the conducting condition or the cut-off condition of the anode circuit of a conventional vacuum tube.

Meanwhile, on the muddy shores of Lake Washington, Donald Byrd and Spectrum Dance Theatre are presenting their creepy (and I mean that in the best way possible!) version of Petrushka, a puppet with very adult issues that comes to life at the hand of his evil puppet-maker, is abused and murdered, finally returns as a redemptive power. Spectacular dancing is embedded in a tawdry carnival and freak show, and the audience wanders from scene to scene, witnessing spooky and disturbing vignettes from the short and unhappy life of Petrushka, both as live dance and as live dance captured through surveillance cameras. Dancers speak, moan, and portray carnival characters, both puppet and human. The audience is a passive witness to the puppet-master’s ultimate demise via his sadistic and single-minded sexual obsessions. Another very successful genre-busting experimental show. (And, in a nice bit of serendipity, Byron Au Young created some of the electronic soundtrack featured in this show.)

Meanwhile still, at the Northwest Puppet Theater, the puppeteers are mounting their annual puppet opera. This year, the puppets are collaborating with their human vocal partners and and a band led by Margriet Tindemans to perform Il Girello, an obscure Baroque comic opera that is much improved by the interjection of huge quantities of bathos and improv comedy. Puppet opera is yet another genre-breaking form of theater, in which dramatic flow and character development come from puppet/singer combinations, and in which great musical performances, spoken word, and silly sound effects combine side by side to achieve a surprisingly integrated theater experience.

All three of these performances stretch actors and dancers to portray multiple dramatic roles simultaneously. By using abstract theatrical presentation realized as rap, dance, and puppetry, they amplify and focus the human traits that are featured in the stories they tell. They are interesting, experimental, and, I hope, a good indicator as to where theater is headed. Both Petrushka and Il Girello are still playing in Seattle. Check them out – I particularly recommend seeing them back-to-back! And go to see Stuck Elevator when it premiers in San Francisco at ACT next year. It will be worth the trip.

As I say on that page: “A. Turing: Automatic Elegy will be a ritualized performance set within a biographically-focused art installation. The installation will contain artifacts from Turing’s life, arranged as a series of small altars, and the performance itself will include spoken word, music, visual images, and movement, all based upon three central mathematical and philosophical structures from Turing’s own mind: the Turing machine, the Turing test, and Turing patterns. Although the music, visual images, and choreography that occur as part of the ritual can be fully enjoyed without previous knowledge of Turing’s intellectual output, the pieces will nonetheless directly reflect his work on the Riemann hypothesis, on algorithms, on cryptanalysis, on group theory, on digital quantization, on x-ray crystallography, on logic, and on mathematical biology.”

I will post relevant source materials and thoughts to the new page as we make progress. Not much there now, just a starter from one of his programming manuals, but I’ll try to update regularly.

(Non-alpha!) character mappings for the Mark I

I’ve marked-up this quote taken from the system programming manual written by Alan Turing for the Ferranti Mark I computer. (The manual is available from the Turing archive.)

10001110111010001001100011100101010101101100100110

10001 11011 10100 01001 10001 11001 01010 10110 11001 00110

Z"SLZWRFWN

Here is Cory’s post. I guess that this makes the project real!

The event will be held as part of the Wayward Music Series at Seattle’s wonderful venue for experimental music, the Chapel Performance Space. Details will be forthcoming, but I plan on presenting a number of musical pieces, poetry that paraphrases a proof by Turing in the style of Dr. Seuss, the work of several visual artists, small vignettes from Turing’s life, and possibly some dance and/or theater. When the final program has been finalized, I will post it here. [If you are an artist and think that you have something that belongs on this program or in the installation, by all means, contact me!]

2012 has been designated as the Alan Turing Year, and it is certainly appropriate to remember Turing, both for his tremendous mind and the tragic demise that was visited upon him. His work has had significant impact in diverse fields, including cryptology, pure mathematics, computing, biology, and philosophy. In this concert we will try to touch on all of these, presenting pieces inspired by both his life and his suicide. It should come as no surprise that his groundbreaking ideas are as interesting and relevant now as they ever were.