Saturday, November 21, 2009

Golden Punched Cards

One of the outstanding examples of continuity, between Cryptonomicon and The Baroque Cycle, involved the set of thin, square pieces of gold. Each of them held 1024 binary digits of information, in the form of a 32 x 32 array of points at which holes could be punched or not punched. The process by which Daniel Waterhouse produced them in 1714 was described in The System of the World (pages 412-423), as well as their transfers to Gottfried Wilhelm Leibniz (pages 801-804) and to the Leibniz-Archiv (pages 873-875). Their subsequent appearances were all mentioned in Cryptonomicon. During World War II they were transferred successively to Rudolph von Hacklheber (hinted at on page 626), to Otto Kivistik’s ship Gertrude, and to the U-boat V-Million (both on page 1058). About 50 years later, thousands of them were recovered from the wreck of V-Million, by Douglas and America Shaftoe and their coworkers of Semper Marine Services (page 569), and transferred to Randy Waterhouse and his coworkers of Epiphyte (2) Corporation at Kinakuta (page 1074).

During the production process, the gold stock was carried on a leather "skid" between work stations, apparently to protect it from damage. The ultimate pieces were called variously foils, leaves, sheets, and cards, names which have connotations suggesting different thicknesses. These statements should raise the questions: What were the mechanical properties of these pieces? In particular, how massive and sturdy were they? They were asserted to be made of "Solomonic gold". In the absence of detailed information about the properties of such material, let us assume that the specific gravity was 19.3, the same as normal pure gold.

The only dimensions given in The System of the World were the thickness, as "thinner than a fingernail", and the edge length, as "about a hand-span" (pages 412, 413). In Cryptonomicon, the dimensions were in mixed units: "maybe eight inches on a side and about a quarter of a millimeter thick, with a pattern of tiny neat holes punched through it, like a computer card" (page 569). Thus each had a volume of about ten cubic centimeters and a mass of about 200 grams. They were indeed heavy and valuable. Each contained enough gold to make more than two dozen gold guinea coins of Queen Anne (see Wikipedia). At $400 per troy ounce ($12.86 per gram), as was current in Cryptonomicon (page 655), they provided long-term data storage at about $20 per byte.

These overall dimensions allow an estimate of other dimensions of the cards. The maximum spacing of the grid lines for the holes would be 6 millimeters, leaving about 8 millimeters between the outer grid lines and the edges. The holes could be 2 millimeters in diameter, thereby looking small, if not "tiny", relative to the spacing between them. This diameter is indeed rather small, because the sensing rods in the ultimate mechanical "Logic Mill" would have to be even smaller in diameter, to ensure free passage despite possible misalignment. The rods could almost be called 'needles'. On the other hand, if the holes were much larger in diameter, they might seriously weaken the cards.

For diameter 2 millimeters, each "bit" of gold punched from a card would have a volume of about 0.8 cubic millimeters and a mass of about 15 milligrams. That would be easy enough to weigh on a balance scale of the day, but another factor should be considered, in using total weight to count the bits (page 423 of The System of the World). The problem arises because Stephenson does not state, specifically enough, how many bits were to be weighed at once. If the bits from only a single row were weighed, the problem does not arise, because the bits would almost certainly have been the same mass within a few percent. However, if all the bits from an entire card were weighed together, there might be hundreds of them. It is not clear that the technology of 1714 could have guaranteed that all bits had the same mass within a fraction of a percent. Let us start by estimating how many how many holes might have been in an average card.

The information punched into the cards resembled random prime numbers (page 711 of the Confusion). One row of 32 binary digits can hold any integer between 0 and 4,294,967,295. From the leading term of the prime-number theorem, that range includes about 190 million primes. It seems unlikely that Daniel Waterhouse had assembled that many logical concepts, during his years in North America.

An alternative approach is to consider the number of cards in the set. There were "thousands" of cards, which were in five crates when transferred from Gertrude to V-Million. If we assume 1000 cards per crate, each crate would contain 200 kilograms of gold, for a total of one tonne. For a very small crew in Gertrude, that might have taken "a whole day to load in." Either they had to empty and refill each crate, carrying the cards in small batches, or they had to work slowly and carefully, so as not to drop a full crate through Gertrude's bottom.

These 5000 cards could hold 160,000 prime numbers. This seems rather on the small side for Daniel Waterhouse’s years of work. If those primes had few duplications or omissions, the largest must be about 1.9 million, from the leading term of the other form of the prime-number theorem. It would take 21 binary digits to express these largest primes, and the average number of significant binary digits over this entire set of primes is more than 19. These significant binary digits would average about half ones and half zeros, so that a typical card should yield about 300 bits.

Note that doubling the number of cards would double the amount of physical work for anyone who ever had any contact with the cards. However, it would increase the maximum and average numbers of binary digits by only about one. The only way to use all 32 binary digits would be to change the nature of the information they were recording. Either they were not prime numbers, or else Daniel Waterhouse skipped over many more primes than he included.

The bits could have different masses for two reasons: the 32 punches in the machine at Bridewell could have different effective diameters; and different cards could have different thicknesses. The difference in diameters might tend to average out for different patterns of holes across a row, but a thin (thick) card would consistently seem to have fewer (more) freed bits by weight than by actual count. For simple weighing to be correct, the card thickness (about 250 microns) would need to be reproducible to better than 0.8 micron. The hand-cranked rolling mill in the "Court of Technologikal Arts" (described on page 412 of The System of the World) demanded discussion about its usage, even without this concern about the consistency of its output. It may not be enough that Daniel Waterhouse claimed "perfectly uniform thickness" (page 424).

It is extremely unlikely that a single pass through any mill could reduce the thickness and increase the area of the original plate by as factor of about 12 (from an eighth of an inch to a quarter of a millimeter). If the mill could be made to feed at all, it would probably just tear pieces off the leading edge. Even a modern rolling mill, working hot metal, typically reduces thickness by perhaps ten percent at a pass, so that many passes would be required. For the total reduction wanted for this gold, 16 passes would each be -15 %, 20 passes would each be -12%, or 24 passes would each be -10%.

I consider multiples of 4 so that the stock could be rotated 90 degrees between each pass, thereby keeping the stock roughly square, if it started that way. A rolling mill mainly increases the length of the material, with much less effect on the width. If Daniel Waterhouse’s original "squarish plate" was 16 inches x 18 inches (one third of a full plate from Minerva) (see below), it would end up about 57 inches by 64 inches. If the rollers weren’t long enough to handle this size, some of the crosswise passes through the mill would have to be replaced by extra lengthwise passes. The frame of the skid to carry the rolled gold would indeed be "the size of a dining table" (page 412).

Yet another complication is that this cold-rolling would work-harden the gold. It would have to be annealed rather often, perhaps after every two passes, in order to restore its ductility. Eventually this would involve a rather large oven. At any rate, rolling out the gold would require resetting the rollers of the mill many times over several days.

Achieving the desired accuracy for the final pass through the mill has two separate phases. First of all, the cylindrical surface of each brass roller would need to be made with radial run-out of a fraction of a micron, over its entire length and circumference. Otherwise, the sheet of gold could not possibly come out with the same thickness throughout its entire area. This is hard to believe for the technology of 1714. However, if that could be done, it might thereafter be possible to reset the final spacing between rollers accurately enough. The rollers could be closed down on a feeler gauge, until a specified force (measured with a spring balance) was needed to pull it out. That measurement should be repeated at many points on the rollers, to verify that the small run-out has been maintained. The final resetting of the rollers might take more time than the actual rolling of the last pass.

Note that the use of feeler gauges would make setting the intermediate spacings rather easy, even without knowing the exact thickness of any gauge. For example, to achieve a reduction of thickness by 20%, one need only require that a stack or four identical thicker gauges had the same total height as a stack of five identical thinner gauges. Those intermediate passes need not be set so carefully as the final pass.

This emphasis on uniformly reproducible thickness may be overkill for counting bits by weighing. However, it would help insure that the mechanical Logic Mill could shift its cards around without jamming. Even an "Electrical Till Corporation" card reader demanded cards of uniform thickness. The area of each golden card was easier to control by the technology of 1714, than was its thickness. The fence on the "shearing-machine" that cut the cards could possibly be reset to an accuracy of 0.1 millimeter (one part in 2000 of the edge length), so that every card had the same area, to one part in 1000. However, the jaws of the shear would have to be quite rigid, in order to control distortion over their length of several feet. The average thickness of an unpunched card would be proportional to its mass, to sufficient accuracy. Thus one needed only to compare the total mass of the bits to the initial mass of the particular card from which they came.

The mass of the cards was also involved in their packaging. The inner container resembled "... a hat-box, about a foot in diameter and half that in height" (page 801), so that each had a typical volume of about 11 liters. They were said to "... float, at least for a little while" (page 802), so that the contents could not exceed 11 kilograms. One must allow several kilograms for the wood shavings, so that perhaps only 8 kilograms of gold could be put in each box. Those 40 cards would be a stack only 1 centimeter thick, easy to be "all wrapped in paper". The outer barrels were rather small, with each holding only six hat-boxes. Something as mundane as salt cod, the intended disguise for this gold (page 804), might be shipped in larger barrels. Note that the 5000 cards we have assumed, needed only 125 hat-boxes and 21 barrels.

Pieces of almost any metal, rolled or beaten to this thickness, can be mutilated rather easily bare-handed. Because this gold is so heavy, one should ask whether it can be handled at all, or whether it 'mutilates itself' under its own weight. We could pose almost any simple problem involving one of these cards as a beam, and calculate the resultant deflection.

The minimum model calculation requires equilibrium at each point along the beam, between the external torque applied by loads and reactions, and the internal torque across a section of the beam, due to elastic stresses arising from the curvature of the beam. At the inside of the curve, the material has been shortened from its free length, and there is a compressive stress in it. At the outside of the curve, the material has been extended, and there is a tensile stress in it. Again we will admit ignorance of the mechanical properties of "Solomonic gold", and assume the value 79 giga-pascal for Young’s modulus, the same as for normal pure gold.

As a particular simple beam problem, let us consider placing one of the cards across a smooth horizontal rod. The center of the card face would be touching the rod, and the edges of the card would be parallel and perpendicular to the rod. (This problem is mathematically identical to that of cantilevering half the length of the card out of a horizontal clamp.) The effect of punched-out holes upon the response of the beam would depend upon the pattern of the punching, so we will consider only an unpunched card.

The solution for this problem (to be verified by the interested reader) is that the free edge of the card would be 6 millimeters below the support, and the slope of the card there would be -0.08. That value for the slope is small enough to validate the linear approximation used, i.e., that the horizontal position of any point in the card is indistinguishable from the distance measured along the curve of the card. It also means that overall tension in the card can be ignored as helping to hold it up.

It is possible to model the 'feel' of one of these cards, by using some different material cut to the same edge length. The expression for the deformation of the card incorporates the information about the material in the combination: Young’s modulus x square of thickness / mass density. I found that an 8 inch square of cardboard, cut from a file folder labeled "11 pt stock", behaves rather similarly to the golden square calculated above, each under only its own weight. It is obvious that the cardboard has an initial 'set' and a 'grain'. The deflection depends upon which side is up, and which edges are parallel to the rod. It averages about 4 millimeters, so that this cardboard acts somewhat stiffer than the gold.

This square of cardboard can then be handled in various ways, and should assume about the same deformed shape that one of the golden squares would. Of course, the mass to be held up is only a few grams, rather than 200 grams. The square can be picked up between the fingertips of both hands, at opposite edges. (You can’t do that with a piece of paper that size.) It can be held by one hand in many different positions, without bending appreciably. It seems appropriate to call one of these golden squares a 'card'; it acts like a card. (It also happens to be close to the thickness of an IBM card. A dial caliper showed my last remaining IBM card to be 1/5 millimeter thick [0.oo8 inch].)

An obviously dangerous maneuver is to try to hold the cardboard square approximately horizontal, by grasping a single corner between thumb and forefinger. It can be made to assume a curve with radius as small as about one centimeter, next to the fingers, but it flattens out again when released.

For a very ductile material like gold, the yield point is apparently poorly defined, or at least, hard to find online. Wikipedia does give an "ultimate strength" of 100 mega-pascal, which would occur at a strain of about 1/8 percent. For the thickness of 1/4 millimeter, this would be a radius of curvature of about 10 centimeters. A gold card, bent to a radius of one centimeter like the cardboard, would almost certainly be permanently deformed in this maneuver.

Several other points about these golden cards were suggested by the description of the arrival of Tsar Peter "the Great", accompanied by Solomon Kohan (pages 601, 602 of The System of the World.) One of these was that assay samples had been cut from corners of the cards of the original (incomplete) set, which had been sent to Leibniz at the Tsar’s court.

An important omission from the functionality of the cards was an orientation marker. These square cards, with a square pattern of hole locations, have eight-fold symmetry, but only one proper way to be inserted into the Logic Mill. Of course, it would be possible to determine the orientation by a close inspection of each card. The punches probably raised burrs on the backside of the card, which could be seen or felt. If the data indeed had the many place-holding zeros as mentioned above, it would be obvious which way to read across the rows.

In comparison, Hollerith/IBM cards are rectangles, which offer four-fold symmetry. The proper orientation for them is indicated by a cut-off corner. (I have never seen ETC cards, but I assume that they were rather similar.) The same sort of asymmetric corner cut would similarly show at a glance that golden cards in a stack were all oriented correctly. A related problem would be to show that the golden cards were in the proper order.

The other problem is most charitably treated as a typographic error. One of the golden plates, from which Daniel Waterhouse produced the cards, had just been carried out of Minerva by a barefoot seaman, as a burlap-wrapped bundle. "The package was perhaps a foot and a half wide, four long, and an inch thick." The last dimension should read "..., and an eighth of an inch thick." Even Isaac Newton had learned that the Solomonic gold entered England as hand-hammered sheets of that thickness (page 145 of The System of the World). The difficulty is obvious: a full inch thickness of gold with that area would have a mass of about 270 kilograms. That would be about double the mass of Peter "the Great", but with his great strength he might have been able to handle it. However, it would be about four times the mass of the barefoot seaman, and beyond his capabilities. One eighth of that mass could be a one-hand load for Peter, or a two-hand load for the seaman.

This eighth-inch plate of gold offers another example for beam problems, to determine how sturdy it was. However, a far more interesting question concerns the nail holes in it. They must have been there, although not mentioned at this point in the story. This plate, and its mates, had been the sheathing on the hull of Minerva (page 796 of the Confusion). To hold such a plate in place, against its weight, the drag of the water, and the working of the hull planks as the ship pitched and rolled, would require at least dozens of nails. As a piece of such a plate was passed through the rolling mill, did its nail holes tend to disappear, or did they tend to become larger?

Personally, I feel that both Leibniz and Daniel Waterhouse showed some conceit when they argued for using gold as the storage medium (page 711 of the Confusion and page 424 of The System of the World). Indeed gold is ductile and does not tarnish, but the information is in the holes, not on the surface. Pure silver and copper are also rather ductile, and 1/4 millimeter is probably not too thin to roll out those metals. Silver may turn black and copper may turn green, but either of them would preserve the information of the holes for many decades, or even centuries.

In fact, the information of the holes was effectively lost in less than three centuries. Daniel Waterhouse had spent years writing out paper cards, each with a number in binary notation along one edge, which represented the information on the card in other formats. That number was what Miss Spates, or other operators, punched into a golden card. Those paper cards were surely included as part of the paperwork, which accompanied a set of golden cards into their hat-box. That paperwork did not survive, becoming lost either during transfers or in the sinking of V-Million.

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