The Blacksmith’s Forge: An Extension of Euclidean Geometric Construction as a Model of Computation

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Models of Computation: The Church-Turing Thesis and Geometric Construction

The Church-Turing Thesis posits that the equivalence class that includes the Turing machine, and is also the basis for modern digital computing, is the most powerful model of computation. And it hasn’t been proven, but when people have checked out other models of computation, every one has turned out to either be equivalent to the Turing machine, or become lesser.

Quite probably it may be impossible to construct some useful computer by this model; quite possibly for that matter its greatest usefulness may come through simulations by digital computer, in which case its simulations will automatically not exceed Turing machines or digital computer by its power. However, even if is a failure at scaling some of the highest peaks, it seems an interesting and provoking possibility to explore.

Standard Euclidean geometric construction is a model of computation. It is not usually presented as such, but you start with a diagram and all that may be inferred from it, and you have two tools, a compass and straightedge, as well as a pen or other implement to draw with. And the solution to a construction is to come back with an algorithm that will go through computational steps that start with its initial state represented in a diagram, and use your three tools to create the desired end result.

Both models figure into the model of computation discussed here, but the model of computation is different.

The model of computation described here is like a blacksmith’s forge. I have read that one of the first things a blacksmith makes, is a pair of real tongs. And a blacksmith is not just turning out nails and other things for other people, but tools used in the forge. The core insight here is that a blacksmith can create tools, much as a computer programmer may make customized tools for their own work. This is at its core a geometric model of computation, with a more obvious debt to geometry, although the tools should be sufficient to implement a Turing machine. One person made the interesting suggestion that it is applying recursion to geometric construction.

The blacksmith’s forge

The main tools the blacksmith’s forge works with are as follows; the first three are taken from geometric computation:

  1. A compass, that can be used to draw circles.
  2. A straightedge.
  3. A marking implement.
  4. A jigsaw. The geometric plane is conceived not to be one point thin, but a uniform distance thick. When the blacksmith’s forge has constructed the closed outline of the shape, the shape can be cut out.
  5. Pins, equal in length to two or more (whole) times the thickness of the plane. If one pin goes between two shapes one on top of the other, and the shapes are not otherwise constrained, they will be able to pivot around the pin with respect to each other. If two or more pins go through, then the two positions will be rigid in how they are joined.
  6. Pieces cut out with the jigsaw, possibly joined by pins.

Idealized Physics in the Blacksmith’s Forge

The blacksmith’s forge has an idealized physics. The pin and jigsaw are parts of this idealized physics, but another part of the physics is that pieces do not tip over: any number of stackings that would immediately fall over in the real world are assumed to simply stand upright, the pieces resting on top of any other piece immediately beneath them for some positive areas. There is friction, and pieces pushed to where one entity crosses another, for instance, will immediately stop moving if they are no longer being pushed. Items touching each other can be pushed past each other, but only so far as they are pushed. This does not exhaust the physics, but if you think of the physics of ordinary geometric construction, you should be close to the mark.

Three Classic Problems

Trisecting an Angle

Consider the following diagram:

That is half of it. Take another one, rotate it by one notch and pin it to the bottom one, and you have forced equality for the angles between adjacent arms:

Take this constructed device, rotate it so point A is at an angle’s vertex, B is in the angle’s clockwise side, and expand or contract the accordion-like device so that C is at the angle’s other side. Angle DAB is now one-third of the (trisected) original angle.

Doubling the Cube

Consider the following modification of the previous diagram:

Take a circle, and draw a concentric circle at twice the radius. Then place the constructive device so point A is at the center, and expand out or collapse in so that B is on the initial circle. Then collapse or expand the device so that it is on the the new “double radius” circle. Point E will have a distance to the center equal to the original radius times the cube root of two.

Squaring the Circle

Cut out two circles, and a tall, thin rectangle. Put the circles snugly and squarely so that the line between them and the rectangles is perpendicular to the rectangles’ long dimension. Put pegs through the circles’ centers through the perpendicular rectangle, and mark (A) both the first circle and the rectangle where they meet. While holding the first circle squarely, push on the outer circle until it wraps the long rectangle around the first circle, and mark on the tall rectangle where it touched the circle’s mark.

You now have a distance marked out on the tall rectangle that is 2π times the radius of your circle. Getting the square root of π is not terribly difficult; you can draw two subsegments of a line segment, one equal to the original circle’s diameter in length and one equal to circle’s circumference, and then draw a long line segment perpendicular to the first segment starting where the two meet. Take one of the corner-like squares above, place it so that it touches both endpoints of the line segment, and while continuing to hold it tight to the ends of the segment, move it so its inner corner lies on the perpendicular line segment. The distance from that point along the line segment to the center is equal to the square root of π times the length of the original circle’s diameter:

(And though this would be laborious, I see no reason why such calculations could not emulate a Turing Machine.)

Foul!
You’re Using Extra Privilege!

I am indeed using extra privilege, but may I point something out?

There is a bit of a historical difference between now and ancient Greece. We now have a number of branches of mathematics, and though there may be likes and dislikes, it is something of an outsider’s question to ask, “Which is right: real analysis or modern algebra?” There is a general sense that as with board games, if you want to play chess you play by the rules of chess and if you want to play go you play by the rules of go.

The three questions neatly and easily answered are the standard three famous problems which it was subsequently proven to be impossible to construct with Euclidean geometry. And these were not simply mathematical chess problems; I don’t know the stories for all of them, but legend has it that there was a plague killing many people and an oracle stated that the plague would be stopped if a cubic altar were built that was twice the volume of an original cubic altar. This was not one where people only used Euclidean construction because they decided they could only play by the rules of Euclidean geometry. This was a “by any means necessary” matter, and it should be understood as much. The attitude of “This is the set of rules for this particular game” is anachronous; people would be very glad to have an extension to Euclidean construction that would allow solution of at least one of these problems.

And it is not clear to me whether this is any sense of useful model of computation. (I personally think, out of my second master’s thesis, that the human brain can do things no Turing-approximant style of computers can do. Some people have said, “A year spent in artificial vision is enough to make one believe in God,” and there are some basic things, like making sense of an I can read book, that most humans do well but are so far insurmountable to computers; one writer wrote of an embodied AI robot “Cog,” “The weakness of Cog at present seems to be that it cannot actually do very much. Even its insect-like computer forebears do not seem to have had the intelligence of insects, and Cog is clearly nowhere near having human intelligence.”

But I think this model of computation is interesting, whether or not it proves useful.

The Administrator Who Cried, “Important!”

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Revisited after some time

Someone said that a memo is written, not to inform the reader, but to protect the sender.

There is something wrong when employees receive so much allegedly mandatory reading material that if they were actually to sit down and read it as told, they wouldn’t get other work done. And it is entirely inappropriate to demand that people without significant legal acumen claim to have read and understood a contract. Really, contracts are rightly understood only if you understand the tradition surrounding how they are interpreted. That means that unless (or possibly even if) you are a lawyer (or else a hobbyist who may not legally be licensed to practice but who is fascinated at learning how law works), you don’t understand the contract. This is, incidentally, why there’s the website tosdr.org (“Terms of Service – Didn’t Read“).

That much I still believe. However, I believe there was some nasty pride in expecting the business world to meet what I consider reasonable. The normal way of dealing with things is to not read, or to read just enough. And that is why in my first job with over a quarter inch of daily allegedly mandatory reading, I should just have listened to a colleague gently tell me that I didn’t have to read that.

I’ve worked on humility a little bit since then.

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Once upon a time, there was a new employee, hired fresh out of college by a big company. The first day on the job, he attended a pep rally, filled out paperwork concerning taxes and insurance, and received a two page document that said at the top, “Sexual Harassment Policy: Important. Read Very Carefully!”

So our employee read the sexual harassment policy with utmost care, and signed at the bottom indicating that he had read it. The policy was a remedial course in common sense, although parts of it showed a decided lack of common sense. It was an insult to both his intelligence and his social maturity.

Our employee was slightly puzzled as to why he was expected to read such a document that carefully, but soon pushed doubts out of his mind. He trotted over to his new cubicle, sat down, and began to read the two inch thick manual on core essentials that every employee needs to know. He was still reading core essentials two hours later when his boss came by and said, “Could you take a break from that? I want to introduce you to your new co-workers, and show you around.”

So our employee talked with his boss — a knowledgeable, competent, and understanding woman — and enjoyed meeting his co-workers, trying to learn their names. He didn’t have very much other work yet, so he dutifully read everything that the administrators sent him — even the ones that didn’t say “Important — please read” at the top. He read about ISO 9001 certification, continual changes and updates to company policy, new technologies that the company was adopting, employee discounts, customer success stories, and other oddments totalling to at least a quarter inch of paper each day, not counting e-mails.

His boss saw that he worked well, and began to assign more difficult tasks appropriate to his talent. He took on this new workload while continuing to read everything the administration told him to read, and worked longer and longer days.

One day, a veteran came and put a hand on his shoulder, saying, “Kid, just between the two of us, you don’t have to read every piece of paper that says ‘Important’ at the top. None of us read all that.”

And so our friend began to glance at the first pages of long memos, to see if they said anything helpful for him to know, and found that most of them did not. Some time after that, he realized that his boss or one of his co-workers would explicitly tell him if there was a memo that said something he needed to know. The employee found his workload reduced to slightly less than fifty hours per week. He was productive and happy.

One day, a memo came. It said at the top, “Important: Please Read.” A little more than halfway through, on page twenty-seven, there was a description of a new law that had been passed, and how it required several jobs (including his own) to be done in a slightly different manner. Unfortunately, our friend’s boss was in bed with a bad stomach flu, and so she wasn’t able to tell him he needed to read the memo. So he continued doing his job as usual.

A year later, the company found itself the defendant in a forty million dollar lawsuit, and traced the negligence to the action of one single employee — our friend. He was fired, and made the central villain in the storm of bad publicity.

But he definitely was in the wrong, and deserved what was coming to him. The administration very clearly explained the liability and his responsibility, in a memo very clearly labelled “Important”. And he didn’t even read the memo. It’s his fault, right?

No.

Every communication that is sent to a person constitutes an implicit claim of, “This concerns you and is worth your attention.” If experience tells other people that we lie again and again when we say this, then what right do we have to be believed when we really do have something important to say?

I retold the story of the boy who cried wolf as the story of the administrator who cried important, because administrators are among the worst offenders, along with lawyers, spammers, and perhaps people who pass along e-mail forwards. Among the stack of paper I was expected to sign when I moved in to my apartment was a statement that I had tested my smoke detector. The apartment staff was surprised that I wanted to test my smoke detector before signing my name to that statement. When an authority figure is surprised when a person reads a statement carefully and doesn’t want to sign a claim that all involved know to be false, it’s a bad sign.

There is communication that concerns the person it’s directed to, but says too much — for example, most of the legal contracts I’ve seen. The tiny print used to print many of those contracts constitutes an implicit acknowledment that the signer is not expected to read it: they don’t even use the additional sheets of paper necessary to print text at a size that a person who only has 20/20 vision can easily read. There is also communication that is broadcast to many people who have no interest in it. To that communication, I would propose the following rule: Do not, without exceptionally good reason, broadcast a communication that concerns only a minority of its recipients. It’s OK every now and then to announce that the blue Toyota with license place ABC 123 has its lights on. It’s not OK to have a regular announcement that broadcasts anything that is approved as having interest to some of the recipients.

My church, which I am in general very happy with, has succumbed to vice by adding a section to the worship liturgy called “Announcements”, where someone reads a list of events and such just before the end of the service, and completely dispels the moment that has been filling the sanctuary up until the announcements start. They don’t do this with other things — the offering is announced by music (usually good music) that contributes to the reverent atmosphere of the service. But when the service is drawing to a close, the worshipful atmosphere is disrupted by announcements which I at least almost never find useful. If the same list were printed on a sheet of paper, I could read it after the service, in less time, with greater comprehension, with zero disruption to the moment that every other part of the service tries so carefully to build — and I could skip over any announcements that begin “For Married Couples:” or “Attention Junior High and High Schoolers!” The only advantage I can see to the present practice, from the church leadership’s perspective, is that many people will not read the announcements at all if they have a choice about it — and maybe, just maybe, there’s a lesson in that.

As well as pointing out examples of a rampant problem in communication, where an administrator cries “Important!” over many things that are not worth reading, and then wonders why people don’t believe him when he cries “Important!” about something which isimportant, I would like to suggest an alternative for communities that have access to the internet. A web server could use a form to let people select areas of concern and interest, and announcements submitted would be categorized, optionally cleared with a moderator, and sent only to those people who are interested in them. Another desirable feature might let end receivers select how much announcement information they can receive in a day — providing a discernible incentive to the senders to minimize trivial communication. In a sense, this is what happens already — intercom litanies of announcements ignored by school students in a classroom, employees carrying memos straight from their mailboxes to the recycle bins — but in this case, administrators receive clear incentive and choice to conserve bandwidth and only send stuff that is genuinely important.

While I’m giving my Utopian dreams, I’d like to comment that at least some of this functionality is already supported by the infrastructure developed by UseNet. Probably there are refinements that can be implemented in a web interface — all announcements for one topic shown from a single web page, since they shouldn’t be nearly as long as a normal UseNet post arguing some obscure detail in an ongoing discussion. Perhaps other and better can be done — I am suggesting “Here’s something better than the status quo,” not “Here’s something so perfect that there’s no room for improvement.”

In one UseNet newsgroup, an exchange occurred that broadcasters of announcements would be well-advised to keep in mind. One person said, “I’m trying to decide whether to give the UseNet Bore of the Year Award to [name] or [name]. The winner will receive, as his prize, a copy of all of their postings, minutely inscribed, and rolled up inside a two foot poster tube.”

Someone else posted a reply asking, “Length or diameter?”

To those of you who broadcast to people whom you are able to address because of your position and not because they have chosen to receive your broadcasts, I have the following to say: In each communication you send, you are deciding the basis by which people will decide if future communications are worth paying attention to, or just unwanted noise. If your noise deafens their ears, you have no right to complain that the few truly important things you have to tell them fall on deaf ears. Only you can prevent spam!

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