The word lotta, on the other hand, will likely take longer to make it. It’s dialect, and writing ‘lot of’ even well after anyone stops saying it wouldn’t be atypical of English generally.

In the last post, we talked degrees and radians. Radians are less arbitrary in some sense, but less intuitive also, at least for most people, and I believe this to be more than just training.

Here’s an example. If you want to know what one radian is in degrees, it’s c. 57.2957795 of them. This does a weird one on our intuition. When we divide a circle into various angles, most of the easy ways give us nice numbers: six ways is 60 degrees, ten ways is 36 degrees, twelve is 30 and so on. Why would the relationship between a single radian and a single degree be irrational?

The reason is Pi, which is itself irrational. A whole circle of arc is defined as 2 * pi radians, because the radian is defined in terms of the relationship between a circle’s radius and its circumference. This is known as circular reasoning.

There are eminently good reasons for this definition, as those who retained their trigonometry know. But when using angles for real work in the world, the value of this mathematical abstraction is not always obvious, and the degree is used by reflex. The real conversion between degrees and radians is 1 degree = 2pi/360 radians; it’s not rational, or pretty, but at least it’s exact.

I propose we simplify our lives, and define a new unit, which I will call the metric slice. A slice is defined as such: 1 slice = 2 pi radians.

Picture a steaming, delicious pi, fresh from the oven. Slice it in half, and each side is half a slice. Slice it into thirds, and each is a third slice. Slice it into twenty-thirds, and everyone will have one-twentythird slice.

Or if you prefer, your .5 slice could be Pi radians. Your seventeenth slice could be pi/8.5 radians, or 2pi/17 for the homies in the know. Conversion is simple, you divide 2pi by your slice. Since you have to remember 2pi to work with radians in the first place, this is free.

Conversion to degrees is also simple: a one-eighth slice is 360*slice, in this case, 45. I predict this will not be hard to remember.

Fun fact: the number of slices cut by a harmonic oscillator in a second is its frequency in Hz! Beat that, radians!

The reason this is so much abundantly simpler is that the natural unit of the angle is the circle, and should be 1. There’s irrationality built into the way the universe is composed; we should tuck it as far out of the way as possible, to ease stress on our poor wet minds.

There’s also a unit of solid angle, which we may as well call the ‘halo’. The concept is so obvious as ato require little elaboration. The steradian, by contrast, is a headache-inducing extension of the idea of the radian into 3 dimensions. To convert, multiply halo by 4 pi, making 1/4 halo equal to pi steradians as expected. If you’re willing to guess that a solid wedge projected through one face of an icosahedron has 1/20th halo of solid angle, you’re well on your way to understanding the utility vs 1/5 pi steradians.

I’m convinced that the use of halo and slice units can significantly simplify and aid understanding of trigonometry and conic sections, among other areas of interest, perhaps even offering a simpler link between these disciplines and symmetry theory. I haven’t had the time to significantly pursue this; in the meantime, I offer the slice and halo as conceptual aids.

The standards, natural dimensions and scales we choose have consequences, in the physical world and in our intellectual world also. Since a standard is a concept we apply, not an object with independent existence, applying different standards to the same measurement can and typically does have important effects on the ways in which the standard is used.

Our last discussion of standards dealt with the pure numbers of dimension, namely, base and scale. Lets turn our attention to angles.

When we start to measure the dimension of length, angles will quickly leap to the foreground. Any two points in space have a length between them, and that’s all we can really say. Any three points in space have three lengths defined by their distance. There are also three angles, which are also defined by their distance.

There are a lot of different ways of referring to those angles. The most common here in North America is the ‘degree’. The degree is an example of preferred numbers being baked into the system. In this case, the bakers were the Sumerians, and the degrees were literally baked into tablets of clay.

The Sumerian culture, and their Babylonian successors, used a counting system based on the number 60, called sexagesimal. There’s a reason for this, namely that 60 is the smallest number evenly divisible by the numbers 1 through 6.

This makes fractions easy to work with. We know all about this, actually, when we tell people we’ll see them in ten, fifteen or twenty minutes, a sixth, quarter and third of an hour respectively. Our time system is inherited from the Sumerians, ultimately, and reflects their counting base.

It was the Sumerians also who divided a circle into 360 degrees. 360 is also close to the ratio days:years, which I am not convinced is a coincidence; 360 is a very special number and Mercury, for example, has a precise 3:2 ratio of days to years.

I think the convention was adopted for a simple, practical reason: 360 and 60 are both highly composite numbers. This means they have more divisors than any number smaller than them. This makes them better than any smaller choice for dividing into even fractions, which is one of our favorite things to do with circles.

So 360 makes sense. The metric system, early on, made various abortive attempts to divide a circle by 400 degrees rather than 360, but this never caught on. 400 allows for easy right-angle math, but sucks for anything involving 3, hence 6 and 12 are also awkward.

The metric system, and math in general, has moved on to a more rational system for representing angles. It is the radian, and it gets to the heart of what an angle is: a relationship between a circle and its radius.

We all know that c = 2 pi r, that the circumference of a circle is its diameter times pi. This is the basis of the radian.

Lets think about the face of a clock, and the minute hand. The minute hand is a radius of a circle, and sweeps out one circle every hour. In that hour, the tip of the minute hand will have travelled the distance of the radius times 2pi. It will also have swept through 360 degrees of arc.

In the time it takes for the tip of the minute hand to travel the exact distance of the radius, it will have swept out one radian of arc.

Or, in one hour (360 degrees, one circle) the minute hand will have swept out 2pi radians of arc. 180 degrees is pi radians, 90 degrees is pi / 2 radians, 45 is pi / 4 radians and so on.

This is a better standard, for some purposes. Doing actual math involving angles is utterly dependent on it, for example. It is able to express any rational subdivision of a circle as a whole number divisor, for example, pi/56 radians is 23 even divisions of a circle, which is 15.652…. degrees.

The radian, however, is confusing and counterintuitive, and a lot of smart people don’t really get it. It makes people uncomfortable. There’s a reason for that; my next post will cover that, and propose another way of talking about angles.

Chances are you’re surrounded by them, from the light over your head to the pipes and beams below. You are clothed by the products of machines. Kept warm by them. Transported by them. They cultivate your food, keep it cool, and help you cook it.

Where are these machines? Have you met them? Do you know the humans who tend them? What are their lives like? What would happen to you if they stopped tending those machines, or if the machines stopped running because the oil was gone?

Where would you get shoes, once the old tires were gone? How could you keep your rags together once you broke your last needle? You know windmills exist, boilers and steam engines and mirrors. Could you build them, and provide power for everything we take for granted?

You could if you had the right machines. And if they were just the right machines, they could make you most of what you need and a whole lot of what you want. They could even make more of the right machines…

RepLab is a project to build the right machines for the job. Serious industrial tools that can shape metal, assemble circuit boards, print in plastic, ceramic, and glass, and otherwise build most of what we need for high-tech living from scrap metal and specialized components like computer chips. All open source, and human scale: nothing you’d need a mortgage to build. Actually, all you’d need is some sweat equity, scrap metal, specialized electronic components, and a RepLab.

RepLab, as the portmanteau would suggest, draws from both the FabLab idea of a laboratory where you can build anything, and the RepRap project’s dream of a self-replicating desktop factory. We aim to design and build a suite of machines for general-purpose production of designs in most of the materials we take for granted: metal, plastics, wood, ceramics, and ultimately glass and thread as well.

The project is practical in emphasis. The call to action for RepRap was penned by Marcin Jacubowski, a visionary builder working to create resilient, high-technology community at the agrarian village level. His concept of permafacture combines the agricultural sustainability of permaculture with a high-tech open source ecology of technologies which can be replicated on a scale of about a hundred people, anywhere in the world. His lab and home, Factor e Farm, has produced impressive results, including an open-source compressed earth block press, an open-source modular tractor for cultivation and construction, and a computer controlled table that cuts steel with plasma.

The last project, called RepTab, is being adopted and modified as one of the first RepLab projects. The table is made mostly of angle steel, which can be cut and drilled with the plasma torch head of a RepTab. The RepTab can make most of its structural pieces, automatically. That’s a recursive machine, and it’s the very beginning of what RepLab is all about.

We’re trying to be fairly precise about our goals, so that we’ll know when we’ve achieved them. For example, we avoid the term ‘self-replication’ in reference to our machines. Self-replication, as practiced by cells, includes self-assembly. This is a hard problem, and not a project goal.

We speak instead of recursive manufacture, and general replicators. Recursive manufacture is simple: it is building a widget using the widget. It is important, but not particularly impressive in and of itself. An example is a kiln, which is simply a pile of bricks and a fire. Since bricks are made in a kiln, and trees grow themselves, a kiln next to a forest and a vein of clay is a fully recursive technology. It is also manual, arduous, time-consuming work.

Recursive manufacture is the rule, not the exception. Everything is made on machines, including the machines that make things. In fact, the traditional factory is split between the production floor, where machines make products, and the machine shop, where machines make tools for making products.

What has changed is the nature of those machines. Consider the single case of writing. To replicate a text, a scribe needed tools: a reed pen, ink, a grinding stone, and papyrus, a knife to trim the reed. Without the tools, the job is impossible. With the tools, the job can be done.

One thing to notice is that the human is working, and the tools are helping. A faster tool, a pen rather than a chisel, makes faster work, as does practice. But all work is human work, and it shows: in particular, skill becomes an all-important factor in the quality of the work and how much can be performed. This kind of replication scales poorly, favors one-of-a-kind items, and rewards lifetime specialization. Call it craft; we call it Type 1 replication.

The printing press was one of the most disruptive technologies of all time, and the reason is that it was one of the first examples of a new kind of replication. Now, by investing significant time in building a machine, casting type, and laying out a page, a human can replicate a text hundreds or thousands of times, just by pulling a lever. Now we have a technology for replication that favors mass-production, scales well, and allows for the exploitation of unskilled labor. This is Type 2 replication, characteristic of the Industrial Revolution.

Type 2 replication results in an explosion of basically identical, cheap products, a lot of jobs in the cities, and a social and industrial ecology dominated by capitalists, who have the money to enjoy the benefits of large-scale manufacture. It remains the dominant form of manufacture, and it shapes our way of life, right down to the concepts of 9-to-5 and the weekend.

The Type 2 tools have evolved in a couple directions, which converge on our goal, the Type 3 replicator. As printing presses evolved, they became fully automatic in operation. A large early 20th century newspaper press still had to be laid out; once this was done, the machine was turned on, the paper rolls started spinning while the ink vats emptied, and bound stacks of newspaper shot out the other end. This is Type 2a replication, ‘a’ for ‘automation’. It displaces Type 2 wherever it is developed, since one no longer has to pay workers to pull levers.

The other direction is exemplified by the early Xerox machine. It could make a copy of any template provided, quickly and automatically, but needed human tending to change the template or do anything other than spit out identical copies, such as collation. For anything but a single sheet, an old-fashioned Xerox is a Type 2b replicator, ‘b’ for ‘bespoke’ or ’boutique’. They favor customization, and scale poorly, but they give good results to unskilled operators. Type 2a replication gave us the magazine; Type 2b replication gave us the ‘zine, which were culturally important without ever displacing the more professional media that inspired them.

The breakthrough to Type 3 came with the laser printer. A laser printer can print whatever template you want, accurately and quickly. It is functionally just as easy to make 1000 different pages than to make 1000 of the same page. The largest laser printers take a variety of raw goods in one end, and out the other end comes books, printed, cut, collated, bound, wrapped in a book jacket, packed in a mailing envelope and stamped. Each book different, each book ready to go. Humans write the books, order the books, empty the hoppers, and refill the feedstocks. The rest is all machine.

Type 3 replicators, in a variety of media, are the goal of the RepLab project. They scale well, favor customization and human empowerment over mass production and serfdom, and tend to economically displace Type 2 technologies when they emerge.

The computer is the best Type 3 replicator ever invented, in fact I would argue it is impossible to have a Type 3 replicator that is not also a computer. It is the computer, not the laser printer, which is displacing the magazine and the newspaper, through its ability to replicate any text in any language for the cost of a sip of electricity. Vinyl records are a Type 3 replicator with a Type 2 production process for the templates. Mp3 players are Type 3 through and through, and that’s the single biggest difference.

So lets get down to brass tacks, which we hope to eventually print with brass powder and a laser. The RepLab project has big ambitions and an open-ended project list, but we’ve already whittled down to a few promising and existing technologies to adapt and pursue.

The first one is a RepRap type printer, with similar function and different design goals. The RepRap aims to make a small printer for 3-d plastic objects, which can make as much of its chassis as possible. Our printer will be based on the RepRap extrusion platform and source code, but modified to be robust and automatic, so it can print plastic parts, one after another, for weeks at a time, with someone stopping by from time to time to empty the hopper and feed it another roll of plastic filament. Recursive manufacture is a goal for the whole RepLab, not each of the machines in it; we will use printed plastic where it makes sense in the design, and other approaches elsewhere.

The second one is based on the plasma cutting RepTab described before, but modified to be capable, with modular improvements, of routing, plasma cutting, and possibly laser and water jet cutting. Add a feed for 4×8 sheets of metal, wood and foam, and an automatic hold-down, and you have a robot that can make almost anything that’s flat, with a minimum of adult supervision.

The third goal is to combine the work of the CandyFab project with that of Open3dp to produce an open-source powder-bed printer. With some development, this machine will be able to make ceramic and opaque glass objects and bind metal powder for heat sintering. Combined with a simple furnace/kiln, and there are shelves full of free designs for those, this offers the promise of a RepLab that can print everything AND the kitchen sink! Plus molds for metal casting, thermal insulators, and many other useful and wonderful things.

There’s more we’ll need, and there’s more we’re planning to build, in particular a system for automated circuit assembly. Even these three technologies, in tandem, would provide a facility which could produce many economically significant goods, including a fair capacity for recursive manufacture.

The project is in the very early stages. We’re building a crowdfunding platform and working on moving forward with our core designs so we can build them as rapidly as practical. To join the conversation, stop by our google group, follow us on Twitter or identi.ca at @RepLab, and look for our project blog coming soon at http://www.replab.org/.

Better yet, get some friends, get a space for a RepLab, and start hacking! We intend to build and replicate these technologies all over the world. We hope you’ll join us.

There is a growing interest in self-replicating machines. Beginning with the RepRap project, and now continuing into RepLab, the open source FabLab, there is a serious effort to build machines which can build themselves.

It is a laudable goal. A machine which can make itself can also make an unimaginable variety of other machines, each unique if desired, and promises an era of material abundance and freedom from scarcity. Pursuing that goal, however, has shown the goal itself to be somewhat unclear. This is an attempt to remedy this situation.

What is replication? The production of a replica, but what is a replica? The original meaning was that of a duplicate of a work of art, properly one produced by the original artist.

I propose we define replication precisely, as the transcription of a template into a physical form. This definition is perhaps more general than we are used to, but I believe it accurately captures the use of the word.

Let’s take the test case: an artist producing a replica of her own painting. The original painting may have taken many months, while the replica may take only a day or two at most. The artist is using tools, a brush and palette, and materials, canvas and paint, to make a copy of a template, the original work.

How about DNA self-replication? We have a toolchain of proteins, from DNA Transcriptase on, which transcribe a template, the DNA, into a copy of itself, using materials (ATP, other nucleic acids, caffeine if there’s some in the organism in question) both extrinsic to the cell and fabricated within it.

Note that in the first case the template resembles the result, while in second case, if we consider the result a new cell rather than new DNA, the result doesn’t resemble the template at all. This is immaterial to the act of replication; a high-resolution inkjet printer and a scanner could replicate the artist’s original work, faster and more accurately than she could, and in this case the template would be stored as bits within a computer.

The rest of this discussion will focus on replication in the context of human-tool interaction.

This means replication of an object, from a template, through the combined efforts of at least one human with at least one tool. We call this a ‘tool’ rather than a ‘machine’ for greater generality.

The earliest form of replication was a human using a tool to make a copy of an object. Our paradigm case for this is a scribe copying a scroll. A human can’t do this without the materials (paper, ink) and tools (quill, sharpener) required for the act. With those tools, a human can copy a scroll in a certain amount of time.

A few observations here: the tool enables, the human works. A faster tool, let us say a pen rather than a chisel on stone, allows for faster work, but more copies will always take more human time. We will call this Type 1 replication. It is characterized by modest gains in efficiency for mass production and poor scaling characteristics.

A second form of replication came about when humans began using tools to automate aspects of production. By investing time in setup, many copies can be replicated much faster. The work is still done by humans, with tools; but by doing the work intelligently, many copies can be made with comparatively low effort. We call this Type 2 replication.

The paradigm case for Type 2 is the printing press. By investing considerable effort in building a press, casting type, and laying out a document, one is rewarded by the ability to print as many copies as one wants, with speed and reliability which increase over time. Type 2 replication has good scaling characteristics and rewards mass production. It is characteristic of the Industrial Age.

Type 2 production next evolved in two directions the first of which we’ll call Type 2a: significant setup time followed by fully automated production. Newspaper presses work in this fashion, although the most recent ones are Type 3. Type 2a has significantly better scaling characteristics than Type 2, and mostly displaces it when developed.

Type 2b replication is where there is significant tool investment, but negligible setup time and non-automatic production. The matching paradigm for this is a Xerox machine, though like newspaper presses these have become increasingly Type 3 over time. An early Xerox would make a copy of anything that fit on the glass, or even a hundred copies, but a human had to stand around feeding it sheets and doing things like collation by hand. This has poor scaling qualities, being labor intensive, but it favors customization over mass production.

Type 3 replication has negligible setup times and fully automatic production. Our paradigm case here is a laser printer. Even a desktop laser printer can print replicas of stored data, each different, for many hours without needing attention. The factory scale printers can make entire books without human intervention. Type 3 replication is characteristic of the Information Age, or Industrial Revolution 2.0.

Type 3 has good scaling characteristics and rewards customization over mass production. One can make an arbitrarily large number of copies off any given template, but there’s little to no advantage over using many different templates.

These are the essential distinctions:

Many FabLab technologies, such as laser cutters, hobby-scale 3d printers, and CNC mills, are Type 2b. What’s interesting about 2b replication is that it’s often a small step away from being Type 3, needing only, for example, a small conveyor belt. I hope that by drawing attention to the radical difference in scaling quality between Type 2b replication and Type 3 replication, I can encourage open-source hardware developers to strive for Type 3 tools whenever possible.

A Type 2b tool is a creative enabler, while a Type 3 tool is both a creative enabler and an economically disruptive force, because it can compete effectively with 2a mass-production technologies. It also frees human labor for other pursuits, which is an important goal for many.

So this is replication; what of self-replication? This is the second concept I hope to make clear with this essay. Self-replication is not different from replication, except that, in the context of human-tool interaction, that which is being replicated is a part of the tool itself. I refer to this as recursion, because that’s what it is.

Note that there are several Type 1 replicators with a high degree of possible recursion. The lathe is a classic example: one can hand-turn many of the components for a lathe on itself, and even bootstrap a lathe by building the spindle and using it to make other components. A kiln is another example, since a kiln can be used to fire refractory bricks to make another kiln. In fact I might argue that a kiln exhibits the highest recursion of any Type 1 replicator, in that a kiln can be nothing but stacked bricks and a fire, and the bricks can all come from a kiln.

Note, however, that this still has poor scaling qualities, though it favors custom production. The Dave Gingery approach of contributing massive human inputs to a bootstrapped machine shop is noble, but it will never be an economic force. It is simply too easy to identify the main shapes needed, turn them over to Type 2 or 2a production, and swamp the market.

The RepRap project faces exactly this dilemma. RepRap knocked the bottom out of 3-d printing, but it did this mostly by making the MakerBot possible. The RepRap is a 2b replicator, which requires human tending during the entirety of the recursion process. This makes it easier and cheaper to simply make a chassis on a faster 2b tool, namely an Epilog laser cutter that NYC Resistor happens to already own, and sell that. If the RepRap were a Type 3, with an automatic conveyor, it might be a different story, since a RepRap could then print its chassis parts at a rate of a few complete sets a week with minimal human inputs. For that matter, a MakerBot with a conveyor belt would be a Type 3 replicator, albiet one exhibiting less recursion than a Darwin or a Mendel.

Let me conclude by suggesting that the RepLab project needs to place priority on Type 3 replication, not on a maximum level of recursion for the whole system. Recursion, I think I’ve shown, is no great trick if it takes a human hand-holding the machine through the whole process, since this is how machine tools have always been built, on other machine tools.

The power of Industrial Revolution 2.0 lies in developing machines which exhibit Type 3 replication. If the outputs of those machines are sufficiently general than the toolset as a suite can exhibit economically significant recursion. An example would be an open-source toolchain for automated printing and assembly of PCB boards. Such a suite would be able to print its own control hardware in addition to any other circuit that can be designed and mounted with a pick-and-place, which is most of them.

The RepLab project should focus on tools exhibiting Type 3 replication for a variety of economically significant goods. Recursion is an emergent result which can lend exponential momentum to the deployment of RepLab technology. By approaching the project goals in this order, we have a rational approach to an open source hardware platform for Industrial Revolution 2.0.

It’s a great platform and I want to see it taken to the next level. They’re working on a 3-d scanner, which is key; I’ve been musing about a way to hack the MakerBot platform to make for even more awesome.

So here’s the idea: a texture printer. Takes an ABS 3-d model and puts a texture right onto the surface. This is doable, like could be done in 2010 doable.

ABS dissolves into acetone quite readily, and acetone evaporates fast. A five-axis machine should be enough freedom, given the pretty firm limitations on overhangs with the MakerBot.

The ink’s no problem, and there are plastics that resist acetone. The printer would need a reasonably sealed enclosure with a fan and a filter disk, so as to not breathe sweet, sweet acetone all day. One of the great things about this is that the acetone on the surface of the model would smooth out a lot of the minor irregularities left behind by the printing process, and the application of a texture would go a long way towards mitigating the rest.

ABS ink powders are used in the latest tattoo inks, so there’s a wide range of vibrant colors. This is a non-trivial project, clearly, but the results…

We would need software that can put the head tangent to the surface of a model. The ink heads on the market today deliver the number of drops you want, when you want them to, we pretty much just have to point them at the right places on the model.

There are plenty of questions that need answering but the basic picture is fairly clear. We’d use a MakerBot style X and Y axis, with one axis about twice as long so the piece can go under the head as well as along the side. The print head would have to be on a Z axis that moved out of the way so that the work piece could pass underneath it, and be accessed from the side. The W axis would be a turntable, and the V axis a servo to control the angle of the print head between XY parallel and Z parallel. That would pretty much put a texture on anything you can print.

There are other control geometries but my feeling is that this would allow maximum leverage of existing code and parts, which is a Good Thing.

The acetone would have to be evaporated fast enough to not drip. The simple way is to run the cabinet around 55 degrees C; the acetone will cool on the way out of the print head and evaporate from the stored heat of the model. This will require either robust electronics or a separate enclosure or both; acetone loves nothing more than eating plastic.

A side effect of this is that no parts in the enclosure can be printed or made from ABS, nor etched from polycarbonate. Polypropylene is resistant as are all metals; there’s a way to make this work while keeping to the ‘build it with your own robot’ approach.

It’s also possible that some better solvent can be found that will still chemically bond the ink with the surface. There are plenty of resins that withstand acetone however and dissolving plastic is kinda the point here.

A wood enclosure isn’t going to work for anything but proving. There’s no practical way to seal it and it absorbs acetone anyway, potentially leading to the growth of mold or bacteria, and no one wants their TextureBot smelling like cheese.  2mm polypropylene panels with Teflon cuffs would slot into MakerBeam with an acceptable seal; the cuffs could simply be PTFE plumber’s tape.

The print head should be easy, if there’s one small enough. Haven’t had a lot of luck looking into that yet, and it’s important. They make really tiny print heads for portable photo printers and one of those might use solvent based inks.

The big software part is bringing the print head tangent to the model. Mapping textures to models, I needn’t add, is a well solved problem, so picking your pixel once you have a clear shot is comparatively easy.

The body, if I’m putting this all together right, is a longer MakerBot, with the turntable functionality needed for the 3-d scanner, plus a servo lever to control the inkjet head angle.

This would be made of awesome and should totally happen. Anyone with thoughts on how, please get in touch.

How do we choose a number, then? Well, there’s a standard way of doing that too. All numbers are created equal, and are an ultimately arbitrary assignment of scale; once that scale is assigned, there are preferred numbers within that scale. How this works is interesting, and provides insight into the nature of standards.

The first preferred numbers are the numbers of the scale itself.  Already we’ve made a decision, and that decision will effect how we proceed. The scale provides a unit, which is a degree of difference (be it length, temperature, mass or otherwise) that is considered equal to a single integer. The scale also provides a base, and in many ways, this is more important.

The SI system of units comes with two bases in the present form, though it was born with one. It is fascinating that there is a binary base for SI, but not particularly meaningful in this discussion as it is seldom used for anything but bits and bytes. The decimal base is far more familiar, and is the system we all learned in school: centi-, milli-, micro-, kilo-, mega-, giga-, and everything in between, and past in both directions.

This is simple enough. A liter has a thousand milliliters in it, a thousand liters is a kiloliter, and Bob’s your uncle. What of a gallon? Well, a gallon is four quarts, each of which is two pints. Each pint is two cups, each cup is 8 fluid ounces,  and each ounce is two tablespoons. Each tablespoon is three teaspoons. Got all that?

This may appear totally insane, but realize: We have defined a gallon as 256 tablespoons, 128 ounces, 16 cups, 8 pints and 4 quarts. These numbers should be familiar: they are powers of two. So what?

Here’s what. If you pour from a full quart bottle into a pint bottle, filling it, you have a pint left in the quart bottle also. This is the standard way to divide liquids for trade, and this system arose in a culture where that kind of convenience was important, far more important than the ability to do arbitrary calculation. The imperial volume system has the preferred numbers built in. What this gives us is easy conversion where it counts most, assuming you’ve learned all the quaint terms for the various units.

What we lose is the ability to do math on our units without driving ourselves bonkers from conversion factors. It’s not worth it, and SI is better.  So we’re going to sacrifice baked-in preferred numbers for the decimal scale. How do we get our preferred numbers back?

Say we have a decimal scale, Graham units, abbreviated Gr, and we are going to devise a range of widgets that provide from 1 milliGraham (mGr) to 1 kiloGraham (kGr) of awesome.  The first numbers we pick are clear enough: we want widgets in 1 mGr, 10 mGr, 100 mGr, 1 Gr, 10 Gr, 100 gR, and 1 kGr.

This produces a logarithmic scale: the ratio between units is a constant 1:10, just as the imperial system keeps a more-or-less constant ratio of 1:2 between units. But keep in mind: 10 times as much awesome is a lot more awesome. It’s like if we were selling awesome by the Altoids tin, the bank box, and the shipping container. Sometimes people might want an intermediate amount of awesome. How do we give it to them?

First instinct: lets provide even intermediates of awesome. Instead of just widgets that deliver 1 Gr and 10 Gr., we’ll make units at 1 Gr, 2.5 Gr, 5 Gr, 7.5 Gr and 10 Gr, so on, up and down the line. This is an easy guess for how to do it, and dead wrong: I’ve had professors and bosses tell me to do it this way and they’re wrong too, and I got an ISO standard and some math to prove it.

The nice thing about our units, before, was the constant ratio between them. This is a good thing in a standard, as it means the preferred numbers are distributed ‘evenly’ throughout the standard. The reasoning, basically, is that a small amount of a small measurement matters, but a small amount of a large measurement doesn’t.

Here’s what this means. 1 Gr and 2.5 Gr have a ratio of 1:2.5, and 2.5 and 5 Gr have a ratio of 1:2. 5 and 7.5 have a ratio of 2:3, and 7.5 and 10 have a ratio of 3:4. In decimal, we’re going from 0.4, to 0.5, to 0.67, to 0.75.

This is horrorshow; our ratio has gone from a nice constant 0.1 to fluctuating all over the map. Remember, if we were selling awesome by the cup, we’d have a nice even 0.5 ratio with the occasional 0.25 thrown in to spice it up.  Something must be done; enter Charles Renard, stage left.

Charles Renard designed a standard for use with decimal scales; a way of subdividing them by 5, 10 or more such that the ratios remain close to constant. Therefore, following Renard, we’re going to sell widgets delivering the following levels of awesome: 1 Gr, 1.6 Gr, 2.5 Gr, 4 Gr, 6.3 Gr and 10 Gr.

Now what are our ratios? In decimal: 0.625, 0.64, 0.625, 0.635, 0.63.  This is good: six and change all the way up and down the scale. If we need finer scales, these are defined also, and we can divide our decimal scales into 10, 20, 40, or 80 distinct gradations.

What does all this mean? If you are designing something, from an experiment to a widget to a presentation, choose preferred numbers. Like magic, your data will have a meaningful spread, your parts will be more likely to be interchangeable, and your ratios will have a mysterious, aesthetically pleasing quality: the Golden Ratio is 0.618, and is the basis of the series of preferred numbers. The Reynard numbers are a compromise between the aesthetic and mathematical ideal of harmonic proportion, and the realities of actually measuring and building stuff.

Basically, if you work with decimal scales, burn these numbers into your heart: 1.6, 2.5, 4, 6.3. Make up a little song. Have dreams about them. Use them liberally and wisely. These are your father’s preferred numbers; an elegant weapon for a more civilized age.

Right now I’m interested in standards. How they come into existence, when they help us and when they don’t, and on what merits a standard should be judged.

A standard isn’t an object. Even standards which are defined in terms of an object, such as the kilogram, are not themselves objects; they are an idea of what an object should be. More specifically, they are an idea of what an object should be, in order to be considered part of a particular class of objects. There are standard actions and events, as well; this concept of a standard applies with little change to these.

The international standards-setting body is called ISO. Many people are under the misapprehension that ISO is an acronym; when asked, they will usually identify it as standing for International Standards Organization. It doesn’t; the name in English is International Organization for Standardization, and in French Organisation internationale de normalisation.

The name ISO is taken from the Greek word ἴσος, meaning ‘equal’, and serving as a prefix in many international words such as isomer, isomorphism, isothermic etc. This is a beautiful choice for a variety of reasons.

It is descriptive, for one. Standards seek to make sure that an object, an action or an event is in some important way equal to others of its same class. If it’s a standard widget, we want to be able to reach for it knowing that it will behave like every other widget in its family. If it’s an action, such as administering a test, we’d like confidence that the results of the action can be relevantly compared, and if it’s an event, such as a world record, we would again like to know that the circumstances surrounding it allow comparison to other such events.

In physics, and all quantitative sciences, the ur-formula for data is

(aspect of reality measured) = (measurement) * (unit)

The unit is a standard. The equals sign is ISO.

It is also a compromise. The French are notoriously insistent that their language be used in international matters, and to be fair to them, they did come up with the SI (the, *cough*, International System of Units). IOS and OIN can’t really be jiggled around to provide a single acronym; ISO suggests both names, in both languages, without overtly favoring either.

Standards are often compromises, and this is a good thing. Standards are seldom completely arbitrary, but by their nature there is often room for several standards that serve the same basic purpose. A good example is rail. The various gauges for full-size rail vary between 1 to 1.5 meters; smaller and you seriously limit the amount of tonnage you can carry, while making much tighter curves and higher banks possible; larger guages are only used to move large equipment small distances, due to the unacceptably large land clearances needed.

Most of the track in the world, and all in North America, is 1435 mm apart, the so-called standard gauge. The standard gauge is just that, standard; the most amount of rolling stock is made for it, making it the cheapest to deploy. The Russians are the big holdout in the Northern Hemisphere, using a gauge of 1520 mm. Finnish rail, incidentally, is defined as 1524 mm; this difference is not substantial, and in practice this is identical to Russian gauge.

Another thing to know about standards: identical standards don’t always look the same. This too is part of the fine art of compromise, as anyone aware of the history between Finland and Russia is no doubt aware.