An Argument For Dozenalism

Note: Writing about dozenalism always presents some semantic complications. When discussing the natural numbers up to twelve, I have opted to spell out the numbers in English, since this is a clear and base-neutral way of representing them. After trying several different systems, I have settled on writing larger numbers in decimal. Unless otherwise specified, "10" means ten, not twelve, et cetera. When I use dozenal notation, and for clarification purposes elsewhere, I have prefixed the radix as an abbreviation before the number. Thus, "dec. 360" means decimal 360, and "doz. 260" means dozenal 260. "360" by itself, unless otherwise specified, refers to the former. Larger numbers spelled out, where it is stylistically appropriate to do so, will always be given in decimal. My first inclination, of course, was to put all numbers here in dozenal, but on further reflection I see no value in confusing people needlessly. (Confusing them for a good reason though is fine.)

Contents

• WTF is dozenalism
• Dozenalism and cryptodozenalism throughout history
• Why it matters
• What can be done
• Conclusions
• Additional sources of information

WTF is dozenalism

The dozenal or duodecimal system is a positional notation system using twelve as its base, or radix. It is an alternative to the decimal system, using ten as its base, which is the universally-understood system for representing numerical values in the modern world. Whereas in decimal there are ten digits, 0-9, in the dozenal system there are twelve digits, including the ten of decimal notation, plus two more representing the numbers ten and eleven. For the purposes of this article, the two additional digits will be represented as "A" for ten and "B" for eleven. Various systems of notation exist for dozenal glyphs, and it is a topic of much discussion amongst dozenal advocates. There are very good reasons not to use "A" and "B" in the long run, but for now it is convenient and simple to do so.

The number twelve is a highly composite number—that is, it is a natural number with more factors than any integer smaller than itself. Twelve can be divided by two, three, four, and six. Compare this to ten, which can only be divided by two and five. One does not have to think too deeply about it to realize how useful a radix divisible by three or even four would be—in everyday life, we are used to encountering situations where ten or a multiple of ten has to be divided into thirds, and we end up with messy results such as "3.3333." Dividing ten into quarters is a little easier, but you still end up with a fractional value (2.5). The trivial and routine calculations most of us perform every day would be vastly simplified by using a positional notation system that allowed for convenient division into thirds and quarters.

Though we often find ourselves dividing things into fifths, it will be noted that, more often than not, we are only doing so because we are using a base ten system. We are used to quantifying things in terms of groups of tens, and as a result we naturally quantify things in groups of five as well, but there is in general no particular reason why this needs to be the case. Certainly most people would agree that, simply in the realm of practical computation, it is necessary or useful to divide things into thirds and quarters more often than into fifths. Obviously, there are always situations where one needs to divide by particular numbers, and equally obviously not all division is particularly hindered or improved by the radix used, but certainly to the extent that we naturally group things into numbers convenient to base ten (we often round off values to the nearest ten, hundred, or what have you), division by thirds or fourths are often at least slightly inconvenient.

Mathematically, divisibility by three is often more useful than divisibility by five simply because the latter is less common in the prime factorization of numbers. That is, one out of every three integers is divisible by three, while only one out of every five is divisible by five. When breaking down large numbers into their prime factors, five is simply less common, and therefore a base that is divisible by five is less useful.

Compare the divisibility of the dozenal radix to almost any other base that has been used for positional notation (* indicates prime factor, numbers in decimal):

• Octal (8): 4, 2*
• Decimal (10): 5*, 2*
• Dozenal (12): 6, 4, 3*, 2*
• Hexadecimal (16): 8, 4, 2*
• Vigesimal (20): 10, 5*, 4, 2*
• Tetravigesimal (24): 12, 8, 6, 4, 3*, 2*
• Hexavigesimal (26): 13*, 2*
• Sexagesimal (60): 30, 20, 15, 12, 10, 6, 5*, 4, 3*, 2*

It will be noted that, of all the bases listed, only dozenal, tetravigesimal, and sexagesimal offer divisibility by the critical prime factor of three. Sexagesimal (which will be touched on later in this article) is somewhat impractical, due to its size, and tetravigesimal does not add any additional prime factors to dozenal, being merely two times twelve. Overall, in fact, it is striking that not only is dozenal the most obviously useful radix listed here, it is also very close to ten. Given the proper intellectual climate, people could transition from thinking in tens to thinking in twelves without too much difficulty, compared to moving to a system such as like sexagesimal, which is simply harder to conceptualize—though we do use it, in a sense, in our reckoning of minutes and seconds.

The only other base worth noting is binary, which of course is used digital computing. The advantages of binary are quite distinct from those of prime factorization, and it is hard to imagine digital computer technology using any other base for the foreseeable future, giving the nature of Boolean logic and of transistor-based computing circuitry. Though with the advent of optical computing, cellular automata logic, and so on down the road, perhaps our technology will at some point transition away from binary. At any rate, it will be noted that digital binary and "human" decimal have coexisted for over 60 years at this point, and binary values will be no harder to work with under the dozenal regime. One slight complication is that, while 2^10 (1,024) is very close in value to 10^3 (1,000), providing the basis for the Greek prefixes used terms like "megabyte," et cetera, there is no similar power of twelve that meshes well with binary magnitudes. This is obviously a fairly trivial concern, but it is worth mentioning.

Dozenalism and cryptodozenalism throughout history

Despite the decimal tendencies of modern society, our culture is replete with relics of dozenal thinking from days of yore. The persistence in the English-speaking world of the word "dozen" itself, though originally derived from some variant of the Latin "duodecimal," is a testament to the ongoing usefulness of twelve as a grouping unit. The 24-hour day, divided into two groups of twelve, was an ancient custom of dividing the periods of light and darkness into twelve hours each. Thus, the ancients would not divide the day into ante meridiem or post meridiem as we do, but they would speak of the "first hour" of sunlight, "fourth hour of night," et cetera—this is the sense of "hour" originally meant by the term "eleventh hour." Likewise, we divide the year into twelve months. This was certainly convenient to the length of the lunar cycle, but it is also notably useful for dividing the year into four seasons, et cetera. Even the French revolutionaries, probably the most deranged decimalist thinkers in history, did not eliminate the twelve-month year.

A related matter to the length of the Earthly year is of course the 360 degrees of arc in a circle. The use of this value goes back at least to the Babylonians, and probably has at least something to do with the length of the year. The fact that degrees continue to be used to this day suggests that people throughout history have found it to be a useful way of dividing circles and measuring angles. Like twelve, 360 is a highly composite number, meaning again that it has more factors than any number below it. Thus it is possible to divide a circle of 360 degrees by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180. Which is all very useful.

The Babylonians not only gave us an excellent system for measuring angles (mathematically not as useful as the radian, but certainly more convenient for everyday use), they also gave us something resembling the earliest positional notation system, using base sixty. The Babylonian sexagesimal system was not an exact analogue to modern positional notation, and it included some pseudo-decimal elements, but it provided the framework for some fairly advanced mathematics at the time. Sixty is, of course, a somewhat dozenal number itself, being a multiple of twelve. It is also a highly composite number, and is therefore more convenient for mathematical operations, particularly division, than any number below it. A particularly useful property of sixty is that it integrates divisibility by five with divisibility by twelve. A sexagesimal system is probably impractical for modern purposes, since we are used to conceptualizing numbers in smaller groups, and it would be difficult to come up with sixty truly unique glyphs to construct a strict, glyph-based positional notation system, but it was an interesting and worthy contribution to the early development of mathematics. The Babylonian sexagesimal system is the origin of not only the 360 degree circle, but also of the 60 minute hour, and by extension the 60 second minute, et cetera. Every time you conveniently divide an hour into halves, thirds, quarters, fifths, sixths, tenths, or twelfths, remember the sexagesimal system and how helpful it has been to our ability to conceptualize and divide time.

Beyond this, the twelve inches of the imperial foot is another obvious example of dozenal thinking in earlier times. There were twelve ounces in a Roman pound (libra, lb.), and the term "ounce" itself is derived from the Latin "uncia," meaning twelfth. If you look closely, history is full of examples of subtle dozenal thinking, which all serve to demonstrate that, despite the ubiquity and hegemony of decimalist thinking, the usefulness of dozenalism has had a consistent appeal through the ages. Most people throughout history probably could not have precisely defined what a highly composite number is or why it might be useful, but they could clearly see that dividing things into groups of twelve was more useful for everyday purposes than dividing them into groups of ten. We would do well to remember their example, and learn from it.

Why it matters

Culturally, humanity is very entrenched with decimalism. Outside of a handful of indigenous cultures (see: Piraha Language on Wikipedia), everyone in the world uses base ten notation, everyone understands Indo-Arabic numerals, everyone learned basic arithmetic using base ten, and nobody wants to learn a whole new way of counting. This is understandable. But consider our situation: Humanity is, possibly, on the verge of a glorious and infinite future. The rise of the metric system is institutionalizing the decimal radix not just for counting, but for measuring and quantifying the world itself. With the advent of modern computers and the internet, we are potentially setting in place structures and systems that will define our relationships with information and information technology for countless generations to come. It would behoove us to pause and reflect on whether or not this is really a system worth further embedding into the fundamental fabric of our civilization.

The metric system in particular is a tragic case. In terms of its intent, it makes perfect sense, and pursues an admirable goal—the systematization and regularization all units of measurement into a consistent and easily-computable system. But in practice, we are replacing some very dozenal systems of measurement—for instance, again, the twelve inches in the imperial foot—with misguided and mathematically limiting decimalism. In my view, the metric system presents the most compelling and urgent reason why dozenalism must be taken seriously and addressed quickly in our society, because it is rapidly becoming so entrenched in technology and science that it will be difficult to extract ourselves from it if we go on too much longer. I am not in any way defending traditional imperial measurements—twelve inches in a foot makes a lot of sense, but 5,280 feet in a mile, less so. But if we as a civilization are going to consciously adopt a new and mathematically consistent system of measurements, we ought to at least put some thought into it.

The origin of the metric system, of course, lies in the French Revolution. And one cannot discuss French revolutionary contributions to decimalism without noting the appalling invention of "decimal time," which is perhaps the limit case of absurdity by decimalist agitators. I have already noted the utility of the Babylonian sexagesimal system. When coupled with the (possibly also Babylonian) 24-hour day, it produces a highly dozenal—though perhaps somewhat inconsistent—system for measuring time. Days, half days, hours, and minutes, can all be evenly and usefully divided into halves, thirds, fourths, sixths, twelfths, and in the case of hours and minutes, into fifths, tenths, twentieths, et cetera. Why on earth anyone would feel compelled to replace this elegant and convenient system with a decimalized one is truly beyond my comprehension. If the French of the time had spent less time chopping people's heads off and a bit more time putting even the most minimal degree of thought into what they were doing, perhaps they could've come up with a better system. There is room to improve the clock system handed down to us, by making all units of measurement recursively consistent. One could easily imagine a truly dozenal clock, with perhaps twelve hours in a day, twelve such-and-such in an hour, twelve "minutes" in that, et cetera. That would make a lot more sense. The fact that not only the French people but the world at large soundly rejected decimal time is further proof that, however entrenched decimalism may become, humanity will not give up its dozenalist traditions easily. Nor should they.

What can be done

What is ultimately and urgently needed, then, is a dozenalization of the SI base units of the metric system. Before anything else is accomplished in the dozenal reconquest of positional notation, this issue must be addressed. There are at present seven base units in the SI system: the meter, the kilogram, the second, the ampere, the kelvin, the mole, and the candela. All other units of measurement are derived from these seven, and they are defined in various ways with various degrees of arbitrarity. For instance, the kilogram is defined by the weight of a block of platinum in France, which is just bizarre. At any rate, the solution to our problems is to simply define a new set of measurements based on these units using powers of twelve rather tha powers of ten. For instance, since a kilo-anything is dec. 1,000 of that unit, we need a new system of terminology to define dozenal magnitudes of dec. 144 (doz. 100), dec. 1728 (doz. 1000), et cetera.

This raises an issue pertinent to the dozenalization of both numbers and units of measurement: terminology. We have a system for referring to both numbers and units according to both a decimal base and a cubic superbase—that is, we divide powers of ten into groups of three, or cubes, separated in American English by commas, and elsewhere by periods or other characters: dec. 1,000 is a "thousand," dec. 1,000,000 is a "million," et cetera. Conversely, in the metric system and elsewhere, we use Greek prefixes to denote the same groups of three powers of ten: kilo-, mega-, giga-, milli-, micro, nano-, et cetera. What is needed is a unique and convenient vocabulary for describing powers of twelve. For twelve squared, terminology already exists: a "gross" is equal to doz. 100 (dec. 144). In various European languages terms exist as well for twelve cubed, doz. 1000, or dec. 1728. However, what we really need is a system of vocabulary that can be consistently applied over large scales. The English terms for numerical orders of magnitude could be repurposed for dozenal use—i.e., by defining, say, a dozenal "million" as doz. 1,000,000, as a counterpart to the decimal million of dec. 1,000,000, et cetera—but this scheme is not only potentially quite confusing, it is also complicated by the fact that these terms, ultimately, incorporate decimalist elements. Specifically, above one "decillion," or dec. 10^33, terminology reverts to a composite decimal form, and the next cubic order is named "undecillion," or one-plus-ten-illion, and so on. Additionally, the fact that the numerical element in these terms (bi-, tri-, et cetera) do not kick in until after the "thousands" level is confusing and inconsistent. For example, to be consistent, a "decillion" should be dec. 10^30, not 10^33, yet it is the latter because the numbering does not start until a cubic order of magnitude after a thousand, which is arbitrary and pointless.

In order to address this issue of magnitude terminology, let us digress for a moment into simple ordinal number terminology. Since English already has unique terms for ten, eleven, and twelve, there is no reason to stop using them in a dozenal system. (It should be noted that "eleven" and "twelve," like the word "dozen," ultimately derive from a decimal etymology, but as they have, for all intents and purposes, been divorced from their decimal roots at this point, it is probably of little concern.) When counting verbally or in written words, we would continue to count "nine, ten, eleven, twelve" in a base twelve system. After that, I would propose that we add the numbers "oneteen" and "twoteen" to represent dec. 13 and 14, respectively, and then so on to dozenal "thirteen," et cetera, until you reach dozenal "nineteen," at which point you also add "tenteen" and "eleventeen," bringing you to dozenal "twenty," being equal to decimal 24, and so on. We simply append "twenty-ten" and "twenty-eleven" before "thirty," and so on with every radix unit up to ninety-eleven, then we add "tenty" and "eleventy," and go on to eleventy-eleven (dec. 143).

This is where it gets complicated. As mentioned, "gross" is already an accepted and conventional term for groups of a dozen dozen, or dec. 144, so it would certainly make sense to use this term, or some variant of it, as the dozenal equivalent of a "hundred." My particular proposal is that we should use Greek and/or Latin numerical prefixes for all superbases beyond this, up to "deca" for doz. 10^26 (dec. 12^30). At that point we should add two new terms, based on English "eleven" and "twelve," instead of for instance the Greek "hendeca" and "dodeca." For the sake of argument, I will use "eli-" and "dozi-," but I will be the first to admit these sound a bit silly, and particular terms can obviously be worked out later. Thus, we could name all dozenal superbases according the following convention, using a tasteful combination of Latin and Greek etymology (my prefence is for the latter, but I have made adjustments to make it sound less retarded where appropriate):

• 10^3 - miliad
• 10^6 - biliad
• 10^9 - triliad
• 10^10 - tetriliad
• 10^13 - pentiliad
• 10^16 - hexiliad
• 10^19 - heptiliad
• 10^20 - octiliad
• 10^23 - eniliad
• 10^26 - deciliad
• 10^29 - eliliad
• 10^30 - doziliad
• 10^33 - dozmiliad
• 10^36 - dozbiliad
• 10^39 - doztriliad
• 10^40 - doztetriliad
• . . . .
• 10^60 - bidoziliad
• 10^63 - bidozmiliad

I don't know if anyone is following me here—I more or less made most of this up on the spur of the moment. This is simply an example of how a grammatically consistent ordinal counting system can be constructed for dozenal notation. Note that I have restructured the syllables of Greek numerical prefixes and eliminated duplicate consonants, et cetera—again, this is just something I made up literally right now as I am writing this, but I think it is pretty sensible. As with our current decimal naming conventions, one would expect that astronomically large numbers, at a certain point, would simply be referred to mathematically, without the need for "common" names, so there is probably no need to extend the system indefinitely for absurdly large values (as opposed to doz. 10^63, which of course is a number we all use on a daily basis).

What is important here, among other things, is that such a system can be easily adopted for units of measurement as well. That is, we do not need to use terms like "mega-," "giga-," et cetera, we can simply use the same prefixes and terms we use for ordinal counting. Thus, the dozenal equivalent of a "gigagram" might be a "triligram" or some such thing. (Though by the same logic, a kilogram might be a "miligram," which admittedly is sort of confusing—again, the details can be worked out down the road.) Ultimately it would of course probably make sense to completely rename the SI units themselves to avoid confusion, but that is another issue altogether. Presumably, a system of terminology such as that outlined above can be adapted to negative exponential values with equal consistency—whether by adding a negative prefix or changing the "-ad" suffix to something else.

At any rate, the whole issue of terminology is one that should be defined by community consensus. And it is certainly a topic of rather heated debate amongst dozenal advocates. Many useful proposals have been put forward, and I am indebted to those who have contributed to this debate in the past for helping to shape my views on the matter. It may seem like a trivial issue at first glance, but the need for a useful and intuitive semantic system for conceptualizing dozenal notation cannot be overstated. We are asking people to make very fundamental changes in the way they conceptualize and organize numbers, both mentally and expressively. This cannot be done until there are tools in place to facilitate it. We cannot simply ask people to conceptualize numbers as doz. "10^6," or what have you, and expect this to make any sense to them. The conversion of human civilization to dozenalism will only be successful if and when people are given both reason and ability to mentally count in dozenal, conceptualize numbers in dozenal, et cetera.

The other great challenge facing dozenalism in somewhat the same vein is that of glyph notation. As noted at the beginning of this article, I have chosen to express dec. 10 and dec. 11 as "A" and "B" respectively. (Though as you have perhaps noticed, these two digits have not come up often, as I have been largely concerned with powers of twelve or multiples of twelve, which, in the former case at least, do not require these digits at all.) This is perhaps the most widely-discussed issue in the dozenal community, as far as I can tell. Several proposals put forward in the past no longer seem as compelling as they may once have. For example, the Dozenal Society of America used to recommend using the asterisk (*) and pound sign (#) for ten and eleven, due largely to their inclusion on North American telephone keypads as far as I can tell. At this point, this makes no sense whatsover, for the simple reason that both the asterisk and the pound sign can be—and often are—used as mathematical operators in computer programming languages. It would be confusing in the extreme to take a symbol that has conventionally implied either a mathematical operation, or at least some sort of syntax function, and start using it as a simple positional value glyph. Alphanumeric characters are rarely if ever used for such syntatical purposes, which is presumably why letters have traditionally been used for transdecimal radix notation (i.e., A-F in hexadecimal, et cetera).

Various other exotic characters have been proposed for ten and eleven, which are beyond my present scope to review fully. For instance, a dec. 90-degree rotation of the glyphs for "2" and "3"—I have never entirely understood the point of this, but apparently it is popular in Great Britain. Suffice to say, I believe the only long-term solution is to develop an entirely new set of glyphs for dozenal counting—replacing not just ten and eleven, but all digits. "0" and "1" make a certain degree of intuitive sense, but everything beyond that in the Indo-Arabic glyph system is complete crap as far as I'm concerned. We need a consistent, logical, easy to write set of glyphs for the new dozenal age.

That, ultimately, is a battle for another time. The glyph issue is, in my opinion, far less important than the magnitude terminology issue addressed above. As I've noted before, and as is obvious, English already has verbal terms for "eleven" and "twelve" that work quite well, and how we actually write the numbers is probably less important than how we verbally and linguistically conceptualize them. In the interim, I am prepared to use "A" and "B," in the manner that hexadecimal notation uses A-F, as an easily-recognizable standard for transdecimal radix notation. The major complicating issue with this, of course, is that the capital "B" looks an awful lot like "8"—indeed, the two are identical on 7-segment displays. But as long as we are confining ourselves to computer fonts this will probably not be too much of an issue. In written form, perhaps the lowercase "b" would be more appropriate. At any rate, I do not consider this a major stumbling block to the implementation of dozenal notation—any number of options are available. The only thing I would urge my fellow dozenalists to consider is, again, to avoid using special characters that are either overly obscure or that could be confused with syntactical elements in computer languages.

If the world is to be brought around to dozenalism, it must be done with clarity. It will do nobody any good to just start talking in dozenal numbers and assume people will understand them. This can partly be addressed by adopting a unique vocabulary and glyph system, yes. But beyond that, it would behoove us to simply prefix all dozenally-expressed values with an appropriate identifier. For instance, if someone asks you what year it is (for whatever reason), instead of replying "oneteen eleventy six" (though that might be amusing), it might be advisable to say "dozenal oneteen eleventy six." (Which will of course be perfectly clear to them.) As with my use of the prefixes "dec." and "doz." in this article, this will be a time-consuming and arguably completely inane exercise. Yet it may be the only way we will ever make progress in this matter. We must teach people to think in dozenal—starting with ourselves.

I find it an amusing and constructive intellectual exercise to count in dozenal to myself whenever possible. Whether at home or work or where have you, menial counting is a part of everyday life, and there is in my view no better place to start one's personal adaptation to dozenal thinking than on these occasions. It is also possible to adopt various systems for personal number-keeping that reflect dozenal values, without committing to any large-scale lifestyle changes. For instance, conventional Western tally marks are arranged in a base five glyph system—four lines plus a fifth crossing line. A simple quasi-dozenal version of this would be to use a six-lined version of the hash symbol, or number sign. One could start by drawing three vertical lines, then three crossing horizontal lines, for a total of six lines representing six tally figures. This provides an eminently usable equivalent to the decimally-skewed base five tally system, which should fit nicely into our dozenal future.

Finally, I think it is important to simply question decimalism at every opportunity. One of the most disturbing manifestations of the decimal hegemony is the degree to which decimal notation is considered by the general public to represent something "intrinsic" about numbers. It of course does not. One often hears of specific attributes of certain numbers such as, "If you add all the digits together, you get such-and-such." Or, "If you reverse the digits, you get such-and-such." These are of course trivial and superficial properties that do not deserve to be held up as some sort of fundamentally interesting aspect of the number in question. This tendency also arises in many occult numerological traditions, such as gematria, where certain numbers are assigned symbolic values based on their decimal digits, and in various other fields of human activity too numerous to list here. The dozenal community would do well to aggressively and publicly challenge those who would assign some sort of praeternatural significance to the decimal radix. Indeed, if geometry and elemental number theory teach us anything on this matter, it's that it is the dozenal radix, not the decimal, that is favored by the laws of nature and mathematics.

Conclusions

If aliens landed on earth tomorrow—proper aliens, in ships and crap—they would, I believe, be astonished and dismayed to see us using a decimal radix as the basis of our mathematics and of our civilization. They would wonder how we got as far as we did using a counting system derived, ultimately, from the rather incidental number of prehensile outcroppings on our hands. And I have to wonder myself. I am at least mildly embarrassed to be associated with a species so dependent on such a nonsensical and counterproductive system. Dozenalism, in various forms, has had its triumphs throughout history. Next to decimalism, it is by far the most prominent and well-represented grouping value across all fields of human endeavor. We are therefore not starting from nothing—momentum and history are on the side of the dozenalist cause.

It is true that decimalism has never been as entrenched in human affairs as it presently is. Converting to dozenal at the dawn of the modern era, when Indo-Arabic numbers were first gaining a foothold in the West, would have been a relatively simple affair. That window of opportunity, however small it may have been, is now long closed, and we are left with a system ingrained into the fabric of our very thoughts themselves.

Yet it is equally clear that the decimal hegemony will only continue to gain ground unless and until it is challenged rigorously, rationally, and openly by those of us who see a better way. And in our modern age of internet publishing and what have you, we have an opportunity unique in the history of human affairs since the dawn of mathematics to get our message out to a wider audience. I believe the case for dozenalism is compelling and overwhelming. If presented with a reasonable argument, I think people are capable of changing. With the aid of computer software, programming libraries, conversion tools, et cetera, the change need not be as onerous as it may have seemed to previous generations. As noted previously in this article, computers do not care, architecturally, what radix human-readable numbers are presented in. It does not change their underlying binary arithmetic at all. It is true that many low-level functions in modern computer systems would have to be rewritten, but to do so would not be earth-shattering, nor would it challenge the integrity of our technological infrastructure.

I believe that, with a focused and sustained effort, we can make the dozenal radix coequal to decimal in popular culture as well as in scientific and mathematical use by the end of this century. (Or grossury?) It is incumbent upon those of us who understand the argument for dozenalism to present it forcefully, enthusiastically, and clearly at every reasonable opportunity. If this is done effectively, I have no doubt the natural course of human events will trend more and more towards a dozenal future.

Additional sources of information

In this article I have provided only the most cursory summary of the arguments for dozenalism. The corpus of dozenal literature is not vast—this is one of the primary reasons I felt compelled to contribute to it—but it is substantive. Several interesting dozenal-related papers and articles can be found in the Hexnet.org Hexagonal Library. Others can be found on the Dozenal Society of America and Dozenal Society of Great Britain websites. I would also recommend that any serious student of dozenalism read the Duodecimal article on Wikipedia, which is perhaps one of the most mathematically interesting and exhaustive explanations of the principles of dozenalism in the popular literature, at least as of this writing.

A further selection of dozenal links to be found in the Links page.