Dimensions are numbers with strings attached to the real world. In the applied mathematics of the built environment, dimensions are one of the primary tools. They allow us to manipulate the environment first conceptually before doing so physically; we can combine, extend, divide, or multiply with a simple calculation. Easier methods of manipulation will make the design process that much more efficient. The search for a better method of manipulation (a better system of dimensions) can begin as an exercise in finding the most flexible numbers.
Simply, frankly, mathematically, unequivocally, the number ‘10’ is not as flexible or as fungible as the number ‘12.’ Ten has four divisors and can only be divided wholly by 1, 2, 5, and 10. Any and all other divisions will result in a remainder. Twelve, on the other hand, has six divisors and can be divided wholly by 1, 2, 3, 4, 6, and 12. Because of these additional divisors, 12 is a more workable and fungible number than 10; said another way, 12 has more utility than 10. More divisors means a number is able to adapt to more situations, making it easier to respond to a range of needs.
A.C. Aitken, Professor of Mathematics at the University of Edinburgh, described the dozen’s benefits this way:
“[Twelve is] a number divisible by 2, 3, 4 and 6, while its square…144, divides by these and in addition by 8, 9, 12, 16, 18, 24, 36, 48 and 72, with all the consequences of economical and suitable use in parcelling, packaging, geometrical and physical construction, trigonometry and the rest, to which any applied mathematician and for that matter any practical man, carpenter, grocer, joiner, packer could bear witness”[^For the complete paper see http://www.dozenalsociety.org.uk/pdfs/aitken.pdf]. [The metric (base-10) system is] "a notably inferior one; it cannot even express exactly for example the division of the unit, of currency, metrical or whatever, by so simple, ubiquitous and constantly useful a number as three”[^ibid.].
Furthermore, numbers are repeatedly subjected to multiple subdivisions. Halving (and its doubling equivalent) is literally in our DNA. Twelve-hundred and its resultants can be halved four times before encountering fractions (1200 to 600 to 300 to 150 and finally to 75) while 1000 can be halved only three times (1000 to 500 to 250 to 125). But ‘halving’ is not the only common method of subdivision. Thirds and fourths make frequent appearances throughout our daily lives (e.g., measuring cups, clocks, music, money). With this in mind, we have analyzed numbers to see how they behave when subjected to successive iterations of subdivisions by quarters, thirds, and halves. That exercise is shown in the two diagrams below (note that fractional numbers are “censored” out in black; only whole numbers are “passed on” to the next generation).
Generational subdivision is being defined here as subsequent divisions of both a number and its resultant “descendants” (e.g., 600 is a descendant of 1,200, being half of 1200; 600 itself can then be subdivided further). The diagrams above reveal the superiority of 1200 over that of 1000: 1000 only has 5 generations of subdivisions, while 1200 has 14 (i.e., 1200 is almost three times more flexible than 1000). Additionally, the descendants of 1200 are themselves superior to the those of 1000. The numerical fungibility of “twelveness” passes from generation to generation. This is a simple observation and characteristic of mathematics and nothing more; however, it can be readily utilized in planning and architecture as we will show.
One cannot wholly divide 100 dollars or 100 feet into thirds. To do so leaves a remainder of money or land. If three people split 10 dollars, who is to end up with the remaining penny? If three property owners seek equal stakes in a 100-foot lot, where exactly should the lot lines be drawn? While geometry or pure algebra (with repeating decimal places) provide solutions, the applied mathematics of the real world is at a loss. In reality, the decimal ends somwhere, whether that’s 33.33 or 33.3333, there will be a remainder.
This problem is not trivial. To artificially skew one way or another can lead to unintended and undesirable consequences. A system of land subdivision must strive for precision. Numbers that are highly composite are those that contain a high number of divisors, making them perfectly suited for systems that require divisions (such as land subdivision). Highly composite numbers are more properly outfitted for the qualities desired. And they are already heavily used elsewhere.
Our construct of time, for example, is built on highly composite numbers. There are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day, all superstar numbers exhibiting extreme levels of divisibility. And there’s geometry, too, with a circle being divided into 360 degrees. Those degrees are divided into 60 arc minutes and those into 60 arc seconds. Both time and geometry combine together to form our modern geographical coordinate system which divides up the entire surface of the earth into highly composite units.
The built environment, as its moniker implies, requires builders in order to materialize. The builder’s task is to expand the scale of a drawing to full scale and to render a drawing using real materials. It is an intense process of communication between an architect, general contractor, their subs, and their subs’ subs. Given all the complexities that already exist in the act of building, it is best to maximize simplicity and efficiency in the process where possible. It is easier to cut a sheet of plywood in half when its length is evenly divisible by two (half of a standard 96-inch sheet yields two sheets 48 inches wide); the mental math is manageable and the units themselves are highlighted on a tape measurer for easy transfer. Otherwise, half of, say, a 95-inch sheet is 47.5 mm, which requires good mental math or the use of a calculator (depending on the mental faculty of the subcontractor).
Highly Composite Numbers and the Metric System
| Measure | Device |
|---|---|
| 12 in. | One foot |
| 24 in. x 24 in. | Standard ceiling tile |
| 48 in. x 96 in. | Standard sheet of plywood and drywall |
| 16 in. | Stud spacing (courses with plywood) |
Whether one is dealing with ceiling tiles, beams, studs, or sheathing, highly composite numbers like 12 become incredibly useful to work with in the field. Not only could that plywood be cut in half evenly, but it could be cut into thirds, fourths, sixths, eighths, twelfths, and so on. Ceiling tiles measuring 24 inches have a better chance at coursing out in a room of similar number than otherwise (saving the need to cut the tiles).
The task here is to incorporate this extant preference for highly composite numbers into a unified, efficient, and practical system of land subdivision. This is by no means the first time that highly composite numbers have been promoted in this way. Plato, in Book V of his Laws written 360 BCE, sought to apply the number 5040 to a city’s citizenry and land area, stating:
“We will fix the number of citizens at 5040, to which the number of houses and portions of land shall correspond. Let the number be divided into two parts and then into three; for it is very convenient for the purposes of distribution, and is capable of fifty-nine divisions, ten of which proceed without interval from one to ten. Here are numbers enough for war and peace, and for all contracts and dealings. These properties of numbers are true, and should be ascertained with a view to use.”
As he wraps up his book, Plato extends this power of number and arithmetic to every facet of life:
“There is no difficulty in perceiving that the twelve parts admit of the greatest number of divisions of that which they include, or in seeing the other numbers which are consequent upon them, and are produced out of them up to 5040; wherefore the law ought to order phratries[^A tribal subdivision (per Merriam-Webstier.com).] and demes [^A unit of local government in ancient Attic (per Merriam-Webstier.com).] and villages, and also military ranks and movements, as well as coins and measures, dry and liquid, and weights, so as to be commensurable and agreeable to one another. Nor should we fear the appearance of minuteness, if the law commands that all the vessels which a man possesses should have a common measure, when we consider generally that the divisions and variations of numbers have a use in respect of all the variations of which they are susceptible, both in themselves and as measures of height and depth, and in all sounds, and in motions, as well those which proceed in a straight direction, upwards or downwards, as in those which go round and round. The legislator is to consider all these things and to bid the citizens, as far as possible, not to lose sight of numerical order; for no single instrument of youthful education has such mighty power, both as regards domestic economy and politics, and in the arts, as the study of arithmetic. Above all, arithmetic stirs up him who is by nature sleepy and dull, and makes him quick to learn, retentive, shrewd, and aided by art divine he makes progress quite beyond his natural powers.”
A bit more recently than Plato, the Lexicon of the New Urbanism references the utility of six in a proposal to use six-foot rods for lot widths.[^https://www.dpz.com/wp-content/uploads/2017/06/Lexicon-2014.pdf]
For our purposes here, we are proposing that the dimensions of all the elements of urbanism (blocks, streets, buildings, rooms, etc.) be functions of highly composite numbers (namely being divisible by six or 12). This helps to ensure, at least to within a half unit, that designs will be efficient and sustainable through the mathematical properties of coursing, packing, and ease of calculation. A six-foot grid could be draped upon a landscape, much like the American Land Ordinance of 1785, with boundaries, buildings, streets, parking spaces, and bedrooms all snapping to their corners.