Nodes by hexnet

This article was transcribed from a PDF file originally hosted on the Dozenal Society of America site, reprinted from The Listener, 25 January 1962.

On the case against the system of decimalization

Many people will be wondering why a mathematician should argue against the proposed change-over to decimal currency and the metric system of weights and measures. Surely, they will think, everyone is agreed that the decimal system is the most practicable system in which to carry out both small calculations and large ones? I wish to point out certain reasons why this is not so, and to show that there exists a better system, which in Britain we have in part already, and which, unwisely in my opinion, we are preparing to throw away.

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.)

The following video came to my attention recently. It presents, in my view, a perfect example of the sort of world-class hexagonal education we once provided our children in that bastion of cultural exceptionalism known as the 1980s, and which seems sadly lacking from today's undoubtedly clever yet somehow less challenging children's programming:

In this image we see the symmetry group D₆ of a regular hexagon. The hexagon can be rotated six ways, and reflected six ways. Note that any combination of two or more of these operations will still result in one of these twelve configurations.

In this diagram we see that four of the five platonic solids can be projected in two dimensions as hexagonally-symmetric figures. The hexahedron, octahedron, and icosahedron can all be orthographically projected as regular hexagons, and the dodecahedron can be so projected as a somewhat lopsided yet equal-angled hexagon (not shown), or as a hexagonally-symmetrical dodecagon (shown here). The tetrahedron—or "freak" polyhedron if you will—can of course be projected as an equilateral triangle, which although not truly hexagonal is still of the same general angular family.

The following is a brief survey of some elemental properties of hexagons, and why they might be useful. It is not intended to be a comprehensive treatment of the subject. My specific concern here is with the mathematical properties of hexagons, and, to an extent, their role in the natural world. I have avoided discussing hexagons as they pertain to human culture, religion, history, and other "local" concerns, though there are many fascinating instances of hexagonality and sixness in these areas, and they will no doubt be treated more fully elsewhere at another time.

In the above images we see Christ in Majesty within a vesica piscis, ostensibly from a medieval manuscript, and an illuminated page from a copy of Ghazali’s The Alchemy of Happiness.

The Saturn hexagon is a persistent hexagonal cloud pattern at the north pole of Saturn. The radius of the hexagon is approximately 8,600 miles, with a rotation period of 10:39:24. While a variety of explanations have been offered for the hexagon, It is generally thought to be some sort of standing wave phenomenon generated by differences in wind speed around the pole.

Screenshots from a Republican National Committee ad (circa July 2008) touting John McCain's supposed environmental/energy cred. Note well that the ad's name is "Balance"—the idea of balance obviously being closely associated with hexagonal tiling, with its six equally-spaced neighbors, and its (accurately) perceived association with organic geometries in general. It is also interesting because McCain himself had another ad a few weeks earlier (called "Purpose") that featured a similar motif but with squares instead of hexagons. Could an ad featuring a three-dimensional tessellation of rhombic dodecahedra be next? Only time will tell.

Carbon nanotubes are allotropes of carbon in the fullerene family, discovered in 1991 (more or less). Rolling sheets of graphene at different chiral angles creates fullerenes with different electrical properties, making them highly suitable for use in nanoscale electronics, as well as other nanotechnological applications. Nanotubes can be either single-walled or multi-walled, and can be joined together with other fullerenes to form a wide variety of structures. Due to their high tensile strength, diverse electrical and mechanical properties, and versatile carbon chemistry, nanotubes—along with graphene and other fullerenes—represent an important hexagonal contribution to the emerging field of nanotechnology.

These two pages are ostensibly from the notebooks of one Leonardo da Vinci, and are identified widely around the interwebs as such, and I am assuming that this is true.

The Montreal Biosphere, formerly the United States pavilion at Expo 67, designed by Buckminster Fuller. The interior was destroyed by fire in 1976, and turned into a museum back in the 90s or something.

Sexagesimal glyphs from the Babylonian numeral system of yore. Although perhaps only indirectly related to hexagons at first glance, this highly sophisticated counting system demonstrates the ancients's respect for the superior factoring abilities of multiples of 6 such as 6, 12, and 60 (doz. 6, 10, 50). Note well that the 60 numerals of the Babylonian system live on in our 60-minute hours, 60-second minutes, and 360-degree circles—the circle itself consisting of 6 "sides" of 60 units each.

The James Webb Space Telescope is a planned successor to the Hubble Space Telescope and the Spitzer Space Telescope, scheduled for launch in 2018. The telescope employs next-generation hexagonal architecture to exceed the technical abilities of its circular antecessors.

"Graphene is a flat monolayer of carbon atoms tightly packed into a two-dimensional (2D) honeycomb lattice, and is a basic building block for graphitic materials of all other dimensionalities. It can be wrapped up into 0D fullerenes, rolled into 1D nanotubes or stacked into 3D graphite."

An assortment of hexagonal mandala and yantra images.

This diagram illustrates both the hexagonal close-packing (left) and face-centered cubic (right) systems for the close-packing of spheres in Euclidean 3-space. Note the hexagonal symmetries of both arrangements. Both can be assembled using the same hexagonally-packed layers—they differ only in how the layers are stacked together.

In each hexagonally-packed layer, there are gaps left between every three spheres. Spheres from the next layer are placed in these gaps. In any given layer, however, one has a choice of which gaps to fill with spheres—only half of the gaps can have spheres in them, since a sphere placed in any particular gap precludes a sphere from being placed in any of the three gaps immediately adjacent to it. Thus, in the hexagonal close-packing system, layers are stacked such that the spheres in each layer align with the spheres two layers below it. In the face-centered cubic system, layers are stacked such that the spheres align with the spheres three layers below it.

A selection of known hexagonal crop formations from recent decades.

"The Crystal," by Sulamith Wulfing.

Various & sundry crop formations and crop circles of unknown provenance.

Interesting article from Science's popular science outfit about the Saturnine hexagon. While it is certainly an intriguing development, and a step forward from the well-known "spinning bucket" experiments, they should get back to us when they've made a hexagon 15,000 miles wide that lasts for 30 years. (Which, for practical purposes, one wouldn't expect any time soon.) Fluid dynamics can be vastly different at different scales. I would also like to know what the viscosity is of this "water" they speak of relative to the atmosphere of Saturn.

The IX monogram was an early Christian symbol, supposedly derived from the Greek initialism for Jesus Christ (Ιησους Χριστος), but which may also be a cryptohexagonal symbol. Though common in early Christian iconography, it was ultimately supplanted by the Chi Ro, perhaps due to the latter's prior history as a pagan-era symbol. Both symbols, were, of course, ultimately marginalized by the far less hexagonal Christian cross.

Modified from a transcription by Charles H. Smith originally published at: http://people.wku.edu/charles.smith/wallace/S083.htm.

My attention has been called to the paper in the 'Annals' for June last on the above subjects, the author of which seems to me to have quite misunderstood and much misrepresented the facts and reasonings of Mr. Darwin on the question. As some of your readers may conclude, if it remains unanswered, that it is therefore unanswerable, I ask permission to make a few remarks on what seem to me its chief errors.

"An adventure through time and space on a voyage of the HEXAGON, nature's perfect shape."