CORD & SPAR SUNDIALS

William S. Maddux

Here is an approach to sundialing that is sparing of materials and of construction labor. The dial examples pictured with this article would by their appearance be best suited to a campground, or perhaps to a "rustic" garden. However, the same design principles might be applied for more highly-finished dials, to be placed in more formal settings. 1, 2

In order to obtain photographs to go with this piece, I constructed two specimen dials, suspending them from masts erected especially for each.
Although these dials required very little time and effort to build, they could have been completed still more quickly and easily, had suitable pre-existing supports been available. (Such as a wall, a pillar, a sturdy pole, etc., situated to afford a firmly fixed, elevated anchorage point for each dial's northern end.)

Consider an equatorial dial as in Fig. - 1. Imagine that after marking off a slender horizontal strip --- and placing time divisions and hour labels upon it as shown --- we were to cut away and discard all of the shaded-tone portions of the equatorial disk.

So long as it stayed in the same position relative to the axis/style, the remaining strip would continue to correctly register its sunlit hours (here, 7 a.m. to 5 p.m..)
Of course we could accomplish the same result without mutilating an actual equatorial dial, simply by placing graduations along a strip or bar, using familiar methods as for a polar dial. (Think of a polar dial-plate that has been reduced in its north-to-south axial dimension to the thickness of the bar.)
Thus, we may consider these as either equatorial --- or as polar --- dials to suit our convenience or cast of mind.

To position such a sundial-bar, or "spar," in correct three-dimensional relation to its axial style, we may suspend it by "cords." 3
The style (see Fig. - 2; Photo 1) may itself be a taut cord, stretched between fixed north and south attachment points NcP and ScP within the local meridian plane, where the ratio of the vertical distance of NcP above ScP, divided by the horizontal distance from ScP to NcP, must be made equal to the tangent of the site's latitude angle. (That is, the style must be aligned parallel to the earth's axis.)

Photo 1: Single Spar

Two equal-length cords Sl - N, N' attached to the common point NcP extend and are fastened to the spar at separated points as shown in the "PLAN" view of Fig. - 2.

Similarly, equal-length cords Sl - S, S' are fastened to the spar and to the common point ScP.
When under tension, each pair of equal cords completes an isosceles triangle, with the segment of spar between their attachments serving as base for both the northern and the southern triangles, whose respective apexes coincide with the ends of the style, NcP and ScP. Hence the spar's long dimension is perpendicular to that of the axial style (i.e., the spar lies in an equatorial plane.)
With the sum of the altitudes of the two isosceles triangles made greater than the taut style's span, (length ScP to NcP) the spar is slung distant from the style by the measure of their mutually perpendicular prime normal PN. (See Fig. - 2, "ELEVATION" view.)
Provided that the suspension cords are maintained under tension, the steric geometry of the spar and its suspension will be kinematically determined. The cords and spar will then together act as a single rigid system, whose sole spatial degree of freedom is one of rotation about the axial style, as defined by the two "pivot points," NcP and ScP.

In an installed dial, the required (tensile) maintaining forces are provided by the weight of the spar, augmented with vector components exerted by the taut control cords, or "lanyards," L and L'.
When secured to a fixed point, or points, the lanyards work in mutual opposition to constrain rotation about the axial style, Fig. - 3: (A).
The lanyards may be conveniently adjusted, Fig. - 3: (B), to choose to set the spar dial (by appropriate rotations about its style) for Local Apparent Time, or for longitude difference, the equation of time, and to show Standard --- or Daylight hours.

General note: If we invoke the hour plane (or shadow plane) concept, we see that assuming the construction is such that the cord-and-spar relationship is stable, the manner in which the hour planes meet the spar will not alter with the sun's changing declination.
Also, if the spar should be a trifle skewed, relative to the assumed equatorial plane, or if the spar is not straight, but is somewhat irregularly formed, once the time divisions have been empirically projected (using a watch and the predictable sun itself) and marked in place, for so long as the suspension and spar elements do not afterwards alter in any of their dimensions and attachment points, the markings will remain "true" under rotation about the style.
This will be especially helpful when building a dial with perhaps a tree-sapling segment as spar, and/or with less-than-exact equality of paired cord segment lengths, spacings of their spar fastenings, etc. (Of course, the validity of applying the hour plane notion will still depend upon correct alignment to the local meridian's direction, and upon setting the style to the proper vertical angle for the site's latitude.)

Spars: The range of potentially usable spar materials is very broad. Working loads are relatively light, and primarily compressional, so strength requirements are quite easily met.
A spar could be a trimmed and peeled sapling, or be fashioned from dimensioned lumber, wood composites, plastic or metal tube, or one of the many aluminum alloy pre-forms available for structural beams and girders, i.e., "box," "T," "L," or "V" sections, etc.. Dimensional stability, stiffness, durability, light weight, and suitable surface(s) to allow for the hour markings, are desiderata.
(Some thought should be given to spar aerodynamics, as wind induced oscillations could possibly cause problems, especially for large installations.)

Cords: Strength, durability, and ease of working are each important, but stability-in-length is paramount. Unfortunately --- as a class --- natural fibers characteristically are hygroscopic, and when twisted into strands and made up into cordage, will change length as an inverse function of their moisture content. This effect can be quite large, and so such materials are not good choices for the suspension elements of these designs. Non-hygroscopic synthetic ("plastic") fibers, and metal wires (multi-stranded or, in some cases, monofilaments) appear to be the major practical options.

The corrosion resistance of metals generally seems reflected in their price; for some of the commoner synthetic fibers, cost may not track with suitability. For example, braided nylon is pleasant to handle and does not change much when wet, but it is quite elastic. It is a good choice for the taut-line axial style. I also like to use nylon for the adjusting lanyards, although its elasticity may perceptibly affect the "feel" of making angle-setting adjustments. However, for the standing parts of the suspension, other --- often lower-cost --- synthetic polymer types may be less elastic, and so can here perform better than nylon would.

For moderately-sized dials, there are available forms of inexpensive "clothesline" with a tubular plastic outer cover surrounding a fiber, or steel wire, core. These resist stretching, and are quite durable overall, although with time they may begin to deteriorate from the cut ends inward. (Perhaps a dab of caulking compound on the ends might help.) Apparently, most sorts of synthetic polymer fibers tend to break down under prolonged exposure to sunlight's ultraviolet radiation, but data about specific products' susceptibility to this seems hard to come by. For a pragmatic sundial builder, life expectancy of metal wires may be more reliably predictable.

Among metals, high-chrome alloys ("stainless steels") are a likely first choice. Wire cable made from "red metals" (bronzes, and their kin) is less widely available than stainless steel, and offers few particular advantages, except perhaps for "eye appeal." Copper electrical wire is a bit too ductile for this application. "Antenna wire," which is stranded copper, made "low-stretch" by steel reinforcement, might serve. ( Check with a "ham" re local sources.) There are various plastic-coated wire cables (as used for boat steering-controls, dog tethers, etc.) that might prove usable. Metal rods, arranged by the same kinematic design principles, should work well as dial suspension elements, if provided with suitable swivel fittings at their end connections.

Local hardware stores and builders' supply outlets offer various coated wires and cables, as well as other possibilities, such as lightweight stamped-link steel chain (commonly with a fairly rust-resisting metal coating, that takes paint quite well.) Marine hardware suppliers are likely sources for durable cordage and wire-rigging materials --- including various end-fittings, thimbles, cable clamps, turnbuckles, swivels, cleats, and so on.

THE DIAL EXAMPLES:

While making these dials, I had in mind a "teaching" model. Accordingly, I tried to use methods and materials appropriate for a tutorial exercise. Both dials were made entirely from items found at hand about my home. The dials' spars, and their north-end support "masts," were backyard-grown. (Weather-seasoned bamboo culms, recycled after prior years' service in our vegetable garden as supports for pole beans, cucumbers, etc..) The plastic cords and twine were leftovers from supplies purchased for use in the garden or for general household purposes.

One-Spar Version: The first example uses a single spar. (See Fig. - 2.; Photos 1, & 2.) Because it is so simple to construct, I would recommend this design for a campers' activity, and/or teaching project.
It has, however, the same limitation as any conventional polar dial --- it cannot practically accommodate much more than a 10 hour time-display range. (As with any dial-building project, it is important to first gather as much information about the site as possible.
At least within my experience of the arboriferous eastern mid-latitude U.S.A., there are many prospective dial locations where a view of the sun's horizon-to-horizon path is obstructed for some low-altitude period in early morning or late afternoon or --- very commonly --- both.
Observations made on a particular site may well reveal that the maximum daily time span during which direct sunlight can actually reach a dial is less than 10 hours.)

Photo 2: Single Spar

To begin construction, I first "stepped" a 6 -1/2 ft. bamboo mast by lashing its base to that of a metal stake with attached wooden batten, as visible in Photos 1 & 2. (The stake, with its wooden extension, had been left on the site from previous experiments with shadow plane dials.)
I ran two guy lines (wire-cored plastic clothesline) from the mast's top to stakes bearing roughly 60° east --- and west --- of north. The 1/4 inch diameter braided nylon style-cord was temporarily hitched to a south stake, to help keep the mast upright as I readjusted the two northward guys to bring the mast vertical in its east to west aspect, and to leave it inclined slightly to the south to afford clearance for a plumb line hung from the cord attachment location NcP. I then drove a plumbed reference peg directly beneath NcP.

The next step was to place another reference peg due south of the one beneath NcP. (By watch-timed observation at Local Apparent Noon --- as predicted by the NASS "Dialist's Companion." )

Establishing the southern anchor-point ScP was slightly complicated by the fact that the ground at the site slopes downward toward the north and west. The ground surface near the base of the mast was about 5 -1/2 inches lower than the surface 70 inches to the south.
I drove a stake several inches north of the dial axis' projected intersection with the ground, and adjusted the height of ScP upon the stake until the style/hypotenuse's length (NcP to ScP ) corresponded to that of a right triangle with the latitude's tangent ratio values for height and horizontal length N to S, as measured relative to the plumb line from NcP.
(Instead of this style-length procedure, a hand level and a marked piece of stiff sheet-material --- e.g.: plywood or fiberboard --- could have been used to set the style angle. See Fig. - 4. )

An 8 -1/2 foot bamboo spar (painted matte-white with latex exterior house paint) was suspended from NcP by equal-length cords Sl - N and Sl - N'. The cords Sl - S and Sl - S' were then adjusted in their common lengths to ScP, to make PN (the shortest distance between the style and the spar) equal to very nearly 12 inches.
Since PN is the perpendicular height of the style above the spar, I had reckoned a distance of the shadow from PN along the spar as: 12 inches x tan (5 hr. x 15°) = 44.8 inches. This meant that my fairly straight, 8 -1/2 foot long spar could (when centered in its suspension) register 10 hrs. of time, with about 6 inches to spare at either end of its 102 inch length.

On a subsequent sunny morning, as I set about placing time-markings upon the spar, the sun did not emerge above the last obstructing boughs of a large White Pine to the eastward of the site until about 9:40 a.m. E.D.T..
Employing my newly made cord & spar instrument as a "Space-Time Machine," I adjusted the lanyards to rotate the spar about the style-axis until the now visible shadow of the style fell about 5 -1/2 inches from the spar's westward end. I then reset a "hack" watch to read 7:00 o'clock when the shadow was on a first penciled mark, placed about 6 inches from the spar's tip. Thanks to a low-tech and not-very-modern "miracle of science," and to the "First Law of Dialing 4,"
I had "virtually transported" myself and the nascent dial to exactly the proper longitude westward from my site's meridian for the dial to then register 7:00 a.m. as its translated local apparent time. For the next six hours --- at each hour and each half-hour as gauged by the hack watch --- I lightly pencil-marked and labeled the position of the style's shadow.

At about 10:45 a.m. E.D.T. on the next (sunny) day, I reset the lanyards so that the style's shadow fell a bit short of the previously penciled 12:30 p.m. mark, then undertook to set the hack watch to the spar dial, so that each would read 12:30 p.m. at the same moment (and then verified this translated setting when 1:00 p.m. was indeed next indicated synchronously by hack watch and dial shadow.) Resuming the marking process, I placed the last (5:00 p.m.) mark 4 hours later.
Thus I now had the dial-spar empirically calibrated for 7:00 a.m. to 5:00 p.m. L.A.T., (if rotated to be horizontal) even though the sun, because of obstruction by trees present eastward and westward of the site, had not directly illuminated the dial before about 8:40 a.m. E.S.T. --- or after about 3:30 p.m. E.S.T. --- on the days of the marking procedure.
(It will be obvious that by proper choice of the readily adjustable PN length --- and by application of similar empirical methodology --- a single spar's length may be optimally graduated and labeled for any specific site's maximum illuminated time span, up to the circa 10 hours limit.)

Two-Spar Version: The second demonstration dial employs two spars, which overcomes the daylight-hour-limitation of a single spar, with 16 continuous daylight hours (sufficient for latitudes up to roughly 47°.
A narrower vee angle could go to even higher latitudes, as could the ring form sketched at right in Fig. - 5)
Since the vee approximates a plane figure, a third attachment point and suspension-cord set is needed to constrain it in space. 5 Of course, this is also necessary --- and sufficient --- to support other planar configurations, such as the annular equatorial dial-plate sketched at right in Fig.- 5.

Photo 3: Two Spar (Vee)

In Fig. - 6, the two spars are shown joined by a lashing at Lf, which acts as a fulcrum in the unequal "X" - truss. MP, the main "preventer" stay, keeps the vee from "scissoring" open more than 60°. Either the east preventer Pe, or the west one Pw, might appear sufficient to keep the vee from scissoring narrower. However when the lanyards L, L' are drawn taut, their component vectors are "crossed" so that they exert a "closing" force on the vee. This force is communicated to --- and opposed by --- the combined preventer-stay system (Pe-MP-Pw) so that the second lateral stay has a role in relieving the spars from bending moments.

For this dial, I first lashed two shorter legs to the north-end mast to form a tripod. As I was working alone, it helped that the mast would thereby immediately stand on its own. (In afterthought --- for photographic purposes this was not a good idea --- for it added potentially confusing visual complication to the images.)

The vee spar dial was graduated by empirical procedures similar to those used for the single spar dial.

Photo 4: Two Spar (Vee)

As mentioned earlier, using the timed apparent motion of the sun as a protractor (of some 93 million miles radius!) has the advantage of automatically compensating for many sorts of minor inaccuracies in less-than-perfect construction. (For a project involving youngsters who have not yet studied trigonometry, this method also offers an opportunity to parcel out the scheduled tasks of timing and placing the hour marks, so that each student may actively participate in creating the working dial right through to its completion.)

Almost any "quartz digital" watch will have a mean solar rate error of less than a second per day. However, the apparent solar rate varies continuously, as described by the equation of time. Perusal of a tabulation of its typical daily noon values (such as "The Equation of Time," in Waugh --- appendix --- Table A 1, p. 205 ) will reveal that from say the 10th of January until the 1st of December, the change in value of the Eq. of T. from noon to next noon (first-difference of the tabulated values) does not exceed + or - 24 seconds of time, relative to mean solar time. That is, the two rates at worst differ by no more than about a second per hour, or less than 1 part in 3600. Further examination of the table will show that this rate-difference may be greatly reduced if the measurements are done on days selected from within the year's many "runs" of successive days whose differences are much smaller than 24 seconds/day.

After the dial is completed, since watches are far more handily available than Eq. of T. tables, it seems to make sense to simply use watch time to readjust the lanyards' settings every day or so, in order to keep the dial's time indications near agreement with standard time. (In the real world, this seems to be the way that many --- if not most --- conventional equatorial dials are set for mean time, despite their provision with tables, or graphs, and scales intended for this purpose.)

Of course for the dial to display Local Apparent Time, simply set the spar's 12 noon mark plumb beneath the style-cord, and snugly secure the lanyards to keep it there.

[end of text]

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[notes]

1 The term "dial" (from Latin dies = day) seems to have long been used for any planar instrument face that indicates measured variable quantities --- and particularly for a disk that bears a circular graduated scale. It has also encompassed graduated-flange "knobs" for controlled adjustment of a measured quantity. Its usage has, during years past, been extended to include radio "slide rule tuning dials," L.E.D. array "bar graph" indicators, etc., so it seems no impropriety to apply the term where solar time-scales are marked off along a spar.

2 These sundials manifestly embody three-dimensional interplay of vector forces, functionally related, and with a considerable element of bilateral symmetry. Moreover, they also exhibit essential asymmetries, which relieves them from monotonous over-regularity. I believe that, simply as artifacts, they offer inherent visual interest and esthetic possibilities.

3 As used here, the term "cord" is meant to include any cordage, wires, chains, rods, and so on, that might be used to suspend a spar or spars. Likewise, "spar" is applied generally for structural members of various materials, cross-sections, etc., whose surfaces can be graduated and labeled to register a style shadow's time-dependent movements.

4 The "First Law of Dialing," or "General Theorem," states that similar dials at any two locations on the earth will indicate the same time whenever both are illuminated by the sun, provided that their corresponding parts are in parallel relation to one another. This is, of course, in recognition that the solar parallax (Earth's equatorial radius / AU) is less than 9 seconds of arc. The effect of parallaxes up to this limit is quite unnoticeable in practical sun dialing.

5 It might be thought that since a spar must have finite cross-section, it too would require a third set of cords to prevent its rotating about its own long axis. For the relatively small-diameter light-weight bamboo canes that I used, there were no noticeable motions of this kind. Any minor disturbing torques that may have been present were presumably opposed by tangential resistance provided by the clove hitches and/or seizings I had used in attaching the suspension cords and lanyards.

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[acknowledgement]

I wish to thank Mac Oglesby for his help in preparing photographs, and reading draft text. Without his interest and encouragement, along with that of Fer deVries and Tony Moss, as expressed during our 4-way e-mail conversations, I quite possibly might never have resurrected these many-years-old "minimalist dialing" ideas from their long-unvisited, dusty storage within my cluttered memory.

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