Some Portable Folded-Paper Flat-Dials
I'd just like to say a few more things about the portable folded-paper flat-dials. What interests me about them is that I like the Regiomontanus, as my favorite portable dial, and the most easily-built sundial of any category. But it's of interest how easily-built can be a portable-dial that doesn't lose accuracy near noon. ...because that could matter for modern urban applications, where public-transportation departures, appointments and events might be near noon. . I spoke of a 1-fold dial with a Horizontal-Dial on one surface and a vertical Polar-Dial on the other surface. I said that it wouldn't work in the manner that I'd described. I'd expected it to work with the top-edge of the vertical section as the gnomon, when use tipped-up as a Polar-Dial with both surfaces as polar-dial surfaces. . But it would work fine using a nodus at the middle of the connecting-thread (the thread that secures the right-angle fold). In that manner, it could work as a Horizontal-Dial and a vertical Polar-Dial, or, tipped-up, it could work as half of a Box Polar Dial. . A 2:1 paper rectangle is folded at the middle of its long-dimension, to make two mutually-perpendicular square surfaces. A thread connects the middle of their outer edges, and that thread has a nodus at its middle. . I call it the "corner-configuration" or "half-box configuration" . With one square horizontal, that square would have a Horizontal-Dial, and the other would have a vertical Polar-Dial. . Or the dial could be tipped up to make both surfaces polar-dials. . --- . To summarize the 3 folded-paper dial-configurations that I've described: . These 3 configurations each forms half of a right-square-prism, cut in half by a plane through its axis (analogous to the half of a right-circular-cylinder comprising a Cylindrical-Eauatorial Dial. . With the 1-fold configurations, the user's thumb, resisted by the connecting-thread, can hold the 2 surfaces apart at a right-angle. ...which worked fine for all of my corrugated-cardboard Tablet-Dials. . Orientation about the vertical axis is by means of the declination-lines, labeled by declination or (preferably) by the approximate date for each 1/12 of the year's ecliptic-longitude variation. . 1. Corner-Configuration or Half-Box configuration: . As described immediately above. . With one plane horizontal, it mignt not be necessary to have a plumb-line for leveling. With both planes polar, a plumb-line could orient the dial about the NS & EW axes. . 2. Box-Configuration: . A 2:1 rectangle is perpendicularly folded-up a quarter of its length from each end. . A thread connects the middles of the top-edges of the up-folded ends. The nodus is at the middle of that thread. . Disadvantage: . The 3 folds complicate things for a folded-paper dial. ...in the matter of holding the up-folded ends perpendicular to the base. That adds a whole qualitatively different design problem to a folded-paper dial. . Advangage: . Potentially convenient use, by its resemblence to Cylindrical-Equatorial and especially to a Box-Polar (...which is what it is when tipped up as a Polar-Dial). . Orientation with respect to EW & NS axes is as for the corner-configuration. . 3. V Configuration: . A 2:1 or 1:1 paper square is folded at-middle into a right-angle V. ...hand-held, with the fold-crease in the plane of the meridian, with the two dial-planes tipped equally from the plane of the meridian. . As with the Corner-Configuration, the right-angle is held by a thread connecting the middles of the outer-edges of the two sections. The user's thumb holds the two planes apart, resisted by the connecting-thread. . - . Each of these 3 configurations can be used in either of 2 orientations: . Horizontal: . One plane, or at least some of the edges, is/are horizontal. . Polar: . All of the planes are polar, parallel to the Earth's axis. . . Those combinations of configuration and orientation make 6 possible portable folded-paper flat-dials. . I'm not claiming that others aren't possible. . . The 1-fold configurations would be much easier to build, given the problem of keeping the box-configuration's two up-folded ends perpendicular to the plane between them. . In the horizontal-orientation, the corner-configuration looks easier to use than the V-configuration, because the corner-configuration has a horizontal surface, giving it easier leveling even without a plumb-line. (My Tablet-Dials were all 5-minutes-accurate without a plumb-line or spirit-level.) . Especially for the polar-orientation, the V-configuration has appeal. ...though, even there, the corner-configuration might be easier to orient about the polar-axis, due to the ramp-like orientation of one of its planes. . --- . Of course, for the dials in polar-orientation the line-construction is much easier to explain. In fact, it practially doesn't need any explanation. .
Some Portable Folded-Paper Flat-Dials
I'd like to correct a few errors I've made, and make a few more comments. . --- . Just as the cross-section of a Cyindrical-Equatorial Dial is a half-circle, the ideal cross-section of a Box-Polar Dial or the Horizontal/Polar combination that I proposed, would be a half-square. . So the piece of paper starts as a 2:1 rectangle, with its end-quarters folded up at a right-angle (and secured there by a connecting-thread with a nodus at its middle). ...resulting in a half-of-a-square cross-section. . The dial that I called "V-Twin Reclining-Declining" (could also be called "V-Twin EW-Reclining")also has a cross-section that's half of a square. The difference is that, with the Horizontal/Polar, the half-square sits on one of its edges, and with the V-Twin, the half-square sits on one of its corners. . If their axes were tipped up, to be parallel to the Earth's axis, both of those would become square Polar analogs of the Cylindrical-Equatorial. . - . Any non-declining Flat-Dial (including the Horizontal-Dial) could have those two Vertical-Polar dials added to its sides. If those Polar-Dials' top-edges are horizontally-trimmed, on a plane passing through the nodus, then declination will be shown whenever the Sun is up. ...the advantage of Horizontal-Polar over ordinary Horizontal. . Of course the same all-day hours-range is acheived with a Cylindrical-Equatorial, if its top-edge is trimmed horizontal on a plane passing through the nodus. . Because it directly shows solar-time and solar-declination, and doesn't need any explanation, the horizontally-trimmed Cylindrical-Equatorial is my favorite stationary-dial. . Similarly-trimmed top-edges would allow a Box-Polar Dial, too, to likewise tell time and declination whenever the Sun is up. . - . Box-Polar Dials (Polar-Dials with added sides perpendicular to the usual polar surface)are sometimes made universal, by having pivot for tipping the bottom polar-surface to an angle equal to the latitude. Of course then it isn't possible to have the horizontally-trimmed top-edges, and so such dials can only tell time and declination from 6 a.m. to 6 p.m. . Of course the same would be true of a similarly universal Cylindrical-Equatorial. . In their defense, of course a lot of locations are shaded outside of that hours-range anyway. . - . Of course then a portable Box-Cylindrical-Dial, too, could be universal, with either a plumb-line attached to the connecting-thread, or to each of the vertical sides, or by aligning your latitude on a latitude-arc with the horizon or a distant point whose height is equal to that of the dial. . Similarly, just as Horizontal-Dials made for a particular latitude were sometimes made universal by tipping, the same could be done with a Horizontal-Polar dial. ...the tipping being achieved as described in the previous paragraph. . In both instances of course, with the top-edges no longer horizontal-trimmed, both of those dials would only read from 6 a.m. to 6 p.m. . . In a recent post I mentioned a 1-fold version of Horizontal/Polar. It wouldn't work in the manner that I described, but it could work. But it would be troublesome. . My favorite of the non-universal folded-paper flat-dials is Horizontal-Polar: . A 2:1 paper rectangle is folded up perpendicularly at its outer quarters, to make a half-square cross-section. ...secured by a thread connected the the middles of the top-edges of the up-folded sections. ...with a nodus at the middle of that connecting-thread. . The top-edges of the up-folded sections are horizontally-trimmed on a plane passing through the nodus. . The dial on the bottom-surface is a Horizontal, and the dials on the upturned sections are Vertical-Polar Dials. Hence the name, "Horizontal/Polar Dial". . I prefer Horizontal-Polar, or Universal Horizontal-Polar, or Universal-Box-Polar, as the portable dial for when near-noon accuracy is needed. . When near-noon accuracy is needed, it seems to me that nothing matches the merit of Regiomontanus. . ..except that Universal Box-Polar has easier explanation. But the altitude-formula, on which Regiomontanus is based, is useful in so many sundial applications (declination-lines, altitude-dials, co-ordinate-transmormations for Reclining-Declining Dials), that it's a very sundial-worthwhile explanation. . - . If universal-ness is desired for a folded-paper flat-dial, then I like the Box-Polar Dial, adjustable for latitude as described above. . ...or Universal Horizontal-Polar, made universal in the same manner. . A folded-paper (with only 1 fold) Universal Disk-Equatorial, as I mentioned in an earlier post, would be another possibility for a universal portable dial, but, as I said, it can be a bit awkward in winter, when it's necessary to read the bottom-surface of the equatorial section. Having tried it, I'd pefer the other designs that I've suggested, which can be read
Some Portable Folded-Paper Flat-Dials
I should add that the Folded-Paper Horizontal & Vertical-Polar Dial can also be made with just one fold: . A 2:1 paper rectanble is folded at its middle in a right-angle. . The fold is held by a thread connecting the middles of the top-edges of the two squares. . A nodus (bead or tied-short-thread) is at the middle of the connecting-thread. . For morning or afternoon, the dial is held with one square vertical and the other horizontal. For both morning and afternon, of course the horizongal square needs a horizontal dial drawing on it, and the vertical square needs a vertical-polar dial drawn on it. . So, it would be necessary to have separate pairs of dials drawn on the two sides of the rectangle, one for morning and one for evening. . That would necessitate adjusting the connecting-thread. The connecting-thread could be in the form of a loop, so that the fold could be reversed, using that same thread-loop. That thread-loop could have a nodus on each of its two opposite sides, so that the noduc could be correctly-positioned in morning or afternoon. . Of course, with both sides of the paper needed for the dials, a separate piece of paper would be needed for the tabulated EqT, and the tabulated decination or ecliptic-longitude. . 48 Su November 24th 2230 UTC . Michael Ossipof --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Some portable folded-paper flat-dials.
Some Portable Folded-Paper Flat-Dials . Paper flat-dials with one or two right-angle folds are easily-built and easily-carried. Some wouldn't be very difficult to use. . First let me comment on one or two other portable dials: . The Regiomontanus card altitude-dial is surely the most easily-built portable dial, and none is easier to use, other than the difficultly-built Ring-Equinoctial. . Yes, of course the disadvantage of altitude portable dials is their lower accuracy around mid-day. I suggest that there's usually no reason why that would be a problem. . The Romans must have just had a shared-understanding to not expect puncuality at mid-day appointments. No problem. . And, if you're in the wilderness, then why would you need precise mid-day times? The important times for that application would be when sunset is near. Of course altitude-dials' accuracy problem is only around mid-day. . And altitude dials have the great advantage and convenience of telling you when sunrise and sunset will be. ...especially useful in wilderness applications. . But suppose that you want a portable dial for modern urban use, when appointments and bus-departures often occur around mid-day? . The Ring-Equinoctial is accurate around mid-day, and is easy to use, but is difficult to build. . There are various folded-paper flat-dials that are (at least in principle) accurate around mid-day, and easy to build. They use a single rectangular piece of paper, with a right-angle fold at its middle. . In a book, I ran across such aproposal, with a pre-constructed dial to Xerox and cut out. It was a right-angle-folded paper Disk-Equatorial. (Though the surfaces were square, of course the degree-arcs were circuar.) . I copied it and made the dial. It was easy to build, but a bit awkward to use. It required a separate little tab, attached to the dial via a slot, to cast a shadow on a declination-scale, for horizontal-orientation of the dial. . Vertical orientation(about the EW axis) was intended to be by aligning the horizon, at sea, with your latitude on a circular latitude-arc. I instead used a plumb-line on that arc. But I also tried substituting, for the horizon, a point that is at the same height above the ground as the dial. . When that dial is awkward is in winter (negative solar-declination), because of course it's necessary to read the bottom surface of the equatorial-section. . It made me feel quite conspicuous holding it up in that manner. Something that can be held lower, as if you were using a smartphone or a book, would be much more comfortable and convenient for me. . Here are a few such possible designs for a portable right-angle-folded paper flat-dial: . 1. Rectangularly-Twice-Folded Horizontal & Vertical-Polars Dial: . The paper is a 3:1 rectangle, with the ends folded up at a right-angle, box-like, forming 3 squares. That position is held by a piece of thread connecting the middles of the top-edges of the vertical sections. . At the middle of the string is a nodus, consisting, for example, of a small bead, or a short piece of thread tied there. . So a Horizontal-Dial is drawn on the middle horizontal square, and Vertical-Polar Dials are drawn on the two vertical squares. . It tells time sand declination whenever the Sun is above the horizon. . Horizontal orientation is by the declination-lines. On the back of one of the squares could be a table of EqT. On the back of another of the squares could be a table of declination. . So of course one would horizontally orient the dial so that it reads the correct declination for the date of use. . Leveling? Of coures there have been Tablet-Dials with spirit-level or plumb-line. But the portable tablet-dials that I made didn't have a spirit-level or a plumb-line, and they were all accurate within 5 minutes (one of them 3 minues). . This Twice-Rectangularly-Folded Horizontal and Vertical-Polar Dial would, with its flat square likely be as easy to level as the Horizontal Tablet-Dials that I made. . But it could also have a plumb-line hanging from the connecting-thread; or have two plumb-lines attached to the vertical squares (anywhere would do). . 2. V-Twin Reclining-Declining-Dial: . This would have simpler folding--just one fold--but would require drawing hour & declination lines for Reclining-Declining Dials, which is what would be on the upper-surfaces of the squares. . A 2:1 rectangular paper is folded at its middle, into a right-angle V, with two square sections. The fold is held by a connecting thread between the middles of the top-edges of the two squares. A nodus (bead or tied short thread)is at the middle of that connecting thread. . Reclining-Declining Dials are drawn on the upper surfaces of both squares. . On the back of one square is the table of EqT. On the back of the other square is the table of declination. . Horizontal orientation is by alighning the dial so that the declination-reading is correct. . Keeping the crease level likely wouldn't be d
