A dividing plate has circles of evenly-spaced holes used to measure out precise divisions of the circle for machining purposes such as cutting gear teeth; by using close-fitting physical contact between hard materials with minimal thermal expansion (for example, a hole in a steel plate and a brass dowel pin shoved into it), they can easily achieve precisions far better than we can achieve by eye. “Cliff” aka “Clickspring” has speculated that such objects might date back to Hellenistic times or earlier, since you’d need some way to lay out the gear teeth in the Antikythera Mechanism, and a compass (without even a straightedge!) is sufficient measuring equipment to construct them. Modern dividing plates are normally used with a dividing head, which gears down the angles by some constant factor to increase the number of possibilities.
But if you had no gears and wanted to minimize the number of holes you had to drill, and thus the opportunities to introduce error, you could get by with a relatively small number of plates stacked on a common axis.
To divide a circle into 6 equal sectors, you can use one plate with two holes 180° apart and a second plate of the same diameter stacked atop it with two holes 60° apart. By aligning each of the four possible pairs of holes in these plates with a dowel pin (several of which seem to have been present in the Antikythera Mechanism, both as gear pivots and as rivets), we achieve four orientations of the top plate relative to the bottom, adding four positions to the two achievable with only the bottom plate. Even if both plates are present, the dowel pin can stick through the top plate, so we can bump our straightedge up against that dowel pin instead of whatever dowel pin we have stuck in the top plate. (The other side of the straightedge might, for example, run through the center of the shaft, as in Clickspring’s ingenious construction.)
A 120° plate would work in precisely the same way as the 60° plate. You can think of the 120° plate as giving you the option to either add or subtract 120° from either of the two reference angles (0° and 180°).
With the 180° plate and a 90° plate, again stacked with the same diameter, we can divide the circle into 4 equal sectors. If we then use the 60° (or 120°) plate from the two 180° and two new 90° positions, we can now divide the circle into the other 8 of 12 equal parts.
Adding a fourth plate, again stacked with the same diameter, we could increase this from 12 equal divisions to 36; the possible angles between the two holes on the rim of this fourth plate are 10°, 20°, 40°, 50°, 70°, 80°, 100°, 110°, 130°, 140°, 160°, and 170°.
If at this point we wanted to continue in this balanced-ternary groove, we would add an angle of 3°20' + 10°n for some integer n and get 108 equal divisions of the circle, but for many purposes it would be more useful to be able to divide the circle by multiples of 5, so a plate with three holes instead of two (at 0°, 2°, 4°, all plus 10°n, thus allowing us to reach 2°, 4°, 6° (10° - 4°), etc.) would be useful.
Note that at this point we are suffering from symmetry: although there are three positions in which we can position this new plate relative to the previous one, the center position of these three offers us no additional dividing power. We’ll come back to the theme of suffering from symmetry below.
So at this point, for 180 equal divisions, we have five plates, four with two holes each (plus the shaft hole) and a fifth with three holes, for a total of 11 holes, or 16 holes if we count the shaft hole. A sixth plate with two holes brings us to 360 equal divisions, 6 plates, and 13 holes. This is substantially simpler than drilling 360 precise holes into a plate. It might be less convenient to use, but when switching between angles you simply leave some of the plate pairs immobile to drop out the factors they contribute, so you can divide the circle by any of the divisors of 360: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
(Perhaps the Babylonians and the Vedic sages stopped there because you can’t construct a regular heptagon with a compass and straightedge.)
It might be desirable to use plates of different diameters, stacked like the discs of the Towers of Hanoi, each one referenced to the disc below it with a dowel pin at its rim. At first I thought that to make this work without increasing the number of plates, though, we’d need more holes in each plate, since you can’t switch the “input hole” referenced to the previous plate and the “output pin” the next plate (or final output angle) is referenced to, so if there are only two holes in the previous plate, there are only two positions for a given plate, so with only two holes a plate always moves you either clockwise or counterclockwise — you don’t get to pick. So I thought you needed three holes per plate, plus the center hole, and one of them can have a reference pin permanently installed in it.
But then I realized you can flip a plate over if the reference pin sticks out of both sides. And that way you can either add or subtract. We were suffering from the reflection-plane symmetry of the plates without even noticing it!
So, by this method, to get to 360°, you still need six plates, each with a dowel pin permanently pressed into one of its holes, five with two holes and a sixth with three, a total of 13 holes, plus the shaft holes in the center.
Incidentally, the width of the dowel pin itself can be calibrated to give us a particular angle, so we can double the number of angles available by bumping our straightedge up against one side or the other of the dowel pin.
But what if we go back to movable dowel pins all at the same radius, and exploit this new possibility of flipping the plates over?
Our first plate, which I will assume is clamped down to a table or something, has two holes 180° apart, as before. Our second plate now has three holes, at -60°, 0°, and +90°. With these two, and the possibility of flipping the second plate, we can reach 0°, 60°, 90°, 150°, 300°, 270°, and 210° from the 0° hole on the first plate, plus 180°, 240°, 270°, 330°, 120°, 90°, and 30° from its 180° hole. So we get to 12 equal divisions of the circle with only 2 plates, 5 holes, and one dowel pin, instead of (as previously) 3 plates, 6 holes, and 2 dowel pins. But maybe we could do better than this, because of our 14 configurations, only 12 are unique — we can reach 270° and 90° in two different ways.
What do we gain from a third flippable plate with three irregularly spaced holes? We could use, for example, -10°, 0°, and +5°, or perhaps some variant that spaces these out by some multiples of 30°. This gets us to 72 equal divisions of the circle in 3 plates, 8 holes, and two alignment pins. I think we could still do better than this, though, because the 15° increment here doesn’t buy us anything.
A fourth flippable plate with holes at -1°, 0°, and +2° gets us to the traditional 360°, in 4 plates, 11 holes, and 3 alignment pins.
We could try to exploit the possibilities inherent in this scheme more fully. Suppose that our second plate, instead of having its holes at 0, -2/12, and +3/12 as before, instead has them at 0, -2/14 and +3/14? As before, this allows us to measure 2, 3, or 5 divisions in either direction from either of our two initial reference holes, which are themselves 7/14 apart. But this doesn't actually work the way we hoped: instead of getting 14 equal divisions, we get only 10 distinct positions, because we have two different ways to reach +2/14 (0 + 2/14 and 7 - 5/14), and simiarly for 5, 7, 9, and 12. By trying to be less clever, and putting the second plate’s holes at 0, -1/14, and +2/14, we do in fact achieve an equal division into 14 parts with 2 plates, 5 holes, and 1 dowel pin. If we divide the first plate into thirds instead of halves, and put the second plates holes at 0, -1/21, and +2/21, we can achieve an equal division into 21 parts with 2 plates, 6 holes, and 1 dowel pin. Adding a third plate with holes at 0, -1/147, and +2/147 gives us an equal division of the circle into 147 parts with 3 plates, 9 holes, and 2 dowel pins.
All of this has a flavor rather similar to the note on the 6 Trit Variac, but with angles rather than voltages.
If the plates are perfectly round and consistent in diameter, the central shaft is strictly speaking unnecessary: you could line the plates up by the feel of your fingers running over the edges. This is perhaps less implausible than it seems, since we know that lathe technology goes back to Old Kingdom Egypt.
Metals are not the only reasonable materials for such discs, shafts, and pins; jade would work well, as of course would various kinds of concretes and sintered ceramics, perhaps even including fired clay, particularly if foamed to improve its machinability. Granite might also be an option. Glasses such as fused quartz would be more challenging to cut without chipping, but might be feasible.
Tom Lipton of Ox Tools has demonstrated a modern alternative to dividing plates, using two plates each containing an identical circular row of identical bearing balls, which are pressed against one another to give as many divisions of the circle as there are balls in each plate. The plates are constrained to move with the balls, rather than rolling on them as in a ball bearing. These balls are routinely made spherical to submicron tolerances, and the errors that do exist are averaged over the whole row of balls, permitting enormously closer tolerances with this mechanism than with the holes bored in a conventional dividing plate.
A sort of hybrid approach that avoids the use of shafts entirely would align adjacent pairs of discs with kinematic ball-and-V-groove mounts rather than entire rows of ball bearings or dowel pins. Each disc (perhaps except the bottommost) would have three balls on its bottom side, spaced evenly 120° apart around its rim, and (perhaps except the topmost) six radial V-grooves on its top side, in two sets of three 120°-apart grooves. The angle between the two sets of grooves would determine the contributions of this disc to the angle of the total stackup. So a single disc pair, where the bottom disc's six V-grooves are all 60° from the previous one, could divide the circle into sixths. A third disc, with its V-grooves at 0°, 120°, 240°, 90°, 210°, and 330°, bumps that up to twelfths — but not 24ths, as you might hope. The third disc adds the possibility of incrementing the angle by 90°, but not decrementing it, but since we already had 180°, we don't need it.
There are a couple ways to try to improve that situation before adding more parts or features. If we put V-grooves on both sides of a single disc, we can flip it over, giving us the possibility of either adding its angle, or subtracting it, as initially — but without the possibility of zero. If we alternate double-V-groove discs and three-ball discs (with the same three balls protruding from both side of the disc) then we could delete a pair of discs from the stackup to get a 0 angle, but at the expense of changing the stack’s thickness, which may be a problem.