The Antikythera Mechanism 8: That's all for now

I've wrapped up work on the 3D printed Antikythera Mechanism for now. There are a few parts that are incomplete, and I would like to come back to it in future and redesign it. For now, I am out of patience and energy. So let's take a look at what worked and what didn't.

Some things which worked

Overall: the mechanism runs fairly smoothly and without excessive friction. Win!

Frame design: the skeletonized and split frame works well. It is rigid enough and has the right geometry, but without taking an excessive time to print, and it can easily be split into pieces which fit on the print bed.

The splits for the frame and the large gears are also good. Using M3 screws to hold the parts together makes them rigid enough, and adding 2mm dowels as alignment pins also helps. Some of the fixings are a little ugly, especially the brackets to hold the halves of b1 together.

Thin supports for the larger gears where they wobbled on their axes: this eliminated most cases where the mechanism was seizing. I also tried supports for b3/b4, but later found I didn't need them.

Press fit hex fixings for joining gears on the same axis.

Push-on shaft collars for gears where the axle was held only at one end. These worked quite well, I saw them slip off occasionally, and for long term use they should be glued in place. The shaft collars worked particularly well in the cases where I could bury them within the hub of a gear, saving vertical space. Similarly the fixing holes consisting of a hole and a slot worked well. It is possible to get pins into them without excessive force, while having a firm fit which won't work loose.

Most of the axles and pins are standard lengths. In past projects, I've often cut them to length. It's easy with brass and a harder with steel. Standard lengths, by which I mean ones I can easily buy, are better.

The gear sizes are mostly right. I discussed the choice of modules and the exact separation in an earlier post, especially as they deviate from the ones in the HAM.

Some things which didn't

Gear sizes for output: the modules I chose for the output gears meant that they had to sit quite far from the center. It pushed the spirals out while still not leaving much space for the Games, Callippic and Exeligmos dials. With a smaller module they could be moved further in, freeing up space. This is not hard to do but requires reworking the backplate geometry, and I ran out of enthusiasm to do it. Similarly, I never completed the output spirals and a change to the geometry to allow more space for them would make them easier to design.

Some of the shaft collars are not tight enough and slip off. It might work to make them longer where there is enough space. Otherwise, the only solution is to glue them onto the pins.

The planets stages use more vertical space than I would like. I tried thinner gears (doesn't save much, makes the mechanism more unstable), and I looked to see if I could interleave the gears in some way, but didn't find any arrangement that worked.

Finally, as I previous described, the a1 support isn't very stable and could do with a redesign, as part of generally redesigning the mid and back frames.

The Antikythera Mechanism 7: Input and Output

 

The mechanism up to this episode is working, with the only remaining elements being the input and output. The input comes from a gear known as a1, which drives the large b1 gear. a1 is at right angles to b1, but because of the difference in sizes (48 teeth for a1, 223 for b1), a bevel gear doesn't work very well. Instead, I made b1 as a regular flat gear, and then added a flat 48 tooth gear for a1. A set of bevel gears is mounted on top of a1. This assembly was a bit of an afterthought, and so I had to make a bracket which extends the existing back and mid planes. I decided I preferred a simple knob rather than a crank. The result looks like this:

It's a little wobbly but works well enough.

The planets output consists of pointers attached to the nested brass tubes. Most reconstructions use concentric rings. Freeth points out that this is better as it avoids parallax when reading the positions, and also allows the rings to be annotated with synodic events such as opposition and conjunction. By this stage in the project, I was cutting some corners and so went with the simpler scheme. Spencer Connor's version takes a similar approach. The upper output stage also accommodates a bevel gear set driving the moon phase output.

Instead of a 3-D printed dial, I printed the dial onto card. Here it is before mounting on the frame. I again followed Spencer Connor's version by using English month names, with displays for the zodiac and the actual spans of the constellations. The actual Antikythera Mechanism appears to have had a more complex output including an outer calendar ring which could be moved. The diagram was constructed as an SVG using a Python program.

The back outputs consist of the Metonic and Saros outputs and three smaller pointers. The Metonic and Saros should consist of a spiral groove which a pointer moves in, driven by the corresponding output gear. An alternative, with some advantages, is in the Spencer Connor video linked above. The spirals are large and my original plan was to laser cut them into pieces of acrylic or MDF. Unfortunately, my local makerspace (https://web.makespace.org/) has been closed to new members for some time and so I have had to defer this part of the work. I could have followed the model on cults by printing the panel in multiple separate parts. In the end I decided the shelve the back outputs for now. As we will see in the next installment, I've learned a lot by getting this far and may do a redesign for a new version.

The Antikythera Mechanism 6: Jupiter, and Beyond The Infinite

The mechanism is now really taking shape. I did more work to stabilize the back gear. One change was to add a rim on the top of the midplate which the b1 gear can rest on. This made a lot of difference and once I had done it, the mechanism began to move more smoothly. Both the inferior planets and the superior planets have a gear that must be fixed and so some work was needed to anchor them indirectly to the midplate and backplate. It's a bit hacky as present, as I hadn't thought ahead about how to do it, and so I used extra bits of framing that could be attached to the midplate. A redesign will follow.

I also made the decision to use brass tubes for the outputs instead of printing them. The thinnest tubes are the longest ones and so hard to print accurately as well as being fragile. The brass tubes I selected have diameters from 4mm to 11mm in 1mm steps, and they have walls 0.45mm thick. They are a snug fit but able to move freely. Some of the tubes came from K&S Metals, others from Albion Alloys, and I bought them from Macc Models in Macclesfield, England. Here's how they look (the middle 4mm one is not yet cut to size):

You can also see that the top frame had to be split into two to fit the printer. The superior plate is just underneath it, and with a bit of creative truncation it will just fit on the MK4 print bed.

Here is the mechanism running without and with the output pointers. You can see the retrograde motion.

The long "dragon" hand looks odd to me, and I'll check that later.

The next step will be to add the moon phase. At this point I can cut the 4mm brass tube and the (currently hidden) 3mm brass rod that connected with b3 to size. Freeth argues that the hands should have their tips in the same plane to avoid parallax; I might also do that later. The back pointers with the spiral Metonic and Saros dial are also still waiting.

The Antikythera Mechanism 5: Back Gears

The back gears lie between the large b1 gear and the outputs on the back, consisting of the Metonic and Saros spirals and the Callippic, Olympiad and Exeligmos dials. There are two slightly different versions of the configuration, from the Freeth 2012 and 2021 papers. I used the more recent one, which differs from the other only in swapping the position of the gears on the n-axis. The back gears are better supported than the planet gears, as most of them are on axles running between the midplate and the backplate. However, as we will see, there are some issue with the stability to deal with.

The following diagram, S20 from Freeth 2021's supplement, shows the arrangement of the gears.

Some co-axial sets of gears such as f1 and f2 can often be combined into a single printed object. The f-axis is also an example of an axis with no output. The diagram shows its axle as not extending to the backplate. By extending it and adding a spacer, we can make it more stable. Similar considerations apply to the h and p axes. The m axis can't be extended in this way as there isn't enough space around n3. Some axes extend through the midplate, for example d and m. For these, the gears are printed as separate objects, connected by a hexagonal extrusion on one which engages with a hexagonal socket on the other. I experiments with setting the size of the extrusion the same as the socket and 0.1mm smaller. Very close tolerance like this are a bit of a guess with 3D printing. Different printers, slicers and filament may end up with the fit being a bit different. If they don't fit, they can be sanded down until they do, and if they are too loose, they can be glued. I have this value set as a variable in Fusion, so it can be changed across all the parts that use it. I also noticed that the fit varies from one orientation to another. When I initially printed the m1 and m2/3 gears, they fit was quite loose, but rotating them 30 or 60 degrees relative to each other gave an adequately tight fit. In the end all the gears with hex joins seemed to work fairly well, but for the m and d axes, I allowed for a M3 screw to help hold them together, if it should be needed.

The axes which come through the backplate to form the output have a hex socket on the end, which a pointer will plug into. Most of these axes consist of two parts: the gear with an extended hub meeting up with the midplate, and an axle plugging into the gear and extending through the backplate. The n-axis is a bit different due to an extra gear, and the i-axis has the axle going to the midplate rather than the backplate due to the tight space. In some cases, I added a ring round the axle of hub to keep it stable. Here is an example of the entire n-axis in cross section to illustrate:

In this version, the hub of gear where it passes through the midplate is printed. In a later revision, I replaced this with a dowel pin.

Both the midplate and the backplate are large, much bigger than will fit on my print bed. The design by FizzyChickens splits these into smaller pieces which are clipped together and glued. It's still a lot of material to print, and large flat pieces like this are prone to warping. I decided on a different design, where I made a skeleton with just the necessary fixing points and enough mechanical support to make them rigid. These still have to be printed in two large parts, plus a circular bracket on top and two smaller brackets. The result is quite robust and also takes a lot less print time. Here is what it looks like:

A few M3 screws hold it together. Alternatively, it could be glued, maybe using M3 screws temporarily to keep everything aligned until the glue takes effect. There are two small brackets, intended to go underneath the frame. I later got rid of these as I don't think they are needed and they take up valuable space. The external back panel will need to be made from a complete panel. Maybe laser cutting it would be best.

The e3 gear is too large to fit on the print bed. Initially, I intended to split it up the same way that I split the large gear in the Swingtime clock, by dividing it in half and gluing the parts together with metal pins to help the alignment. The e3 gear is attached to another large gear called e4, and this is small enough to print as a single piece. So I decided to split e3 in two and attach it to e4 with screws along the spokes, plus a few extra ones on a tab. I arranged the position of the split so that it does not divide the hub in two, which makes for smoother movement. The three pieces look like this:

In a later revision, I also added some horizontal pins to the halves of e3 for alignment and extra stability.

Did it work?

The prototype back gear assembly worked somewhat, but not very well, as you can see in this video:

The calendar and eclipse prediction gear trains work well individually. They are supposed to work together, connected via the e-axis gears, and driven from b2, which is attached to the back of b1. In the last segment of the video, you can see that the mechanism seizes up. This is caused by (I think) a combination of friction and the larger gears tilting. In particular the very large e3/e4 gear and the d2 and m1 gears tilt a lot. The friction can be reduced a bit by lightly sanding the gear teeth as well as the parts of the gears that run through the frame.

The gear tilting is the bigger problem. The historical Antikythera mechanism includes supports near the rim of these gear which help keep them level. Although this increases friction locally, the net effect is to keep everything in alignment. I decided to try this out in two ways. For e3/e4, I added small brackets to the frame which lightly hold the rim of the gear. For d2 and m1, I added a thin flange to the gears themselves. In the HAM, supports attached to the midplate are used for both. The results are a bit better:

It still has a way to go, but is getting closer.

More changes

I made most of the gears 3mm thick in this version with just a few either 4mm or 5mm. I think it's better to make more of them at least 4mm. For the gears with a hub through the midplate, but which do not have a gear on each side (for example the n-axis), I had made the hub 10mm in diameter. There isn't really a need to do this, and for the next iteration, I replaced the hub altogether with a 3mm dowel for most of these. The e3/e4 brackets in the video hold the gear on both sides, when one side will do.

I broke the b3 gear tube during testing. It will eventually be much longer and go all the way through the planet mechanism. I'm not even sure this can be printed reliably and so I will probably replace it with a brass or steel rod. Similarly, I might replace some of the printed tubes in the planet mechanism with brass tubing. I've not decided about this yet.

The Antikythera Mechanism 4: Superior Planets Prototype

The prototype for the superior planets (Saturn, Jupiter, Mars and the true sun) uses similar design principles to the inferior planets. It's hard to video it in operation as there are parts that won't stay in place until it is combined with the front plate and the inferior planets. You can see the pin for the Mars gear slipping out its slot. I'll use a longer one in the final version.

The video shows first the Saturn and Jupiter trains and then the Mars and true sun ones. In the assembled version they will be linked by the sp1 gear. The superior planet plate is truncated so it will fit on the print bed. For the final version, I'll need to either find a larger printer or split it into parts.

There is only one new design element here: the use of an axle to hold the pair of eccentric gears used for the epicyclic motion, for example like this in the Jupiter gear train with the axle highlighted:

The two circular hole are for screws to hold the axle to the superior plate, and the remaining hole is for a dowel pin on which the eccentric gear will run.

I used black non-silk PLA for some parts of the prototype. Getting the dowels pin in is much harder with this filament, probably due to slight differences in how much the filament expands or squishes when printing and because it is less slippery than silk PLA. I filed out the holes in some cases to make pins fit more easily.

Next, to the back gears, a larger and more complex part of the mechanism.

The Antikythera Mechanism 3: Inferior Planets Prototype

The inferior planets mechanism is a part of the Antikythera mechanism which models the motion of Mercury and Venus as seen from Earth, as well as the mean position of the sun and part of the information about the phase of the moon. This part of the mechanism was not found amongst the recovered fragments and so has to be hypothesised from the required motion and some hints provided by fixings on the parts that were recovered. There are several versions, with the most recent one in Freeth's 2021 paper. The previous version was also proposed by Freeth, in a 2012 paper.

I decided the make the inferior planets mechanism as a prototype for the design. I cut some corners, in particular only printing part of the large b1 gear and printing short versions of the tubes which will ultimately carry the output pointers. Here is a very shaky video:

It moves fairly smoothly, so the dimensions and tolerances are about right. Some things I want to change or experiment with:

• the two dark coloured gears in the middle are wrong. After printing them and the b1 fragment, I noticed that they should have a tube which runs through b1 and is fixed to a (not yet implemented) back plane. See Freeth 2021 supplement section 6.2.6 for details.
• the idler gears attached to the "strap" (upper plate) wobble too much.
• co-axial gears such as the purple and grey mercury gears (called mer1 and mer2, following Andronis) can be joined in a simpler way.
• some of gears can probably be made thinner. My version is scaled from the original by roughly a factor of 2 in the XY plane, but I needed to almost triple the Z direction size. Freeth reports the strap and b1 as an estimated 16.2mm apart while my version has them 41mm apart. I'd like the reduce this.
• there are a few places where the mechanism is sticky. I need to see if this is due to printing or a design issue. I have done very little to clean up the parts and reduce their friction, so it's more likely the former. A couple of places where there really was interference have already been corrected.

Reducing wobble

As you can see in the video, some of the gears wobble a lot. They skew away from the vertical, sometime causing the mechanism to bind. This happens when either the arbor passes loosely through a thin component - in this model either the gear itself or the strap. In addition, you also need something to stop the arbor just dropping out of the loose hole. In the video you may be able to see some shaft collars on top of the strap to do this. You can improve this a bit by making sure the hole is just large enough for the arbor to run smoothly and not more. Thickening the part or adding a hub/bush also helps when the spacing allows it. The idea is really to have the arbor anchored securely at each end. The geometry does not always allow this. Some earlier versions (like Freeth 2012) has arbors with a bracket to hold them steady, but Freeth 2021 largely does away with this.

I tried a redesign with a larger diameter arbor, 5mm instead of 3mm, and found it didn't make a lot of difference. Another thought I had was the have the arbor contain a bearing like a miniature ball bearing. It might help if the arbor is tight in the bearing, and the bearing is tight in its mount (for example the strap), though you might still need a shaft collar.

One other experiment I did was to use print-in-place bearings, for example in the middle item here:

In this case, the arbor sits tightly in circular bearing in the middle, which then rotates freely in the plate. There is about 0.2mm or 0.3mm spacing between them. The bearing widens up inside like this:

With a good enough printer, this prints quite well and you can move the bearing easily once you have broken it free by turning it a few times. It still does not work very well for preventing wobble, so I won't push this any further.

Gear wobble or tilt is an issue which came several times in the prototype of other sections, and I will come back to it with further solution later on.

Second prototype

For the second prototype I changed the gear heights to be 3mm apart from a few which are deeper for clearance purposes. All arbors are now rigidly fixed in either b1 or the strap and the gears run loosely on them. As a consequence, the arbors now also need a shaft collar or some other way of stopping the gears from falling off the shafts. I wasn't very happy about this at first as it costs vertical space. However, I found a nice solution, which is to hollow out the gear hubs and embed the shaft collar into the gear. You can see this on the video at the 15 second mark, for example. For the tall gears such as mer1/mer2, I printed them with a hexagonal piece on one which fits into a recess on the other. It holds together well with just a friction fit, but could be glued for extra strength.

Here is a view of the shaft collar, and of the space it fits into. The angle of the top of the recess means that the part can be printed without support.

 

The Antikythera Mechanism 2: Design Principles for 3-D Printed Version

I plan to have a go at making a 3-D printed adaptation of the Antikythera Mechanism. See the previous episode for background.

A side note: from now on when I want to refer to the original mechanism, I'm going to refer to it as the HAM (Historical Antikythera Mechanism), to save some typing.

My guiding principles for the design are:

• make something that works well.
• stick the original mechanism for the overall configuration, but not for the precise dimensions.
• make practical decisions guided by the limitations and advantages of 3-D printing.

For example, the HAM used gears with triangular teeth. It is a consequence of the tools available when it was made. Triangular teeth don't print well and there are better modern alternatives. Another example: the gears of the HAM, as measured from the imaging data, use a variety of different module values. The gears are if we change the modules to more consistent values, with some consequent changes to the geometry.

Another way of saying this is that I am aiming for functional equivalence with the HAM rather than strict adherence. So now here is a list of some design considerations. The notes here were made at a fairly early stage in the design and some of the details were changed later.

Choosing the gear sizes

The largest gear in the HAM is referred to as b1. This name, along with the others that I'll use below comes from the Freeth 2012 and 2021 papers. The gear has 223 teeth and an outside diameter of 65mm, giving it a module of 0.578. For an involute gear, the teeth would be about 1.3mm from base to tip. Teeth this small won't print well and are likely to be fragile, and it would be better to scale it up. One option is to fit the whole gear on the bed, constrained by the 210mm dimension, another is to split the gear in two and fit the diameter on the bed, constrained by the 250mm dimension. Splitting a gear into parts and then gluing them together (or some other way of attaching them) does work. I gave some information in a previous blog post, and did it in my build of the Swingtime clock.

Here are some numbers for different choices:

• Fit to bed. Module = 0.93, diameter = 209, tooth depth = 2.09, scale = 1.61.
• Split and fit half to bed. Module = 1.1, diameter = 247.5, tooth depth = 2.475, scale = 1.90.
• Split and fit on a diagonal of the bed. Module = 1.156, diameter = 260.1, tooth depth = 2.601.

The last one is tricky. With a bit of careful positioning, you can a gear split in two fit on the Prusa print bed like this:

The modules of the original gears vary between roughly 0.42 and 0.48. I decided to simply use a module of 1.0 wherever possible and adjust the geometry to fit. For the large gear b1, it only has to mesh with one other gear (a1, which drives the whole mechanism), and I decided to stick with module 1.1.

Another gears, the combined e3 and e4, is also very large. I'll return to how to split it into pieces in a later episode.

A few of the other gears need slightly different modules to fit correctly. This applies to gears on the back part of the mechanism, driving the Metonic and Saros cycles and other outputs. The gears on the front side, driving the planetarium were not recovered and so their configuration and sizes are conjectured. I elected to use a module of 0.950 for the outer planets, which is close to twice the module used in Andronis's amclock version. For the inner planets, this didn't quite look right and so I changed it to 0.954, except for a few gears which needed a slightly smaller module.

Gear teeth profiles

The gear teeth in the HAM are triangular. It is the easiest shape to make with hand tools. However, it is not an efficient shape for transferring power through the gear train, and triangles do not print very accurately. The alternative choices are to use involute gears teeth or cycloid gear teeth. Involute gear teeth are the most common type. Cycloid gear teeth are commonly used in clocks due to lower friction and wear. They can also print better, as the sides of the teeth are more or less parallel for most of their length.

Involute and cycloid gear teeth

I don't think it makes a lot of difference in this design. My design tool is Autodesk Fusion, which has an add-in for generating involute gears. There isn't one for cycloid gears (though I have recently found, but not tried, a third party one), and alternatives such as Rainier Hessmer's program for generating SVGs are not very convenient to use. I'll use involute gears.

There are a few places where HAM uses a crown or contrate gear, that is a normal gear driven by another one at 90 degrees to it and with teeth standing upright. The modern alternative is bevel gears. This won't work in all cases, as sometimes the angle of the bevel makes the teeth intrude into the interior of the gear. The 223-tooth b1 gear has additional posts attached to it, and the beveled teeth came too close to them in a prototype. In such cases, a crown gear can be used with involute teeth.

Laying out the gears

I laid out the gears by constructing sketches in Fusion, with a circle for each gear. The diameter of the circle was the pitch diameter plus a small amount called the depthing, of which more in a moment. I then added Fusion constraints:

• tangent constraints for gears that mesh.
• coincident constraints for gears on the same arbor.
• horizontal and vertical constraints for some gears that should be aligned, for example the centers of the b1 gear and the arbors of the Metonic and Saros pointers.
• a few distance constraints for the eccentric epicycle gears. The distances came from the formulae in Freeth 2021.
• the angle of the "strap", a small platform mounted on b1, which Freeth says is at 11 degrees.

Most of the gears positions ended up fully constrained. A few could still be moved and I adjusted their positions to look similar to published layouts.

Gears should be set so that their pitch circles just touch, but in practice a small amount of extra space is sometimes needed for smooth running known as the depthing. I used a Fusion variable to specify the depthing, with the initial value set to 0. Later on, I made a small test print of two gears mounted on a base and found that they did not move very smoothly, and so adjusted it to 0.2mm. Fusion recalculated the positions and updated the sketches. I also created a component for each gear and positioned them in the right places to make sure they looked about right.

The sketches look like this:

(Superior planets, from above; back gears, from below; inferior planets, from above.)

Here is the gear tester:

Miscellaneous practical issues

Arbors and connected gears

Arbors are the shafts on which the gears turn. In many cases they are also used to connect gears which have to turn together. HAM has many examples where the shaft is circular at points where it passes through a bearing, and square or some other regular shape where gears need to fit rigidly to it so that they can turn together. My preference is to use 3mm steel or brass shafts or in some cases a printed shaft where it can be a larger diameter.

Where the arbor serves to connect gears together, they can in some cases simply be printed as a single object instead. This isn't always possible, for example two gears which are on opposite sides of a plate. For these cases, the two gears can be connected using a hexagonal socket on one and a corresponding extrusion on the other. Episode 5 will say more about this.

I like to try to use standard sizes for the arbors, so that you can buy pre-cut dowel pins instead of having the cut them myself.

Tubes

The front side (planetarium) outputs are carried on a set of ten concentric tubes. These could be brass: tubs with many diameters and thicknesses of either 0.5mm or 0.2mm are widely available. Alternatively they can be printed. Printed tubes are weaker and may have more friction. The strength doesn't matter much as they don't carry a lot of load, and the friction can be kept down by not making them too tight and by sanding them inside and out to eliminate layer lines. Based on some prototypes, ones with at least two perimeters (0.8mm thick) and at least 0.2mm spacing work fine (though 0.5mm may be better). For the innermost one, driving the lunar output, a solid brass arbor might be best as it does take a little more load.

The full set of tubes is different in the Freeth 2021 version and the ones before it. The 2021 has, in order from innermost to outermost, Moon, Mean Sun, Nodes ("dragon hands"), Mercury, Venus, True Sun, Mars, Jupiter, Saturn and Date.

(A series of tubes.)

Part thickness, hubs and bushes

The gears in the HAM are mostly 1 to 3 mm thick. For the 3-D printed version, the minimum is 3mm. Ideally gears which drive other ones should be thicker. If the gears are too thin, then any tilt when under load will risk them slipping apart, and the thickness also helps reduce tilt. Figure 6 of Amabile 2022 (reproduced from Lin & Yang's book) illustrates which gears should be thick and which thin.

Some gears in the HAM have no vertical spacing between them even though they run independently. The faces of the gears rub against each other (at least, I think this is the case, from some of the publications about the HAM). With plastic gears, the friction that results is a more serious problem than with metals ones, where the surfaces can be polished. I extended the vertical space, usually by adding a hub to the gear or a bush to whatever it is mounted on.

Discs and spokes

Most gears in the HAM have solid discs (b1 is a rare exception). I prefer more open designs with a hub and spokes. It makes the mechanism more visually interesting, and takes a bit less material and printing time. It can also make the prints better: sometimes I've found that prints with a full disc are more prone to warping.

Fasteners

Parts of the HAM were riveted or soldered together, or held in place with a pin through an arbor. Clickspring's videos show many examples. In ancient Greek times, this would have been because other forms of fastening either hadn't been invented or were not commonplace. I use screws in most places instead.

Print settings

The base print setting is PrusaSlicer's 0.15 structural. Some changes may be useful:

• more perimeters, to add extra strength. Sometimes this makes the slicing look less good, for example little blocks in the gear rim.
• aligned seams. It easier to locate the seams and then file them down when they are aligned.
• Arachne slicer in most cases.
• scarf joint seams if they are too obvious.
• maybe extra top and bottom layers for strength.
• 25% infill, cubic or gyroid.
• elephant’s foot reduction increased to 0.25mm or 0.35mm. It's important to get no elephant's foot where gear engage; sometimes I see it with the standard setting.

It may also work to vary the layer heights between 0.15 and 0.2mm so that pairs of gears which engage do not have the same value. This might reduce friction slightly.

The Antikythera Mechanism 1: Introduction

The Antikythera mechanism is an artifact from ancient Greece, discovered in a shipwreck off the coast of the island of Antikythera in 1901. Investigations and analysis have shown that it is an analogue model of the solar system, used to predict positions of astronomical bodies and eclipses, together with a calendar and with several additional indicators. There are many popular and scholarly publications about it on the web. Two good introductory articles are the one in Wikipedia and Tony Freeth's article in Scientific American from 2022. The heart of the mechanism consists of several gear trains, ultimately driven by a single crank. A person would have turned this crank, and the gear trains translate it into the motion of various outputs: the Metonic cycle (a sun-moon calendar), the Saros cycle of possible eclipses, the positions of the planets, sun and moon as seen from Earth, the phase of the moon, and others. Of the many videos on the web, I recommend Freeth's one and Jo Marchant's Darwin lecture.

The scholarly research is wide ranging, and the parts of it that I find most interesting are the ones which deduce the mechanical characteristics. There is a great deal of detective work. CT scans and image processing have allowed the size and configuration of many of the gears to be determined, although some parts of the mechanism are damaged or missing entirely.  Comparison of gear ratios with astronomical cycles led to an understanding of what parts of the mechanism are for. A "user manual" in the form of inscriptions on the external surfaces of the mechanism provides further information about its purpose and operation. In some of the more recent work, hypotheses have been put forward for the missing parts of the mechanism, notably the gears that drove the planet positions, by looking at possible mechanisms and deducing what would fit the physical constraints of the surviving parts. Some of the research has been largely settled for a while, such as the gear trains for the Metonic and Saros cycles, while other parts have been updated more recently, such as the planet mechanism. The astronomical model is centered on Earth, and so it has to account for the motions of the planets, sun and moon as seen from Earth. They appear to speed up and slow down and even to reverse direction. This motion can be modelled to some degree of accuracy with two gears, one off center with respect to the other, coupled by a pin and slot. The references above give more detail.

(Fragment A recovered from the Antikythera shipwreck. Image attribution: Giovanni Dall'Orto., Attribution, via Wikimedia Commons)

Several published papers include detailed diagrams of the gear trains and tables listing the number of teeth and size of the gears (or equivalently, their modules). Two of Tony Freeth's articles are particularly useful:

• Freeth, T. and A. Jones (2012) The cosmos in the Antikythera Mechanism. ISAW Papers 4, available at http://dlib.nyu.edu/awdl/isaw/isaw-papers/4/. Includes gear schematics. There are measurements for the planet gear trains, but not for the Metonic/Saros part of the mechanism.
• Freeth et al.  (2021). A Model of the Cosmos in the ancient Greek Antikythera Mechanism. Nature (Scientific Reports). https://www.nature.com/articles/s41598-021-84310-w. Presents a revised version of the planet mechanisms. The supplement (linked from the same page) has videos showing the assembly, and a table with the sizes of many of the gears. A very valuable resource.

There have been a number of projects to reproduce the mechanism, either as computer models or actually manufactured. Michael Wright made a replica in the 1970s, and Mogi Vincentini made a computer model from it. Freeth has used computer models to illustrate his published papers and to show that his proposed mechanism fits together. There is a lengthy series by Clickspring in which he builds a replica using original tools; at the time of writing it is not yet complete. A bronze version was made by Nicholas Andronis. He has also made a scaled up wooden version. Spencer Conner has another version made out of brass. He made a number of changes to modernize the design. There is a 3-D printable version, for which plans can be purchased, and a rather magnificent wooden version. I have also seen videos generated from other CAD models which may or may not have been made physically, and a 3-D printed version which even goes so far as to reproduce the damage to the casing.

There are plenty of YouTube videos with misleading information. It did not come from aliens or time travel. It's not an ancient mystery. It is an archaeological relic and scientific enquiry by thoughtful people has elucidated its structure and functioning. If you see a video which is an episode of someone's podcast or which employs AI, you should probably skip it. Unless you like that kind of thing, of course.

(Back panel of a reconstructed mechanism, showing the Metonic and Saros dials. Image attribution: Gts-tg, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons)

(Front panel of a reconstructed mechanism. Image attribution: Chris Olszewski, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons)

Dr. Andronis's site at https://www.amclock.net/ deserves further mention. He has detailed instructions and engineering drawings for his bronze version. The instructions include assembly and operating instructions, and many notes on practical issues such as adjusting the size of the parts to make them fit. He includes a table giving the tooth counts and sizes of all the gears, pulling together information from multiple sources. There are a couple of errors: some entries in the table quote the diameter of a gear under a column marked as radius and then also have an (incorrect) radius which is half of this, while others have the correct values; and there is one gear which I could not find on the engineering drawings, but can be reconstructed from the table.

I have been looking for a substantial new 3-D design and printing project, and the Antikythera mechanism is it. At the time of writing, I have prototypes for most of the mechanism, but still have a way to go before it operates smoothly. Perhaps I will succeed, perhaps it will end up in the Closet Of Abandoned Projects. What I want to make is a functional reconstruction: it should work, and use modern design techniques when that makes it work better, even if this means departing from the original mechanism in appearance.

The manual on Dr. Andronis's site says Warning: The only thing you learn from designing and building Antikythera Mechanisms is how to build a better Antikythera Mechanism next time. Well, perhaps, but as in many of my previous projects, the journey may turn out to be the destination.

Printing holes for a snug fit

I often want to design and print parts which fit on a metal shaft with a diameter of a few millimetres. 3D printers don't accurately print holes. The diameter of the hole in the printed part is typically less than designed for, due to factor such as the plastic squishing and expanding horizontally or the plastic expanding as it leaves the nozzle. If the part is intended to run freely on the shaft, it's not a problem. Just drill the hole out, for example with a pin vice, and maybe sand or file it if needed. I have also tried using a reamer to improve the finish, though I am not sure it makes any difference.

It is a little more of a challenge to make parts with holes that provide a snug fit. This would be for the case when you want to the part to be rigid, or at least firm, on the shaft. You can design the part with a hole for a set screw. I have often done this with 2mm and 3mm screws. I usually design the part with a hole 1.8 or 2.8mm for the screw and then use a M2 or M3 machine screw. It generally works quite well. I have had less success using grub screws. I think their thread isn't deep enough to cut into the plastic. Tapping the hole works. It is still not always a satisfactory solution as some parts don't have a good surface for placing a hole or for getting a screwdriver. Think of a gear with a hub the same height as the teeth, for example. Also, there are cases where you want a fit that will hold under normal use, but where the part can move on shaft if enough force is applied.

I decided to try experiments with a few different ways of making a snug fit. In each case, the shaft is a nominal 4mm in diameter and made from stainless steel. The measured size was about 3.95mm. Usually I find steel shafts are slightly under the nominal size and brass shafts are slightly over. The obvious method is just a simple hole, which I did at 4.0, 4.1 and 4.2mm diameter. I also used a 6mm hole with ribs in it. The ribs are 0.4mm wide. You have to use the Arachne slicing algorithm for them. Finally, tried a 4.0mm hole with a 10mm by 1mm cut out. The designs look this like:

Slicing for the ribbed version shows that there will be a single line of filament:

The tips of the ribs are designed to be 4mm from the center. Note that it prints well, but not perfectly, with some stringing between the ribs.

One preliminary comment before the results. In the past, I think we would have worried about things like whether the printer was square, calibration of the motion system and extruder, whether the filament was precisely 1.75mm diameter and so on. In my view, these are not really concerns for current printers  and filaments.

The shaft would not fit at all in the 4.0 and 4.1mm holes and was a bit loose in the ribbed version. It was fitted OK in the 4.2mm hole but was still a bit loose. The fit for the cutout hole was good: I could get the shaft in with a little force, and then it held very firmly. I could make it slip when turning the part on the shaft only with a lot of force.

These tests were with Hatchbox black PLA, using PrusaSlicer's 0.2 structural setting and the filament profile for Hatchbox PLA, Arachne slicing, and some extra elephant's foot compensation. The latter was because the first one or two layers sometime squish a bit more, and may be a tight fit for these layers but not for the others.

With Amazon Basics Gold Silk PLA using the Generic PLA Silk filament profile, the results were more or less the same. The only difference is that the 4.1mm hole was usable and was a tight fit. This wasn't what I expected. Maybe it is because silk filament is sometimes a bit more slippery.

There are no big conclusions to draw here. I hope it will provide some guidance for my designs in future.

Ways of finishing text on 3D printed object

As part of a project to make a 3-D printed Curta calculator, I've been looking at various ways of creating and finishing text on a 3-D printed object. In the Curta, this is used for the results dials and the digit selectors, and for some text on the body.

Some possible techniques are:

• make a mask or stencil using a Cricut and apply paint.
• cut the text into vinyl using a Cricut and stick it to the surface.
• print the text onto self-adhesive paper and stick it on.
• print the text onto paper, glue it on, and then spray-coat with lacquer.
• deboss the text into the surface and fill with the characters with paint.

There are undoubtedly other possibilities. For example, I saw one video using silicone caulk, with washing up liquid to help mask the surface. And if you have a multicolor printer, you have more options.

For the painted versions, there is a further question of whether the prepare the surface by sanding it smooth, so that the paint does not track into the layer lines. It seems obvious that you should do this. With some filaments, the act of sanding makes the surface take on a distressed appearance. In this case, you may be able to paint the surface to restore its color first.

Most of my experiments are with the debossed text technique. I stipple paint into the text, then wipe off the excess with a dry paper towel. When the paint is partially dry, I clean the remaining excess from the surface with alcohol on a paper towel. It worked best when I did this several times, as the alcohol makes the paint that it still in the text run a little more. It helps to clean a patch then dry it, then clean the next patch and so on. I used an acrylic paint. It's better straight out of the tube rather than mixed with any water, as it is more solid and doesn't run as much. I also tried a solid marker (basically very thick paint) and this works as well.

Over time, I refined the technique for this, to minimize the amount of extra paint. When I wiped away the excess with alcohol, I also tried to dry it immediately to limit extra leeching of paint from the digits.

Here are some example results.

The first thing to say is that none of these look as good as the results that Marcus Wu, designer of the 3-D printed Curta, obtained by preparing and painting the surface, then using a stencil. However, he also invested a lot of time in it, and wanted to get high quality for when he gave away his model to a Famous Person. I'm willing to compromise.

Key:

• row 1
• a. grey filament straight off the printer (unsanded).
• b. black filament after sanding with grades from 150 to 3000 and cleaning up.
• c. text printed on self-adhesive paper.
• row 2
• a. grey filament after finishing, with silver paint.
• b and c. grey filament after finishing, with red paint.
• row 3
• a. black filament after finishing, with white paint.
• b. black filament after finishing, with white paint from a solid marker.
• row 4
• a. unsanded blue filament, with red paint. Unlike all the other painted examples, I did an extra round of cleaning up several days after painting it, and this removed a little more paint from the layer lines without causing any additional running from the text.
• b. unsanded blue-grey filament, with white paint.
• c. blue-grey filament after finishing, with white paint.

Observations and comments:

• 1b shows how sanding can distress the surface of the print. In 3a and 3b, some of the white or grey marks between the text are the result of sanding, rather than paint.
• 2b and 2c show that you can get different results even with the same materials and finish. I think 2c is worse because the paint may have been a little more dilute. I was still experimenting with whether to mix any water in with it.
• 3a and 3b are similar in how much the paint spread. So the more solid paint from the solid marker does not make much of a difference. It might fill the text a bit better, though the comparison is not exact, as I think I debossed it by 2mm for 3b, compared to 1mm for all the others.
• the text version in 1c is one of the best. I do have some concerns about the longevity of the glue. Spraying it with lacquer would help, though to do this well you need to be able to let the lacquer dry without dripping. You can also get a seam where the paper wraps round, not shown in this photo.
• 4a and 4b show that you can get good results even if you don't sand the surface. 4b is better than 4c.

Here is a more recent print, using variable height layers, and after a bit of refinement of the technique.

My general conclusion is that printed paper gives the best results of the techniques I tried, provided it holds up over time. I think the painted debossed text works OK if you want something functional but with imperfect appearance, and that printing without sanding down the layer lines is as good or better than sanding them. However, it's also worth noting that I am not very skilled when it come to doing this sort of thing, and some more adept could probably do better painted versions that I did.

The Curta Type 1

One of my most treasured possessions is a Curta Type 1 mechanical calculator. It's a beautiful and compact device for addition, subtraction and indirectly for multiplication and division. There are many web sites and videos devoted to it, so I won't recapitulate the details here. I recommend curta.org as a starting point, or wikipedia for a summary.

This is mine:

It originally belonged to my father. He left it to me, along with some unusual slide rules which I might write about another time. I can remember him using it in the early 1970s, sitting in his armchair with a pile of experimental results to analyze. Some time around 1973 or 74, he got a HP-35 and then later a TI-57, and he stopped using it.

I put a short clip of it on YouTube to show how smooth the mechanism is.

Subtraction is done by adding the ten's complement (with a special case). You can see this in the second half of the video. The fact that it is so smooth is an indication of how little friction there is and how precisely it is made. There are two excellent videos which explain how the mechanism works: part 1, part 2. I really recommend watching all the way through, as it gives a detailed explanation of how subtraction works, as well as many points of detail which make it so nice to use. The underlying approach is not unique to the Curta, and in fact originates with Leibniz. You can see similar principles in this video about the Arithmometer. The genius of the Curta is to make it compact, lightweight and easy to use.

The serial number tells me that the manufacturing date is February 1962. It's odd to think that when I remember my dad using it, it was only about 10 years old, and it's now over 60. It is only a few months younger than I am.

The reason to write about the Curta now is that I am thinking of making a 3-D printed one, using Marcus Wu's 3:1 scaled design. I've looked at this before, but decided it was beyond my capabilities and the affordances of my printer. The latter has improved; the next few weeks or months may provide whether the former has advanced enough.

Some notes on Igus I151 filament

Igus is known in the 3D printing world for their bushings, which are sometimes used instead of linear bearings. I recently noticed they sell printer filament as well, see https://www.igus.com/3d-print-material/3d-print-filament. The filament is expensive; for example I150 is $72 for 750g, about 4 times what you would typically pay for PLA. They have several filaments and they seem to be quite different in characteristics. I've written a lot about making clocks, where low friction is critical to some parts of the gear train. By the time you get to the escapement wheel and the gears nearby, they are operating at very low torque and a tiny amount of frictional force could be enough to stall the whole gear train. Steve Peterson, in the build notes for his SP13 clock, comments that the weight of a house fly landing on the escapement wheel would be enough to stop it. The Igus I150 filament looks interesting, as it is said to have low friction as well as being easy to print. There is a posting on Steve's forum from someone who used I150 and seemed to get good results, as shown by needing a lower than typical weight to run the clock.

The Igus web site has a link to request a sample of the filament. Unfortunately, there sample link for I150 does not work. I emailed to ask about this, and a very helpful sales person told me they don't have any samples of I150 at present, and asked if I was interested in I180 instead. It looks difficult to print and requires an enclosed printer, which I do not have. As an alternative, they offered me I151. I am not sure quite how I151 and I150 are related. Comparing the technical specifications side by side, I151 has a higher Shore hardness and density, and a lower flexural strength. I assume that the similar numbering of the filaments means they have similar composition, and this video (https://www.youtube.com/watch?v=xD5_0mWAmZo) seems to confirm it.

A few days after the email discussion with Igus, a FedEx package arrived containing two samples of I151, each about 40g.

Igus provides profiles for the Prusa MK4, and I used these as a starting point. The profile has a layer height of 0.3, which I reduced to 0.2. Note that the I151 and I150 profiles are a bit different, for example, I150 has a higher maximum volumetric flow rate. All my tests use the I151 profile as a starting point.

The first test was to see how easy it was with standard print surfaces. Typically, I start with a textured PEI sheet. A filament such as PETG can stick too well to smooth PEI and damage it (or damage me as I try to dig it off with a scraper). It printed easily on the textured PEI with no warping. The test object was a 20x20x10 cuboid: not a very tricky test, though one which will trip up difficult to print filaments such as Nylon. I dried the filament in a filament drier for a few hours first. The default print temperatures in the profile are 240 on 85. On smooth PEI, the adhesion was very strong, close to what you get with PETG. The only time I saw any warping was on one print where I accidentally 210 on 60, after selecting the wrong profile. One print stuck so hard that the bottom layer cracked when I pulled it off the bed. As with PETG, glue stick on smooth PEI is a good compromise: the print sticks well, but not too well, and there is no warping.

I looked for a couple of things in the print quality. The first is to examine the vertical edges. On a good filament such as Hatchbox PLA, they will be completely straight. Silk PLA often shows some bulges, which may mean that gear teeth printing using the filament won't engage very cleanly. It's not usually a showstopper, but can reduce the efficiency of the gear train. All of the test prints I did looked good in this respect. However, I was a bit surprised to find that the surface texture of the sides of the print was a little rough. It wasn't, as already noted, distorted, and there were no visible pits or zits, it just felt slightly grainy. It's interesting that the raw filament also feels like this out of the package. You can see the texture if you look closely (I don't have a microscope to try to take a picture of it). It's like a less pronounced version of what you get when printing with the fuzzy skin setting. I don't think I have ever known other filaments with this feel, except for some of the exotics such as wood fill. So perhaps I151 has some extra filler in it to provide its special properties and this is what I am observing.

I tried a bunch of things to try to fix this, including copying the extrusion widths and infill settings from a PLA profile and reducing the extrusion multiplier. The default extrusion widths in the profile seem a bit odd: they set the first layer to 200%. The only change which really made a difference was dropping the temperature to 220 (240 for the first layer). It reduced the feeling of graininess, but did not make it go away altogether. The best print in this respect was the one where I accidentally used my PLA setting (210 first layer, 200 after), though the low first layer temperature led to some slight warping in this case, and I am worried about getting a nozzle jam at such a low temperature.

The card that comes with the filament says "Part cooling should be adjusted to the minimum necessary temperature". The supplied profile sets this to 20% min/80% max. As a (rather goofy) experiment, I turned cooling off altogether and unsurprisingly ended up with a squishy mess. I then also tried changing it to same settings as PLA (100% after 3 layers and some other changes), based on the thought that I was seeing better surface textures with lower temperatures. This made the surfaces rougher.

For a final test, I printed one of the gears from Steve Peterson's SP5 clock. There isn't enough in the sample to print a full set, so this is just for visual inspection. The print quality is good: the gear teeth look precise with no bulging, and no lifting from the print bed. The hole for the arbor was undersized and needed to be drilled out, and there is some stringing, but that's similar to many other filaments. The print has the same surface texture issues and I think would not do well when low friction is needed. I did this both with the unmodified profiles from Igus, and with the reduced temperature and other changes I made. It seemed to be important to set elephant's foot correction to 0.

This picture shows the I151 print and one in gold silk PLA. The bright sunlight emphasizes the surface texture, and show how rough the I151 is.

I am unsure what to do next. Videos and forums posts comment on how smooth the surface of I150 prints feels. So there are several possibilities: I151 is different from I150; there is some sweet spot in the settings which I haven't found; and that this is a bad batch. It's hard to believe that the rough surface texture would result in anything other than more friction. For a cheaper filament, I'd just by a small roll and discard it if it didn't work out (I did this with some Nylon recently). But at $72 for the smallest roll they sell, I really have to think about it.

At last, a 3D printed watch

I've recently completed the 3D printed watch with tourbillon, designed by Christoph Laimer and published on Thingiverse in 2016. Here it is in operation:

You might notice that it does not remotely keep time: a minute on the watch is only about 43 seconds in real time. I'll say more about that later. A few more pictures:

In brief, the watch works like this. There is a mainspring in the base, which drives the minute train. The minute train is linked to the tourbillon, which locks and unlocks it and so provides the timing. The minute train drives the bronze (greenish) ring gear. Another gear takes the movement from this ring gear into the hour train, which ends up driving the gold ring gear.

I got my first 3D printer in mid-2015, and when this design came out in January 2016 I decided to give it a go. It completely failed to run, and I set it aside. This is not a surprising outcome. The printer was not very accurate, and I had no idea how to debug clocks and watches or even really much understanding of how they work. Time has passed, and now I have a Prusa MK4. It much more precise than my first printer, and also a lot faster. A consequence of the speed is that I am willing to print at much finer layer heights than before. On my first printer, I was probably using 0.3mm for large parts and 0.2mm for smaller ones. With the MK4, I nearly always use 0.15mm without worrying about how long the prints will take.

On the debugging side, I now have a better understanding of how clocks and watches work, and this helps in building up the mechanism in stages and do partial tests. The tourbillon itself is a self contained unit and I first tested it in isolation. Then I checked that the minutes train worked smoothly, separate from the tourbillon, then the minutes and hours trains together, and finally the whole mechanism.

Most of the parts are from the original design. The tourbillon spring is the medium strength one from A flight of hairsprings. I used the Massey pin in the balance wheel mechanism. It makes the  balance wheel less prone to jam against the fork. The bridge on the top is modified from the logo-less version from the torque modification of the design. I decided not to use the hands from the original design as they don't attach very well, and instead printed the black notch you can see above directly into the ring gears.

Now to why the timing is off. When I first put the tourbillon together, I found that it would seize. The reason for this is that the balance wheel was swinging a long way on each tick. While it was at the ends of its movement, the fork could flop around. When the balance wheel swung back, the fork might not be in the right place for the balance pin to engage with it and everything locked up. Using a stiffer spring limits how far the balance wheel swings and so avoids this problem. However, it means the timing is no longer right. For a real timepiece, this would be an issue. But in my case, I intended it more as an objet d'art: something to look at. It isn't really a practical clock, as you can't tune the timing or even set the hands. If it ticks at the wrong rate, that's OK.

I modified a few of the parts. Many interior holes were too small. For the ratchet and ratchet bushing (check the thingiverse page if you want to know which parts these are), I slightly opened them up by editing the STLs in blender. For all of the gears in the minutes and hours trains, I could have drilled out the holes. Instead, what I decided to do was to modify them as follows. All of these gears run on 2mm arbors, so I made the axis hole slightly larger than this for the 2mm or so at each face of the gear (in Blender, again). The rest of the axis hole I made 2.4mm diameter. This means that when I drilled out the holes, I only needed to cut through the 2mm part at each end, so there is not much chance of the drill going askew. The rest of the interior of the axis hole does not touch the arbor, thus reducing friction. Steve Peterson uses this in his clocks, and I've mentioned it in a previous post.

The teeth on most of the gears are tiny, only about 1.5mm from tip to base. You need accurate printing followed by a close visual examination for any blobs or wisps of filament for them to work smoothly.

The filaments are Flashforge burnt titanium PLA for the body, gold and bronze silk PLA for the gears and moving parts, and PETG for the main spring. The burnt titanium filament looks very nice, but is not so good to work with. You get a lot of stringing and blobs, and the printed surfaces are slightly rough. The ring gears have a large contact area, so you really need something smooth and with low friction. Silk PLA is ideal for this.

The design also calls for a rather bizarre range of small screw sizes. I think many of these could be replaced with more standard sizes (or at least to use one or two sizes throughout) with minor design changes. I didn't try this, though it's noteworthy that the design included the original CAD model in Fusion 360 format, making such modifications easier.

How long will it run? From a full winding, I can get 20-25 minutes. I expect this will decrease over time as the mainspring weakens. Here is a 10x timelapse starting with a fully wound spring and letting it go until it stopped. With a nudge it will run for another minute or two.

And also a look at the mechanism in slow motion:

This is a remarkable design by M. Laimer (aka TheGoofy). It was one of the first 3D printed timepiece designs and it outclasses many more recent designs in the care and thought that went into it as well as the attention to its visual appearance. I would definitely rate it higher than, for example, the Tourbillon Mechanica, which just looks messy to me, or the many published tourbillon design which just don't run well. Very nice.

Tips and Tricks For Printing Small Gears

It is sometimes difficult to print small gears. Typical problems are teeth pulling up or distorting or simply the whole thing coming loose and sticking to the nozzle. I don't believe there is any universal solution to this, so here are a few tactical things you can do. Obviously start by making sure that your printer is well calibrated, the print surface and nozzle are clean, and that the bed is levelled. I usually clean the nozzle by heating it to just under the print temperature then removing any stuck on filament by lightly brushing it with a brass wire brush. It helps to put a mirror on the print bed so you can see what you are doing. I have a concave shaving mirror that also magnifies. My Prusa MK3S is set up for levelling using a slight modification together with an associated process known as "bed levelling without wave springs" (it's a misleading name reflecting the history of how the author of the process got there), and if you put in the time with this you can get a very accurately levelled bed.

The problem with small gears is that you can get small segments in the teeth which result in a lot of nozzle moves and retractions. Due to the viscosity of the filament, the move may pull the feature it has just printed off the print surface or plow through a small feature which is isolated from other features and so doesn't have much attachment area. Generally once you are past the first layer, things will go OK. You might get some stringing or blobs, but you can fix these up when the print is complete.

A first thing you can do is examine the features in the slicer preview and adjust settings to try to avoid such features. For example if you see something like the small triangles and dots in this example, you may be in for a problem:

There are several parameter changes which may help. It's not possible to say that any of them will definitely improve the slicing. The best thing is try them, look at the result and see if it appears better:

• increase or decrease the number of perimeters. Stick to a minimum of 2. More perimeters give you greater strength and rigidity. If the gear doesn't take much load (for example, the gear train from minutes to hours) then 2 will be strong enough.
• reduce or turn off elephant's foot compensation. It makes the first layer smaller, so making it more prone to these small features. If your printer is well-calibrated, you may not need it anyway.
• switch between the Arachne and classic slicing algorithms. Sometimes one just does better than the other.

If you have access to the model you might also be able to tweak it, but that's a bigger task and doesn't always help.

Another setting that can be useful is external perimeters first. This won't change the slicing, but guarantees that you have a single solid outline for the remaining tracks to stick to.