Factorio Space Exploration: Arcosphere Cookbook with explanations
I found the arcosphere puzzle intriguing but poorly explained and clunky, and when looking for advice I found a lot of other frustrated people who were brute-forcing the puzzle, using giant combinator fields, and/or copying entire solution blocks from other people's blueprints. I have further complaints, like the strict spoiler policy and the F:SE wiki deliberately avoiding this info, so I'm going to talk about arcospheres and try to make this be something like the reference I wanted. I will try to explain principles, not just solve, to help people understand. This will be long, and have colored diagrams.
Contributing to my frustration is that Arcospheres are a finite*, accidentally losable resource that are required for victory. On the Cruelty Scale from interactive fiction, Arcospheres put Space Exploration somewhere around Nasty. Having a spoiler policy on top of this is rude!
So. When you first get spaceballs, they are bland grey. You can get some from looting ancient ruins and the Interburbul game on other planets, but the main source is from the deep void, by launching arcosphere collectors from deep space asteroid fields. These have diminishing returns, so start with about 5 from each field. Make sure to launch the collector, not the arcosphere, I've seen one guy lose his spheres that way because he was outputting silo spheres into the same container that collectors were loading from, and the loader loaded arcospheres instead.
Then you put them through Arcosphere Polarization. This takes in 4 grey balls and puts out a set of 4 colored arcospheres: either [Lambda, Zeta, Epsilon, Gamma] or [Xi, Theta, Phi, Omega]. Do some of both recipes to get all 8 types. A machine set to run continuously will randomly flip between them - this comes up later - but you can manually set it for your first few spaceballs.
Arcospheres are unusual in that they're not consumed like most ingredients in Factorio, instead they are merely transformed between the different polarities. The recipe for a Naquium Tesseract takes in 3 arcospheres [Lambda, Xi, Zeta] and outputs 3 arcospheres [Theta, Epsilon, Phi].
Productivity modules are banned here, naturally.
*Finite: Arcospheres are hard to produce, with arcosphere collection from space having logarithmically diminishing returns, which means at some point it would take a lifetime of play to get another set. Make sure to keep them well protected from biters, meteor strikes, spaceship landings, and so on.
Since the full names are heck to diagram and the greek letters are heck to type, I'm going to refer to the 8 polarizations by their English-language initials: L, X, Z, T, E, P, G, O. That's the order the items are sorted ingame, don't blame me.
Next it's useful to visualize the various arcosphere folds and transforms. Since there's 8 of them, we can map them to the corners of a cube, and a cube is relatively simple to diagram as two squares. I will label the corners in semi-clockwise order, which makes several of the folds easier to visualize the patterns for, and color them correspondingly. Starting with the Arcosphere Polarization, you can see it produces either inner (LZEG) or outer (XTPO) spheres.
Next is the Arcosphere Inversion basic recipes, it would be a nuisance to memorize exactly which four invert, but with the diagram...
...it's swapping the diagonals. Inversion is invertible.
Then there's 8 kinds of basic Arcosphere Folding. The first one is [LO] -> [XT] and we can think of this as X-centric. Each fold is centered on a corner: it takes in the "previous" (counterclockwise) and the parallell sphere, outputs the chosen corner and the "next" (clockwise) sphere. Each fold takes one inner and one outer sphere, outputs either two inner or two outer. Here's the X-centered and the L-centered fold.
Here's all eight of them in a block. Stare at it, feel the patterns.
With this, I find it much easier to think about recipes in terms of "I have..." and "I need..." when I know they move sorta-clockwise, and are output in adjacent pairs.
Also, each column as presented forms a combination to build from. If you have [OL,XG] you can run the two leftmost recipes and output [XT,LZ]. Reorder these and it's [LXGO] -> [LXZT]. We have used LX as a catalyst to turn GO into ZT.
This generalizes by rotation. If you have one of the half-diagonals such as [LX], [ZT], [EP] or [GO], you can convert it into the opposite half-diagonal this way. This is the 1st combination recipe.
Here's what this looks like in practice in Factorio, with the pair from the fourth column that combines [GP]->[OX] and [EO]->[GL], with a net result of [EP] -> [LX].
[EP] comes in at left, is taken up by the first set of filter inserters. [GO] is a catalyst, passed between the two gravimetrics machines indefinitely. [LX] is output to the right.
In addition to turning a half-diagonal-pair into its opposite with 2 catalyst spheres, any two of a single polarity can be turned into their opposite with 1 catalyst. (Opposite corner cubewise, so L becomes P, not E.) This is the 2nd combination recipe.
You may want to copy the Eight Folds diagram to use as a reference for the individual steps here. Starting with for example two Lambda, we use an Omega as a catalyst, and perform the following folds, converting the pair marked in brackets at each step.
L [L O] [L T] X E [Z X] [E T] P O P P
And here's that built in Factorio:
Place an [O]mega catalyst in the upper left machine once the very first time, then insert many L in top chest. They will be converted to P, and the O catalyst returned to upper left after each pass.
The topmost filter inserters from the L chest are wired to take only when (Lambda > 1) to ensure even splits in my demo, you could use a splitter belt or something else for the same effect. The middle red long inserter uses a nonobvious Factorio feature: when pointing into a machine, inserters will only take ingredients needed in that machine's recipe, so it will only take the E and not the Z from the top-right machine to the lower-left machine.
The full set of 8 opposite-corner chains:
Now we've got the Inversion recipe that switches full diagonals of 4, turning a diagonally adjacent pair into the other half of the diagonal, and turning a single ball into its opposite corner, and. That leaves one more building block of useful folding combinations:
Turning an opposite-corner pair into the other two on its long diagonal. This phrase sounds clunky even to me, so here's the visual: starting with a bunch of L, generate LP, and from there XLEP.
The [LP] -> [XE] transform is accomplished with only one catalyst ball, but needs more balls to be converted. You need to start with LLPP, and use T as catalyst into XXEE, with subelements of [LP]->EE and [LP]->[XX]. This is the 3rd combination recipe.
The specific steps: [TL] P E [ZP] EE G and then use that G as intermediate catalyst for [GP] L X [OL] XX T giving you the T back.
This takes a little bit of sorting input to build the machines. Factorio build with similar 4-machine shape to the above, note again the wired inserters set to (L>1) and (P>1), can be replaced with belt splitters or other method of even distribution. The whole system can be rotated and rearranged. Think of this as a Lego-like toolkit, not a definitive solution. (That's why I'm posting diagrams, not blueprints.)
Place a single catalyst T in the top-left [LT] machine to start, and the system will turn sets of [LLPP] into [XXEE], returning the T.
The four sets of this are: T as catalyst to turn [LLPP] into [XXEE] O as catalyst to turn [XXEE] into [LLPP] P as catalyst to turn [ZZOO] into [GGTT] X as catalyst to turn [GGTT] into [ZZOO].
Let's revisit the diagram.
Starting from a bucket of one single arcosphere such as L, you now have the steps and components to:
Turn it into the opposite corner. (-> LP)
Turn those opposites into the rest of the diagonal. (-> XLEP)
Invert the diagonal. (-> OGZT)
Turn two diagonally adjacent balls into the opposite diagonal. (OG < - > ZT.)
and the rotations of these.
Combining these steps now gets you from any ball to any other ball, in sufficient quantities. For example, if you're running the Naquium Processor recipe that outputs five Lambdas, and you want to turn your stack of L into E:
L -> LP with opposite-corner LP -> XE with on-diagonal X -> E with opposite-corner.
With the building-block theory done for converting arcospheres, let's look at productive recipes featuring spaceballs. I'll start with the first data card from Deep Space Science 3: Space Folding Data.
The recipe for this card comes in two flavors: each takes Naquium Plate, Significant Data, -273C Thermofluid, and arcospheres [LX]. One flavor outputs ZT, and the other flavor outputs EP as our arcospheres. You can manually choose which the first time, but Factorio is all about the automation, and a machine set to automake Space Folding data may randomly swap between recipes each time a crafting has completed! So you need to plan for getting a mix of ZTEP out, unpredictably. Let's consult the diagram again. White for input, but grey and black for the two sets of output.
Well the EP is easy: just convert it back to LX, using a set of OG as catalysts.
The ZT output is a little harder. Convert half of it to OG (using a set of EP as catalysts) giving you the OGZT diagonal, then invert that into XLEP. Then filter the XL back into the data card maker, and and the EP can convert to XL.
How this might look in practice, zoomed out and labeled:
Zoomed in and cropped so it's easier to see filter settings:
Tracing the path through this system, start with the machine on the right. Insert some LX. If it comes out as EP, it takes the upper belt leftwards, and gets converted back into LX, easy peasy.
If it comes out as ZT, the two lower filters put that onto belts, and split them so that some ZT goes to become OG at the bottom, while the other ZT goes to join the fresh OG in the OGZT inversion machine on the left.
Space Warping Data is very similar and straightforward. You input [EP], get out OG and/or ZT, you convert as necessary until you have an OGZT set to turn into XLEP, and turn the extra XL into EP.
Space Dilation Data is a little harder. You input [ZO], and get out either LL or PP. Conditional wires on your inserters start to be useful here, as does having more arcospheres for a buffer when the random number generator is streaky. Hope for LLPP; but plan to convert LLLL or PPPP into LLPP. Turn some of the LP into XE, then a whole XLEP diagonal inverts to OGZT, and GT can be converted into OZ.
Space Injection Data gets tricky. You input [GT] and get out either ZZ or EE. For the first time, these aren't on the same diagonal. Oh well.
When you get Z, convert half of it to O with opposite-corner, then ZO to GT.
When you get E, convert half of it to X with opposite-corner, then convert half of the resulting EX mix to LP, giving you a full diagonal. Invert the XLEP diagonal to OGZT, and then convert OZ to GT. This requires five sets of folding combination machines, and it sprawls quite a bit and wants a large buffer of extra spheres.
I will note one more data card later in tech tree: Wormhole Data, [LZEG] -> [XTPO]. No randomization here.
This is a sneaky recipe, because the diagonal steps described above can be used but it gets much larger than necessary because they're a poor fit.
Notice instead how this takes inners and produces outers. What turns outers back into inners? The basic Arcosphere Foldings.
Produce a set of XTPO, and keep a spare set of LZEG as a catalyst. Run the lower four basic foldings from the Arcosphere Folding block once each, like [XG]->[LZ], producing [LZ]+[ZE]+[EG]+[GL], or two sets of LZEG.
For the other recipes, like Naquium Tesseract, hopefully these visualizations and combinations have helped you understand what's going on and how to build your own design from parts. Or you can download someone else's solved blueprint. You don't need to bash your head against a wall of obfuscated math to play with spaceships.
You are welcome to copy, annotate, share and edit this guide as you like.
