m. c. de marco: To invent new life and new civilizations...

Today’s (more like this year’s) Big Dumb Object (BDO) comes from Eric Brown’s two-book series Helix (2007, reissued in 2023) and Helix Wars (2012).

There’s a tiring amount of framing of the human side of Helix, in which a severely depopulated Earth sends out a lone ship of 4,000 sleeping colonists plus a skeleton crew of six. Eventually, the crew awaken to a crash landing on what they at first think is their target planet. Soon enough, the sun rises over something quite different, a helical planet that the author never quite describes to my satisfaction.

The Helix is a long, skinny megastructure wrapping eight times around a central star to form a single-stranded helix. It seems to be rotating around its cylindrical axis the way a topopolis can, but it also seems to be a solid planet, at least until book 2. The Helix contains approximately ten thousand worlds, divided by ten thousand seas, like a necklace of alternating green(ish) and blue beads.

There are no seasons, though the outer tiers are colder than the central ones. What happens physics-wise at the two ends of the structure is not addressed. The question of whether a solid ring or helix can rotate like a topopolis is moot because Helix Wars reveals that at least some world-beads rotate at different rates, so it must be jointed although no joints are ever shown.

The diameter of this tubular planet isn’t specified until well into book 2, nor is the length of the individual world-beads. The seas are described as a thousand miles wide, i.e., long. Though they seem to serve as buffers between the different atmospheres and geographies of the individual worlds, how this works is also never described. Most of the travel in the books is by spaceship, at an unspecified speed. So in book 1, the biggest clue to the dimension of the world is a statement that “there’s sufficient landmass in the entire helix to contain oven ten thousand planets the size of Earth.” The distance between the tiers (later called circuits) of the Helix is never discussed.

In Helix Wars, a tale of interworld conflict, a few humans get to see the “spine” of the Helix, a tunnel two hundred kilometers (125 miles) in diameter running the length of the Helix, within a wall ten kilometers thick. There’s some unnecessary artificial hollow-earth-style gravity to counteract the shell theorem. The tunnel isn’t really necessary for either the plot or the structure, though gigantic machines within it are somehow supposed to be maintaining everything from a distance of “almost 8,000 kilometers” (4971 miles) from the surface. In addition to this radius, a length for the Helix is also finally provided: 200,000,000 kilometers (124,275,000 miles). These are big numbers, but they’re not big enough; see the calculations below for details.

Both novels provide entertaining adventures involving a reasonable number of worlds and species of the Helix; they’re very much a typical example of the genre rather than one of those novels that provide only a glimpse of the BDO in the last chapter. The lack of physical detail about the Helix in the first novel interfered with my sense of wonder, as did the reliance on magic technology once details were revealed (e.g., the unnecessary artificial gravity in the unnecessary tunnel through the center of the Helix), though my personal BDO construction principles probably don’t matter to the average reader.

Helix
Helix Wars

Spoilers

Helix starts out slowly, with a remnant of post-ecopocalyptic humans living a hardscrabble life on Earth. Enough government survives to send out a colony ship to a nice-looking sun a thousand light years away, and enough ecoterrorists survive to necessitate secrecy. It’s never clear whether terrorism or accident causes the ship to crash in their target system, but it is eventually explained that the Builders of the Helix concealed it from Earth’s view. If you’re likely to be annoyed by all the eco, this may not be the novel for you; I thought it worked well enough in the context of a BDO story. The human characters are serviceable, if not particularly memorable.

The main alien characters are a bit more interesting. A race of otter-like humanoids living on a cold, cloud-shrouded world-bead have developed an unnecessarily insular religion in which the universe is a sea of gray with their city in the middle and not much else. There’s as much bickering about religion in their chapters as there is eco-handwringing among the humans. If you’re likely to be annoyed by all the atheism, this may not be the novel for you; I thought it came close to working well enough in the context of a BDO story but the situation felt more assumed than explained.

Eventually some otters stumble across the spaceship of a lone representative of a third alien species, sent to recover technology stolen by the religious fanatic otters. In the process they rescue the humans and escape to another world, Phandra, inhabited by a species of peaceful, apparently primitive but psychic aliens who have been waiting for their arrival. They show the humans and otters (the lone alien doesn’t make it) to a crashed ship of one of the engineers that manage the Helix. In this way they discover where the world of the Builders of the Helix is, fly to a world neighboring it, and use native transportation (living flying carpets) to reach the Builders' world undetected by the pursuing religious otters.

In the best of BDO fiction fashion, they meet the Builders (after a fashion), discover the purpose behind the Helix, and settle down for the long haul. Oddly, the two impossibly estranged otters get back together.

Helix Wars gets going faster, with more adventure and fewer Big Issues. Humans have had two hundred years to settle into their new colony and new interworld peacekeeping role. As is all-too-common in science fiction, all their futuristic technology disappears when it comes time to give a character a tragic backstory involving the death of a child, but otherwise the humans are doing well on the Helix…until an upstart alien world (Sporell) invades its next-bead neighbors (which happens to be Phandra, from book 1).

Sporelli shoot down a human shuttle that flies by at the wrong time, and only the pilot, Jeff, survives. Humanity’s bureaucracy doesn’t seem up to the task of rescuing him, never mind actually fulfilling their peacekeeping duties, but coincidentally he has an alien friend Kranda from the engineering race (the Mahkan) who owes him a favor debt of honor and goes after him. Before she can rescue him, the locals, Phandrans from book 1, find him, treat his injuries, and try to get him home. They fail, but the Mahkan succeeds, and there’s a brief interlude on New Earth before Jeff and Kranda reunite to rescue some captive Phandrans now being held at the next world and target along the line, D'rayni.

They are only partly successful in this mission, and a setback sends them fleeing to the tunnel at the center of the Helix. They travel by its advanced magic car system to Sporell itself, where we get a dose of this novel’s Big Issue, fascism bad, but is it bad enough to justify killing Hitler? This particular cloud-covered world is extremely gray and fascist, and its dying Supreme Leader is planning a surprise to extend his domains and his life indefinitely.

While the surprise is a neat plot twist, the vast scope of the Supreme Leader’s ambitions makes his minor incursions on the surface of the Helix, heretofore a large part of the plot, feel irrelevant. Our lone heroes handle the new threat with aplomb, magic builder technology, and too much arguing about killing Hitler. Only off-screen do the peacekeepers finally get off their collective butts and restore the peace on Helix.

There’s another meeting with the Builders, and a denouement that would have had more impact with more believable relationships. (The two impossibly estranged humans get back together.)

Calculations

The radius of the Helix (at an access mountain on the D'rayni world) is almost 8,000 kilometers, or 4971 miles, more than the radius of the Earth (3,963 miles). The length of the Helix is 200,000,000 kilometers (124,275,000 miles), so about 20,000 kilometers (12,428 miles) per world. Minus the thousand miles of ocean, this is about 11,427 miles per world. Each world would have a surface area of 2π times the radius times its length, or 356,900,000 square miles. This is closer to twice the size of the Earth, so the Helix is about the size of 18,000 Earths, not 10,000 Earths. This, however, is not the major error in the BDO.

Without knowing the distance between the tiers, we can only approximate the distance of the Helix from its central sun. Judging from the only slightly uninhabitable ends, the cover art, and the ziggurat transport system that the characters use on one occasion in book 1, the pitch of the helix is minimal, with a distance between tiers of perhaps as little as ten times the diameter of the beads. For a guess let’s call it 100,000 miles.

With that and the length of the Helix we can approximate the radius of the Helix—the distance between it and its central sun—at 2,500,000 miles. Unfortunately, this is not a reasonable distance from a G-class sun, being about one thirty-seventh of an astronomical unit (92,955,807 miles). Tweaking the distance between tiers is no help; the Helix’s goose is cooked. In other words, the Helix is an object lesson on not pulling numbers out of your arse.

An 8-tiered helix fitting the parameters of the book but positioned a habitable 1 AU away from its sun should be 4,672,468,555 miles long, with room for 376,000 worlds, not a mere 10,000. Another option would be to place the original Helix in orbit around (but outside of) the sun rather than being a spring with the sun at its center. But such a helix would have seasons, and the temperature would vary across the diameter of it, unlike the weather of the novels.

I applied for non-commercial access to BoardGameGeek’s XML APIs (there are two) earlier this year, at some point well after the announcement of the coming of the APIcalypse. I was also less than prompt in updating my BGG tools to authenticate against the soon-to-be-secured API. This week the APIcalypse arrived and I finally updated my tools to get them working in the post-apocalyptic hellscape era.

I was thinking that, due to the interesting implementation of the new API authentication, I’d have to send my sekrit in the clear from my purely client-side BGG apps. But then I remembered that all my API requests are going through my personal CORS proxy, in order to deal with some other interesting design choice at BGG (the gory details of which I’ve since forgotten). So I could just hack my proxy to also authenticate my requests without revealing any secrets. The hardest part was getting PHP working locally on my Mac in order to test my changes.

This is a future post to my BGG blog, 40 Graphs. Check there for the boardgame photographs, but for all the diagrams, look here.

In the Kingdom Builder images at BoardGameGeek back in May, Jack Wingard added a fifth board on top of the other four. This inevitably led me to wonder how many such boards there are, a question I promptly forgot about for a few months because the thread was hidden in the images (until I linked it in the forums).

When it came up again recently, the idea was still intriguing enough for me to convince a normally unwilling opponent to play a game this way. The fifth board did unbalance the game in ways expected and not: by creating both smaller and larger territories than the designer designed (which I’d expected) and by bringing locations closer together (which I should have expected).

Once I had the pile of boards out I could figure out the number of possible layouts, and this blog post was born.

There are Four Boards

To recap the basics of Kingdom Builder unique board counting from my Winter Kingdom Builder Math post:

A Kingdom Builder modular board section has two possible positions, which we can call right side up and upside down based on the orientation of the location hexes. The base game comes with 8 board sections. You choose four of these sections for the base game, in order, so there are 8 x 7 x 6 x 5 (1680) possible ordered sets of boards. Each modular board has two positions, so within each set there are 2 x 2 x 2 x 2 (16) possible layouts, for a total of 26,880 possible game boards. However, for the purposes of the game, a game board is the same as the upside-down version of itself, so we need to divide this total by two to handle the symmetry, leaving 13,440 boards. Each of the four expansions comes with four more sections, so if you have all the expansions (as in the bigger of the Big Boxes), the number of possible boards is 24 x 23 x 22 x 21 x 16 / 2, or just over 2 million possible board layouts.

Note that “right side up” refers to the reading direction of the printing on the front of the board; the reverse side of KB boards is not used.

And then there were five…

When adding a fifth board, you have four to choose from in the base game, or twenty with all the expansions, so the possible boards are increasing to at least 13,440 x 4 = 53,760 (base) or 2 million x 20 = 40 million (with expansions). But that’s just the beginning, because the number of potential orientations of the fifth board is much greater than with the first four.

Consider the fifth board oriented right-side up (so that the locations aren’t upside-down). In this orientation, there’s a protruding hex in the upper left corner. We can enumerate all possibilities for the fifth board by counting up the possible positions (including orientations) of this first hex.

It’s easiest to separate the calculations into the three possible board orientations that line up (hex-atop-hex) with the underlying boards, and that are not just the upside-down version of another orientation (which we can incorporate by multiplying by 2 later on). I’ll call these orientations N, NW, and W, according to the compass direction the first hex appears to be pointing in.

Unskewed

The easiest direction to visualize and count is NW; this is when you orient the board horizontally as if it were one of the regular boards:

There are a hundred hexes on the first (upper left) board which the designated hex on the fifth board can fit over. There are also 10 hexes in the top row of the lower left board where we can place the fifth board without going outside the border of the underlying boards. There are another 10 hexes in the first column of the upper right board, and there is a single hex on the lower left board.

However, four of the hexes we’ve counted represent our new board completely replacing an old board, reducing us to a four-board layout:

We do want to count this ersatz normal game, but only once, and we will do that by adding 13,440 (or the two million) to whatever total we come up with of genuine 5-board permutations. There are 99 + 9 + 9, or 117, non-degenerate cases, and in these cases when we invert the fifth board, we get another unique layout, so the factor is 117 x 2 = 234 possible horizontal placements of the fifth board.

So in the NW orientation, the overlapped board count for the base game is 53,760 x 234 = 12,579,840, and for all expansions, 40 million x 234 = 9.36 billion.

Skewed

The other two orientations are a bit harder to visualize; it helps to slide the boards around to see how far they can go.

In the N direction, there are nine hexes horizontally by six hexes vertically where the first hex can go without the fifth board going outside the borders of the underlying board. The fifth board can still be flipped, so our factor is 9 x 6 x 2: 53,760 x 108 = 5,806,080, or 40 million x 108 = 4.32 billion.

In the W direction, there are 8 horizontal hexes in each of 4 indented horizontal rows, but 9 horizontal hexes in each of 3 out-dented horizontal rows, for a factor of (8 x 4 + 9 x 3) x 2: 53,760 x 118 = 6,343,680, or 40 million x 118 = 4.72 billion.

To add these all together, we can add up all our factors (NW degenerate + NW + N + W): 1 + 234 + 108 + 118 = 461, so 53,760 x 461 = 24,783,360 boards for the base game, or 18.44 billion boards with the expansions. (The precise number for the expansions is: 24 x 23 x 22 x 21 x 8 x 20 x 461 = 18,810,570,240.)

If you don’t want to include the original boards or the horizontal orientation, just use the N + W factor, 108 + 118 = 226: 53,760 x 226 = 12,149,760, or 40 million x 226 = about 9 billion.

Adding the Goals

At this point, it’s traditional to multiply by other variables in this extremely variable-setup game: n choose 3 goal cards, plus optional items like Crossroads' tasks and/or the mini-expansions Caves, Capitol, and the Island. My excuse for leaving this exercise to the reader is that not all goal cards are simple to use with a fifth board: Lords and Farmers are particularly challenging because they refer to quadrants of the board that are either obscured by the fifth board or unbalanced by it.

Naming the Layouts

You can describe a regular Kingdom Builder board as is done in the randomizer, by listing the four modules in a fixed order and also indicating whether each board is inverted. For example, the game on the back of the box could be described as Paddock Tavern Farm Oasis reading left to right across the two rows. Note that it could also be described as the upside-down version OasisInverted FarmInverted TavernInverted PaddockInverted.

To add a fifth board, list the new board/location type, whether the new board is inverted, the direction/orientation (N, NW, or W), and give coordinates for the placement of the first cell of the fifth board on the underlying grid. For coordinates I use letters (A–T) for the rows and numbers (1–20) for the nth hex of each row.

For example, the game pictured here could be described as:

Oracle OasisInverted Barn TavernInverted + Farm West L8

Note that it could also be described as:

Tavern BarnInverted Oasis OracleInverted + FarmInverted West I13

AI Again Ineffectual

I tried to use AI to generate the diagrams for this post, but it couldn’t get past the first step. I asked it to make an outline of a Kingdom Builder board segment, for which I already had the svg. It kept telling me it had done so, really I mean it this time, while handing me various rectangles with a perfunctory indent or two, or perhaps half a side’s worth of a zigzag. So I had to make them all myself like an animal. The AI mass delusion revolution continues to pass me by.

Once again, I haven’t blogged much but I have been up to stuff. I’ve been playing and programming some Decktet and other games at Abstract Play. Rather than devise a Piecepack game as I was thinking last time, I devised a domino game, Hegemino. It’s the easy-to-play Kingdomino-style game that Personimo never really was.

I haven’t blogged much but I have been up to some stuff. I’ve tweaked the draft rules to Darcana, my reimplementation of Dectana, a bit. I’ve been playing some Decktet and other games at Abstract Play, and I’m hoping to find or devise an abstract Piecepack game (besides Alien City).

Lately I’ve been playing a little Hive. I was inspired to implement a randomizer for the solo Hive puzzle Hive in Five, and of course to add my own algebraic notation for Hive to the long list of proposed Hive notation systems.

Maybe I’ll get to play one of the games at Balticon this weekend.