The Library of Babel, again

I'm still messing around with floor plans of The Library of Babel, and if you are asking yourself, "Why is he doing this?" then you and I have that in common.

An idea I had was that if the "gallery" and the "alcove" were both laid out on hexagons of the same size, it makes things pack better -- or at least makes the voids between galleries always be hexagonal, instead of other weird shapes, and that pleases me.

Here are two contradictory phrases that occur in the text "narrow corridor" and "spiral staircase". Spiral staircases are huge. They have to be. That's kind of their thing. So I tried to pack things down by putting the staircase off-center, walled off, and that works out ok, except it adds packing constraints: the stairway no longer lines up if the alcoves above or below this one has been rotated. So that severely limits the variety of layouts available.

And here are a bunch of them, all aimed in the same direction, because the two doors on each gallery are opposite each other:

So maybe that wasn't such a good idea. So I went back to a central, symmetrical spiral staircase. Here's what that looks like, with every room having a 101110 layout, creating six-gallery loops:

And here are three floors of it: to make this connect, each spiral staircase would span two floors:



If you want to play around with other layouts, here's my Sketchup file: babel.skp.

Update: Now with smaller and more treacherous stairs, narrower hallways, and hallways that turn!

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43 Responses:

  1. Matthew Exon says:

    I was reading this and laughing, "haha, jwz is such a maniac, but if I had spare time I could imagine myself worrying about this problem."

    And then...

    Suppose three kinds of hexagon, all packed: gallery, vestibule, and empty. On each plane of hexagons, alternate columns consist of just empty hexagons, or galleries and vestibules. Each plane of hexagons is rotated 60 degrees to the one below. Look down an air shaft in a gallery and you can see galleries every third floor, the others only being empty hexagonal spaces. You can walk infinitely far in one direction on one plane. To get left or right you need to go up a stairway, cross over, and go down again.

    That gives you a crystal structure. But the text says that each gallery is identical, it doesn't say each vestibule is identical. They could go straight, bend to the left, or bend to the right. This allows for a satisfying maze structure.

    Probably. Literally the only blank paper in the house is a post-it pad. And when I fired up inkscape just now I remembered that I don't, in fact, have time for this.

  2. I wish I could add something meaningful to this, but I can't. This is brilliant!

  3. db48x says:

    Why do your corridor vestibules only have a doorway on one end?

    • jmags says:

      To match the description. Only one side of the hexagon has a doorway, and one side is unspecified, so it could be empty.

    • jwz says:

      jmags's retcon is nice, but the real answer is, I forgot.

  4. UnlikelyLass says:

    I've been assuming the stair was going to be like the library one in My Fair Lady, which is mostly compact. Something like this:

    <img src="http://juniperbooks.com/wp-content/uploads/NoMad-Spiral-Staircase-in-Library.jpg

  5. David Hoover says:

    Here are two contradictory phrases that occur in the text "narrow corridor" and "spiral staircase". Spiral staircases are huge. They have to be. That's kind of their thing.

    That's not necessarily true; these folks will make ones that take up only 42x42 inches. You could make it an even tighter squeeze/steeper climb if you didn't care how annoying it is for the librarians to traverse.

    I think we can assume whatever could/would build a near-infinite library is apathetic if not downright malicious toward the pesky little insects inside, so hunching over to tip-toe up way too small for comfort seems plausible.

    • Epsz says:

      I love the idea of the hexagonal vestibles. I feel the stair steps could afford to be twice as narrow though. The few spiral stairs I've seen in Buenos Aires are uncomfortably small.

    • jwz says:

      Ok, stairs updated to be much more treacherous.

  6. Epsz says:

    What do you mean by " each spiral staircase would span two floors"? It's very clear in the text that the stairs seem infinitely high.

    • jwz says:

      I meant that you would have to go up two floors to reach a landing, rather than there being a stop on every floor.

  7. Philip Guenther says:

    It's unfortunate that the library is devoid of other resources, else we could posit the presence of infinitely descending slinkies on the stairs.

    (Challenge for the makers in our reality: a slinky that works on spiral stairs)

    • J. Peterson says:

      Rather than a special slinky, perhaps you could just slightly angle the steps so the slinky naturally travels in a spiral pattern.

  8. Matt Ridenour says:

    I eagerly await your rendering of The Aleph once the Library is finalized.

  9. When do the wind-up booze bots get unleashed to waddle through this?

    Why do none of the staircases end in a spike pit or sacrifice pool?

    What if the librarians have to pee?

  10. jwz says:

    I think someone should 3D print about 10,000 of these components and send them to me. In return I will take many photographs.

    That's what I think.

    • db48x says:

      Now what bothers me is that your gallery hexagons are missing a side, leaving beveled edges behind. Your original straight corridor-vestibule seems to fit into that gap, so perhaps it's not exactly a regular hexagon?

      I think it's a pretty good rendition though.

      An alternative though is that all of the hexagons are galleries, but the walls of the gallery are thick enough to accommodate the corridor-vestibules. That leaves plenty of room for a spiral staircase from the layer above to reach the layer below even when the galleries are at different rotations or have different layouts.

      • Matthew Exon says:

        Gah, now you actually made me fire up inkscape. If the staircases spiral around the entire hexagon, it all works quite neatly. It solves the problem of close-packing the hexagons while still being able to reach all neighbours. That is, to get from A1 to B1, you can go via A2 and B2.

        And it sits particularly neatly with "through here passes..."

        • db48x says:

          Ah, that's nice. I really like it!

          And if you prefer a more maze-like layout, steeper stairs will let you vary the layouts of the galleries without intersecting the stairs.

        • jwz says:

          That's a great idea!

        • db48x says:

          I too felt compelled to make a model:

          I've put the source up at https://github.com/db48x/library-of-babel/blob/master/library.scad should anyone want to check it out. It's a parametric model, you can adjust the interior radius of the galleries and the diameter of the stairwells, and everything else adjusts. The galleries are pretty large in this image, with walls 22ft long. The stairs are 5ft wide, so they'd be pretty comfortable.

          • jwz says:

            I like this too! This image does seem to violate the "galleries are identical" rule, though. I interpret that as "every gallery has the same layout of doors."

            • db48x says:

              Sure. I was mostly testing various combinations to make sure everything worked ok:


              gallery(hex2coord(0, 0, 1), hex_inner_radius, hex_height,
              stair_dia, layout=[1, 0, 1, 1, 0, 1]);
              gallery(hex2coord(1, 0, 1), hex_inner_radius, hex_height,
              stair_dia, layout=[1, 1, 0, 1, 0, 1]);
              // etc

              I think that one of the tiling rules that results in circuits is the way to go if you want to have all the galleries be strictly identical. I think a more irregular layout is fine though, even thematically appropriate. The books were maddening, so why shouldn't the layout be just as annoying?

        • gaplant says:

          I really like this one. We're looking at this from above and worrying about the packing and the voids, but Borges is writing more from the view of one of the people in there- he would see the inside shape of the rooms' walls and could guess at the shape of each repeating unit, but couldn't really be sure of anything that was completely surrounded by walls.
          I think it's fine to include voids, as long as they are completely walled off and wouldn't be noticeable. Likewise, having a stair inside the thick walls would be fine.
          These stairs really fulfill the "stair passing through" idea, but they worry me a little in that the phrase "spiral stair" usually means a fairly tight, sometimes self-supported stair, like a metal post stair or a narrow tower. In your diagram, the stair would be more like a stair that went up straight with a kink at the landing, and beyond which you can't see. The original text makes it sound like the viewer can see a long way down the stair, not just half the width of one hexagon...

  11. Dusk says:

    I expect a screensaver flythrough by the time you're done.

    • jwz says:

      The difficult part with that is, you kind of want a game engine for that sort of thing, and finding one of those that is both portable enough to run on X11, MacOS, iOS and Android, and small enough for shipping it with xscreensaver to be practical sounds a little unlikely.

      • db48x says:

        Could do a pseudo-3d version that shows overhead views like we've been making. The animation could be it building different layouts.

  12. dkeiwan says:

    I also like this, since this is pretty much the way the stairs and the general structure of the library works in my interactive 3D version: Library of Babel 3D

    If I made the walls thicker there would be no space between the hexagons, but you don't see that anyway when you walk around in the library.

    • dkeiwan says:

      This was meant to be a reply to @Matthew Exon's post (the wordpress comment system is pretty buggy for me)

    • Matthew Exon says:

      Very nice! And I'm sure all of us find it very validating that someone else spent more time thinking about this than we did.

  13. Ian says:

    My impression has always been that the hexagons posses walls of a nontrivial thickness. Instead of rendering the bookcases back-to-back, render them with (say) a 4' thick wall between them.

    Think of it like, there's a bunch of big hexagons, and then within those are the hexagonal rooms (and then the hexagonal airshafts within those). The wall thickness determines the length of the vestibule and the amount of room available for the bathroom and sleeping chamber (set into voids within the wall material). The width of the inner hexagon's doorway determines the width available for the spiral staircase.

  14. Chris Davies says:

    It seems to me you're too taken with symmetry. The image that springs to my mind is pairs of hexagonal galleries together, long walkways jutting from each on random, non-opposite sides. To me it sounds like the air shafts refer not to voids within the galleries, but the abyssal gloom between them ("vast" is the operative word.)

    I'd suggest that the galleries be open, not walled in. The ceiling visibly supported only by the ornate floor to ceiling bookshelves, and surrounded by a low railing, affording a dim view in the infinite space beyond, both above and below. The corridors too must be open, why else the mirror? The mirror obscures the view of the direction the path takes beyond, and even with careful counting of angles a traveller can't say if he has walked the circuit in his lifetime, or simply lost track as one day blends in to the next.

    • Leonardo Herrera says:

      I thought that too, but the shafts are actually in the middle of each hexagon, and those are the ones with the small handrails. About the hallway, the key word is the original spanish word "zaguán," which is unequivocally a closed (arched) hall.

  15. Michael says:

    I am not a Spanish speaker, but I tried looking carefully at the text (and applying some amount of French analogies and machine translation and dictionaries).

    The first mention of the spiral staircase uses the definite article, and maybe it should actually mean that there is only one spiral staircase.

    In that case «the spiral staircase» can legitimately be huge — even unimaginably so, with its elevation between two adjacent rooms almost negligible (A slight slant of stairs can allow it to still look like a staircase). It probably passes across the vestibule, the entire width of the vestibule being just a part of a single stair's width.

    I also second the observation that there is no mention of walls; and even the vestibules may turn out to be porches.

    Also, if you insist on having multiple spiral staircases, «there is a hallway… to the both sides of the hallway… the staircase passes there… in the hallway» should be a good enough excuse to assume that there is a vestibule with closets to the two sides of it being a part of the hexagon and a staircase passes not inside the vestibule but through the hexagon where the vestibule and the closets are attached.

    Vast ventilation shafts shouldn't be able to fit inside the small hexagonal galleries, and the mystery side should just have a low fence — as all the sides do — separating it from the actual ventilation shaft in the middle between the galleries.

  16. Steve says:

    Love your ideas and thoughts on this project. I was just thinking your original plan, from part 1, would work out perfectly if you made your hallways ~30" longer, just enough to put a door and a small landing into the unused interior void between hexagons then wrapped the spiral staircase around the interior of the "unused" shaft space. (Not the shafts bounded by the "low railing")

  17. Aubrey says:

    There are 2 great books for you: the first, is "The Unimaginable Mathematics of Borges Library of Babel", easy to find even in the web babel libraries of illegal texts. It's not super helpful on this problem though.
    The second is much more apt: "Come costruire la biblioteca di Babele", from Renato Giovannoli, which apparently solves the puzzle. Unfortunately is only in italian, but is definitely the best text I read on the specific issue of the architecture.

    • jwz says:

      Does it present any ideas that weren't discussed here? As I don't read Italian...

      • Aubrey says:

        I should compare your solutions side by side. I read the book last year so I don't remember the details, just the fact that, as of 2015, the author read all the existent bibliography on the issue and analyzed accurately the description in both Borges' versions. He then derived 4 axioms, and proposed different solutions. I will try to see if it's final one is the same as yours.

  18. jwz says:

    ‏@Beschizza: "already regret assigning the new Macbook Pro review to Borges":

  19. Tomislav Ostojich says:

    Hexagons tessellate 2d space, but not 3d volume (although the hexagonal prisms you devised do, at the cost of providing only boring dihedral group symmetry). Borges should have found a shape with interesting symmetry to tessellate 3d space and used that.

  20. Jon says:

    Love this. Still thinking about why you abandoned the beautiful circle of rooms around the railing, which I loved. Here is my contribution after a little research: While the English has "fecal necessities," this seems to be a weirdly specific and inaccurate translation of "necesidades finales." These words "literally" mean "final necessities," but seem to generally translated as "basic needs" -- so I now picture a pantry and kitchenette (mini-fridge). (How can you have fecal needs if there's no food?!) I think the best (Borgesian) translation would be "essential needs" as I think that "essential" needs (food, water, shelter) are more easily identifiable than "basic" needs and it plays off essential/essence/existence. My Spanish is poor, but perhaps a native speaker can tell me if "final necessities" contains a whisper of death -- perhaps this is where the librarians go to die. The idea of a moldy skeleton in every n-illionth closet seems like something that would make Borges chuckle. Keep in mind that we're thinking about translation issues, we really need speakers of Argentine Spanish as varieties of Spanish can vary tremendously!