2891: Log Cabin

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Log Cabin
I'm sure the building inspectors will approve my design once they finally manage to escape.
Title text: I'm sure the building inspectors will approve my design once they finally manage to escape.


Ambox notice.png This explanation may be incomplete or incorrect: Created by a TRAPPED BUILDING INSPECTOR APPROXIMATELY 1.618 METERS TALL- Please change this comment when editing this page. Do NOT delete this tag too soon.
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Golden spiral (approximately) overlaid on the floor plan

This is a comic featuring a floor plan, presumably of a log cabin, and a pun on the word "log". The odd part about it is the right half, which appears to be infinitely recursive copies of the building, a self-similar fractal. The house as a whole represents a golden rectangle with a side ratio of the golden ratio (phi = 0.5 + (1.25 ^ 0.5) ≈ 1.6180339887...), with successively smaller living areas (further subdivided into rooms, to a common but shrunken and rotated plan) being square adjoined by a golden-rectangle of all smaller living areas in the manner of the areas defined by the classic Golden spiral diagram. The joke is that Randall is intentionally conflating the word "log" in the common phrase "log cabin," where "log" refers to the wood the cabin is made of, with a logarithmic spiral, as "log" is shorthand for "logarithm".

Every square subunit of the premises consists of a main combined kitchen/lounge area, with an adjoining shower-equipped bathroom, and a hallway leading to a closet and two bedrooms (a double-bed one with en-suite bath-equipped bathroom, and another with a single bed; both having their own walk-in closet space). Between the kitchen and the dining table is the open doorway that serves as the entry into the next inwards level of the floorplan.

A visitor can also walk in a spiral by using the open doorways in the building plan.

In the title text, Randall says that he is confident that building inspectors will approve the design of his log cabin, assuming they can escape. As one moves deeper into the infinite spiral of architecture, the entire log cabin seems to be a denser labyrinth of rooms and hallways.

It is unknown how the inspectors lost in the inner rooms would shrink in the same ratio, and would only have to head out through the last doorway they walked in through (or two, if they're currently in the en-suite) and then exit each 'main' area in turn until they exited the building itself. If they have any trouble at all (other than rescaling themselves), it would be that there is always a further inward area that they might consider needs exploring to fulfil their inspection routine. This construction could imply folded spacetime. To perform a correct inspection, the inspectors might need sufficient relation that inner rooms are identical to the outer rooms, and to solve algebraic equations for various parameters.

In any case, though, they would not need to physically enter the floorplan in order to approve the design, so any risk of getting 'lost' would come solely from a psychological inability to break off from the infinite recursion. Approval of construction would be a different matter. Whether the floorplan could genuinely be infinitely recursive would depend on how Randall has presented it. Typically, these would be submitted as static images (as it is in the comic), in which case at some point there would be insufficient resolution to continue the recursion, effectively ending in a solid pillar. Randall, however, perhaps channeling his inner Black Hat, may have used a program that dynamically generates further levels as you zoom in to the image.

It could have been that the repetition continued outside of what is drawn here, continuing to grow larger as it moves further out, except that only the top-level external entry has a door: the entries to each lower level only have open doorways marked with no door, and except for the clear marking of external windows on the first four iterations, with no internal ones (and therefore no equivalent windows on the subsequent iterations) precludes that possibility. In fact, the first iteration has windows on the 'southern' edge for the single bedroom and seating area, for which there are no equivalents on any of the other iterations.

Log Cabin is also a classic quilting block, which starts with a tiny square and spirals outwards.


Ambox notice.png This transcript is incomplete. Please help editing it! Thanks.
[A drawing of the top view of a log cabin, which includes two beds, a kitchen, a bathroom and furniture. A smaller version of the log cabin is connected to the main building. An even smaller version is connected to the smaller version, and so on, forming a golden rectangle.]
[Caption below the panel]:
Log cabin

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"The odd part about it is the bottom right corner, which appears to be infinitely recursive copies.."

The whole right side is the left side, shrunk and recursed. Each iteration rotated 90 degrees. The 'shrink' is about 1.616 by my squint, a lot like a "Golden Ratio" LOGarithmic spiral, as NickM says. PRR (talk) 19:49, 7 February 2024 (UTC)

This is a LOGarithmic spiral 19:52, 7 February 2024 (UTC)NickM
It is precisely the golden ratio, assuming the left side is a square Terdragontra (talk) 22:09, 7 February 2024 (UTC)`
Note how the spiral cuts the smaller bedrooms wardrobe in half, intersects the two doorframes of the rooms leading off the master bedroom, the toilet and the sink. The majesty of the spiral! 🌀 11:20, 9 February 2024 (UTC)

Assuming a standard 36" wide front door, then the next "front" door would be 22.27 inches, then 13.78 inches, then 8.53 inches, at which point I doubt the inspector could squeeze through it, though I guess they could still take a peek inside the next recursion. 20:57, 7 February 2024 (UTC)

That's enough for the inspector to verify that the plans are *not* up to code; you can't get a wheelchair into the right-hand side of the house. 11:16, 9 February 2024 (UTC)

Infinite bedrooms, infinite baths, close to schools and shopping. 21:00, 7 February 2024 (UTC)

Government: Your property tax comes up to infinite dollars. -- 21:45, 7 February 2024 (UTC)
In some jurisdictions bedrooms need to have at least one externally facing window. Under this rule there are infinite rooms with beds, but only 8 bedrooms. Erin Anne (talk) 17:07, 8 February 2024 (UTC)
Only 7 bedrooms. 22:16, 15 February 2024 (UTC)
Looks like eight to me. In the first 'set', both 'main' and 'minor' rooms have windows on west wall (plus north/south, respectively). In second set, that west wall is rotated to north (plus north now to east, while south becomes wall-only west). Third set it be becomes an east wall (plus south for Main). Fourth set rotates the shared external wall to south (the main's other wall is now also internal). For the fifth-plus, that w>n>e>s wall is west again, but internalised. So four pairs of bedrooms all have (at least) one wall with windows, then no more. 02:41, 16 February 2024 (UTC)

Interesting comparison with the archetypal "labyrinth". It's actually a fractal version that only avoids being unicursal-with-no-dead-ends due to the off-living-room private spaces being quite trivial offshoots. Which arguably makes it fairly classical in nature. 21:18, 7 February 2024 (UTC)

Should there be a mention that due to the limitations of the image format, it only actually achieves eight iterations? Which makes sense given that construction materials also have limits, and is still enough that the inspectors might be a bit confused if they don't pay close enough attention. 23:21, 7 February 2024 (UTC)

Write it how you want, I just wanted to avoid calling a room with no bath (but a shower) as a "bathroom", especially when I was mentioning a 'bathroom' with an actual bath in it so soon after. Not that there's a completely unambiguous term for the room with the toilet/lavatory/whatever in it. (For reference, for me it's "the toilet", despite that also being the porceline item itself, and even that is derived from a hairdressing cloth, through a string of euphemisms. But knew that wouldn't be accepted by the wider readership.) 01:49, 8 February 2024 (UTC)

In the USA, any room with a toilet and sink is typically called a "bathroom" whether or not there is a literal bath within. 09:33, 8 February 2024 (UTC)
Don't think it's particularly a US/UK thing - it's pretty commonly called a 'bathroom' in the UK too. I think the point of the editor above was the potential for confusion between the two 'bathrooms', and how to avoid it. 09:44, 8 February 2024 (UTC)
It's my experience in the US that real-estate parlance counts a full bathroom as toilet, sink, and tub / shower. It counts a half-bathroom as a toilet and sink only. A house with 1 full bath and two half-baths would generically be considered 3 bathrooms but not listed that way. I don't know what they would call a standalone tub or shower, if any such thing is ever constructed unless it is adjacent to an outdoor swimming pool. Rtanenbaum (talk) 01:37, 9 February 2024 (UTC)
Could you not just call it lavatory? 00:07, 12 February 2024 (UTC)
Using a language that allows to say "S***house" in a nice way (praised be the diminutive!) helps in such cases...Tier666 (talk) 10:15, 8 February 2024 (UTC)
There is a truly astounding range of terms, in English alone, (and I can think of several not included there, even discounting the rather localised overly-vulgar or overly-polite ones). 11:38, 8 February 2024 (UTC)

Are there any non-Euclidean geometries in which you could fit this house without having to shrink the rooms or the people? 16:42, 8 February 2024 (UTC)

The logarithmic scaling should be compatible with hyperbolic geometry. However, the distance metric changes continuously in such a geometry. Here you'd probably need something like a discrete mapping that maps the 1st, 2nd, 3rd, etc. iteration of the outer cabin layout to its proper scaling and rotation. (I vaguely know the concepts but can't do it properly) Transgalactic (talk) 09:21, 9 February 2024 (UTC)
If we're committed to discontinuities (at least in undifferentiatable senses), we can just treat the floorplans as separately just traditional Euclidean but the main doors between 'levels' as a wormhole/portal. Moreover, one which could satisfy the rescaling requirement, (the 'door frame' outward is at the same metric of width/height as the inward one; as well as the inward one of one layer out, which is not experienced as a funnel). In a connected-voxel manner (as one might render it in virtual terms), every level of square-property has the exact same internal consistency.
Save for the windows, which are additional portals limited only to the primary property (present on three walls, all but the 'inward exit' direction), secondary/tertiary ones (two walls, having both lost the 'entry wall' externality that faces their predecessor property) and quaternary (one windowed wall, now enclosed by the prime-property). All these windows similarly portal to the zeroth-level exterior (irregularly spaced, but 'proper-sized'), and of course whilst the 'obscured walls' (including all from quinary-level and beyond) could be not denied windows by being portaled to some surrogate external wall (infinite, necessarily!), in this example they are not.
This renders the plans shown as 'wrong' (door widths are scaled differently at both inward and outward locations, with a single square sub-property, and such windows as exist from sub-levels (all nine window-outers on the external south-wall should be the same size as the nine inner windows from the primary square's outward-walls, not just the two and the door which are the direct identical ones).
Thus rescaling issues as one traverses doors (or windows) go away. It does leave dissimilar distances inside and out (speed-of-light communications out of a Level 4 window and into a Level 1 one could be faster than any 'direct' through-wall one, if allowed, and of course diffracted propogation only via door-portals would be slower), but we're already talking of wormholing between/across domains, so our only real issue is how/if arbitrary knocking-through of holes through walls can be allowed (once the building is somehow BS Johnsoned into existence, perhaps at some point necessitating at least a countable infinity of builders/decorators). 10:45, 9 February 2024 (UTC)

One way to build this would be with robotics that alter the structure as the visitor moves, like a holodeck. This could give the perception of the visitor shrinking forever, and the robotics wouldn’t let them leave until they rewalked their entry path. Another way would be to just make the rooms get too tiny to enter and hide the missing ones around a corner. 16:58, 8 February 2024 (UTC)

I'm sure it could be done with the Gallifreyan technology that produced the TARDIS. Barmar (talk) 17:39, 8 February 2024 (UTC)
Indeed: [1] -- Jgharston (talk) 12:45, 13 February 2024 (please sign your comments with ~~~~)
That sort of thing happens in nuWho era, too! 13:08, 13 February 2024 (UTC)
maybe somebody will 3d print a tiny one some day. i websearched but didn’t even get any hits for “robotic maze moving walls” ! this is surprising to me in this decade.

Has anyone else thought of a cabin log? The horror story kind, with a spiraling captain? Transgalactic (talk) 09:21, 9 February 2024 (UTC)