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A group of people have stumbled into one of those classic fantasy/horror houses where the usual rules of space don't seem to apply and is very confusing. Bits of the building appear to overlap with each over.

Eg. from the front room you walk forwards ten paces and enter the kitchen. Turn left 90 degrees and walk forward another ten paces. Turn left another 90 degrees, walk forward ten paces into the dining room. Finally turn left 90 degrees and walk another 10 paces forward and you end up in the study, not where you started.

You can back track and get back to the front room. The rooms don't move about, there is no way to detect the bits that overlap. The house is very large. Due to the way the rooms overlap there is more volume inside the building than contained by the exterior walls. All the rooms appear to be on the same plane. There is no climbing through holes in the ceiling, or rooms with right hand wall being the ceiling, from the point of view of the door.

Having checked out the house for monsters and finding none they decide to try and make a map to stop them selves getting lost in future. Of course simply drawing rooms on paper becomes a problem. They started to carefully fold and cut layers of paper so where the building overlaps you can fold sheets out of the way to find the layer you are on. Each time they find a new room they end up having to start from scratch to figure out how to get the map to fold correctly to incorporate the new room.

However this has created a very unwieldy map that is very hard to use. Is there a better way to create a map of an area that "overlaps" its self?

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    $\begingroup$ Your description and comments make it sound like you actually found this house somewhere and are trying to move into it... Most people would be asking questions about how it would affect culture/science/religion/etc but you're wondering how to get internet access in it :O Please invite me over sometime so I can see this place (once you finish the map of course) $\endgroup$ – thanby - reinstate Monica Jun 17 '15 at 19:16
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    $\begingroup$ @thanby ok so I might be a bit of a computer nerd. I can't go with out my internet access. :) $\endgroup$ – Wil Selwood Jun 17 '15 at 19:21
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Create a topological map where it is only the relationship between the rooms and corridors that is significant, not their physical position relative to each other.

So in other words if you imagine a triangle with a room at each corner, going from A -> B -> C and then continuing would normally get you back to A. However a topological map might show A, B, C and D on a line so you know that going from C will lead to D rather than back to A.

If journey distance is important then you can write the distance next to each line ("corridor") to show the distance between each room.

Think of something like the London Underground Map - This does not attempt to show the physical relationships between stations (other than broadly speaking), only the "lines" that go between each station showing which routes you can use to go from Start to Finish by visiting or changing trains at points in between.

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    $\begingroup$ Ha! You beat me to it; this is the answer I was going to propose. $\endgroup$ – LindaJeanne Jun 17 '15 at 15:51
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    $\begingroup$ This technique is popular in the interactive fiction (text adventure) community, where these maps often take the form of boxes with lines drawn between them. They even have specialized software for it. $\endgroup$ – Kevin Jun 17 '15 at 15:52
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    $\begingroup$ @Kevin that's a great extra. Would not have found that. Though now I have to figure out how they are going to network up the house. WiFi feels like a bad/really weird idea. $\endgroup$ – Wil Selwood Jun 17 '15 at 16:03
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    $\begingroup$ If the people have to physically walk the distance to get to a certain room, the wifi signals have to traverse the same distance. If you're hand-waving a mechanism to prevent them from tearing down walls, the walls should also block signals (as a rule of thumb, if a hammer has a tough time traveling through a surface, so does a wifi signal). To be safe they would basically just need an AP in each room and then bridge them all to "thread" a signal through the house. Hooray for metaphysical signal theory! $\endgroup$ – thanby - reinstate Monica Jun 17 '15 at 19:04
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    $\begingroup$ Example image from @Kevin's software. Shows how rooms are linked by arbitrary lines and how said lines can cross over each other. $\endgroup$ – Bobson Jun 17 '15 at 22:08
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I would map this in a more abstract way, essentially you are just looking for a graph of what connects to what.

Draw each room as a rectangle (or whatever shape it is) on the paper and map the contents as usual. For the connections between each room just draw a line connecting the doorway of one room to the corresponding doorway on another room.

The result would be like a flow chart of possible routes through the house.

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If you need a map thats actually "easy" to read and to use you need strings, duct tape and paper. Each piece of paper represents a room and every piece of string represents a door to another room. When you need to get from room A to B you stack all papers, flip until you reach the first room and pinch that paper. Now you flip until you reach the other room and pinch that too. Now you shake the map and look for the path with the least connections, always changing grabbing the next piece of paper as you change rooms.

If you have a computer at hand or a bigger table you could enumerate each room and label each door and use an algorithm to create a list of shortest paths from any given room to another following the door-labels.

As a third solution, to find your way around without a map afterwards you can take the Kreta labyrinth aproach and use paint or strings to lay a path from the very first room to points of interest. You can color code those and maybe attach directions markers so you know which way to go.

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    $\begingroup$ That's a really smart trick with the string, although I'd be worried about it breaking or tangling ¡ $\endgroup$ – Tim B Jun 17 '15 at 20:53
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    $\begingroup$ That's a genius idea for plotting a route. I have enough trouble with the wires for my laptop. My copy of this map would end up un-usable in about five minutes, which sucks because the shortest path route finding "algorithm" is fantastic. $\endgroup$ – Wil Selwood Jun 17 '15 at 21:25
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    $\begingroup$ @WilSelwood: Forget the scare quotes, BTW. This is a proper algorithm, and quite an efficient one at that. Pathfinding might be hard when all you have is a Turing Machine and an infinite roll of tape, but when you have a stack of paper and some strings, it becomes O(1)! $\endgroup$ – Jörg W Mittag Jun 17 '15 at 23:53
  • $\begingroup$ @JörgWMittag The cases where pathfinding is intractable are not going to be solved by this either. There are lots of pathfinding algorithms that solve simple cases fast: in complex cases, the tangle of rope won't unravel above, and you'll get junk. At best, the above gives you a finite amount of parallelism (bounded by how much paper and rope you can lift and shake at the same time). $\endgroup$ – Yakk Jun 19 '15 at 1:31
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This is very similar to other suggested answers but instead of clogging your map up with lines you can use symbols to identify connections.

Draw each room on a map as a square with indications where one room leads to another. On this indicator draw a symbol such as 'A', '1', or '@'. Then on the corresponding indicator on the second room draw the same symbol.

You see something like this on many maps to indicate which staircase from floor 1 ends up on which staircase on floor 2. All we are doing is changing it from indicating staircases to indicating room connections.

The benefit to this would be that is may be easier to read at certain junctions. If the imaginary house has multiple cloisters with 8+ exits out of it, the lines get tangled up very quickly. The downside is that it requires more room on the map and it can be difficult to find matching symbols if there are hundreds of rooms or messy handwriting.

Another option would be to label each room with a symbol, and each indicator with the symbol of the room it leads to. This may let you identify routes more quickly, 'I want to go to D, which rooms have a D on an indicator' instead of 'I want to go to the kitchen, which routes have a D or # or $ or 0 on an indicator' but it doesn't allow you to know which door in the target room you will come out of which can be an issue if the doors in the target room aren't all connected. Such as a broken catwalk connecting the doors to each other.

You could even create common pathways on a 'key' by simply writing out the path such as 'A@b178IU' to get from the entrance to the bathroom in 7 easy steps which is something you can't do with lines connecting each room.

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For the room layout you described, it's easy to draw in hyperbolic geometry:

enter image description here

The Mathematics of Old School D&D explains that concept well (the above image is theirs, and it's exactly the five rooms you described in the beginning of your example).

Slightly different kinds of oddities can be drawn by following the rules of other non-Euclidean geometries.

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If you have some time, you could build a simulation of the house (with a computer).

Real-world rules (such as the idea that buildings should be smaller on the inside) don't have to apply in a simulation.

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    $\begingroup$ Incidentally, geometries like this are a serious problem for most computer game engines, and many don't bother supporting it. In general this has to be implemented by treating each door as a portal. $\endgroup$ – Random832 Jun 18 '15 at 12:20
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    $\begingroup$ I once spent quite a while making the UT2004 engine cry trying to build an environment like this. Never found a way to get projectiles to go through the portals though. $\endgroup$ – Wil Selwood Jun 18 '15 at 17:09
  • $\begingroup$ Ironically, Bungie's Marathon game series (starting back in 1994) had some oddities in its engine specifically around this sort of issue. The infamous "figure eight" sample map to create with the editor (with no elevation change, but the "middle" two cross corridors never actually intersecting) was one such example. $\endgroup$ – Ghotir Nov 16 '16 at 22:42
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I'm going to shout in on the layered map approach. The adjacency graph already being covered. Only do nothing like this if you need local detail.

Draw the map on pieces of post-it notes. They can be easily peeled off. They are also transparent, so you can see under them when you press them down. Use this to mark the positions.

The alternative is to draw it as an isometric image with shifted layers.

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Draw the map in different colours, and view the "layers" of the house through differently coloured lenses to block out the other rooms. The lenses could be salvaged from a 3D movie, or something like sheets of cellophane laid over the map.

For an example - the back of the Declaration of Independence in National Treasure. Separate layers of information existing on the same piece of paper.

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