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A Topological Type Problem—I Think…what Is It


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#1 SaxonViolence

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Posted 02 June 2019 - 12:06 AM

Friends,

 

Imagine a round room. The room has 3 one-way entrances and 3 one-way exits. 

 

Pick an exit at random and it will led you to another round room.This room also has 3 entrances—including the one that you just came through—and 3 exits.

 

Can I hook a number of round rooms together in such a way that they form a closed loop and my path will inevitably lead me back to the original room?

 

Connections can be a wooly-bear worm—so assume that the doorways teleport you. or that some connecting corridors are much longer than others...

 

Or quit being so damned literal and just think of it as an abstract topological space.

 

I assume that if it is possible to turn the rooms into one interconnected net, there should be a minimum number of rooms required before the rooms can become an interlinked network...

 

What is the minimum number of rooms and what is the minimum number of rooms that I must transverse to return to my starting point?

 

Can I make larger networks larger than the minimum number?

 

Just exactly what kinds of equations or visualizations do I need to handle this problem?

 

Thanks.

 

…..Saxon Violence

 

BIG PS:

 

Can the equation be expanded to include a system of round rooms with 5 entrances and 5 exits each? How about 7 entrances and exits?


Edited by SaxonViolence, 02 June 2019 - 12:12 AM.


#2 rhertz

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Posted 02 June 2019 - 02:48 PM

Friends,

 

Imagine a round room. The room has 3 one-way entrances and 3 one-way exits. 

 

Pick an exit at random and it will led you to another round room.This room also has 3 entrances—including the one that you just came through—and 3 exits.

 

Can I hook a number of round rooms together in such a way that they form a closed loop and my path will inevitably lead me back to the original room?

 

Connections can be a wooly-bear worm—so assume that the doorways teleport you. or that some connecting corridors are much longer than others...

 

Or quit being so damned literal and just think of it as an abstract topological space.

 

I assume that if it is possible to turn the rooms into one interconnected net, there should be a minimum number of rooms required before the rooms can become an interlinked network...

 

What is the minimum number of rooms and what is the minimum number of rooms that I must transverse to return to my starting point?

 

Can I make larger networks larger than the minimum number?

 

Just exactly what kinds of equations or visualizations do I need to handle this problem?

 

Thanks.

 

…..Saxon Violence

 

BIG PS:

 

Can the equation be expanded to include a system of round rooms with 5 entrances and 5 exits each? How about 7 entrances and exits?

 

As far as I can imagine it, the simplest network that unites K rooms (stations) is a closed loop six path's (corridors) ring. Three of them are for EXIT only and three of them are for ENTRANCE only. The amount of K rooms (stations) is undefined and limited by propagation delays (walking time) and queuing at entrances (N persons select the same room).

 

Edited post:

 

To preserve the original thought, I left the initial answer as it was.

 

To be honest, I didn't even read about the rooms to being "round". This subject is familiar for me in the topic of "general networking".

 

Logically speaking, it doesn't matter the shape of the room. All of them could share an apex, like portions of an ideal pizza, or being 2D triangles, whatever.

 

Being off-line, I came with a general solution, which is simpler.

 

If you connect three one-way outputs from one "room" to the three one way inputs of the next "room" (station), and do the same for the second room to the first one, you only need three uni-directional paths. The topology of the network is still a ring.

 

The minimum number of "rooms" (stations) is TWO.

For the maximum number of "rooms", the limits of the original post still apply. So, theoretically, this number could approach infinity.

 

In a 2D world, the shape of the rooms is arbitrary. Also the amount of rooms.

 

In the real world, you have to account for traffic and delays (traffic: persons circulating per path per unit time).


Edited by rhertz, 02 June 2019 - 05:47 PM.

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#3 rhertz

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Posted 03 June 2019 - 09:10 PM

I have one simple topological problem for you (or whoever dare to solve it).

It's a very old theorem which, supposedly, has no solution:

 

1) Assume that you live in a 2D world. So, only length and depth are observable.

 

2) Draw six boxes in two rows of three boxes each.

 

3) Assume that one row represent three homes, and the other row represent three public service companies (energy, water, gas)

 

4) The challenge is to connect, point to point, each company with each house.

   

     The interconnecting paths can adopt any form: a rect or a curve.

 

     But it's forbidden that two lines cross at any place.

 

Does this simple problem has a solution?

 

Make as many draws as you want but remember: no lines crossing is allowed.


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