# A thought experiment
~ 2019-03-14T02:37:25+00:00 ~
I recently thought of an interesting question related to mathematics. At first I was just thinking different thoughts and stumbled onto this semi-philosophical / half-random question.
The problem I was thinking about was a scenario – not a realistic one hence why it is more of a mathematical question than anything else. The problem goes like this:
Imagine a warehouse which is infinitely long (d=inf.) and only a finite width (w) and height (h). Inside this warehouse is an infinite amount of boxes each half a unit cubed (l^3; l=0.5). The door to this warehouse is of finite width (w\_d) and height(h\_d) also and it can only support a certain number of carriers(w\_c) across it at any given time (n) and it requires a certain amount of time(t\_xn : i=0..n) to reach the boxes depending on how many have been removed. The boxes though infinite do not magically creep closer each time one is removed – instead they just stay in their original placements into the infinite warehouse away from the door. There are an infinite amount of carriers available (c\_i) who can go for boxes, but only at a constant pace(R) and they can only carry one box at a time. What would the output (x) from such a warehouse be over time?
It is a time series problem that rings of convolution with what appears to be some form of pattern/algorithm (f(x,R,…)) for the order in which boxes are removed. Each order could affect the output making this question open to a few interpretations. For example, if the boxes are removed with the rule of closest first then it will inevitably end up looking like like some form of semi-sphere -> bullet-esque shape – this is assuming you can coordinate everyone to know which order they are in and be aware of how the system is laid out so they can directly approach their box. However, say instead you chose to implement a different pattern such as doing a row by row approach, or a column by column approach.
Suddenly I feel like this is less of a pure math problem and more a complexity problem at this point. For example, what if you implement a more complex algorithm for collecting boxes by building a supply chain of carriers who instead of just picking up a box and delivering it themselves instead pass it on and form a line of successive carriers but only add as many as the carrier before it can handle. Additionally you could have a form of a networked carrier system which perhaps readjusts as it proceeds in some form of organic fractal like pattern to optimize the shortest paths between the boxes and the carriers themselves and to fill the volume as efficiently as possible (ideal).
This begs the question – is this a solved problem that can result in an almost never ending supply of boxes or is this a problem where the inevitable rising costs of the boxes and their decreased output make it prohibitive and costly resulting in it being an inevitable dud? What is the solution?
I think the solution to this must result from finding the key algorithm for the warehouse shape (in this case a rectangle on infinite length). That algorithmic function will power the complexity problem and will provide a means of calculating the output of the warehouse as well it derivatives like depletion in units and volume. So does that suggest that different shapes require different algorithms – is there a shape that has no solution? Or, do all shapes have the same solution? Is there more than one solutions for one or all or many shapes?
Maybe its weird to think these random questions – I always enjoy a good thought experiment though and I feel like this one might be one I ponder on occasionally for a while. Feel free to comment if you see something – is this already a question, is it solved already, is it trivial and I am just over thinking it – let me know.