I recently moved to a city where the streets are arranged in a grid: they run East-West or North- South. To visit a friend South-East of my location, I can, say, head South on the street I'm on for 3 blocks and then turn East onto the street She lives on for 3 blocks. If I had travelled 1 unit of distance South and 1 unit East, the total distance I travelled would be 2 units. If there were streets running in diagonally in the South-East / North-West direction, I need have travelled only 2^0.5 units. Had I gone East from my starting point and turned South at the corner of the first block, then East at the end of that block, then South, then East and finally South, I would have arrived at my friend's location and covered a distance of 2 units. I have tried to include a couple of diagrams to illustrate. Now, I hope you'll agree that no matter how many times each block is subdivided with more East-West and North-South streets, I will always have to travel 2 units to my friend's location. This journey I'll refer to later as the 'travel path'. In the absence of diagonal streets and access to an aerial vehicle, only birds can take the more direct journey of 2^0.5 units. This journey I'll refer to later as the 'bird path'. So, now to my question. If 3 dimensional space was quantised, consisted of very small bits packed together and could not be traversed in an omnidirectional fashion but were more like houses with a limited number of entries and exits, what shape would they need to be such that bird path is 1 unit to any location from my starting location AND is always x units to that same location should I travel there? In other words, a radial distance of x 'in' space and a radial distance of 1 'out' of space (where x > 1)? I imagine that if the travel path was 1 unit and the bird path was 1 unit to any location from a given starting point, then the spacial tessellation of quantised space would have to permit omnidirectional travel. But if the travel path was always greater than the bird path and that travel path was the same in every direction as was the bird path, what kind of tessellation is required? A semi chaotic one? If it turned out that the travel path did vary with direction as compared to the bird path, how would one go about measuring this, assuming that difference was not too small to measure? I realise there are some big 'ifs' in this question, so any thoughts as to why it's not worth exploring are very welcome. Thanks for your response in advance.