(I tried using the Spyder IDE to write a more literal Python program. Output samples needed to be pasted in to the source, but I didn't particularly want many of Jupyters strengths, so tried Spyder together with http://hilite.me/)

**Best viewed on larger than 7" screens. **

# -*- coding: utf-8 -*- """ Created on Sat Jan 30 10:24:54 2021 @author: Paddy3118 """ #from conway_cubes_2_plus_dims import * #from conway_cubes_2_plus_dims import neighbours from itertools import product # %% #Full neighbourhood """ In my [previous post][1] I showed how the number of neighbours to a point in N-Dimensional space increases exponentially: [1]: https://paddy3118.blogspot.com/2021/01/conways-game-of-life-in-7-dimensions.html """ # %% ##Function def neighbours(point): return {n for n in product(*((n-1, n, n+1) for n in point)) if n != point} for dim in range(1,10): n = neighbours(tuple([0] * dim)) print(f"A {dim}-D grid point has {len(n):_} neighbours.") assert len(n) == 3**dim - 1 # %% ##Output """ A 1-D grid point has 2 neighbours. A 2-D grid point has 8 neighbours. A 3-D grid point has 26 neighbours. A 4-D grid point has 80 neighbours. A 5-D grid point has 242 neighbours. A 6-D grid point has 728 neighbours. A 7-D grid point has 2_186 neighbours. A 8-D grid point has 6_560 neighbours. A 9-D grid point has 19_682 neighbours. """ # %% ##Visuals """ Here we define neighbours as all cells, n whose indices differ by at most 1, i.e. by -1, 0, or +1 from the point X itself; apart from the point X. In 1-D: .nXn. In 2-D: ..... .nnn. .nXn. .nnn. ..... # 'Axial' neighbours There is another definition of neighbour that "cuts out the diagonals" in 2-D to form: ..... ..n.. .nXn. ..n.. ..... In 3-D this would add two extra cells, one directly above and one below X. In 1-D this would be the two cells one either side of X. I call this axial because if X is translated to the origin then axial neighbours are the 2N cells in which only one coordinate can differ by either +/- 1. # Let's calclate them. """ # %% #Axial neighbourhood from typing import Tuple, Set, Dict Point = Tuple[int, int, int] PointSet = Set[Point] def axial_neighbours(point: Point) -> PointSet: dim, pt = len(point), list(point) return {tuple(pt[:d] + [pt[d] + delta] + pt[d+1:]) for d in range(dim) for delta in (-1, +1)} print("\n" + '='*60 + '\n') for dim in range(1,10): n = axial_neighbours(tuple([0] * dim)) text = ':\n ' + str(sorted(n)) if dim <= 2 else '' print(f"A {dim}-D grid point has {len(n):_} axial neighbours{text}") assert len(n) == 2*dim # %% ##Output """ ============================================================ A 1-D grid point has 2 axial neighbours: [(-1,), (1,)] A 2-D grid point has 4 axial neighbours: [(-1, 0), (0, -1), (0, 1), (1, 0)] A 3-D grid point has 6 axial neighbours A 4-D grid point has 8 axial neighbours A 5-D grid point has 10 axial neighbours A 6-D grid point has 12 axial neighbours A 7-D grid point has 14 axial neighbours A 8-D grid point has 16 axial neighbours A 9-D grid point has 18 axial neighbours """ # %% #Another way of classifying... """ The 3-D axial neighbourhood around X can be described as: Think of X as a cube with six sides oriented with the axes of the 3 dimensions. The axial neighbourhood is six similar cubes with one face aligned with each of X's faces. A kind of 3-D cross of cubes. In 3-D, the "full" neighbourhood around point X describes a 3x3x3 units cube centered on X. In 3-D: Thinking of that 3x3x3 cube around X: What if we excluded the six corners of that cube? * Those excluded corner points have all their three coordinates different from those of X, i.e. if excluded e = Point(e_x, e_y, e_z) and X = Point(x_x, x_y, x_z): e_x != x_x and e_y != x_y and e_z != x_z Also: |e_x - x_x| == 1 and |e_y - x_y| == 1 and |e_z - x_z| == 1 * We _Keep_ all points in the cube that have 1 and 2 different coords to those of X Another definition of the _axial_ neighbourhood case is * We _Keep_ all points in the cube that have only 1 different coord to those of X This can be generalised to thinking in N dimensions of neighbourhood types keeping from 1 to _up to_ N differences in coordinates (DinC). """ def axial_neighbours(point: Point) -> PointSet: dim, pt = len(point), list(point) return {tuple(pt[:d] + [pt[d] + delta] + pt[d+1:]) for d in range(dim) for delta in (-1, +1)} #%% #Neighbourhood by Differences in Coordinates from collections import defaultdict from pprint import pformat from math import comb, factorial as fact def d_in_c_neighbourhood(dim: int) -> Dict[int, PointSet]: """ Split neighbourhood cube around origin point in dim-dimensions mapping count of coords not-equal to origin -> those neighbours """ origin = tuple([0] * dim) cube = {n for n in product(*((n-1, n, n+1) for n in origin)) if n != origin} d_in_c = defaultdict(set) for n in cube: d_in_c[dim - n.count(0)].add(n) return dict(d_in_c) def _check_counts(d: int, c: int, n: int) -> None: """ Some checks on counts Parameters ---------- d : int Dimensionality. c : int Count of neighbour coords UNequal to origin. n : int Number of neighbour points with exactly c off-origin coords. Returns ------- None. """ # # From OEIS # if c == 1: assert n == d* 2 # elif c == 2: assert n == d*(d-1)* 2 # elif c == 3: assert n == d*(d-1)*(d-2)* 4 / 3 # elif c == 4: assert n == d*(d-1)*(d-2)*(d-3)* 2 / 3 # elif c == 5: assert n == d*(d-1)*(d-2)*(d-3)*(d-4)* 4 / 15 # # Getting the hang of it # elif c == 6: assert n == d*(d-1)*(d-2)*(d-3)*(d-4)*(d-5)* 4 / 45 # Noticed some powers of 2 leading to # if c == 6: assert n == d*(d-1)*(d-2)*(d-3)*(d-4)*(d-5)* 2**6 / fact(6) # if c == 6: assert n == fact(d) / fact(c) / fact(d - c) * 2**c # Finally: assert n == fact(d) / fact(c) / fact(d - c) * 2**c print("\n" + '='*60 + '\n') for dim in range(1,10): d = d_in_c_neighbourhood(dim) tot = sum(len(n_set) for n_set in d.values()) print(f"A {dim}-D point has a total of {tot:_} full neighbours of which:") for diff_count in sorted(d): n_count = len(d[diff_count]) print(f" {n_count:4_} have exactly {diff_count:2} different coords.") _check_counts(dim, diff_count, n_count) # %% ##Output """ ============================================================ A 1-D point has a total of 2 full neighbours of which: 2 have exactly 1 different coords. A 2-D point has a total of 8 full neighbours of which: 4 have exactly 1 different coords. 4 have exactly 2 different coords. A 3-D point has a total of 26 full neighbours of which: 6 have exactly 1 different coords. 12 have exactly 2 different coords. 8 have exactly 3 different coords. A 4-D point has a total of 80 full neighbours of which: 8 have exactly 1 different coords. 24 have exactly 2 different coords. 32 have exactly 3 different coords. 16 have exactly 4 different coords. A 5-D point has a total of 242 full neighbours of which: 10 have exactly 1 different coords. 40 have exactly 2 different coords. 80 have exactly 3 different coords. 80 have exactly 4 different coords. 32 have exactly 5 different coords. A 6-D point has a total of 728 full neighbours of which: 12 have exactly 1 different coords. 60 have exactly 2 different coords. 160 have exactly 3 different coords. 240 have exactly 4 different coords. 192 have exactly 5 different coords. 64 have exactly 6 different coords. A 7-D point has a total of 2_186 full neighbours of which: 14 have exactly 1 different coords. 84 have exactly 2 different coords. 280 have exactly 3 different coords. 560 have exactly 4 different coords. 672 have exactly 5 different coords. 448 have exactly 6 different coords. 128 have exactly 7 different coords. A 8-D point has a total of 6_560 full neighbours of which: 16 have exactly 1 different coords. 112 have exactly 2 different coords. 448 have exactly 3 different coords. 1_120 have exactly 4 different coords. 1_792 have exactly 5 different coords. 1_792 have exactly 6 different coords. 1_024 have exactly 7 different coords. 256 have exactly 8 different coords. A 9-D point has a total of 19_682 full neighbours of which: 18 have exactly 1 different coords. 144 have exactly 2 different coords. 672 have exactly 3 different coords. 2_016 have exactly 4 different coords. 4_032 have exactly 5 different coords. 5_376 have exactly 6 different coords. 4_608 have exactly 7 different coords. 2_304 have exactly 8 different coords. 512 have exactly 9 different coords. """ # %% #Manhattan """ What, up until now, I have called the Differences in Coordinates is better known as the Manhattan distance, or [Taxicab geometry][2]. [2]: https://en.wikipedia.org/wiki/Taxicab_geometry """ # %% #Formula for Taxicab counts of cell neighbours """ Function _check_counts was originally set up as a crude check of taxicab distance of 1 as the formula was obvious. The formulea for taxcab distances 2, 3, and 4 were got from a search on [OEIS][3] The need for factorials is obvious. The count for a taxicab distance of N in N-D space is 2**N which allowed me to work out a final formula: If: d is the number of spatial dimiensions. t is the taxicab distance of a neighbouring point to the origin. Then: n, the count of neighbours with exactly this taxicab distance is n = f(d, t) = d! / t! / (d-t)! * 2**t We can use the assertion from the exercising of function neighbours() at the beginning to state that: sum(f(d, t)) for t from 0 .. d = g(d) = 3**d - 1 [3]: https://oeis.org/ """ # %% ##Taxicab visualisation """ Lets create a visualisation of the difference in coords for neighbours in <= 4-D. (The function is general, but I'm lost after 3-D)! The origin will show as zero, 0; and neighbours surround it as digits which are the taxicab distance from 0. """ def to_str(taxi: Dict[int, PointSet], indent: int=4) -> str: """ Parameters ---------- taxi : Dict[int, PointSet] Map taxicab distance from origin -> set off neighbours with that difference. indent : int indent output with spaces Returns ------- str Visusalisation of region. """ if not taxi: return '' ap = set() # all points neighbour2taxi = {} for taxi_distance, nbrs in taxi.items(): ap |= nbrs for n in nbrs: neighbour2taxi[n] = taxi_distance # Dimensionality dims = len(n) # Add in origin showing as 0 origin = tuple([0] * dims) ap.add(origin) neighbour2taxi[origin] = 0 # Plots neighbourhood of origin (plus extra space) space = 1 minmax = [range(-(1 + space), (1 + space) + 1) for dim in range(dims)] txt = [] indent_txt = ' ' * indent for plane_coords in product(*minmax[2:]): ptxt = ['\n' + indent_txt + ', '.join(f"dim{dim}={val}" for dim, val in enumerate(plane_coords, 2))] ptxt += [''.join((str(neighbour2taxi[tuple([x, y] + list(plane_coords))]) if tuple([x, y] + list(plane_coords)) in ap else '.') for x in minmax[0]) for y in minmax[1]] # Don't plot planes with no neighbours (due to extra space). if ''.join(ptxt).count('.') < (3 + space*2)**2: txt += ptxt return '\n'.join(indent_txt + t for t in txt) print("\n" + '='*60 + '\n') for dim in range(2,5): d = d_in_c_neighbourhood(dim) tot = sum(len(n_set) for n_set in d.values()) print(f"\nA {dim}-D point has a total of {tot:_} full neighbours of which:") for diff_count in sorted(d): n_count = len(d[diff_count]) print(f" {n_count:4_} have taxicab distance {diff_count:2} from the origin.") _check_counts(dim, diff_count, n_count) print(to_str(d)) # %% ##Output """ ============================================================ A 2-D point has a total of 8 full neighbours of which: 4 have taxicab distance 1 from the origin. 4 have taxicab distance 2 from the origin. ..... .212. .101. .212. ..... A 3-D point has a total of 26 full neighbours of which: 6 have taxicab distance 1 from the origin. 12 have taxicab distance 2 from the origin. 8 have taxicab distance 3 from the origin. dim2=-1 ..... .323. .212. .323. ..... dim2=0 ..... .212. .101. .212. ..... dim2=1 ..... .323. .212. .323. ..... A 4-D point has a total of 80 full neighbours of which: 8 have taxicab distance 1 from the origin. 24 have taxicab distance 2 from the origin. 32 have taxicab distance 3 from the origin. 16 have taxicab distance 4 from the origin. dim2=-1, dim3=-1 ..... .434. .323. .434. ..... dim2=-1, dim3=0 ..... .323. .212. .323. ..... dim2=-1, dim3=1 ..... .434. .323. .434. ..... dim2=0, dim3=-1 ..... .323. .212. .323. ..... dim2=0, dim3=0 ..... .212. .101. .212. ..... dim2=0, dim3=1 ..... .323. .212. .323. ..... dim2=1, dim3=-1 ..... .434. .323. .434. ..... dim2=1, dim3=0 ..... .323. .212. .323. ..... dim2=1, dim3=1 ..... .434. .323. .434. ..... """