Introduction

Traversing a 3D array from the center outward is really a question about visit order. You still want to touch every cell exactly once, but the ordering should expand in layers instead of scanning from one corner. The best implementation depends on what "outward" means in your problem: cube shells, Manhattan distance, or Euclidean distance.

Define The Center First

For an array shaped like x_size * y_size * z_size, the simplest center is:

• 'x_size // 2'
• 'y_size // 2'
• 'z_size // 2'

That is straightforward when each dimension is odd. For even dimensions, there is no single geometric center cell. You must pick one of the middle cells or treat a small block of cells as the center region. Most algorithms still work if you choose one anchor cell consistently.

1def center_of(shape):
2    x_size, y_size, z_size = shape
3    return x_size // 2, y_size // 2, z_size // 2
4
5print(center_of((5, 5, 5)))
6print(center_of((6, 4, 8)))

Choosing the center explicitly matters because every later distance calculation depends on it.

Expand In Cubic Shells

If you want a traversal that feels like growing cubes around the center, iterate by increasing radius and visit coordinates whose maximum offset equals that radius. This gives a clean shell-by-shell order.

1def traverse_from_center(shape):
2    x_size, y_size, z_size = shape
3    cx, cy, cz = center_of(shape)
4    max_radius = max(cx, cy, cz, x_size - 1 - cx, y_size - 1 - cy, z_size - 1 - cz)
5
6    for radius in range(max_radius + 1):
7        for x in range(max(0, cx - radius), min(x_size, cx + radius + 1)):
8            for y in range(max(0, cy - radius), min(y_size, cy + radius + 1)):
9                for z in range(max(0, cz - radius), min(z_size, cz + radius + 1)):
10                    if max(abs(x - cx), abs(y - cy), abs(z - cz)) == radius:
11                        yield x, y, z
12
13coords = list(traverse_from_center((3, 3, 3)))
14print(coords[:10])
15print(len(coords))

This visits the center first, then every cell in the surrounding shell, then the next shell, until the array is exhausted. It is a good fit for simulations where influence spreads evenly along axes.

Distance-Based Ordering Is More General

Sometimes you want the visit order to reflect distance rather than shells. The easiest implementation is to enumerate every coordinate, compute a distance from the center, and sort.

1def sorted_by_manhattan_distance(shape):
2    x_size, y_size, z_size = shape
3    cx, cy, cz = center_of(shape)
4    cells = []
5
6    for x in range(x_size):
7        for y in range(y_size):
8            for z in range(z_size):
9                distance = abs(x - cx) + abs(y - cy) + abs(z - cz)
10                cells.append((distance, x, y, z))
11
12    cells.sort()
13    return [(x, y, z) for _, x, y, z in cells]
14
15print(sorted_by_manhattan_distance((3, 3, 3))[:10])

This is easy to reason about and easy to change. For example, you can switch from Manhattan distance to squared Euclidean distance if the application treats diagonals differently.

When BFS Makes Sense

If neighbors are defined as six-directional moves and you want a natural wave expanding outward, breadth-first search is another good model. It guarantees that cells are visited in increasing shortest-path distance from the center.

1from collections import deque
2
3def bfs_from_center(shape):
4    x_size, y_size, z_size = shape
5    start = center_of(shape)
6    queue = deque([start])
7    seen = {start}
8    directions = [
9        (1, 0, 0), (-1, 0, 0),
10        (0, 1, 0), (0, -1, 0),
11        (0, 0, 1), (0, 0, -1),
12    ]
13
14    while queue:
15        x, y, z = queue.popleft()
16        yield x, y, z
17
18        for dx, dy, dz in directions:
19            nx, ny, nz = x + dx, y + dy, z + dz
20            inside = 0 <= nx < x_size and 0 <= ny < y_size and 0 <= nz < z_size
21            if inside and (nx, ny, nz) not in seen:
22                seen.add((nx, ny, nz))
23                queue.append((nx, ny, nz))

This version is especially useful if the array represents a graph-like volume and adjacency matters more than geometric shell shape.

Complexity Tradeoffs

Any correct traversal must visit all cells, so the lower bound is proportional to the number of elements.

• Shell iteration is efficient and avoids sorting.
• Distance sorting is easy to customize but costs extra sorting work.
• BFS is clean when adjacency is the key concept, but it uses a queue and a visited set.

If the array is huge and performance matters, shell traversal is usually the best starting point because it stays close to pure index arithmetic.

Choosing The Right Definition Of "Outward"

This is where many solutions go wrong. "From the center" is underspecified unless you define the metric.

• Use shell traversal when you want cube-like expansion.
• Use BFS when movement rules are neighbor-based.
• Use sorted distance when you need a mathematically controlled order.

There is no universally correct traversal. There is only the order that matches the problem.

Common Pitfalls

• Assuming even-sized arrays have a single obvious center cell.
• Mixing different distance rules and expecting the same output order.
• Forgetting bounds checks when expanding around the center.
• Using sorting for very large arrays when a direct shell generator would be cheaper.
• Claiming a traversal is "spiral" in 3D without defining the exact neighbor or shell rule.

Summary

• Center-origin traversal is mainly about defining a consistent visit order.
• Pick the center explicitly, especially for even-sized dimensions.
• Shell traversal is a strong default when you want layers moving outward.
• BFS works well when adjacency defines distance.
• Distance sorting is flexible, but it adds sorting cost that may be unnecessary.