import math

def rectangular_layout(num_elements, max_dimension):
    grid_size = int(math.sqrt(num_elements))
    spacing = max_dimension // grid_size
    return [
        {
            'position_x': (i % grid_size) * spacing,
            'position_z': (i // grid_size) * spacing
        }
        for i in range(num_elements)
    ]

def circular_layout(num_elements, radius):
    return [
        {
            'position_x': radius * math.cos(2 * math.pi * i / num_elements),
            'position_z': radius * math.sin(2 * math.pi * i / num_elements)
        }
        for i in range(num_elements)
    ]

def diagonal_layout(num_elements, max_position):
    return [
        {
            'position_x': i * max_position // num_elements,
            'position_z': i * max_position // num_elements
        }
        for i in range(num_elements)
    ]

def triangular_layout(num_elements):
    positions = []
    row_length = 1
    while num_elements > 0:
        for i in range(row_length):
            if num_elements <= 0:
                break
            positions.append({
                'position_x': i * 10 - (row_length - 1) * 5,  # Spread out each row symmetrically
                'position_z': row_length * 10
            })
            num_elements -= 1
        row_length += 1
    return positions