Menger Sponge — Definition, Formula & Examples

A Menger Sponge is a three-dimensional fractal created by repeatedly subdividing a cube into 27 smaller cubes and removing the center cube of each face plus the very center cube, then repeating this process infinitely on every remaining small cube.

The Menger Sponge is a self-similar fractal obtained by starting with a unit cube, dividing it into a $3 \times 3 \times 3$ grid of 27 equal sub-cubes, removing the 7 sub-cubes at the center of each face and the center of the cube itself (retaining 20 sub-cubes), and iterating this procedure on each remaining sub-cube ad infinitum. Its Hausdorff dimension is $\frac{\ln 20}{\ln 3} \approx 2.727$ .

Key Formula

V_n = \left(\frac{20}{27}\right)^n

Where:

$V_n$ = Volume of the Menger Sponge at iteration n (starting from a unit cube)
$n$ = Iteration number (0, 1, 2, ...)

How It Works

Start with a solid cube (iteration 0). Divide it into a

3 \times 3 \times 3

grid of 27 identical smaller cubes. Remove the center cube from each of the 6 faces (6 cubes) and the single cube at the very center of the grid (1 cube), leaving 20 cubes. This is iteration 1. For iteration 2, repeat the same subdivision-and-removal process on each of the 20 remaining cubes. At each iteration

n

, the sponge contains

20^n

tiny cubes, its total volume shrinks toward zero, and its surface area grows toward infinity.

Worked Example

Problem: Find the volume and the number of sub-cubes after iteration 3 of a Menger Sponge that starts as a unit cube.

Number of sub-cubes: Each iteration multiplies the number of cubes by 20.

N_3 = 20^3 = 8{,}000

Volume at iteration 3: Apply the volume formula with n = 3.

V_3 = \left(\frac{20}{27}\right)^3 = \frac{8{,}000}{19{,}683} \approx 0.4064

Answer: After iteration 3, the Menger Sponge consists of 8,000 tiny cubes with a combined volume of approximately 0.4064 cubic units — already less than half the original cube.

Why It Matters

The Menger Sponge is a key example in fractal geometry courses for illustrating how an object can have infinite surface area yet zero volume. It also appears in computer science (recursive algorithms) and 3D printing projects that test precision manufacturing.

Common Mistakes

Mistake: Removing only the center cube at each step (1 out of 27) instead of all 7.

Correction: You must remove the center cube of each of the 6 faces AND the single cube at the very center of the grid, totaling 7 removals per subdivision. This leaves 20 cubes, not 26.