Menger Sponge - Formal Definition

Formal Definition

Formally, a Menger sponge can be defined as follows:

where M0 is the unit cube and

M_{n+1} := \left\{\begin{matrix} (x,y,z)\in\mathbb{R}^3: & \begin{matrix}\exists i,j,k\in\{0,1,2\}: (3x-i,3y-j,3z-k)\in M_n \\ \mbox{and at most one of }i,j,k\mbox{ is equal to 1}\end{matrix} \end{matrix}\right\}.

Read more about this topic:  Menger Sponge

Famous quotes containing the words formal and/or definition:

    “That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through formal instruction.”
    —Fred Rogers (20th century)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)