The tetrix is the three-dimensional analog of the Sierpiński sieve illustrated above, also called the Sierpiński sponge or Sierpiński
tetrahedron.

The th iteration of the tetrix is implemented
in the Wolfram Language as SierpinskiMesh [n ,
3].

Let be the number of tetrahedra, the length of a side, and the fractional volume of tetrahedra
after the th iteration. Then

The capacity dimension is therefore

so the tetrix has an integer capacity dimension (which is one less than the dimension
of the three-dimensional tetrahedra from which it
is built), despite the fact that it is a fractal .

The following illustrations demonstrate how the dimension of the tetrix can be the same as that of the plane by showing three stages of the
rotation of a tetrix, viewed along one of its edges. In the last frame, the tetrix
"looks" like the two-dimensional plane .

See also Chaos Game ,

Menger
Sponge ,

Sierpiński Sieve
Explore with Wolfram|Alpha
References Allanson, B. "The Fractal Tetrahedron" java applet. http://members.ozemail.com.au/~llan/Fractet.html . Borwein,
J. and Bailey, D. "Pascal's Triangle." §2.1 in Mathematics
by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A
K Peters, pp. 46-47, 2003. Dickau, R. M. "Sierpinski Tetrahedron."
http://mathforum.org/advanced/robertd/tetrahedron.html . Eppstein,
D. "Sierpinski Tetrahedra and Other Fractal Sponges." http://www.ics.uci.edu/~eppstein/junkyard/sierpinski.html . Kabai,
S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica.
Püspökladány, Hungary: Uniconstant, pp. 159-160, 2002. Kosmulski,
M. "Modulus Origami--Fractals, IFS." http://hektor.umcs.lublin.pl/~mikosmul/origami/fractals.html . Mandelbrot,
B. B. The
Fractal Geometry of Nature. New York: W. H. Freeman, pp. 142-143,
1983. Referenced on Wolfram|Alpha Tetrix
Cite this as:
Weisstein, Eric W. "Tetrix." From MathWorld --A
Wolfram Web Resource. https://mathworld.wolfram.com/Tetrix.html

Subject classifications