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Tetrix


tetrix1
tetrix2
tetrix3

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

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

Let N_n be the number of tetrahedra, L_n the length of a side, and A_n the fractional volume of tetrahedra after the nth iteration. Then

N_n=4^n
(1)
L_n=(1/2)^n=2^(-n)
(2)
A_n=L_n^3N_n=(1/2)^n.
(3)

The capacity dimension is therefore

d_(cap)=-lim_(n->infty)(lnN_n)/(lnL_n)
(4)
=2,
(5)

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.

TetrixRotation

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

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