i.e. 2 copies for 1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For the Sierpiński triangle, doubling its side creates 3 copies of itself. Thus the Sierpiński triangle has Hausdorff dimension , which follows from solving for . The area of a Sierpiński triangle is zero (in Lebesgue measure). The area remaining after each iteration is of the area from the previous iteration, and an infinite number of iterations results in an area approaching zero. The points of a Sierpinski triangle have a simple characterization in barycentric coordinates. If a point has barycentric coordinates , expressed as binary numerals, then the point is in Sierpiński's triangle if and only if for Generalization to other moduli A generalization of the Sierpiński triangle can also be generated using Pascal's triangle if a different modulus is used. Iteration can be generated by taking a Pascal's triangle with rows and coloring numbers by their value modulo . As approaches infinity, a fractal is generated. The same fractal can be achieved by dividing a triangle into a tessellation of similar triangles and removing the triangles that are upside-down from the original, then iterating this step with each smaller triangle. Conversely, the fractal can also be generated by beginning with a triangle and duplicating it and arranging of the new figures in the same orientation into a larger similar triangle with the vertices of the previous figures touching, then iterating that step. Analogues in higher dimensions The Sierpinski tetrahedron or tetrix is the three-dimensional analogue of the Sierpiński triangle, formed by repeatedly shrinking a regular tetrahedron to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process. A tetrix constructed from an initial tetrahedron of side-length has the property that the total surface area remains constant with each iteration. The initial surface area of the (iteration-0) tetrahedron of side-length is . The next iteration consists of four copies with side length , so the total area is again. Subsequent iterations again quadruple the number of copies and halve the side length, preserving the overall area. Meanwhile, the volume of the construction is halved at every step and therefore approaches zero. The limit of this process has neither volume nor surface but, like the Sierpinski gasket, is an intricately connected curve. Its Hausdorff dimension is ; here "log" denotes the natural logarithm,