The computation of the distance between any two points of the Sierpinski gasket with respect to the intrinsic metric has already been investigated by several authors. However, to the best of our knowledge, in the literature there is not an explicit formula obtained by using the code set of the Sierpinski gasket. In this paper, we obtain an explicit formula for the intrinsic metric on the Sierpinski gasket via the code representations of its points. We finally give an important geometrical property of the Sierpinski gasket with regard to the intrinsic metric by using its code representation.

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V 0 = {p 1 , p 2 , p 3 } be the set of vertices of SG and u i (x) = 1 2 (x + p i ) the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations u w = u w 1 u w 2 · · · u w n for any sequence w = (w 1 , w 2 , . . . , w n ) ∈ {1, 2, 3} n . The union of the images of V 0 under these iterations is the set of nth stage vertices V n of SG. Let F : V n → R be any function. Given any numbers α w (w ∈ {1, 2, 3} n ) with 0 < |α w | < 1, there exists a unique continuous extension f : SG → R of F , such thatfor x ∈ SG, where h w are harmonic functions on SG for w ∈ {1, 2, 3} n . Interpreting the harmonic functions as the "degree 1 polynomials" on SG is thus a self-similar interpolation obtained for any start function F : V n → R.

In this paper, we examine the number of geodesics between two points of the Sierpinski Gasket ([Formula: see text]) via code representations of the points and as a main result we show that the maximum number of geodesics between different two points with respect to the intrinsic metric on [Formula: see text] is five.