The paper presents a new interpolatory subdivision scheme for surface design, specifically designed for general triangulations of control points. The scheme introduces a tension parameter that allows for design flexibility, enabling the resulting limit surface to be $C^1$ within a specified range. The authors demonstrate the application of the butterfly scheme and the role of the tension parameter through several examples. The scheme is based on a local four-point interpolatory subdivision scheme for curves, extended to surfaces by transforming each triangular face of the control polyhedron into a patch of four triangular faces. The new points are inserted using an "eight-point rule" called the "butterfly scheme." The convergence analysis and smoothness properties of the limit surface are discussed, with the tension parameter serving as a tool to control the tightness of the surface towards the control points. The paper also includes a software package for implementing and testing the butterfly scheme, along with examples illustrating its performance and the effects of local and directional tension.The paper presents a new interpolatory subdivision scheme for surface design, specifically designed for general triangulations of control points. The scheme introduces a tension parameter that allows for design flexibility, enabling the resulting limit surface to be $C^1$ within a specified range. The authors demonstrate the application of the butterfly scheme and the role of the tension parameter through several examples. The scheme is based on a local four-point interpolatory subdivision scheme for curves, extended to surfaces by transforming each triangular face of the control polyhedron into a patch of four triangular faces. The new points are inserted using an "eight-point rule" called the "butterfly scheme." The convergence analysis and smoothness properties of the limit surface are discussed, with the tension parameter serving as a tool to control the tightness of the surface towards the control points. The paper also includes a software package for implementing and testing the butterfly scheme, along with examples illustrating its performance and the effects of local and directional tension.