Round Surface: 3D Construction with Equilateral and Similar Triangles

This GeoGebra example demonstrates a smooth three-dimensional surface generated by the Surface command, whose shape is controlled by multiple geometric objects including equilateral triangles, similar triangles, and parameters such as a, f, U, V, n, g, s1, s2, s3. The core mathematical concepts involve parametric surface representation, properties of equilateral and similar triangles, the Pythagorean theorem in spatial distance calculations, as well as set theory and hypergeometric distribution from probability and statistics. The construction uses the Surface command to map a 2D parameter domain (defined by U and V) into 3D space, with sliders or variables like n and g adjusting the mesh resolution and shape, while auxiliary objects s1, s2, s3 help build the triangular framework. This example is suitable for high school or university students learning solid geometry, parametric surfaces, and advanced GeoGebra modeling, and it also offers a bridge between geometry and probabilistic concepts.