How Many Hidden Dots? Counting Hidden Vertices via Sampling

This example demonstrates how to determine the number of “hidden” dots within a constructed planar configuration built from polygons (including rhombus-like parts and internal layout). Learners can observe which candidate points become hidden and then compute the total count using a reproducible computational geometry workflow. The key mathematical ideas are plane geometry (polygon structure and embedded regions), discrete sampling along geometric objects, and distance-based decisions using Length; it also connects naturally to arithmetic operations and term-like bookkeeping for counting. Construction highlights in GeoGebra: Flatten is used to turn nested generated data (points/objects) into a single list; Sample performs discrete sampling on selected geometric entities (e.g., polygon edges or paths) to create candidate points; Length measures distances to test geometric conditions that determine whether a point is considered hidden; Polygon builds the involved polygonal regions; and TableText displays the computed variables and the final count in a clear table/text form. Suitable for middle to early university students, especially for geometry exploration with GeoGebra and for activities focused on modeling, rule-based classification, and result summarization.