An apian revolutionary proposes a radical revision of the traditional honeycomb structure. In place of tessellated hexagons, the rebel suggests a configuration of closely packed circular cells.
Every cell has the same radius .
A hive starts with a single cell, called the nucleus. Hexagonal layers of cells are added around the nucleus sequentially, as shown in the diagram below. The layer to which each cell belongs is written in its centre.
Question 1: What is the expression for the number of cells in the hive?
Question 2: What is the expression for the number of gaps in the hive?
Question 3: What is the area of a single gap?
Question 4: What percentage of the hive do the gaps constitute in the limit that ?