1. The problem asks to construct geometric shapes and tilings based on Euclid's equilateral triangle construction. 2. **Constructing a regular hexagon (1.2.1):** - Start with an equilateral triangle of side length $s$. - A regular hexagon can be formed by placing six equilateral triangles around a common center point. - Each side of the hexagon equals the side length $s$ of the equilateral triangle. - To construct, draw a circle with radius $s$ centered at one vertex of the triangle. - Mark six points on the circle spaced by $60^\circ$ angles; connecting these points forms the regular hexagon. 3. **Tiling the plane by equilateral triangles (1.2.2):** - Use the equilateral triangle as a tile. - Repeat the triangle by translating it along vectors equal to its sides. - Arrange triangles so that each vertex is shared by six triangles, creating a tessellation with no gaps or overlaps. - This forms a triangular lattice covering the plane. 4. **Tiling the plane by regular hexagons (1.2.3):** - Use the regular hexagon constructed in step 2 as a tile. - Repeat the hexagon by translating it along vectors equal to two adjacent sides. - Arrange hexagons so that each vertex is shared by three hexagons, creating a hexagonal tessellation. - This tiling corresponds to the dashed lines in the figure. These constructions extend Euclid's method by using the equilateral triangle as a fundamental unit to build complex, repeating patterns covering the plane without gaps or overlaps.