Then two vertices were joined by a line. A line segment.
Then three line segments formed a polygon with three sides. Triangle.
Joined by a fourth line, the polygon became a quadrilateral. Trapezoid. Parallelogram. Square.
A fifth line segment formed the pentagon. Pentagram. Pentacle. Star.
Moving through the seventh grade geometry was about the evolution of shape and its properties as it transformed from a single point to a pentagram. Much like the evolution of the universe from a single particle to the infinite stars.
On a human level, the sacred geometry was discovered in the ebb and flow of the Nile River. The ancient Egyptians used the magical 12-knotted rope to plot perfect right angles for crop fields whose boundaries were washed away with the seasonal floods of the Nile. They discovered the aesthetic quality of the golden ratio to design the Great Pyramids. Centuries later, Thales of Miletus establishes theorems to formalize geometry.
On a more personal level, Leonardo Pisano aka Fibonacci was fascinated - no, obsessed - with numbers all around him. Unlike his up and coming Renaissance counterparts who found beauty through paint and marble, he found beauty in nature's numbers. 0, 1, 1, 2, 3, 5, 8, 13... The same numbers seemed to appear everywhere. Mother Nature revealed herself to Fibonacci - it was the growth pattern of the universe.
The universe, indeed, for the Fibonacci numbers even reflected the approximate lengths of the lines of a pentagram. Each line, in relation to the next longest or shortest line, was close to the ratio of 1.618 to 1, the golden ratio, phi.
The seventh grade students discovered the golden ratio in the star. But they did not have to look to the stars to find it, for in their own fingertips, they held the golden ratio in the lenghts of their bones. The point was driven home once again, we are connected to the universe.
Our study of geometry took us to the introduction of measuring volume in cubes, cylinders, and spheres. In art, we continued to refine our skills in manipulating light and shadow. With an advanced technique of drawing black and white on blue paper, we got some magnificent effects of contrast and realism in the study of a hand holding a cube.