pentagonon

Greek pente meant five and gōnia meant angle or corner. A pentagonon was a five-angled figure — what we call a regular pentagon. Greek geometers were fascinated by the pentagon because it embodied the golden ratio: the diagonal of a regular pentagon divided by its side length equals φ (approximately 1.618), the golden ratio. Euclid devoted significant attention to pentagon construction in his Elements.

The regular pentagon was sacred to the Pythagoreans, who used the pentagram (the five-pointed star inscribed in a pentagon) as their symbol of recognition. The pentagram's diagonals divide each other in the golden ratio, and the intersections create a smaller pentagon — an infinitely recursive pattern of golden ratios. The Pythagoreans reportedly called this pattern hygieia (health), and it became one of the most symbolically loaded geometric figures in Western culture.

The Pentagon building in Arlington, Virginia, was designed in 1941 and built in 16 months — one of the largest office buildings in the world. The five-sided shape was chosen because the original proposed site had five unequal sides constraining the footprint. When the site changed, the shape was retained. The building became so iconic that the Department of Defense itself is now called simply 'the Pentagon.'

The five-pointed star — derived from the pentagon — is the most widely used symbol in national flags: it appears on the flags of the United States, China, the European Union, and dozens of other countries. The Greek five-angle figure has become the world's most common symbolic shape.