πολύγωνον

The Greek word πολύγωνον (polýgōnon) is a compound of πολύς (polýs, many, much) and γωνία (gōnía, angle, corner), with the neuter noun-forming suffix -ον. Γωνία (gōnía) derives from the Proto-Indo-European root *genu- (knee, angle) — the same root that gives Latin genu (knee), English 'knee,' and 'genuine' (via a different path). The connecting sense is the bent joint, the angle made by two surfaces or lines meeting: the knee is the place where the leg bends, the angle is the place where two lines meet, and both are named for the same bent-joint shape. Greek geometric vocabulary was rich in gōnía compounds: τρίγωνον (trígōnon, three-angled, triangle), τετράγωνον (tetragōnon, four-angled, square or rectangle), πεντάγωνον (pentagōnon, five-angled, pentagon), and πολύγωνον for the general class of many-angled plane figures. The word polygon entered medieval Latin mathematics as polygonon or polygonium and reached English in the sixteenth century through the translation of Greek geometric texts.

Greek mathematics had developed a sophisticated understanding of polygons by the time of Euclid (c. 300 BCE), whose Elements systematizes the properties of triangles, quadrilaterals, and regular polygons with a rigor that was not superseded for two millennia. The culminating achievement of Euclidean polygon theory was the proof that exactly five regular convex polyhedra exist — the Platonic solids: the tetrahedron (four equilateral triangular faces), the cube (six square faces), the octahedron (eight equilateral triangular faces), the dodecahedron (twelve regular pentagonal faces), and the icosahedron (twenty equilateral triangular faces). Plato in the Timaeus associated each of the four elements with a different solid: fire with the tetrahedron, earth with the cube, air with the octahedron, and water with the icosahedron, with the dodecahedron representing the shape of the cosmos itself. The regular polygon, through the Platonic solids, became a cosmological category — the shapes underlying physical reality.

The word 'polygon' carries within it the root πολύς (polýs, many), which proliferated across Greek and then English vocabulary in ways that knit together geometric and political thinking. The same polýs root gives English 'polygamy' (many marriages), 'polymath' (many subjects), 'polyphony' (many voices), 'polytheism' (many gods), 'polytechnic' (many arts), and 'politics' itself — via polis (city-state) and politēs (citizen). The citizen of a polis and the polygon share the same root only in the most distant sense, but Greek mathematical language was deeply embedded in civic language: the word for the democratic assembly (politeía) and the word for the geometric figure (polygōnon) were both expressions of plurality, of the many as a constitutive category. Greek geometry was not politically neutral; it was developed by people for whom number and plurality were simultaneously mathematical and civic concepts.

The regular polygon became one of the fundamental forms of European architectural decoration, appearing in Islamic geometric tiling, Gothic window tracery, and Renaissance perspective drawing. The construction of regular polygons with compass and straightedge alone — possible for some polygons, impossible for others — was one of the great problems of classical geometry, and the history of its solution traces major episodes in mathematical development. The Greek geometers could construct regular polygons of three, four, five, six, and fifteen sides with compass and straightedge. Carl Friedrich Gauss proved in 1796, at age nineteen, that the regular heptadecagon (seventeen-sided polygon) could be constructed by straightedge and compass — the first new constructible polygon in two thousand years. The question of which regular polygons are constructible was fully answered only in the nineteenth century, connecting the Greek vocabulary of angles and sides to the abstract algebra of field extensions.