Madame Polygon

Madame Polygon grew up in Tessellation. She was the eldest of three sisters. Her family name was Polygon. Her younger sister, Hexa, specialized in hexagons. Her youngest sister, Octavia, lived in the dodecagon town hall and managed the regional tile-shop. Hexa and Octavia do not appear in this story. They make brief cameos in Kit 13.

Madame Polygon's given name is Polly. The academy children eventually learn this, usually after about three kits, and it always delights them. Polly was elected to the Polygon Council when she was twenty-six. The Council had needed a spokesperson for several years. Their previous spokesperson, a particularly dignified regular heptagon, had retired. He sought a quiet life of seven-fold symmetry and showed no interest in returning. The Council desperately needed someone who could explain regular polygons to the world.

Polly was the only choice. She had been speaking on behalf of polygons since she was seven years old.

This was, in Tessellation, not unusual. The village's children grew up with polygons as their constant playmates. Yet Polly was, even by Tessellation's high standards, unusually good at it. When she was seven, she could explain to her four-year-old cousins why a regular pentagon and a regular hexagon could not tile the same plane together. (The angles, she'd say, simply do not add up to 360° at the meeting point.) When she was twelve, she could derive, on a slate, the interior-angle formula for any regular n-gon. Interior angle equals (n−2)·180° divided by n. She would show her younger cousins how the formula came from cutting the polygon into n−2 triangles by drawing diagonals from one vertex. When she was sixteen, she could explain why a regular tiling of the plane could only use equilateral triangles, squares, or regular hexagons. (These are the only regular polygons whose interior angle divides evenly into 360°.) She could draw all three tilings on a slate without lifting her chalk.

When the GeometryForge academy searched for someone to teach *regular-polygon* properties to children, the Polygon Council unanimously nominated Polly. Polly, who was twenty-seven and had served as the Council's spokesperson for one year, accepted the position. She has been teaching at the academy for fourteen years now.

She arrives at the academy each morning in full Council regalia. This is, she always explains to the children, not vanity. It is pedagogical. Her headdress is peacock-feathered, with eyes patterned as small regular n-gons. (Count them, she invites: there are nine, representing the triangle, square, pentagon, hexagon, heptagon, octagon, nonagon, decagon, and dodecagon.) Her gown is paneled in alternating regular polygons. She carries a folded fan that opens, with a single flick, into a perfect dodecagon. (Twelve sides; she chose twelve because, she says, dodecagons are underrated.)

The children love her. They draw her in their notebooks. Sometimes they even try to make polygon-fans of their own.

In her classroom, Madame Polygon begins every first-day lesson with the same announcement. She enters the room. She lays her dodecagon-fan carefully on the desk. She turns to face the class. Then she says, in her theatrical Polygon-Council voice:

"The Council convenes. Each polygon has its angles, its symmetry, its place. Today we begin with the regular triangle. We will work our way up."

She then teaches the regular triangle. It has three sides. Its interior angles are 60°. Its exterior angles are 120°. It has three-fold rotational symmetry. It also tiles the plane perfectly. She teaches this slowly and ceremoniously. The children, who at first are slightly bemused by the ceremony, gradually learn to enjoy it. By the end of the first lesson, they can identify a regular triangle, square, pentagon, and hexagon by sight. They can also recite the interior-angle sum formula.

The lessons continue. Madame Polygon moves steadily up the polygon ladder. Pentagon. Hexagon. Heptagon. Octagon. She explains symmetry. She explains tessellation. She shows why some polygons tile the plane and some do not. She is patient. She is dignified. She clearly enjoys her work.

When children ask her whether regular polygons are hard, Madame Polygon always says the same thing:

"They are not hard. They are orderly. A regular polygon is the most orderly shape that has more than three sides. Each one has its number. Each number determines everything else. It sets the interior angle, the exterior angle, the axes of symmetry, whether it tiles. You learn the number. Everything else follows."