# Polygon

## Meaning

Informal: A polygon is a two-dimensional (flat), closed shape made only of straight sides. They can't have holes in them, or extra frills, or have lines that "cross" each other, or...

Polygons and not-polygons
• Example A: Polygons can be very wiggly and have no familiar shape, but the line segments that form it must eventually come back to the place where they started. (The shape must be closed.)
• Examples A and B: Polygons must be closed. That is, there must be no gaps in the sides.
• Example C: All sides must be straight.
• Examples D and E: There can be no "extra" lines, no lines that are not part of the "boundary" of the polygonal region. Nothing inside or outside.
• Example F: The region bounded by the polygon must have no other boundary. Here, the brown region at the top has its outside edge as the only boundary. But the non-polygon has a hole in it. No matter what shape that hole is (even if it, too, is a polygon), it is another border of the brown region, an inside border, so the region is not polygonal.
• Example G: The straight-line sides of the polygon may not cross each other. In this example the same four points are connected in two different ways. The top figure, a polygon, has one "inside" and one "outside." If the same four points are connected so that the lines cross, the figure is not a polygon.

Under construction: Puzzle: Which of these is a polygon?

Formal: Under construction: to come soon...

## What's in a word?

poly-, "many" + -gon, "angle"

The -gon is originally "knee" (-gon and knee are cognate, with the g-n of -gon being the kn of knee!)

Bones figured prominently in naming mathematical ideas! Just as the geometric object we now call "angle" was pictured as and named after a bent knee, the "legs" of a triangle were also literally legs. The -scel- in "isosceles" is the "leg" part; our modern English word related to that is "skeleton"! The word algebra derives from Arabic, and uses the image of bone-setting (its original literal meaning) to allude to the way mathematical ideas fit together.

The -gon- part also appears in diagonal, where dia- roughly means "across" or "connecting," to suggest the meaning of a line that connects one "knee" (angle) of the figure to another that is not right next to it, and therefore must run across the figure.