**INTRODUCTION: BIRTH OF THE CONCEPT**

The idea which gave birth to the **Canonical
Polygons** (**CPs**) arose informally while drawing figures on graph
paper. Especially interesting to me was a certain class of polygons whose
sides followed the grid lines or diagonals. Adopting the restriction that
each side should not include more than one square division, whether orthogonal
or diagonal, I arrived at the concept of canonical polygon -- **canonical**
because it is constructed according to well-defined rules, which limit
it as to extension, shape and number.

An alternative way of expressing the concept
of CP is drawing a closed polygonal chain passing through the square grid
intersection points, proceeding from each point to one of the 8 adjacent
to it in the orthogonal or diagonal direction, necessarily changing direction
after each segment.

In January 1977, at the very beginning
of the activity in experimental combinatorial geometry that created them,
I discovered that there exist only 8 convex CPs -- a fact which is intuitively
obvious once one possesses their graphical representations, but which remains
to be formally proved.

**FORMAL DEFINITION**

A **Canonical Polygon** is a polygon
whose sides are straight-line segments proportional to 1 (in two mutually
perpendicular directions) and the square root of 2 (in two other directions,
also mutually perpendicular, at 1/2 right angle to the previous ones),
and whose interior angles are of the form *k*/2 right angles, where
*k*
= 1, 2, 3, 5, 6 or 7. Cases of crossed sides or multiple vertices are excluded.

**INFORMAL DEFINITION**

A CP is a polygon that may be drawn on
a plane square grid in such a way that each one of its sides is a side
or a diagonal of one of the grid's squares; this excludes as sides of the
CP such segments that include more than one side or diagonal of those squares.
The exclusion of crossed sides or multiple vertices is valid.

**IDENTITY AMONG CANONICAL POLYGONS**

Rotations, translations or reflections
in the plane, while maintaining the coincidence (implied by the definition)
among sides and vertices of the CP and those of the grid, are not considered
to generate a CP different from the initial one.

**TERMINOLOGY**

The short and long sides of CPs (whose
lengths are proportional to 1 and to the square root of 2) are called **orthogonal**
and **diagonal** sides respectively.

Two CPs that possess interior angles in
the same order, with **o** and **d** sides interchanged, are said
to be **dual** to one another. If such a transformation generates the
same CP, it is said to be **auto-dual**.

**PROPERTIES**

Below are defined some **morphic**
properties, depending only upon the shape of the CP, and some **metric**
properties, which involve measurements; as the latter have a necessary
connection with the shape, they may therefore be called **morpho-metric**.
The symbols accompanying the names of the properties refer to the *Canonical
Polygon Catalogue* table, where
*n* denotes the number of the CP's
sides and *Nº* is its order among those of *n* sides.

**Morphic Properties:**

* **Interior Angle Formula ( FA)**:
An ordered sequence of the CP's interior angles, expressed as multiples
of pi/4, starting with the largest and in that direction which produces
the largest numerical expression. It is followed by the letter(s)

It is useful for the textual description of a CP (without graphical representation) and in constructing CPs for a given

*

- the number of the CP's sides in the most frequent orthogonal direction;

- the number of the CP's sides in the least frequent orthogonal direction;

- the number of the CP's sides in the most frequent diagonal direction;

- the number of the CP's sides in the least frequent diagonal direction.

*

*

*

**(Morpho-)Metric Properties:**

* **Area ( A)**: Expressed as
a multiple of the area of the fundamental grid square.

*

*

*

*

*

*

A smaller value of

**CONSTRUCTION**

Two approaches aiming at constructing
all CPs with a given number of sides *n* have been worked on:

* **Construction by Segments**: Tries
to draw all closed polygonal chains -- under the canonical restrictions
-- with *n* segments(#).

* **Recursive Construction**: Constructs
CPs of *n* sides starting with those of *n*-1 sides, adding to
or subtracting from them conveniently located canonical triangles.

**THEOREM TO BE DEMONSTRATED**

* It is possible to construct all CPs
of *n* sides starting with those of *n*-1 sides by the recursive
process, for every *n* > 4.

**CATALOGUE**

The *Canonical Polygon Catalogue*
contains:

* A table showing
the morphic and (morpho-)metric properties of the CPs (up to *n* =
9) ordered by their ** FA**;

* The graphical representation (up to

* axes and centres of symmetry;

* duality.

(#) Quandt*et
al.* -- Universidade Federal de Santa Catarina, Florianópolis
/ SC / Brasil

See also: Eric
Weisstein's World of Mathematics

Correspondence
on CPs