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.
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.
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.
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.
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.
* 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) o (and/)or d, according to whether the segment following the largest interior angle, in the sense defined above, is orthogonal (and/)or diagonal.
It is useful for the textual description of a CP (without graphical representation) and in constructing CPs for a given n without duplications.
* Directional Formula (FD): A sequence of four integers representing in order:
- 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.
* Duality (Du): +, - or A according to whether the CP is dual to the following one, to the preceding one or auto-dual.
* Symmetry (Sm): Ay, if the CP has y axes of symmetry (axial); C, if it has only a centre of symmetry (central).
* Number of Concavities (K): A concavity is a polygon bounded by the CP and the minimal-area convex polygon which circumscribes it (its convex hull).
* Area (A): Expressed as a multiple of the area of the fundamental grid square.
* Perimeter (P): Expressed as a multiple of the side of the fundamental grid square.
* Diameter (D): Maximal distance between CP vertices.
* Area/Diameter Relation (A/D): Quotient of A and D in the units above.
* Perimeter/Diameter Relation (P/D): Quotient of P and D in the units above.
* Convexity (C): Ratio of the CP's area to that of its convex hull.
* Square Fraction (f): Fraction of the CP's area constituted of grid squares.
A smaller value of f denotes a more "filiform" CP.
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.
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 n = 9) normalised (the direction of rotation, considering angles in the order of the FA, is clockwise; the first side of the first angle points to the east or north-east), indicating:
* axes and centres of symmetry;
(#) Quandtet al. -- Universidade Federal de Santa Catarina, Florianópolis / SC / Brasil
See also: Eric
Weisstein's World of Mathematics
Correspondence on CPs