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) 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).
(Morpho-)Metric Properties:
* 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.
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 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;
* duality.
(#) Quandtet al. -- Universidade Federal de Santa Catarina, Florianópolis / SC / Brasil
See also: Eric
Weisstein's World of Mathematics
Correspondence on
CPs