Newsletter: 3 Sunday 18 January 2026

 Symmetry_of_Quadrilaterals.pdf

Let us keep things simple by considering closed, linear, geometrical shapes or forms drawn by straight lines going from point to point and arriving back at the starting point after a number of steps. We can move around a form, say in a clockwise direction.  If we start at a point the activity alternates from moving along a line, then turning/rotating at a point, repeating. Distance > angle > distance > angle and so on.  A line or lateral, edge or side joins two points or vertices.  A joint, knee [-gon {genu}] or ankle [-angle] links two lines at each vertex.

I am emphasing the polarity between line [lateral, finite] and point [vertex, infinitesimal].

To describe a form, we could count the number of steps, the length of each line between points and the angle of rotation between the lines at each point.  In the following diagram of a random, irregular pentagon, a clockwise rotation is measured as positive (+), anticlockwise as negative (-).  A complete turn is 360º.

The names of these figures are derived from Greek and Latin (see chart of words with numbers in them) so we start with 3 – triangle, 4 – quadrangle, quadrilateral or square, 5 – pentagon, 6 – hexagon, and so on.

Let us look at one of the simplest forms: TRIANGLES.
If we look at triangles, we can categorise them according to the degree of symmetry they show:

• All 3 sides/angles equal – Regular = Equilateral + Equiangular,;
• 2 sides equal – Isosceles [same legs];
• No sides equal – Scalene.

• The Regular Triangle has THREE lines of symmetry;
• The Isosceles Triangle has ONE line of symmetry;
• The Scalene Triangle has NONE.

You will also notice that:

• the longest side is opposite the largest angle
• the shortest side is opposite the smallest angle

Triangles can also be categorised by the size of their largest angle:

• Angle < 90 – acute (sharp)
• Angle = 90 – right
• Angle > 90 – obtuse (thick)

This distinction is very important in the famous Pythagoras Theorem, which we will look at later.

The Regular Triangle [tripoint, trilateral, triline, ...] is of special interest because it is unique in its shape, as it is both equiangular and equilateral. 

Here are the first six Regular Polygons: Triangle [3 sides] to Octagon [8 sides]

When we nest them all together they look like this:

Odds and Evens - what do you notice?

Next week we will look at the various Centres of a Triangle and the Symmetry of Quadrilaterals.

RB