Angle
&
Symmetry

You know,

when people talk about angle and symmetry

it can be simply about what you get when two lines cross,

or a line crosses two parallel lines,

but that's basic theory,

and it's mostly because they are dealing with different kinds of shapes

and want to understand them.

There are the interior and exterior angles of polygons,

with the regular polygons being particularly easy to deal with,

but also the other shapes,

like the rectangle, the rhombus, the trapezium

and the different kinds of triangles,

and also of course

that shape without a straight line

but linked to the rest in all sorts of interesting ways,

namely the circle.

Now, the way to calculate the angles,

perimeters and areas of all these shapes

is the field of maths that is usually known as geometry,

and the ways to do these things

for these various shapes

are mainly done using simple formulae.

It can also mostly be done for complicated shapes,

by breaking them down into simpler shapes

from which they are constructed.

One of the most important simple shapes

is the right angled triangle,

with the properties associated with the squares of its sides,

and by using the relationships between the lengths of those sides and the angles between them

in the really useful type of maths we call trigonometry,

and all the stuff that springs out of Pythagoras theorem

and its variations.

Of course everything gets a little bit more complicated

when you add an extra dimension to make solids,

and need 3D geometry to calculate volumes,

surface areas and nets,

but just a few formulae will still do the trick.

Another aspect of the geometry of shapes

is how to handle the four transformations,

translation, enlargement, rotation and reflection,

and then again,

there is the wonderful property of symmetry to be considered,

an aspect of shapes that people find a source of great beauty,

but which can also cast a light on deep properties of number.

Symmetry.

Humans usually see it as beautiful.

It is obviously balanced.

But things can be symmetrical in 3D as well.

A solid can turn about a line through its centre

to end up looking the same way as it started,

and sometimes a solid can do that in lots of different ways.

Just look at the number of different ways

that you can twist a tetrahedron or a cube.

So different polygons have various different orders of lines of reflected symmetry,

and rotational symmetry,

and planes of symmetry. But a magical property of symmetry is that it can be applied

not just to 2D and to 3D solids,

but to shapes in even higher dimensions.

There is so much more to the creation

than this simple world we seem to live in,

but no matter how vast and complex we can imagine it to be,

we can be sure that God is Greater.