Linear Algebra II - Scalars, Vectors and Matrices ( An Introduction )

Islamic Geometry WIP in Houdini 16

Islamic Geometry WIP in Houdini 16

In order to demonstrate how Vectors and Matrices can be used, I'm going to run through how I constructed the artwork on the previous post called "Al-Jabr".

This piece was inspired by Islamic art which uses many different concepts that are fundamental to Linear Algebra -  vectors, vector spaces and transformations.  I have been fascinated by the beautiful style of artwork ever since I visited the Alhambra in Granada, Spain. There I learned that one of my favourite artists, M.C. Escher was also inspired by the amazing mosaics found around the complex to create his famous explorations of symmetry and tessellation.

I will be demonstrating how both the linear mappings of vector spaces or in more familiar terms the transformations of vector quantities are used to create these geometric patterns. As stated in my previous posts, I won't be going into the fundamentals in any detail, the links found in my first post cover all that we need to know. I will, however, share some insights that I feel may be helpful and that I haven't always seen in educational literature.

Before we begin, it is worth checking out the mosaics at the Alhambra and their connection to Escher as some inspiration.

If you are not comfortable with the basics of scalars, vectors and matrices, I suggest you visit the links mentioned above. If you are happy with these ideas, lets discuss a few more that may be useful to us in our comprehension.

Scalars - One thing that is not always clear is that scalar values are not just numbers in the abstract sense. They are denominate numbers that represent the  "value", "scale" or "magnitude" of a specific measured quantity and therefore require a unit of measurement.  Some examples could be speed, temperature or position along one axis of a coordinate system. Each of these would require a unit of measurement such as Meters, Celsius, Meters/Second. It is important to note that units of measurement also have a dimension, for example:

m^2\\ denotes a two dimensional quantity - area

m^3\\ denoting a three dimensional quantity - volume

ms^{-1}\\ or m/s denotes a relation between two units of position and time which we refer to as speed.

These unit dimensions can be manipulated as with any other algebraic quantity using a technique known as dimensional analysis.

Vectors - When Vectors are introduced in most courses, it is not always in the most formal of terms. We generally only learn about Geometric vectors which contain both a magnitude and a direction so can represent a larger range of quantities than scalars. Examples include a position in multiple dimensions, a directed line segment, a velocity, force, acceleration, or even the normal direction of an infinite plane centered around the origin.

Aside from Geometric Vectors however,  there are also other abstractions that fulfill the axioms of vector spaces so can be manipulated using the same operations as geometric vectors. These include sets of numbers  \mathbb{R}^{n}\\ and also confusingly any polynomial or function. This is a bit crazy and we will return to that later in this series as it is a very powerful realisation.

Matrices - Vectors of higher dimension are known as Matrices, They have many interesting and powerful properties that we will be exploring in several posts. For now we will just be looking at a specific kind of matrix known as an Affine Transformation Matrix but will certainly be looking at other types of Matrices such as the Jacobian later in the posts about Vector Calculus. Affine Transformation Matrices are used widely in computer graphics as they can represent a wide range of operations: translations, rotations, scales and shears.

We will be focusing on demonstrating the use of these powerful constructs over the next few posts as we breakdown the geometric creation of the Islamic style art work above

Before I finish this post, it is well worth knowing that all of the above constructs can all be abstracted into a single class known as Tensors. These are well beyond the scope of this series but fascinating nonetheless and worth exploring.