TIMES TABLE ART
I have recently been watching some very interesting YouTube videos from Mathologer. I don’t claim to understand a lot of what he says because maths is not my subject. I am keen to learn more, though, because there is something deeply fascinating about how numbers are the foundation of all creation, and the laws that govern the universe and everything in it are based in mathematics. Mathologer is a charming and entertaining presenter who has some pretty cool animations. Even if you don’t understand it all, they are very intriguing to watch.
The other day I watched one of his videos, “Times Tables, Mandelbrot, and the Heart of Mathematics” which had me absolutely riveted. I remember the fashion a good few years ago of thread-and-nail pictures which always intrigued me, creating curved patterns out of straight lines.
What Mathologer did was to create visual representations of times tables in this way. I decided to have a try myself.
My first efforts
In my large Rhodia design drawing book I did a lot of practising, gradually refining the process. I drew a series of circles and divided them up, first of all into 5 degree increments. Mathologer used a computer for his animations and could make the divisions really tiny, but I didn’t have that luxury, and anyway I wanted to start relatively simply.
The x2 table
This is how I began to map out the x2 table.
The idea is to number all the marks around the circle, and then draw lines between them according to the table in question. To begin the x2 table, I drew a line from 1 to 2 – “One two is two” as we used to chant at primary school when learning out times tables! Then I joined 2 to 4, 3 to 6, 4 to 8, and so on, around the circle. In the picture above, you can see that the end of each line is two numbers apart, but the beginning of each line is at each consecutive number on the circle.
Progressing with the drawing
You can see that as I move around the circle, the curved line created by all those straight lines is beginning to turn towards the beginning.
The completed x2 table drawing
As I completed the circle, you can see that the curved line has produced a very interesting shape.
This mathematical shape is known as a cardioid (i.e. “heart-shaped”). You can create a cardioid by rotating one circle around another of equal size; a mark on the moving circle will trace out the shape of the cardioid.
The x3 table
I really struggled with this, until I worked out a better way of avoiding making mistakes.
This time you go 1 to 3, 2 to 6, 3 to 9, 4 to 12, and so on, around the circle. This produces a different pattern.
Here are my early fails!
(Completed x2 table at top left.) The following three drawings all contain errors, resulting in an asymmetrical pattern.
At this stage I thought perhaps I’d bitten off more than I could chew, and should start with a simplified version. I drew some more circles, and divided them tinto 10-degree increments instead of 5. It was easier to see, but it did not produce the density of lines required to make a really pretty pattern. The following picture shows another x3 table but again this is asymmetrical, and underneath, my first attempt at the x4 table. With the marks being so far apart, it was hard to see the pattern very well at all.
Back to the drawing board, and returning to 5-degree increments.
My first successful x3 table drawing. Symmetry at last!
Making plans
I realised that if I was going to do anything with these drawings and make some art out of them, it was going to be a very laborious process. I therefore decided to create a starting grid on the computer and print it out, which would save all that work drawing in the degree marks around the perimeter each time. I did this in Inkscape, and it was a pretty lengthy process involving tiled clones and what not. The result wasn’t perfect as the outline of the circle was clunky and uneven. I messed about with this for hours and then decided to save what I’d done, and edit it further in another program – my ancient Serif PagePlus desktop publisher. I overlaid another ring over the existing drawing, just exposing the tick marks for the degree increments, and printed out several copies of this image.
I printed these on Neenah Bristol Vellum paper – I bought a pack of 250 sheets of this a while back – great drawing paper, but thin enough to go through theh printer.
Further thoughts
I was able to use these printouts for my “fair copy” drawings, which turned out well. No errors apart from a few which I was able to correct as I went along. I worked in pencil first, and then inked over the lines.
At this point I thought that I could save myself further work in advance of any art on these patterns, if I scanned my completed drawings and printed them out. There’s a huge amount of work involved in plotting the times tables in this form, and if I had a stock of blank grids, I could use these and save myself time and effort.
Here is an example of a printout of one of the drawings on full-sized paper. There is plenty of room for adding embellishments or decorative text or whatever, around the finished art work if I want.
Before scanning, I erased the pencilled numbers from around the circumference. I increased the brightness and contrast in my photo editor as the fine pen lines were a bit indistinct.
I made a series of images from the set I had drawn, cropping them down to a square. I will be able to use these in a variety of ways. Here’s a montage of them.
Information about these images
I think you will agree that these make extremely attractive images. There are some intriguing points of interest about them.
As stated before, the x2 table produces a cardioid. The x3 table produces a shape known as a nephroid (kidney-shaped). This image can be traced by rotating a circle around another one twice the size. The x4 table produces three lobes, the x5 four lobes, and the x6 table produces five lobes – one lobe less than the number of the times table in each case, which is intriguing. I decided to stop at the x6 table as the number of times you have to go round the circle gets into the 400s.
Solving the problem of struggling with accuracy
It was very complicated drawing the lines in the correct place, especially as the design progressed, and also especially where the lines were drawn very close and almost parallel to the circumference. In the end, I wrote out the pairs of numbers for each times table, going from 1 to 72 (the number of marks at 5 degree increments around the circumference). Running down the list as I drew the lines made the whole process a lot easier and I was able to complete the latter designs a lot more quickly than the earlier ones. I did everything in pencil to start with, and once the design was complete, I went over it with a fine permanent black marker.
The Mandelbrot Set
For those not familiar with this, it is a highly complex fractal design discovered by Benoit Mandelbrot back in the 1980s. The equation for plotting the graph to create this looks very simple – z = z2 + C – I can’t go into all the details here and anyway I don’t really understand it – but for our purposes, it is interesting that when this equation is plotted, the main part of the Mandelbrot Set is a cardioid.
If you substitute a 3 for the 2 in the equation, the resulting Mandelbrot set will have a nephroid. Exactly like our times table charts – the x2 table produces a cardioid, and the x3 table produces a nephroid. Most intriguing. The Mandelbrot set is infinitely complex. You can zoom in continuously and it never comes to an end, and never becomes less complex – in fact it becomes even more complex. Some people have called this a mathematical monster but it is incredibly beautiful and intriguing. Look up “Mandelbrot animations” on YouTube and there are plenty of examples of zooming in through it. Benoit Mandelbrot did not have access to our modern computers and the only way he could see the results was to write some code and then print it out on a primitive printer (probably even pre-dot matrix). He did not invent this fractal but discovered its existence and revealed it to the world. Like all mathematics, this is a concept rather than a physical reality – as one YT presenter said, “You cannot stub your toe on a number 3 – it’s conceptual rather than a physical reality.” We all know what “three-ness” is – we recognise three apples, for example. You can eat the apples, but not the three! Numbers pre-exist everything. There are examples of fractals everywhere in nature, from ferns to Romanesco broccoli, from coastlines to spiral galaxies – all following particular equations and all containing self-similar patterns repeating across different scales. In nature eventually the zooming-in feature breaks down but they follow the fractal rules. These natural forms also follow the Fibonacci series in their structure – maths is everywhere.
An area I find particularly intriguing is the presence of the Fibonacci series concealed in the Mandelbrot set. I need to study the Fibonacci series further as it relates directly to the Golden Rectangle so prevalent in great art as well as in nature along with the Fibonacci spiral. The series is formed by adding the two previous numbers in order to produce the next one, i.e. 1 + 1 = 2; 2 + 1 is 3; 3 + 2 is 5, and so on – 1 1 2 3 5 8 13 21 and so on. In one of his videos on the subject of the Fibonacci series, Mathologer was wearing a t-shirt that read, “This Fibonacci joke is worse than the sum of the previous two.” Haha! All great fun.
All these wonderful geometric concepts seem to be related and the universal physical laws would not work without their foundational mathematics.
I was hopeless at maths at school. I struggle with these concepts and feel as if I am grasping at something just beyond my reach, but what I see in all this is order and beauty. Who would have thought that our humble times tables learnt in our primary school years could contain such beauty and symmetry.
