Recent creations

Back in November, I posted a collection of some of my favourite Desmos creations up to that point. Having made quite a lot of material since then, including some using different software and media, I thought it was about time to collate some of the highlights again…

After a colleague mentioned Dudeney’s dissection, I showed a 3D printed hinged version to some pupils, who seemed captivated by it. This inspired me to explore further. First up, I challenged myself to use Processing (in which I am relatively inexperienced) to create an animation showing a fluid transition between the two states:

I then got the idea that it might be possible to make a tiling pattern based on this dissection, and it was very satisfying to see this come to life in Desmos:

After this, I wondered if something similar might be possible with other dissections, and eventually I developed this sliding hexagon-square tiling:

Having seen an image of a tiling on a Poincaré disk, I started to play with this, but only really got as far as being able to draw any circular arc within the disk that meets the boundary at right angles:

A piece of geometric art by Regolo Bizzi inspired several of us on Twitter to experiment, and it was interesting to try to generalise it:

This trace and zoom version, based on the same idea, was quite challenging to work out:

Occasionally, others on Twitter have specifically asked for me to make something. This creation was the result of one such request:

This animation began as a straight attempt to copy something I had seen elsewhere, but it evolved as a result of a couple of ‘what if’ questions: What if it spins? What if there are multiple arms?

Having seen an intriguing animation shared on Twitter (original source unknown), I felt obliged to make my own version so I could play with the parameters and see what would happen:

The following came out of an exciting impromptu discovery during a lesson. I posed a question to the class about the number of ways to pick two teams from an even number of people. Their approach was completely different to mine, but entirely valid, and resulted in a conjecture that we subsequently proved. Here are the details, followed by a representation of the problem in a 3D print:

I went through a bit of a modular origami phase (warning: it was quite addictive!):

With Desmos, I found a way to shade alternate regions bounded by a set of curves:

Just before Christmas, Taylor Belcher shared a static image of a dabbing snowman in Desmos. It was practically begging to be animated!

@3Blue1Brown shared a brilliant video showing how it is possible to solve the Towers of Hanoi by counting in base 3. This meant it should be possible to programme a solution, so I gave it a go:

After a recommendation from Roy Wiggins, I played with some software called Fragmentarium, which is great for making high-definition images of regions defined by inequalities:

@konkretegifs is very new to Twitter. His style of hand-drawn mathematical art animations has already inspired many others to have a go, including me:

Most recently, I have been collaborating with Annie Perkins about 2D representations of 3D polyhedra. She has been doing some explorations on paper, constructing diagrams using compasses and a ruler, whereas I have adapted some of her ideas to make animations in both Desmos and Geogebra:

Banoffee pie: mathematical musings

This evening I made a banoffee pie, and at every stage I wondered about the mathematical questions I could be asking about the process…

Background: One of my Sixth Form classes has a rota declaring whose turn it is to bring in cake each week. A few weeks back, I made a banoffee pie that was decorated with a mathematical chocolate chip pattern (specifically, a dissection of the circle into congruent pieces that do not all meet at the centre).


Since then, the mathematical theme has escalated. Pupils have brought in, amongst other things, a Rubix cube, a Penrose tiling and a cakeulator:

Now it’s my turn again, and I spent a while trying to find some new ideas (the empty cake was high on my list of possibilities, having seen it as an entry in a MathsJam baking competition). Eventually I settled on making almost exactly the same thing as before, but with added mathematical questioning:


1. At what angle did I take this photo?



2. Use this image to find an approximation for the value of pi.



3. How thick will the layer of caramel be?



4. How many slices of banana will I need?



5. Why is there this curve of light in my cream?



6. What is the scale of this photo? Give the approximate area in terms of banana slices.



7. If I were to keep making stars inside stars in this way, what would be the total length of the edges? (unit: chocolate chip widths)


Not-so-random walks

Back in December, I saw this post by John Golden, where he was exploring a visual representation of functions using modular arithmetic:


I was curious to look into this, so I set up the same thing using Desmos. It works as follows:

  • Set n to be some positive integer (the base for the modular arithmetic).
  • Divide the circle into n different directions, all equally spaced, numbered 1 to n anti-clockwise.
  • Choose a function (odd powers generate particularly interesting patterns).
  • Start at the origin. Then begin to draw a sequence of connected line segments of length 1 by using your function. For example, if you have chosen x³, your first line segment will be along direction 1³=1. At the end of this, you will draw another line segment along direction 2³=8, and you will continue in this fashion. If the output is larger than n, you will use its remainder when divided by n (i.e. for output y, use y mod n).

Click here to experiment with this yourself in Desmos.

I have documented some of the results for the function f(x)=x^5. Below you will find images for every integer value of n from 1 to 150, followed by a further selection up to n=400.

You will notice:

  • The majority of the patterns form a continuous loop.
  • Where there is not a continuous loop, the pattern repeats but is translated horizontally (what is special about these values of n?).
  • Where there is a continuous loop, there is always at least one line of symmetry, but in some cases there are many more (which values of n give rise to high degrees of symmetry?).

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Mesmerising curves

Parametric equations with periodic functions can produce some beautiful results. The initial idea here was to see the effect of combining two polar roses, out of phase. In the end, I introduced an additional variable on top of this – a polar rose would have parametric coordinates

(cos(mt)cos(t), cos(mt)sin(t))

but I adjusted it to

(cos(mt)cos(t), cos(nt)sin(t))

to allow for more variety in the results.

All 9 gifs at the end of this post were made with the same parametric form:

parametric form

The ‘acos(kt+b)’ part represents a polar rose whose size depends on a, number of petals depends on k and phase depends on b. By animating variable b, effectively we are rotating this rose over time.

Overall, we see the combined effect of the first curve plus the rotating rose. For example:


The original Desmos version that I used is here.

Hover/click to see the values of m, n, a and k for each one (b is the variable used for the animation).

Snowflake Tessellation Art

Previously, I had created a snowflake design with draggable points using Desmos (click on image to interact with it using Desmos):

My 3 year-old daughter asked if we could make some ‘colourful snowflakes’, so I copied the image from Desmos into Microsoft Paint, duplicated it multiple times, and let her choose the colours:


As we were doing this, I began to wonder how it would look if it joined together neatly in a hexagonal tiling pattern so, having adjusted the initial snowflake so that a lot of the points would lie on a hexagonal boundary, I tried again:

snowflake for tessellation

In Paint, the ‘transparent selection’ tool worked really well for lining the snowflakes up next to each other.

snowflake tessellation

Finally, I decided that I’m not too handy with colouring pencils, so I just used the fill tool in Paint to colour it in. Given the six-fold symmetry of the snowflake, I settled for six different colours and what I thought was a pleasing arrangement. It looked pretty good when some of it was still blank, so here are two versions of the end product:

snowflake tessellation coloured

snowflake tessellation coloured 2