Author: mholtham

Mathematical colouring

Click the thumbnails below for images to colour (N.B. in some browsers, you might have to swipe the images across the page to see the next one). A printable PDF document containing all of the images is further down the page.

I made all of these images in Desmos over the past couple of years, but it was only after a conversation with a colleague today that I realised the potential for colouring!

If you would like more knots/links images, click here for a version with adjustable dimensions. Desmos links for all of the other images are available via my Twitter feed (@ghsmaths) – if you would like one but can’t find it, just let me know.

 

 

 

mathematical colouring images (printable PDF)

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Behind the mathart…

My most recent mathart animation was the latest in a series looking at the regions formed by rotating overlapping shapes:

Mary Gentry contacted me to ask for some insight into how it was created:

This particular graph involved quite a few different techniques which I had used in previous work. These are all strategies that I have developed through many hours of play and experimentation with Desmos, as well as by collaborating with and being inspired by other Twitter Desmos artists, such as Dan Anderson, Suzanne von Oy and Luke Walsh.

Follow the links below for details of some of the steps in the process:

Creating an array

Rotating a graph about a point

Shading regions defined by multiple inequalities

Expressing polar graphs in Cartesian form

The final graph

The animation that I posted was then made by adding frames individually on Gifsmos.

Finally, my number one tip for creating art in Desmos is to set yourself challenges and start playing!

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).

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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:

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1. At what angle did I take this photo?

 

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

 

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3. How thick will the layer of caramel be?

 

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4. How many slices of banana will I need?

 

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

 

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

 

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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:

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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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