Suppose you want to find the area of a dodecagon inscribed in a circle of radius 1. One way to do this is to divide the dodecagon into 12 isosceles triangles with two sides of length 1 and an angle between them of 30°. Then we can use the formula for the area of a triangle $$A = \frac12ab \sin C$$ to give us an area of $$ \frac12 \times 1^2 \times \sin 30 = \frac14$$ Multiplying by 12, the area of our dodecagon is therefore 3 square units.
There’s another way to get this result which doesn’t rely on knowing that $\sin 30 = \frac12$. I first came across this dissection of a dodecagon when working at NRICH and looking for ideas for the Wild Maths site, which was about exploring mathematical creativity. Take a look at the GeoGebra construction below.
Move the slider, or click the play button to start the animation. There’s a little bit of thinking to be done to convince yourself that the pieces fit nicely together with no gaps, but I hope you agree it’s a very pleasing way to show that the area of the dodecagon is 3 square units!
If, like me, you stocked up on loo roll at various points during last year, and, like me, you don’t always remember to check the bathroom on recycling bin day, you may have ended up with a mountain of cardboard tubes that just cry out for a Blue Peter style craft project. Here’s something you could try. I saw a completed model using this technique ages ago on social media, and immediately decided I needed to reverse-engineer it and make my own models. I started making a dodecahedron and then ran out of loo roll middles and abandoned it in a corner to gather dust. It turned up when I was having a New Year Declutter and I realised I had plenty of raw materials (seriously, I don’t think I remembered to throw out a loo roll middle for the entirety of 2020), so I had a go at making something.
The pictures above show how to get started. Cut several toilet rolls into thin loops, around 3-5mm across should be fine. Then interlock three of them so that each loop goes over one and under the next, and then pull it tight as shown in the third picture. Now it’s decision time – what solid shall we make? I decided, rather ambitiously, to begin with a football, which (as Matt Parker famously got quite upset about once) are made up of a mixture of pentagons and hexagons. In fact, there are 12 pentagons and 20 hexagons, giving a total of 90 edges (or 90 little bits of cut up loo roll middle!) Mathematically speaking the football shape is a truncated icosahedron, one of the Archimedean solids.
I’m pretty pleased with the finished product. If you put a light in the middle, you get cool shadows too, but that’s really hard to take good pictures of, so you’ll just have to imagine it (or make your own!) As with other mathematical makes, I felt as though I understood the structure of the shape better as a result of figuring out how to make it. If you do decide to make your own, it’s quite a challenge to figure out how to get every piece but one into place (unfortunately the last piece does need to be cut and taped in order to close off the shape). Of course, if you’re doing it just for fun and you’re not so worried about using some tape, you could cut more sections for ease of assembly.
One of my favourite origami collaborations with Fran Watson last term was our Christmas Decorations webinar that we put on in December. Between us we demonstrated how to make hanging decorations, stars and a little box to put presents or treats in. The folding pattern that Fran showed in the webinar was one that I had learned many years before from a book but had forgotten about until Fran re-introduced it to me. Here are some online instructions.
The nice thing about this box is that by adjusting the folding pattern very slightly, you can make a box with a slightly larger footprint that can be used as a lid, or you can adjust the model and fold by thirds instead of quarters to get a much taller cubic box. I decided to use my mathematical skills to calculate the paper size needed to make a box with a lid to wrap a Christmas robin that I crocheted for a family member’s Christmas present. As the robin was about as tall as he was long, the cubic box seemed the most appropriate, but I wasn’t sure how big a square I needed to start with, so I folded a box from a 15cm square (which I knew would be too small), and measured the dimensions of the completed box. Then I could scale it up and cut a square of wrapping paper which would definitely fit my robin.
For the lid, I wanted to make a normal box rather than the cubic box, as I wanted the lid to be much shorter than the box, but slightly larger in cross section than my cube. Again, I worked out the relative sizes of the paper and the completed box, and used this to work backwards and calculate the paper size I needed to begin with. I cut the appropriate square from the same wrapping paper, and my robin’s home was finished.
When thinking about how I want mathematics classrooms to be set up, one of my goals is to make sure that children leave school with the confidence to be able to handle questions of scaling, ratio and proportion like my robin box problem. So many people would be anxious to attempt the mathematics I needed to do, or would lack the confidence to try. This is another good reason to include origami or other hands-on construction in mathematics lessons. Setting a challenge like my robin problem gives children opportunities to experiment, conjecture, explain their reasoning to others, make mistakes, revisit their thinking and finally experience success, in a way that a dry and dusty textbook exercise of abstract ratio questions might not.
Another “notice, wonder” memory from lockdown – I got into the habit of taking a photo of any mathematically interesting morning snack that I enjoyed with my coffee for a while, and posting it on Twitter to provoke mathematical discussion. Back in April I was eating up some miniature cookies that were left over from a musical carousel biscuit tin that I had bought in the post-Christmas sales (musical carousel biscuit tin is every bit as awesome as it sounds, I promise you). They were small enough that it was difficult to estimate what should count as a proper portion, so I experimented with some different ways of arranging them on the plate:
Ten normal-sized biscuits would be a bit much I think, unless I’d been for a ten km run immediately before, which is something that has never happened to me, but ten of these tiny cookies seems like a reasonable fuel to accompany some coffee and then power my brain to write high-quality maths content…
It’s interesting to me though that ten cookies can be arranged in these two pleasing ways. In the first picture, the cookies show 10 as a triangular number, 1 biscuit on the top row, 2 on the second, 3 on the third and 4 on the fourth. The second picture steps up into the third dimension and represents 10 as a tetrahedral number (remember those from this post?) – a triangle of 6 (1+2+3) on the bottom layer, 3 (1+2) on the middle layer, and 1 on the top layer.
If you’re anything like me you’re now asking yourself which other numbers are both triangular and tetrahedral. Well 1 is a rather boring example, but how far after 10 do we have to go before we find another? And will it be a reasonable number of biscuits to eat in one sitting? I decided a quick way to find out would be to create a spreadsheet showing the triangular and tetrahedral numbers in Excel and just look down until I found a number that appeared in both columns. Thankfully the next example is small enough that I could just eyeball it, but I also remembered a neat trick for checking whether a number is a triangular number or not; if T is triangular, then 8T+1 is always square, so I got my spreadsheet to work out √(8x+1) for my tetrahedral numbers, and then looked for the ones which were exact answers rather than lots of decimal places. This gave me 1, 10, 120 and 1540, the first four numbers which are both triangular and tetrahedral according to the Online Encyclopedia of Integer Sequences. I think 120 or 1540 biscuits might just be too much for one person, so until Covid restrictions are lifted for long enough for me to share them, I will stick to just eating 1 or 10 biscuits at a time!
Back in the autumn, the excellent Aperiodical website ran a competition to design “fractal bunting”. I had done a fair amount of sewing over the summer, making face masks for me and my loved ones so that we could protect ourselves and others when out and about while also looking stylish in a variety of fabrics that had been sitting in my stash for years waiting for the perfect use. When I saw the “fractal bunting” prompt, it seemed like a good excuse to use the sewing machine for something a little different. I had never tried making bunting, but as it’s usually triangular, it seemed like the Sierpinski Triangle would be a good starting point for my design.
I found a bunting pattern and cut out a paper template, and cut out four red triangles. Then I folded the top corner of my template down to meet the midpoint of the base to make a new template, a triangle with half the side length and a quarter of the area. I cut out three yellow triangles using this template. I did the same again to cut out six green triangles and then finally cut out nine tiny blue triangles. Then I figured out how to use the applique settings on my sewing machine to attach all the different sizes of triangle as shown below:
My bunting shows four stages of evolution of the Sierpinski triangle. I was right at the limits of being able to sew round the smallest triangles which is why they look a bit wonky – if I wanted to do five stages I’d have to make my bunting significantly bigger! However, I think there are lots of “What do you notice? What do you wonder?” questions that the image above might provoke: If I made 5 pieces of bunting (with one extra fractal layer), how many of each triangle would I need? The stitching represents the perimeter of each triangle added, so how much extra stitching is there on each new layer? Which of the four flags took longest to make? How much longer did it take to make than the previous one?
I found that as with so many mathematical ideas, actually making a physical model helped me understand the idea in new ways. Sewing round all those little triangles gave me a different appreciation of the fractal nature, and coming to the physical limits of the material I was working with made me reflect on the theoretical nature of limiting processes. When we talk about infinity and limits, we are talking about something we can’t hold in our hands or see. I can’t physically make the Sierpinski triangle, I can only make an approximation to it. But making that approximation helps me to think about the properties of the theoretical mathematical object in a different way.
Back in September, when lockdown had lifted sufficiently to visit neighbouring towns and countryside for a walk and a look at slightly different scenery, I visited the town of Thetford with my partner. As we waited to cross a road I happened to glance up and notice this window:
There are many mathematical “What do you notice? What do you wonder?” questions that might be provoked by this octagonal window and the decorative brick pattern around it. Of particular interest to me was the way the octagon was made up of the smaller panes, and I wanted to see if I could recreate it using paper.
These photos show how I used three squares of paper (there was some paper left over) to recreate the octagon. The photo on the right shows how I rearranged the pieces to help me calculate the area of a regular octagon. Let’s suppose the octagon has side length 1 unit. That means the large square in the middle has side length 1, and the square at the top of the right hand picture has side length √2. Each of the pale rectangles has a long side of 1, so the rectangle in the right hand image has dimensions √2 and (√2+2). So the area of the regular octagon is √2(√2+2) which is 2+2√2.
Back in April, starting to get a bit of cabin fever from lockdown, and inspired by a photo sent to me by a friend, I decided to get the Lego out and build a tower all the way up to the ceiling. The video above shows a timelapse of the build. Some questions you might wish to ponder as you watch the video: How long did it actually take? Or alternatively, how much has the video been sped up? How many bricks do you think I used? (If I used this as a prompt in the classroom I might ask children to tell me a number they know is definitely too low and one they know is definitely too high – giving a wrong answer is much easier than giving a right answer as there are so many more of them!) Are there any patterns to the colours that I used? Which colour did I use most? How tall is the tower? If we scaled the tower up by a scale factor to make the door big enough for a real person rather than a lego person, how tall would the tower be? How does this compare to actual skyscrapers in the real world? If you have answers to any of these questions or a question of your own, leave a comment!
I hope you enjoy playing with my little snowflake-generator above – move the five points on the right to create snowflake designs on the left! Made of course with GeoGebra – if you haven’t already heard me wax lyrical about this fab bit of software, that means you didn’t read Monday’s blog post.
The snowflake generator was made in a slightly different way from the method I used in my GeoGebra workshop before Christmas – I created a polygon, reflected it in the y-axis, and then used the spreadsheet to quickly rotate each polygon through multiples of 60°. By changing this angle from 60° to 45° you could make designs with eight-fold symmetry (but don’t call them snowflakes or you’ll incur the wrath of the anti-snowfake campaigners!)
If you create any pleasing snowflake designs, do share a screengrab with me, and if you have other ideas for GeoGebra things you think I should create or you’d like me to run a GeoGebra workshop for you, get in touch!
2020 has raised many challenges, and has required incredible patience from all of us. Patience as we wait for life to get back to normal, patience as rules change suddenly and plans are cancelled at the last minute, patience with technology as we cope with frozen screens, muted microphones, and unreliable broadband. There has also been solitude – my last day in an office with colleagues was March 16th, and by the time restrictions lifted to allow co-working again, I had moved on from my job and become a full-time freelancer, so I now spend much of my time enjoying my own company. It turns out that time and patience are the two ingredients needed to make an origami model I’ve had my heart set on making for many years…
I first came across the Five intersecting tetrahedra model in “The Origami Handbook” by Rick Beech, which I bought when I was a student in around 2002 or 2003. The model was designed by Tom Hull, and you can read more about it on his website. When I first saw the model I wanted to make it but having made one tetrahedron I abandoned the attempt because my paper was too flimsy and I did not think all five would stay together. I didn’t return to the model for more than a decade.
The tetrahedra crossed my path again at Electromagnetic Field in 2018. Matt Parker had a harebrained scheme to create a lit-up version of five intersecting tetrahedra made out of wood, and a bunch of us helped hold bits of the framework to try to get the lines going over and under each other in the right order. This was where I first appreciated that in the model, you can pair off tetrahedra using different colours to help you; if you have a red tetrahedron and a blue one, then the point of the red goes through the base of the blue and the point of the blue goes through the base of the red. In the completed model, every face of each tetrahedron has one of the other four colours going through it. This helps with the construction, though it wasn’t enough for me to get my head round the 3D geometry.
My next encounter with FIT, as it’s known to its friends, was at Talking Maths in Public 2019. I was part of a collaborative effort to make the model, under the guidance of Philipp Legner, founder of the excellent Mathigon website. There are instructions on Mathigon for making the FIT modules, and I’ve referred to these in making my own models.
I was delighted to be part of a group that successfully made the model but I was still determined to make my own some day. I thought that day might come in time for MathsJam 2019, when I gave a talk about dodecahedra. One way of understanding the five intersecting tetrahedra is by looking at the twenty vertices of a dodecahedron and partitioning them into five groups of four. I successfully created a 3D GeoGebra construction to help me better understand the geometry but I still couldn’t manage to put the origami model together – every time I tried I would get two tetrahedra together and part of the third and then become frustrated with which parts went over and which parts went under, and end up with battered and creased modules that were only fit for the recycling bin. MathsJam 2019 had to make do with a photo of the Talking Maths in Public model.
Fast forward to November 2020. I had spent a big chunk of the autumn doing origami; Fran Watson and I had put together webinars for Maths Weeks in Scotland and in England. I had also found that it was an ideal lockdown hobby – modular origami with its soothing repetitions kept me calm in a chaotic and frankly scary world. I had recently made a beautiful though complicated star from instructions on Paolo Bascetta’s website, and this somehow gave me the confidence to go back to the model that had become my nemesis. If I didn’t have the patience and dexterity to do it now, then perhaps I should make my peace that I never would…
I cut the pieces from some A4 paper – the model is folded from 3 by 1 rectangles. Each tetrahedron is made from 6 edges, so 30 rectangles cut from 10 squares of paper are needed in all. I folded the modules gradually in spare moments until all 30 were ready to assemble. Then I put together the first two tetrahedra and got stuck again. It was the evening before Big MathsJam, and I thought to myself how lovely it would be if I could start my 2020 talk tying up the loose end from 2019. That gave me the motivation I needed. I read Michal Kosmulski’s assembly tips, and then sat at my desk manipulating paper and comparing it with the diagrams. I got a third tetrahedron in, then a fourth, and finally, all five were together! I then proceeded to tell everyone I could think of, including Twitter!
So, what advice would I offer someone else who wanted to make this model? Well, first of all, it’s much easier to make it once you’ve already made it! Having an actual 3D real life model to compare it to made the assembly of my second, third and fourth attempts much smoother than the first one. (Yes, I’ve got an origami problem. Yes, I’ve stopped making them now. Yes, I would make another one given half a chance.) For your first model though, it really does help to look at someone else’s diagram, so here’s a photo of one of mine with some notes that might be useful.
If you’re a fan of modular origami and you have the patience to tackle a complicated model, (or the three-dimensional thinking to make it easier to visualise the assembly), I think it’s definitely worth making at least once. I made a version from shiny paper which I suspect will adorn my Christmas tree for quite a few years to come. Origami is a great tool for helping young people understand the importance of care, precision and perseverance, and while I wouldn’t necessarily teach this model in an origami workshop, I will certainly use it as an example to inspire!
Earlier this month I ran a couple of workshops to help people get started with the amazing free software GeoGebra. If you’re not familiar with GeoGebra, it’s a bit difficult to sum up in just a few words as it does so much! I first knew it as a dynamic geometry package but it also does 3D graphics, has a built-in spreadsheet, a computer algebra system and many more features – the more I use it, the more I discover.
In my workshops we looked at how to create snowflakes (coming up in a future post) and stars. In this post I’m going to share another way of creating stars in GeoGebra which we didn’t cover in the workshop.
To make the animated 13-pointed stars shown above, first I specified a point which I called “B” with the coordinates (1,0). Then I created the slider “a” which gives integer values from 1 to 6. The rest of the star was done with one (rather long) command. I used the Polygon command and then specified the 13 points using 12 Rotate commands. The first part of the long command was: Polygon(B,Rotate(B,a*360°/13),Rotate(B,2a*360°/13),Rotate(B,3a*360°/13)…) Each of the “Rotate” commands multiplies the position in the list by whatever the slider a is set to, and of course when the angle goes over 360° it just wraps around! This is a much neater method for drawing the polygon than my previous method which used modular arithmetic, although typing out all 12 rotations was a bit tedious. I think if I combined this with the method I shared in my recent workshop, which used a spreadsheet to create a list of points, I could make a very nice interactive star exploration tool.
Some mathematical questions to consider: This family of stars has 13 points, and the slider went up to 6. What would happen if the slider went beyond 6? What if I made stars with fewer or more points instead? I initially started with 9 points but some of the stars were quite boring – what do you think the star looked like when I set “a” to 3?
If you’ve never played around with GeoGebra before, I urge you to have a go. https://www.geogebra.org/classic gives you everything you need to start exploring in your browser, and you can also download versions for desktop. I even have the phone app for emergency geometrical diagrams on the go! I’m hoping to run some more GeoGebra workshops in the spring so look out for news of them on Twitter, or if you would like me to run a bespoke GeoGebra workshop just for you and your school or organisation, get in touch!