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If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

3D Printing with Desmos: Stewie Griffin

Using Desmos or Geogebra to design a picture or pattern is quite a nice exploration topic – but here’s an idea to make your investigation stand out from the crowd – how about converting your image to a 3D printed design?

Step 1

Create an image on Desmos or Geogebra.  Remove the axes and grid pattern.  This image is a pre-drawn image already on Desmos available here.

Step 2

Take a screen capture image of your picture (jpeg, gif, png).  We need to convert this to a SVG file.  You can convert these for free at sites like picsvg.

Step 3

Lastly we need to use a 3D editing site .  You can join up with a site like Tinkercad for free.

Step 4

Making our 3D model.  We import our SVG file and we get the image above.  We can then resize this to whatever dimensions we wish – and also add 3D depth.

Lastly I would then save this file and send it to a 3D printer.  You can see the finished file below:

So, if we printed this we’d get something like this:

3D printing the Eiffel Tower

Let’s use another Desmos art work. The Eiffel Tower above was a finalist in their annual art competition drawn by Jerry Yang from the USA.

This is then converted to the SVG file above.

And this is the result on Tinkercad when I add some depth and change the colour scheme.  Let’s see what that would look like printed:

Pretty good- we’ve created a cheap tourist souvenir in about 5 minutes!

Mathematical art

I thought I’d have a go at making my own mathematical art.  I started with using some polar coordinates to create this nice pattern:

Which then creates the following 3D shape:

This topic has a lot of scope for exploration and links with art, design technology and engineering.  Thanks to our ever resourceful ICT wizz at school Jon for assistance, and also thanks for this excellent method which was posted by Ryan on Thingiverse. You can also explore huge numbers of ready made 3D templates on the site.

Essential Resources for IB Teachers

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

Maths of Global Warming – Modeling Climate Change

The above graph is from NASA’s climate change site, and was compiled from analysis of ice core data. Scientists from the National Oceanic and Atmospheric Administration (NOAA) drilled into thick polar ice and then looked at the carbon content of air trapped in small bubbles in the ice. From this we can see that over large timescales we have had large oscillations in the concentration of carbon dioxide in the atmosphere. During the ice ages we have had around 200 parts per million carbon dioxide, rising to around 280 in the inter-glacial periods. However this periodic oscillation has been broken post 1950 – leading to a completely different graph behaviour, and putting us on target for 400 parts per million in the very near future.

Analysising the data

One of the fields that mathematicians are always in demand for is data analysis. Understanding data, modeling with the data collected and using that data to predict future events. Let’s have a quick look at some very simple modeling. The graph above shows a superimposed sine graph plotted using Desmos onto the NOAA data.

y = -0.8sin(3x +0.1) – 1

Whilst not a perfect fit, it does capture the general trend of the data and its oscillatory behaviour until 1950. We can see that post 1950 we would then expect to be seeing a decline in carbon dioxide rather than the reverse – which on our large timescale graph looks close to vertical.

Dampened Sine wave

This is a dampened sine wave, achieved by adding e-x to the front of the sine term.  This achieves the result of progressively reducing the amplitude of the sine function.  The above graph is:

y = e-0.06x (-0.6sin(3x+0.1) -1 )

This captures the shape in the middle of the graph better than the original sine function, but at the expense of less accuracy at the left and right.

Polynomial Regression

We can make use of Desmos’ regression tools to fit curves to points.  Here I have entered a table of values and then seen which polynomial gives the best fit:

We can see that the purple cubic fits the first 5 points quite well (with a high R² value).  So we should be able to create a piecewise function to describe this graph.

Piecewise Function

Here I have restricted the domain of the first polynomial (entered below):

Second polynomial:

Third polynomial:

Fourth polynomial:

Finished model:

Shape of model:

We would then be able to fit this to the original model scale by applying a vertical translation (i.e add 280), vertical and horizontal stretch.  It would probably have been easier to align the scales at the beginning!  Nevertheless we have the shape we wanted.

Analysing the models

Our piecewise function gives us a good data fit for the domain we were working in – so if we then wanted to use some calculus to look at non horizontal inflections (say), this would be a good model to use.  If we want to analyse what we would have expected to happen without human activity, then the sine models at the very start are more useful in capturing the trend of the oscillations.

Post 1950s

Looking on a completely different scale, we can see the general tend of carbon dioxide concentration post 1950 is pretty linear.  This time I’ll scale the axis at the start.  Here 1960 corresponds with x = 0, and 1970 corresponds with x = 5 etc.

Actually we can see that a quadratic fits the curve better than a linear graph – which is bad news, implying that the rates of change of carbon in the atmosphere will increase.  Using our model we can predict that on current trends in 2030 there will be 500 parts per million of carbon in the atmosphere.

Stern Report

According to the Stern Report, 500ppm is around the upper limit of what we need to aim to stabalise the carbon levels at (450ppm-550ppm of carbon equivalent) before the economic and social costs of climate change become economically catastrophic.  The Stern Report estimates that it will cost around 1% of global GDP to stablise in this range.  Failure to do that is predicted to lock in massive temperature rises of between 3 and 10 degrees by the end of the century.

If you are interested in doing an investigation on this topic:

1. Plus Maths have a range of articles on the maths behind climate change
2. The Stern report is a very detailed look at the evidence, graphical data and economic costs.

Essential Resources for IB Teachers

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

Plotting Stewie Griffin from Family Guy

Computer aided design gets ever more important in jobs – and with graphing software we can  create art using maths functions.  For example the above graph was created by a user, Kara Blanchard on Desmos.  You can see the original graph here, by clicking on each part of the function you can see which functions describe which parts of Stewie.  There are a total of 83 functions involved in this picture.  For example, the partial ellipse:

when x is bounded between 3.24 and 0.9, and y is bounded as less than 1.5 generates Stewie’s left cheek.  This is what he looks like without it:

By clicking on the various functions you can discover which ones are required to complete the full drawing.  Other artwork designed by users includes:

A minion from Despicable Me

See if you can create some designs of your own!  This could make an interesting maths investigation for anyone thinking about a career in computer design or art – as it is a field which will grow in importance in the coming years.

You might also like to look at a similar post on using Wolfram Alpha to plot the Batman and Superman logos.

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