Functions are fun to play with. Just watch kids sitting around a graphing calculator. The more math you know the more fun you can have. Even better with the power of computers you can play with ideas that you do not (yet!) understand. Especially when you can engage actively so get to it…(click below to head to the desmos graphing calculator)

As you play…wait you did click and play didn’t you? Go back and do so at once!

As you play there are two interesting creative directions to try, exploring and mapping. In the first you just try things to see what will happen, in the second you try to gain control. For the first anything goes, for the second you often way to try the simplest thing you can. Talking of which ready to play again?

This time you were playing with a function from the plane to the plane. We can’t draw a standard graph for those without seeing four dimensions. I have tried and tried but can’t do that. So we have to be more creative, there are a few ways, but this works by showing how an understandable image changes.

You might have noticed that this does not always work, for some functions you enter the image goes wrong. For example try

and

can you work out why, and how this might be avoided (or used to advantage)?

To finish lets introduce one more chart. This one plays with parametric equations where both the x and the y positions of a point are described by functions. If you investigated the first chart you might have seen them already. Parametric equations often have a point moving at different speeds along the line, which you can see here, the function thinks all the circles are equally spaced:

Another example to try here is:

These notes, explore further working through some classic functions and starting to control the image, to me the balance between wild play/exploration and control/understanding creates the space where art can happen.

If you find functions you really like with the second method send me the details and I will create a laser cut version (demand permitting).

Notes

I have been working on the exploration of functions found here for some time, the ideas were originally developed along with David Celento and Brian Lockyear for a workshop at  Acadia 2012. Sam Shah also used the ideas for student projects.

The main problem was that they were only implemented in Mathematica. Telling people “Here’s this cool idea, but you need expensive software to run it” just did not seem to work for some reason. Coming across the beautiful Desmos finally got me to create a more accessible version. Though there is no reason that I could not have made on in Geogebra and I plan to. Let me know if there are other formats that might be useful. I am also looking for help from teachers to connect these ideas to the classroom.

Collegiate typography parsed into a fractal, with the theme of lots of parts coming together to make the whole.

That’s the corporate design spin on the new logo for the Twitter Math Camp, but for an audience of mathematics geeks a little more detail might be appreciated. So here goes.

The image of the logo uses a technique to visualise functions from the plane to the plane. We are familiar with visualising functions from one dimension to one dimension. This can be done with a line plot, the y axis value of a point giving the value of the function at that point, for example :

You should go and play with this in desmos. Functions from two dimensions to one dimension, for example can be shown using three dimensions. In this case we look over a range of possible values of x and y and create a surface where the height is given by the value of the function.

Here we hit a snag. Our eyes and brains have never been trained to see beyond the three dimensions our reality provides. This is really quite unfortunate, as a lot of mathematics only really starts to get going in three or four dimensions. Luckily space is not the only dimensional thing we can perceive. There is also colour. We can redraw the previous image colouring by height, from yellow at the bottom to red at the top:

Now the information is in the colour we can remove the height, just showing a flat image with the function determining the colour at a point:

We can now take advantage of the dimensions of colour for example a second function can leave the colour the same but make it lighter. So we can use a new function that takes in two numbers and gives two numbers back the first number gives the colour as above, the second lets the colour fade with the function creating a fade as we go further from the bottom left corner:

So we have solved our problem, by using two spacial dimensions and two colour dimensions we can visualise functions that take in two number and give two numbers out. In fact we can go further as our perception of colour is a three dimensional system, with independent amounts of red, green and blue. These actually do not model colour at all, as in the world each wavelength of light is independent so colour is infinite dimensional. Our eyes, however, only have three sorts of receptors so the infinite dimensional space is projected onto a 3 dimensional one. It is quite possible to have higher dimensional perception of colour with more receptors. Most birds and some fish and turtles have been found to have four channels of colour perception, leading to an experience of colour far richer than we can imagine (but still far short of reality).

Just dealing with two dimensions though does turn out to be incredibly powerful, thanks to complex numbers, which make many mathematical problems easier and are used throughout science and engineering. They can be expressed as two dimensional numbers over the real numbers we have been using so far, through the Argand plane. So a function that takes a complex number to another complex number is 2d to 2d (over the real numbers) and is precisely the situation we have described. The process of visualise complex functions in this way is called domain colouring, and I have linked to a google image search, though be prepared for intense psychdelia (in the name of math). To explore the topic in more detail the book Visual Complex Functions by Elias Wegert is beautiful and worth getting just for the images.

To get to the TMC logo, we have to leave behind using the dimensions of colour, however, and use a different method to colour our image. We are dealing with a function that takes in two (real) numbers and gives two (real) numbers back. We can use the two numbers we get out to find a point on the plane, a point in an image, and use the colour of that image at that point. The result is called the pullback of the image through the function, as we can consider that the image has been literally pulled back through the function.

For the logo I started with a pattern made up of the letters TMC:     then made the letters fade out as they went away from the centre:with the image in place I could start to explore complex functions through this notion of pullback. Generating huge worlds of potential logos:

For these images, however, the letters TMC are hard to pull out. A better candidate is :The final image used this, though breaking the purity a little the three middle copies of the letters were removed to be replaced with an undistorted version. Of course you can also switch the image, I hope Megan does not mind me using her copy of the iconic image from TMC13 pulled back through : As a final note you might notice that the final logo has a different colour to all the examples shown here. Just as colour can be generated from numbers we can also pull numbers out of a colour, change them and then go back to colour, in this case shifting red to blue.

OMOOS

SOOS

OSOTS

OOOS

SOZS

Linear algebra is one of my favourite areas of mathematics. Its a simplification but you could say that the things that mathematics does well are small numbers and straight lines. The rest is just clever ideas to covert other things into those. As the mathematics of straight lines and flat surfaces, the importance of linear algebra should be clear. Its techniques are also very fast when done on a computer, allowing live motion in video games. This lead to every computer having a GPU, essentially a special chip for linear algebra.

Within linear algebra a central object is the linear transformation (that can be encoded as matrices) and the subspaces it preserves given by the eigenvectors. This gives some powerful tools to break linear transformations into pieces that can be studied more easily. As well as eigenvectors, however, linear transformations that do not have negative real eigenvalues preserve other families of curves. Curves that are taken to themselves by the transformation. These animations show how these curves change as the matrix changes. Giving a glimpse into the detail of what linear transformations do. These are of particular interest in dynamical systems where these images so some of behaviours possible close to a fixed point.

OMOOS

This animation shows a pair of eigenvalues changing from complex (creating a rotation and scaling) into real values. Can you tell where it happen? It uses the matrix

with s running from 1.05 to 4.05.

SOOS

This animation shows a hyperbolic fixed point (attracting in one direction and repelling in another) changes into a attracting equilibrium point, using the matrix

with s running from 1.05 to 4.05.

OSOTS

This animation shows a transformation changing as both the eigenvalues and the direction of the eigenvectors, using the matrix

with s running from 0.05 to 3.05.

OOOS

This animation shows a transformation changing as both the eigenvalues and the direction of the eigenvectors, using the matrix

with s running from 1.05 to 4.05.

SOZS

This animation shows how a shear changes with different eigenvalues, using the matrix

with s running from 0.05 to 3.05.

These are Curvahedra pieces:They can hook together to make all sorts of geometric objects. For example, take three pieces and make a triangle (or something triangle like with wiggly edges)

Taking a close look, each piece has five arms, and they are equally spaced around so the angle between two arms must be 360/5 or 72 degrees.

The interior angles of this triangle are all the same so we have 3*72 = 216. Yet from geometry we know that the the interior angles of a triangle always add up to 180. What has gone wrong?

Here is a 60 degree triangle (note the pieces have six arms, so the angle between neighbours is 360/6), can you see the difference?

Unlike the first triangle this lies flat on the table whereas the first curves away. The difference is clearer if we complete all the pieces around a corner for each.
Going further the five triangles come together to form a ball, while the six triangles would keep on spreading, we won’t be able to complete that sheet.
What we are seeing here is the curvature of the surface we are making. The triangle with 72 degree angles can be said to have an excess of 36 degrees. The greater the excess the more it curves. Look at this triangle with 90 degree angles (for a total of 270 degrees, an excess of 90 degrees), the curvature is very clear:
Completing this creates a smaller ball.
This new ball has eight faces, each with a 90 degree excess. Adding all these together gives a total excess of 720. The first model has 20 faces, with a 36 degree excess, and again a total of 720. Lets think about the model with just three triangles around a corner:
The total angle is 120*3 = 360, so the excess is 180 degrees. If the pattern holds we should need 4 of these triangles to make a ball, and indeed we do:
In fact if you take anything that is like a sphere, take the angle excess on every face you will always get 720. For a more complex example take this model:
This has eight triangles and eighteen squares, and all the angles are 90 degrees. For the square this is normal the total interior angle of a quadrilateral should be 360 and 4*90 is 360. So there is no angle excess. This leaves the eight 90 degree triangles once again giving 720. Also notice in the model the square faces are flatter with the curvature occurring at the triangles. This gives the model the shape of a cube with rounded edges, rather than a sphere.

This result is called Descartes’ Theorem and it is a special case of the Gauss-Bonnet theorem, both are closely related to the Euler Characteristic. These theorems stand at the heart of topology and differential geometry.

A natural follow up to this is to ask what happens with a shape with two little angle (an angle defect). For example the sum of the angles of a quadrilateral should be 360. What happens if we take a square (4-equal sides and angles) with 72 degree angles. The sum is now 72*4 = 288, which is less than 360. This creates a saddle:
The saddle is said to have negative curvature, and connecting up more and more squares, like this does not create a ball connecting up on itself. Instead it gives this wavy surface that grows faster and faster, modelling a hyperbolic plane, all these images are the same object!
Final note: The curvature discussed here is actually called Gaussian Curvature, and is a property of the surface itself not the way it fits in space. For example consider this cone:This is covered with equilateral triangles with 60 degree angles. So although it looks curved the geometry is flat, the triangles all have no angle excess. In other words if you investigated distances just on the surfaces of the model they would be the same locally as those on the locally flat plane of triangles given above. You can only detect the change if you loop back on yourself round the cone. The only exception is the tip of the cone. Here you can see a piece is left hanging.

The same thing happens when you bend a piece of paper, you change how that sheet lies in three dimensions, but not what happens on the sheet. You can even use this to work out how to best hold pizza. On the other hand the Gaussian curvature, discussed above, does change what happens on the surface. The angles of triangles can be measured without leaving the surface. In fact this might have been part of Gauss‘ motivation. He wanted to work out if the earth was a perfect sphere, but did not have access to space. In other words he had to take measurements just on the surface of the earth.

These ideas had even greater importance with the work of Einstein. General relativity assumes that the three dimensional space (or the four dimensional spacetime) that we live in is itself curved. In fact that curvature is related to gravity and explains how gravity acts at a distance. This huge idea fundamentally changed our understanding of the universe yet we can start to appreciate it with a simple toy, which you can get for yourself here. Another way to explore the geometry is with crochet from Daima Tamina’s beautiful book.

Curvahedra can make all sorts of objects, but some of the most satisfying are spheres, like the classic ball itself (here serving as a Christmas ornament).

So what other spheres or near spheres can be made? A good answer from those with a little geometry are the regular polyhedra. These are the 3d shapes where every face, vertex and edge looks exactly the same as every other.These shapes provide the climax of Euclid’s Elements, and the proof that they are the only ones is up there with Hamlet, the ceiling of the Sistine Chapel or the Taj Mahal as one of the greatest human achievements, but you can make this one yourself from paper:There is a reasonable argument that each of these must be close to a ball due to the high symmetry. This is a little stretched for the tetrahedron, but we will give it a pass. Can we do anything else with these pieces? To make a sphere we want to have the same amount of curvature everywhere. This does not require that every shape created have the same number of sides and the same angles at the corners, but that seems to be a useful place to start. Then at least every face has the same total curvature. In addition as all the angles on an individual curvahedra piece are the same, the angles around a corner should be the same. These conditions lead us to the Catalan solids, or Archimedean duals, the shapes where the model looks exactly the same from every face. Thus every face in a Catalan solid is the same. Here is one made with Curvahedra:

but that is not a sphere, what went wrong?

Looking at an individual pieces we see that the problem is, the curvature is not spread equally over the shape, one end is more bent.

The same polyhedron with flat sides shows where this might come from. The edges of the shape are not all the same length (but all curvahedra edges are).That does leave two Catalan polyhedra though, the rhombic dodecahedron and rhombic triacontahedron (with 12 and 30 sides). For these every edge is the same length.

They use 3 and 4 connectors (dodecahedron) and 3 and 5 connectors (triacontahedron) arranged to make rhombs

This gives us two new balls!

The triacontahedron is particularly pleasing, being a large ball,, as you can see when it is compared to the classic ball:

So by reasoning we were able to discover something new, but have we found everything? Maybe other close to perfect spheres are possible, what can you find?

Find your own Curvahedra pieces here.

Last week I taught at the first (year 0) of C&!, the Camp for Algorithmic Math Play. It was a lot of fun working on mathematical play and games with a group of 8 to 12 year olds, who both love math and are incredibly good at it. One of my courses was to use Desmos to help develop thinking on functions and start to get to some of the ideas of calculus (without the need for the algebra). Here are the example calculators that I set up for the course. Click on each one to go to Desmos, animate and adjust to play for yourself! Each calculator/graph also includes additional information about what is happening.

We begin with a list of curvesKeeping the idea of a list we can create circles traveling along another curve you are free to define, varying radius as you go.

The next one does not look so interesting perhaps, as it is just a frame from an animation, click to see the whole thing. It is a version of the dots from the previous image traveling along a track, but now using differentiation to make sure they stay touching. A second animation, uses the direction of the curve at each point.

Bonus Material

Some more calculators doing interesting things, for example distorting a grid of circles:Or showing a vector fieldor showing the first family of cubic equations to have their directions analysed by calculus, from Newton’s tract of 1666: