Spring 2024: I have created some more hinged tiling fractal space filling curves;
writing this up somewhere... link to follow at some point.
Code could be improved; some of these are slow to load.
Might not work on devices other than chrome on mac.
October 2023: I have no time! Too busy to post things over the last
year. I decided I would make an effort to post some shorter sketches,
even if they are not fully developed.
I put a youtube video up to explain the metrics example for my students.
December 2022:
I have been supervising third and fourth year projects on elliptic curves.
The students are good and pretty much teach themselves, but I gave a talk about the associativity
of the
group law on elliptic curves, and wrote some notes on
eliptic curves, so the first picture below links to something based on that.
The idea is to show the space of cubic curves passing through some points. You can move the points and so change the family of points. Probably it makes more
sense as something I could talk someone through, since I'm not sure my instructions are detailed
enough. However, when one of my children took a look, and played around to get different shapes, he
wanted to know what it was, and so I told him about the equation for a circle, x2 + y2=r2, which he had not seen at school, but he does know about graphs of lines and he does know about Pythagoras' theorem, so he got it, and thought it was really cool, so that was nice. So maybe these pictures can be a base for a discussion for other people.
A story about families of cubics ought to be chapter three in a book of stories that starts with a story about lines, so I wrote the second page below (link from lines), which is a sketch for a story about a line and about lines, and is inspired by teaching MA243 (Geometry) this term at Warwick. However, the story is only partly written and ideally requires a voice over and sound.
Writing these out was partly inspired by giving a talk at the
2022 Generative Art conference in Rome, which is third link. I wrote that talk based on the
art work inspired by tutoring first year geometry and motion and multivariate calculus (also link further down the page). The link here is the talk slides. Ideally this work would also be much further
developed, but it's some sort of starting point for a mathematical discussion.
I had wanted to go a bit further with illustrating calculus concepts, but even the osculating
circles pictures can be really nice, though possibly the sin curve is not always drawn in exactly the right place, and a lot is missing without the explanation I gave when giving the talk.
Autumn 2022:
I wrote these inspired by teaching Geometry, MA243. The first is about triangle groups.
In the class I covered reflections (and other transformations) in Euclidean, spherical and hyperbolic
space. The group generated by all reflections in the sides of a triangle is called a triangle
reflection group. Generally these are required to be discrete, which implies the triangles have
to tile the plane (or other space) in a nice way, so there are not so many possible triangles to
use. The link to triangle group here is a sort of game. Click "more" to show the option to draw
a yellow triangle. Then click on the A, B, C buttons to move the pink triangle by reflection in
the bold red, blue, green lines, to get to the yellow triangle. There is a simple algorithm,
which is fun to find. My 9 year old son had fun playing this game for a bit, and trying to work out the algortihm.
The second picture shows a triangle tiling of the Poincare disc.
This was probably not very useful for my students because we covered the hyperboloid model of the hyperbolic plane, not the Poincare disc model, although the relationship between the two was briefly mentioned. So in future I need to rewrite this to show tilings on the hyperbolid model. Also I ought to update this to show the matrix corresponding to the mobius transformation applied by moving the slider. The transformations do not change the hyperbolic geometry of the disc.
The third picture shows a hyperbolic triangle in the hyperboloid model, projected to the plane, but it only shows one triangle, not a tiling. The point was to show that a composition of translations can result in a rotation, but this version of the program is slightly buggy and needs rewriting.
You'll see the difference between the hyperbolic plane and Euclidean plane
better when you change to a larger scale. Moving the vertices will change
the triangle, but moving the centre point just applies a translation to the
triangle.
Spring 2022: I have been helping teach Algebra 2, second year algebra course on groups and rings. So I made a little puzzle. If I ever have time I may make more of these.
Winter 2021/ January 2022: Conways game of life inspired pictures and still life drawing.
There are two different generative art projects here; one is to use a variation of Conway's game of
life to generate pictures with a resemblence to Life, but with continuous variation.
I wrote about this in the "life, about", link below,
which is supposed to be an elementary description
of a basic 2D shader. Because the version with lots of different versions
takes too much GPU for a mobile, I put up a second version with just a few options.
I only test these on either Chrome on a macbook pro, or on a mobile with Chrome on andriod,
so they may look bad on other devices.
I used the underlying method for generating still life pictures, using the
addition of a mask.
As with pretty much all my shaders, these could all do with more work, so are classified as in progress!
You have to manulally alter parameters to get the still life to terminate (third program). The first
program could be rewritten to use less GPU.
The following was written in Autumn 2020 / Spring 2021, together with
Imogen Breeze.
This includes work on the complex plane, fundamental domains and Farey symbols
The following rows are a few of the p5.js works I wrote in Spring/Summer 2020,
before switching to Webgl.
This is art work inspired by hyperbolic tilings. These are not interactive.
Some are supposed to illustrate some properties of hyperbolic space, and some are just pretty.
On the page, click on the image to download a screenshot.
Here are another few p5.js things.
A couple of webgl at the end. Some of these come out upsidedown on Chrome vs how they appear on safari... on my mac book. Don't know why. So these
may appear upsidedown, depending on your browser.
The following were written in 2019 / early 2020. These are mostly SVG things.
I have been writing other
interactive animations since Spring 2020, but putting videos of them on my instagram page @havcircles instead of here. The first row is not really circles... in fact I am diverging away from just circles.
