JW's Pictures and Patterns Site

Welcome

Hello,

Welcome to my site.

This is where I showcase the various images I've created over the years pursuing my hobby of using maths to create patterns. Wherever possible I've tried to provide explanations of how the patterns are generated and in some cases I've provided source code that you can download and adapt.

The site also contains some photos from some of my holidays, artwork by my brother and the fruits of my brother's butterfly hunting around the United Kingdom and in the West Indies. He's one butterfly short of the full set.

News

June 2021: I recently stumbled across a YoutTube video by Numberphile, which discussed finding integer solutions for the expression x^3+y^3-z^3=k, where k=42. The solution found has x, y and z values with 16 digits. Currently the only known way to find solutions to equations like this is to search all the possibilities looking for matches, and testing 10^[48] values isn't currently feasible. The search can be reduced to order n^2 (a still impractical 10^[32] values) by noticing that for small 'k' and large 'x', 'y' and 'z', z~=−(x^3+y^3)^[1/3]. What made the video interesting is that they had an algorithm that they claimed was order 'n', provided you focused on a specific 'z'. The video didn't go into details of how this was implemented so I thought I'd have a go myself. The results can be found on my Searching for Cube Sums page.

A few things to note:

• My page only describes how the algorithm works, in includes demonstrations written in Javascript, which will allow you to search for some numbers, k=72 and k=83 work quite well, but as the algorithm involves calculating z^3 you won't be able to go beyond z=2^[18] (the cube root of the maximum safe integer).
• I could rewrite it using unlimited length integers (I have a Javascript class that does that), but then you have to consider that the Numberphile people ran their calculation for days on a rack of computers with 512 cores, probably using something that runs faster than Javascript in a browser.
• My implementation involves finding the prime factors of z^3-k. This isn't linear in 'n' so the whole calculation isn't quite of order 'n', The video didn't discus this, maybe they have a clever way around it.
• As 42 modulo 9 is 6, the solution can only include 'x', 'y' and 'z' values that are '2' modulo '3', eliminating 2/3 of the possibilities for 'z'. This is explained on the page.

June 2021: I've recently given up full-time working so I'm hoping I'll have more time to maintain this site.

May 2021: I've redone the Newton Fractal page, hopefully making the user interface easier to use, especially when creating animations.

October 2019: I've redone the affine transformation pages using javascript, making them interactive. You can find the new page here.

June 2016: I've created an interactive version of the colouring-in pattern generator to replace the one that used to run on Google's AppSpot. You can find it here.

Featured Links

Neil's British Butterfly Photos

A collection of pages showing the results of Neil's mission to photograph all types of butterfly in Great Britain.

Neil's Index Page

Links to Neils computer generated artwork, photoes and other pages.

Artwork

These pages contains a selection of artwork by my brother Neil.

Holiday Photos

The image on the left shows the domes of the Eden project (as they were in 2005). Click to see the rest of my Holiday Photos.

Poems

A collection of poems by various relatives.

My PhD Thesis

This exists only on paper so I am slowly digitising it. The graphics were generated using programs I created.