Hi Philip, thanks for this very clear article & code. I used your code to simulate the 8 figure solution of three points with mass 1/3 and found it is stable for only 1 period, then it diverges into chaos. Is this normal (due to physics?) or is this due to numerical issues in plain python/numpy? (I used the initial conditions from Eq8 from https://arxiv.org/pdf/1707.06462.pdf at the bottom of page 2. thanks, Peter.


with Gurobi as the Model Solver

As a consultant to FlandersMake, I get to help the Flemish Industry with innovation. In particular, the FM CodesignS department offers mathematical optimisation experience to calculate optimal solutions to industry problems. We present two examples from different projects for different sectors, where we find optimal or near-optimal solutions to geometric puzzles.

Packing circles in the desert. This rectangular grid is not the most dense packing possible though. A hexagonal grid would be, assuming all radii are equal and space is unlimited. Bees know that … or do they?

Optimization of a Production Process

The manufacturing industry needs to plan (1) which resources (operators, robots and other tools) need to be deployed in the different production steps that are part of the entire production process towards one or more end products. (2)…


In 2002 Stephen Wolfram published his 1197 page book called “A New Kind of Science” which, generously, is completely available online here and from which we took some screenshots for this article.

Stephen Wolfram’s 1197 page thick book, published in 2002.

It’s 2020 now. So in retrospect, let’s shed a new light on this book. Did it influence or even revolutionize science? Here is my summary.

1. Complexity From Simplicity

In the book, Wolfram mainly proposes and defends the thesis that, with very simple rules, simple programs running those rules, a lot of visual patterns can be generated, some of them showing high complexity. To know if this is true, it is important…


If you have not yet studied topology before, you are in for some very nice surprises when you read David S. Richeson’s book “Euler’s Gem, The Polyhedron Formula and the Birth of Topology” . These revelations come not only in the shape of pretty pictures, but you’ll also get deeper insights and see new connections between mathematical disciplines.

You can of course pick up the book yourself if you want to avoid me spoiling your surprises. If you prefer to ride the fast lane, here follows my personal account of what I discovered while reading the book.

1 Regular Polyhedra or Platonic Solids

1.1 What are Regular Polyhedra?

From reading some…


“The person furthest from the fence is the only one feeling the electric shock!”… Debunked!

Who gets hurt the most in a series of people giving hands when the first person touches an electric fence? It’s also quite shocking what some people believe! :)

1. The Discussion

Dining last night with some neighbours, neighbour X came with the following story. “When we were kids”, he said, “we used to give hands to each other, left to right, in a long chain and then the first one used to touch an electric fence. None of us would feel any electric current or shock, but the last one felt it all!


Master Mind, Programmed on the HP-15C Calculator, Game On!

Once upon a time, long time ago, an exquisite-yet-useful object came into life, called the HP-15C. Millennials will not remember, even can likely not imagine that there was such a time as the pre-internet era, and a bit further back, even a pre-personal-computer era. The HP-15C was born in those years, 1982 to be exact. I got mine on September 23rd 1987, exactly the time I needed a more advanced scientific calculator, because that’s what it was.

Proud owner of the HP-15C, serial number 2707A25331, since 23/09/1987

I had been the almost-as-proud owner of a HP-11C already for 3 years, but my dad — who, being an electronics engineer, went…


Onion Peeling the very first Algorithm you have run in your head as a kid.

Which was the first algorithm you have ever run as a kid?…

If no cookie, cry? That has an if, and so depends on external input, so yes.

Something with a loop? While no cookie, keep crying? Ok, that counts.

Open box of toys, select toy, play and put back toy? That’s a multi-step algorithm, also toy-in-the-box dependent, be it without a condition or loop.

So now think of one with a loop, a condition, and multiple steps …

For me it was scoring my…


We take a chess puzzle challenge from 1985, published in Kijk magazine and — since we solved it to optimality before — devise a 2020, harder ‘concurrent’ version of it and solve it again to optimality, again with Integer Programming. Finding the optimal solution to the 1985 version of the puzzle was still doable for some humans (like chess masters). This 2020 version of the puzzle is a lot harder, so maybe grand masters only can manage. Mere mortals will find a feasible but maybe still far from optimal solution. …


1 Introduction

Photo by Piotr Makowski on Unsplash

1.1 The Challenge

As a 14 year old boy, so in the year 1985, my parents donated me a subscription to “Kijk”, which on its cover, in Dutch, called itself “Populair Wetenschappelijk Maandblad”. This translates as “popular science magazine” rather than “popular scientific magazine”, as it wasn’t so much trying to boast about its high subscription levels but was aiming to awaken interest with readers in general scientific and engineering topics. As far as I am concerned, it managed to do so.

One month, I must have been 15, Kijk published a challenge that intrigued me. It was about trying to place as…


1 The well known Sudoku Puzzle

1.1 Introduction

The Sudoku version played on a 9 x 9 squares board, is a well known puzzle game. The goal of the puzzle is for the player to put a digit from 1 to 9 in each free square, so that: (a) in every row of 9 squares, no digit occurs more than once (b) in every column of 9 squares, no digit occurs more than once (c) in every marked 3x3 subsquare, no digit occurs more than once. A given 9 * 9 Sudoku puzzle typically contains some squares that already contain a number from 1 to 9. …

Peter Sels

Interested in all things beautiful, especially Computational ones.

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