The behaviour of quantum objects – tiny particles like electrons or photons – contradicts our everyday intuition. Sometimes they behave like solid particles, colliding and bouncing off each other. Sometimes they display wavelike properties such as diffraction – the same phenomenon that creates rainbows.

The laws of this amazing branch of physics called “Quantum Mechanics” resolve this apparent mismatch by including all the information about the tiny objects in a mathematical description called the wave function. It turns out that from a mathematical point of view, waves and particles are two sides of the same coin.

A wave function is said to describe all the possible states that the object can be in – for example, all possible locations where we could observe it. The evolutions of the wave function over time are governed by a famous equation called the Schrödinger equation.

What you see above is not just a random animation but a quantum system created to provide you with a glimpse of the quantum world. The movements of the shape are derived from solutions to the Schrödinger equation, using the letters of your name to determine the possible states.

Different combinations of letters lead to different shapes and behaviours. Feel free to search for interesting quantum systems by trying out the names of friends and relatives.

Season’s Greetings and Happy New Year from everyone at ThinkTank Maths!

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Modern physics and astronomy have revealed that our relationship with the stars is even closer than the ancients could have imagined. Every atom in our bodies and the Earth itself was formed in the nuclear furnace of a nova explosion — the death of a star fusing hydrogen and helium into the heavier elements of the periodic table. We now understand that stars themselves have a lifecycle, and instruments like the Hubble Space Telescope allow us to look at them right in the eye.

We use a mixture of geometry and optimisation to discover a star pattern that matches the letters of your name as closely as possible. To allow you to have a personal relationship with at least one star, we have constrained the algorithm to ensure that at least one star imaged by Hubble is contained in the constellation. It is highlighted in the image above — click the image to read more about your star.

The Astronomy Nexus HYG Database is used to plot star locations, brightness and colours.

Image credit: NASA, ESA and The Hubble European Space Agency Information Centre.

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The pattern you see is generated by little creatures called Langton ants, originally invented by Christopher Langton, one of the founders of the field known as artificial life. The letters of your name encode the starting positions of three ants, who then weave their trails together to produce a unique pattern.

You can use the form to create patterns for your friends. Clicking on the image will give you a large version that you can print out and use as wrapping paper. An A4 sheet should be just the right size for a small meaningful gift!

A Langton ant is a very simple critter: it starts out by heading in a selected direction and looks at the colour of the square it lands on. If the ant lands on a coloured square, it paints it white and turns 90 degrees to the left. If it lands on a white square, it paints it with its colour and turns 90 degrees to the right.

Although the rule is very simple, a Langton ant can generate quite complex behaviour. Initially it behaves very chaotically. But after a while it may shoot off in one particular direction, as if it has suddenly decided what to do — building a diagonal band structure called a “highway”.

One of the biggest unsolved mysteries about Langton ants is that they always seem to create a highway, even if their surroundings are initially peppered with black squares. But no mathematical proof that this always happens exists!

Langton’s ants illustrate the idea that simple mathematical rules can underlie complex behaviour in nature. Especially in the middle of the stressful holiday period, it is good to keep in mind that beauty can often be found in simplicity…

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Snowflakes are a form of fractal, complex mathematical shapes that are often found in nature.

The fractal pictures here were grown using growth rules that are repeatedly applied. The technique is known as Lindenmayer-Systems (or L-Systems for short) after biologist Aristid Lindenmayer who used them to study plants.

The rules can be very simple - it’s the repetition at different scales that creates the intricate form of fractals. From simple rules, great oak trees grow.

Click on the snowflake above to see them regrow. You can also make snowflakes for your loved one, football team, goldfish, etc. by entering a word here:

Happy holidays! See you in the new year.

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