The *Voyager 1* and *Voyager 2* spacecraft have now left the solar system and will continue into deep space. How did we manage to send them so far? The *Voyager* spacecraft used gravity assists to visit Jupiter, Saturn, Uranus and Neptune in the late 1970s and 1980s. Gravity assist manoeuvres, known as **slingshots**, are essential for interplanetary missions. They were first used in the Soviet Luna-3 mission in 1959, when images of the far side of the Moon were obtained. Space mission planners use them because they require no fuel and the gain in speed dramatically shortens the time of missions to the outer planets.

### Slingshot Orbit to Asteroid Bennu

Published November 16, 2017 Irish Times Leave a CommentTags: Astronomy, Mechanics

### More on Moduli

Published November 6, 2017 Occasional Leave a CommentTags: Arithmetic, Number Theory

We wrote last week on *modular arithmetic*, the arithmetic of remainders. Here we will examine a few other aspects of this huge subject. Modular arithmetic was advanced by Gauss in his *Disquisitiones Arithmeticae*. In this system, number wrap around when they reach a point known as the modulus. Numbers that differ by a multiple of the modulus are called congruent. Thus 4, 11 and 18 are all congruent modulo 7.

### Modular Arithmetic: from Clock Time to High Tech

Published November 2, 2017 Irish Times Leave a CommentTags: Arithmetic, Time measurement

You may never have heard of *modular arithmetic*, but you use it every day without the slightest difficulty. In this system, numbers wrap around when they reach a certain size called the modulus; it is the arithmetic of remainders [TM126 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Modular Arithmetic: from Clock Time to High Tech’

Sometimes the “obvious” answer to a mathematical problem is not the correct one. The case of Malfatti’s circles is an example of this. In an equilateral triangle of unit side length, we must draw three non-overlapping circles such that the total area of the circles is maximal.

The solution seems obvious: draw three identical circles, each one tangent to two sides and to the other two circles (above figure, left). This is certainly the most symmetric arrangement possible. However, it turns out not to be the optimal solution. There is another arrangement (above figure, right) for which the three circles have greater total area.

### Learning Maths has never been Easier

Published October 19, 2017 Irish Times Leave a CommentTags: Education, Ireland, Recreational Maths

Maths is hard: many people find it inscrutable and have negative attitudes towards maths. They may have bad memories of school maths or have been told they lack mathematical talents. This is unfortunate: we all have the capacity to apply reasoning and logic and we can all do maths. Given the vital role mathematics plays in modern society, there is an urgent need to help young people to become more numerate and comfortable with mathematics. With a wealth of online resources, learning maths has never been easier. [TM125 or search for “thatsmaths” at irishtimes.com].

### Pedro Nunes and Solar Retrogression

Published October 12, 2017 Occasional Leave a CommentTags: Astronomy, Time measurement

In northern latitudes we are used to the Sun rising in the East, following a smooth and even course through the southern sky and setting in the West. The idea that the compass bearing of the Sun might reverse seems fanciful. But in 1537 Portuguese mathematician Pedro Nunes showed that the shadow cast by the gnomon of a sun dial can move backwards.

Nunes’ prediction was counter-intuitive. It came long before Newton, Galileo and Kepler, and Copernicus’ heliocentric theory had not yet been published. The retrogression was a remarkable example of the power of mathematics to predict physical behaviour.