Archive for November, 2017

Disentangling Loops with an Ambient Isotopy

Linked-Loops-1

Can one of these shapes be continuously distorted to produce the other?

The surface in the left panel above has two linked loops. In the right hand panel, the loops are unlinked. Is it possible to continuously distort the left-hand surface so as to unlink the loops and produce the right-hand figure? This seems impossible, but intuition is not always reliable. 

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A Symbol for Global Circulation

The recycling symbol consisting of three bent arrows is found on bottles, cartons and packaging of all kinds. It originated in 1970 when the Chicago-based Container Corporation of America (CCA) held a competition for the design of a symbol suitable for printing on cartons, to encourage recycling and re-use of packaging materials.

Recycling-Symbol-2Verions

Original (Moebius) and a variation (3-twist) of the universal recycling symbol.

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Slingshot Orbit to Asteroid Bennu

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.

OSIRIS-REx

Artist’s impression of OSIRIS-REx orbiting Bennu [Photo Credit: NASA]

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More on Moduli

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.

Z12-Addition-Table

Addition table for numbers modulo 12.

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Modular Arithmetic: from Clock Time to High Tech

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].

Clock-Arithmetic

We use modular arithmetic for timekeeping with a 12-hour clock [Image Wikimedia Commons]

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