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Vibrating strings and infinite series

Time to dive into some mathematics again – today we celebrate the birth of British mathematician Brook Taylor (18 Aug 1685 – 29 Dec 1731).

Taylor is best known for ‘Taylor’s Theorem’ and the ‘Taylor series’, a mathematical method for expanding functions into infinite series. In 1715, he published a groundbreaking work Methodus Incrementorum Directa et Inversa, which introduced a new branch of mathematics that became known as the ‘calculus of finite differences’.

Using finite differences, Taylor was able to mathematically express the movement of a vibrating string, reduced to mechanical principles.

The above work also contained what became known as Taylor’s Theorem – this blog is neither the time or place to even try and go into the details of the theorem, but suffice to say it is a pretty significant mathematical construct. Despite being introduced in his 1715 publication, it wasn’t until almost 60 years later that it’s value was fully recognised – in 1772 the great mathematician Joseph-Louis Lagrange termed it ‘the main foundation of differential calculus’.

Taylor employed the calculus of finite differences to mathematically express the movement of a vibrating string.
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Besides being one of the great mathematicians of all time, Brook Taylor was also a keen artist, with one of his particular interests being the principles of perspective – he wrote an essay called “Linear Perspective” on this subject, which also included the first general introduction of the concept of vanishing points.

So to celebrate this day, how about strumming a guitar while staring off into the vanishing distance… or painting perspectives while listening to some soothing guitar (the Majestic Silver Strings, perhaps)… 🙂

Sounds like a good day to me!

Some paradoxical fun on Infinity Day

Today is 8 August, the eighth of the eighth, 8-8.  Or, if you turn it on it’s side, a couple of infinity signs stacked on top of each other… Yep, it’s Infinity Day!

The concept of infinity refers to something that is without limits. It has application in various fields such as mathematics, physics, logic and computing. Infinite sets can be either countably infinite (for example the set of integers – you can count the individual numbers, even though they go on forever) or uncountably infinite (e.g. real numbers – there are also infinitely many of them, but you cannot count the individual numbers because they are not discreet entities).

The wonderful ‘Numbers’ sculpture (artist: Anton Parsons), situated in mid-town, Palmerston North, NZ. While this sculpture does not explicitly deal with the concept, it always reminds me of infinity – from it’s resemblance to an infinity symbol, to the continuous cycle of random numbers. A definite favourite of mine.
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Since infinity is really, really big – incomprehensibly so – it can lead to some amusing paradoxical scenarios; things that don’t make sense, by making complete sense.

An example of this is the Galileo Paradox, which states that “Though most numbers are not squares, there are no more numbers than squares.” In the set of positive integers, for example, the squares (1, 4, 9, 16, 25…) occur with much less frequency than the non-squares (2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24…). So there must be less of them, right? At the same time, however, every number is the square root of some square number, so there’s a one-to-one relationship between numbers and squares. Thus there cannot be more numbers than squares…

And therein lies the paradox… In a finite set, the square numbers would indeed be a minority, but in an infinite set, this is no longer the case.

Cool, right?

OK, here’s another fun one – Hilbert’s paradox of the Grand Hotel, presented by the German mathematician David Hilbert. This one states that “If a hotel with infinitely many rooms is full, it can still take in infinitely more guests.” In other words, let’s assume we have a hotel with a countably infinite number of rooms, all of which are occupied – in other words, each room has a guest in it. Then, since all the rooms are occupied, there can be no more room for new guests, right? Not so – simply move the guest in room 1 to room 2, move the guest in room 2 to room 3, and so on, to infinity. Then room 1 becomes available, so we can accommodate the new guest. And we can repeat this process indefinitely, so a hotel with an infinite number of fully occupied rooms can still accommodate an infinite number of new guests.

And on that note I will leave you to contemplate the concept of infinity. Don’t worry if it’s complicated – you have an infinite amount of time before the end of the day. Before you reach the end of the day, you have to reach the midway time between the current time and that time. And before you reach that midway time, you have to reach the midway between the current time and that time. And so on, to infinity…

It’s going to be a long day!

Disposable products; iconic designs

It’s time to celebrate the disposable, to honor the expendable. On this day, 29 July 1914, Baron Marcel Bich was born – the man who built his business empire on his brilliantly designed and cleverly marketed throwaway Bic pens, lighters and razors.

Whether you’re absentmindedly doodling, or jotting down a shopping list, chances are there will be a Bic pen at hand.
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Bich, together with his friend Edouard Buffard, acquired an old factory where they proceeded to produce a low-cost, reliable ballpoint pen that took the world by storm. Bich based his design on a ballpoint pen manufactured in Argentina by László Bíró, but introduced many design innovations into his pen. His manufacturing innovations also allowed him keep the production costs of the pen down.

Bich invested heavily in advertising, and based on a recommendation from his advertising company, shortened the name of the pen to Bic. The Bic pen was an unprecedented success both in Europe and later in the USA and the rest of the world, and with its slogan of “Writes first time, every time”, it became the main driving force in changing the worldwide market from costly fountain pens to disposable ballpoints.

The industrial design excellence of the the Bic Cristal pen (with its transparent polystyrene barrel and classic pencil shape) has been acknowledged by the New York Museum of Modern Art, where it has been included in its permanent collection. Some of the innovative features of the design is its transparent barrel that shows the ink level of the pen, and a small hole in the barrel to ensure equal air pressure inside and outside the pen.

The Bic pen is said to be the world’s most efficient pen, able to write a line over 2km in length. After more than half a century, the Bic pen remains a top seller, with more than one hundred billion having been sold internationally.

Over time, Bich’s company, Société Bic, diversified its business to include the Bic lighter (1973), followed by the Bic shaver (1976). As in the case of the Bic pen, the Bic lighter has become an icon of modern industrial design, and has also been included in the New York Museum of Modern Art’s permanent collection. The design of the lighter has remained virtually unchanged since its inception.

Getting back to the Bic pen, one of the most impressive artists I’ve come across lately is Juan Francisco Rosas, who creates huge, incredibly detailed, photo-realistic artworks using nothing but Bic pens – mindblowing, and further testament to the iconic, throwaway Bic pen.

Love your camera on Camera Day

Me and my camera; my camera and me.

The photographer and his camera – where does one start and the other end? How much of what you see in an image is down to the brilliance of the photographer, and how much can be attributed to the technical abilities of his photographic tools?

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I am, generally speaking, a supporter of the school of thinking that a great artist will produce great art irrespective of his tools. I have seen photos taken on mobile phone cameras that are significant artistic achievements, and there are movements in photography who go to great lengths to show how great art can be produced by technically “bad” equipment. The Lomographic Society International, for example, owns galleries, etc, showcasing photographs taken with very low-tech LOMO cameras. LOMO, a former Russian state-owned camera manufacturer, produced 35mm compact cameras that have become iconic for producing unique, sometimes blurry images, at times with light leakage, and various other “faults”.

On the other hand, particularly in technical fields of photography, the camera plays a critical role in enabling the photographer – think about fields like macro photography, for example. In some ways the camera also dictates the photographers’ approach to the subject. For instance, the time and effort required to set up a large format view camera to photograph a landscape, will almost by default result in a different stylistic approach to the subject compared to, say, a photo snapped with a mobile phone.

Given my current context (photographing science, technology and industry) my “weapon of choice” is my Nikon D3 DSLR, with a range of lenses for different applications, and I have to admit I love this bulky machine – its reassuring weight, ever willing, ever ready for anything I may throw at it.

That is not to say I am not eagerly eyeing the D4 and even the D800, not to mention the wonderful, iconic Leica M9. And don’t even get me started on some of the glorious medium format cameras out there, just waiting for me to take them in my arms!

On the other end of the technology scale, I’ve recently started playing around with pinhole photography again – in a sense this still remains to me the most magical, wonderfully rewarding field of photography. But more on that in a future post.

Whether you photograph with a mobile phone or a Hasselblad, today is Camera Day – the day to show some special appreciation for your camera, and to take it out and capture the world around you. Wherever you may be – have fun.

Going Green on Sewing Machine Day

It’s Sewing Machine Day, the day to dust off and celebrate the trusty sewing machine, unsung hero of the industrial revolution.

Tracking the invention of the sewing machine is like reading the script of a sensational TV drama – a juicy tale of betrayal and deceit, industrial sabotage, stolen ideas and legal battles. The first patented design dates back to Thomas Saint in 1790, followed by various iterative improvements, but the first commercially viable design came some 60 years later, courtesy of Isaac Merritt Singer who combined ideas from various previous designs.  Unfortunately he borrowed a bit too heavily from a patent by Elias Howe, who promptly took him to court for patent infringement, winning the case and forcing Singer to pay him a fee for every sewing machine sold.

Despite its checkered past, the sewing machine quickly gained popularity, vastly improving efficiency in the clothing and fabric industries. As such it played a key role in the industrialisation of the manufacturing sector.

By the early 20th century, the household sewing machine was a common appliance in almost every home. Most families had one in the house – used to sew new clothes, do alterations, or to mend worn or damaged clothes. This golden era of home sewing lasted almost a century, but with the proliferation of mass produced, super cheap clothes from giant producers like China, the trusty home sewing machine seems to be facing extinction.

Scientific sewing? Even a basic sewing machine offers many more creative opportunities than just shortening a pair of pants.
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So, with today being Sewing Machine Day, perhaps it is high time to dig out the old sewing machine, give it a good dusting and reacquaint yourself with the possibilities it offers. Or if you don’t have one, check out the secondhand stores or online auction sites – perfectly functional machines are going for a song.

Not only is home-sewing an excellent outlet for your inner Chanel or Versace – it is also a positive step towards green living.  Sure, it may be quicker and easier to go out an buy a new $5 t-shirt, $30 jacket or a pair of $20 jeans, but Mother Nature will be so much better off if you rather patch up the elbows and cuffs on your old jacket, mend those torn jeans, and wear them for a while longer.  The environmental impact of a few minutes of home sewing is negligible compared to the impact of creating a new garment in some industrial sweat-shop.

Like 150 years ago, when the sewing machine became a key player in the industrial revolution, it now has the potential to become a surprise hit in the green revolution.

Happy Birthday to Donald, the Mathmagical Duck

On this day back in 1934, the world was introduced for the first time to Donald Fauntleroy Duck, when he made his first appearance in the cartoon “The Wise Little Hen”.  The excitable, short-tempered but lovable duck went on to become one of the world’s favourite cartoon characters, and the de facto mascot for The Walt Disney Company.

So what does this have to do with science, you may ask?  Well, Donald Duck often appeared in cartoons touching on traditionally non-cartoony subjects like politics, religion and, yes, science and mathematics.

Donald Duck is no stranger to the magical world of mathematics
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In the 1959 cartoon “Donald in Mathmagic Land“, Donald accidentally stumbles into a magical land of mathematics – a land where trees have square roots, streams are filled with numbers, and a geometric bird recites the digits of Pi.

In the cartoon, Donald is shown that mathematics is not just for eggheads (his original opinion) and that it’s actually useful and even exciting.  He meets, and plays some music with, Pythagoras and his secret band of Pythagorians, where he discovers that mathematics form the basis of musical scales.  From Pythagoras he also receives a pentagram, through which he goes on to learn about the golden section and the golden rectangle, and how these appear in architecture (the Parthenon, etc) and art, such as the Mona Lisa.

Donald discovers that the golden section also shows itself in the human body and in nature, in flowers, plants and shells.  He learns that mathematics even applies to sports and games, such as chess, baseball, basketball and billiards.

A cool bit of intertextuality in the cartoon comes through the inclusion of some themes from “Through the Looking Glass” by Lewis Carroll, who was himself also a mathematician.

In this cartoon Donald Duck, and with him millions of children, are introduced to the wonders of mathematics in a fun and humorous way, and the cartoon closes with the wonderful Galileo quote:
“Mathematics is the alphabet with which God has written the universe”.

Here’s to you, Donald – Happy Birthday, you grumpy old duck!

It’s Drawing Day – time to get creative

Happy Drawing Day, everyone! (Or perhaps I should say Happy Pencil Day – there seems to be some disagreement on the correct title of this day.)

To participate is easy – simply grab a pencil and start drawing, and most importantly, share your artwork with as many as possible. And don’t fret if you think you’re not artistic – this day is not so much about great art as it is about nurturing your creative side, and about sharing.

You can share your doodles with friends, or you can even go global and upload your drawings on the Drawing Day website.

Come on, drop that daily chore, and get drawing – it’s good for your soul.

Despite the apparent ease of drawing with a pencil, artists often prefer to work with graphite sticks as it allows a greater range of creative expression.
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By the way (this is Sciencelens, after all), did you know that the “lead” inside a pencil that makes it write is not lead at all, but a mix of graphite (a type of carbon), clay, wax, and chemicals?  Coloured pencils work the same, except that they have colour pigment added to the mix instead of graphite.

Pencils are manufactured by mixing ground graphite (or colour pigment), clay and water, and squeezing out this mixture into long spaghetti-like strings.  These are cut into pencil lengths, baked, and then covered with a wax coating to make them write smoothly.

So how do they get them inside the wood? The wood around a pencil may look solid, making it appear as though a hole was drilled through the wood to insert the graphite, but that sounds like some extreme manufacturing.  In fact, most wooden pencils are made from blocks of wood cut into slats.  Grooves, half as deep as the graphite string, are cut into the slats, and the graphite strings are placed in these grooves. Another grooved slat is glued on top of the first, encasing the graphite in the wood.  The slats are then cut into individual pencils, sanded and painted to give it the appearance of a solid structure.

Pencils are sharpened to reveal the graphite inside, and when you write, fiction causes a small amount of the graphite from the core of the pencil to be deposited on the paper, creating your images or words.