## Mathematics Key to 4D Printing

Although some of us are just getting our heads around the amazing potential in 3D printing, the next step is already on the horizon.    A leading mathematician has started working on the formulas required to step into an extra dimension!

Three D printing is already revolutionizing all sorts of areas from manufacturing, medicine to science and engineering.  It’s now fairly simple and inexpensive and has the potential to create all sorts of intricate objects quickly and cheaply.   There are printer parts in our machines and indeed people are having printed body replacement parts transplanted into their bodies with great success.

However there is always a next step, and now mathematicians are working on taking us in to the world of 4d printing.  Just to clarify we are talking about the possibility of fabricating objects with a programmable shape over time.  It’s always been theoretically possible however no-one had really starting looking at working through the complexities involved.

This seems to be changing as Professor Pasquale Ciarletta from Milan has just published a paper in ‘Nature Communications’ where he has started working through the numbers about a specific problem with this.  The professor has been focusing on how to control the sudden nucleation of localised furrows in the soft solids produced in 3d printing.

The advantages and possibilities of these developments may not be initially apparent.  However in addition to the advantages to the field of engineering there is huge potential to have the ability to design and print objects which can morph over time.  The paper related the development to the field of development biology as particular interest.  Here we could look at things like tissue morphogenesis and other areas such as issues in the brain or tumour control.

Ciarletta has acknowledged that there are great complexities behind making this work.  There has already been lots of experimental investigation of the issues involved – the physics behind the concept of ‘creasing’ being particularly challenging.  His study proposes a unique mathematical approach to predicting the experimental conditions required to trigger the onset and how creases change over time.  This is the key to being able to control their appearance on a specific scale and ultimately to be able to print them in 4d.

There are parallel advancements being made in the area of 3d printing too.  You can already sit down and watch the football on Match of the Day live like this on a completely 3d printed television set.  It is also now possible to edit specific printed objects after they have been created.  This is achieved by repeatedly changing the colours of 3d printed objects after thy have been printed.

The concept is currently being developed under the name ColorFab and it involves using a specially created 3d printable ink which can actually change colour under certain conditions – primarily after being exposed to UV light.  This of course has a time delay currently estimated at around 20 minutes, however the researchers are hoping to improve on this substantially in further development.

Further Reading: Available on British TV

## The Life of John Napier

John Napier was born in the year of 1550 at Merchiston Tower in the City of Edinburgh in Scotland.  He dies at the age of 67 on April 4, 1617  in his home town of Edinburgh, Scotland.  If you were to choose a top ranking table of world mathematicians then Napier would almost certainly feature in it.   He spent much of his life including his work as an Alma mater at the University of St Andrews where he was also a Doctoral adviser.  Of course as any school child will probably  know he is most famous for that little book of Logarithms.  However he was a man of many talents and he is also famous for such things as Napier Bones and the introduction of the Decimal Notation. John Napier of Merchistonalso signed as Neper, Nepair, nicknamed Marvellous Merchiston, was a Scottish landowner known as mathematician, physicist, and astronomer. He was also actually the eighth Laird of Merchiston. His Latinized name was Joanne Nepero or Joannis Neperi.  However we know him know as John Napier and is certainly best known as the discoverer of logarithms.

John Napier was featured in the recent BBC’s history of mathematics which you can still get on the BBC iPlayer for a few weeks.  If you’re outside the UK then this article entitled How to Watch UK TV from USA should help, you just need to hide your location and it all should work perfectly.

He also invented the so called Napier’s bones and made common the use of the decimal point in arithmetic and math. Napier’s birthplace, Merchiston Tower in Edinburgh, Scotland, is now part of the facilities of Edinburgh Napier University. After he died from the effects of gout, Napier’s remains were buried in St Cuthbert’s Church, Edinburgh. Napier’s father was Sir Archibald Napier of Merchiston Castle, and his mother was Janet Bothwell, daughter of the politician and judge Francis Bothwell, Lord of Session, and a sister of Adam Bothwell who became the Bishop of Orkney. Archibald Napier was 16 years old when John Napier was born.

As was the common practice for members of the nobility during that time, John Napier didn’t enter schools until he was 13. He didn’t stay in school very long, however. Little is known about those years, where, when, or with whom he might have studied, although his uncle Adam Bothwell wrote a letter to John’s father on 5 December 1560, saying I pray you, sir, to send John to the schools either to France or Flanders, for he can learn no good at home, and it is believed that this advice was followed. In 1571, Napier, aged 21, returned to Scotland, and purchased A castle in Gartness in 1574.

On the death of his father in 1608, Napier and his family moved to Merchiston Castle in Edinburgh, where he resided the rest of his life. Advances in maths – His work, Mirifici Logarithmorum Canonis Descriptio contained fifty seven pages of explanatory matter and ninety pages of tables of numbers related to natural logarithms. The book also has a fantastic discussion of theorems in spherical trigonometry, commonly known as Napier’s rules on circular parts.

Jim Hamilton. Watch BBC iPlayer in Spain

## Deciphering the Fibonacci Sequence

You may have heard the expression, it’s certainly one of the most famous mathematical concepts – show what’s involved with he Fibonacci Sequence?.

The thirteenth Century Italian Leonardo of Pisa, better known from his nickname Fibonacci, was possibly the most gifted Western mathematician of the Middle Ages. Little is known of his life except that he has been this son of a customs official and, as a young child, he traveled to North Africa along with his father. It was here that he first heard about the Arabian mathematics. On his return to Italy, he helped to spread this knowledge through Europe, putting so in motion a rejuvenation in Western mathematics, which had lain largely dormant for centuries throughout the Dark Ages. Especially memorable was that in 1202 he wrote a very influential book called Liber Abaci, wherein he encouraged using the Hindu Arabic numeral system. Here he used the book to describe its lots of advantages for retailers and mathematicians alike across the clumsy system of Ancient Rome numerals then in use in Europe.

Despite its apparent benefits, uptake of this system in Europe was slow, and Arabic numerals were banned within the town of Florence in 1299 on this pretext they were easier to falsify than Ancient Rome numerals. Yet, common sense finally prevailed and the new system has been adopted through Europe by the fifteenth century, making the Ancient Rome system obsolete. The flat bar notation for fractions was initially first utilized in this work. Fibonacci is best known, however, for his debut in Europe of a certain number sequence, that has since become known like Fibonacci Numbers or this Fibonacci Sequence.

There are lots of explanations of this, which although initially sounding quite complicated is actually very simple.  One of the most straightforward ones I’ve heard is to be found on the BBC’s History of Maths programs – you can access this and any other UK TV abroad, from here.

He discovered this sequence – this first recursive numerical sequence known in Europe – although considering a practical problem in this Liber Abaci involving this growth of a hypothetical population of rabbits based about idealized assumptions. He noted that, after every monthly creation, this number of pairs of rabbits increased from 1 to 2 to 3 to 5 to 8 to 13, etc. Soon he also recognized how a sequence progressed by adding this previous two terms, a sequence which could theoretically extend indefinitely.

The arrangement, which had really been known to Indian mathematicians since this sixth Century, has many intriguing mathematical properties, and a lot of this implications and relationships of this sequence weren’t discovered until several hundreds of years after Fibonacci’s death. For example, this sequence regenerates itself in some surprising ways: every 3rd F number is divisible by 2, every 4th F number is divisible by 3, every 5th F number is divisible by 5, every 6th F number is divisible by 8, every 7th F number is divisible by 13, etc.