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Celebrating August Möbius and his crazy, twisty Möbius strips

Today we celebrate the birth of August Möbius (17 Nov 1790 – 26 Sep 1868), German astronomer, mathematician and author, and the man who introduced the concept of the Möbius Strip, one of the more interesting objects in mathematical topology.

Getting into the mathematics behind the Möbius strip is beyond the scope of this blog, but suffice to say that a Möbius strip is a two dimensional surface with only one side. Mathematically it is said to be a non-orientable object.

The one-sided, non-orientable Möbius strip.
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A physical model of a Möbius strip can be constructed by taking a long, rectangular strip of paper, giving it a 180 degree half-twist and joining the ends together to form a loop.  Starting at any point on the surface of the resultant loop, you can draw a line along its centre, passing along both sides of the strip and eventually returning to your original starting point, without once lifting your pen.

Moreover, cutting the Möbius strip along this centre line with a pair of scissors will not result in two separate loops, but rather a single loop, double the length of the original, and with two full twists in it. Thanks to having an even number of twists, the resultant loop will not be a Möbius strip anymore – if you draw a line along its centre, you will remain on one side of the strip, and the line will not traverse both sides.

If you repeat the cutting exercise and cut this new loop along its centre, you will end up with two separate loops, wound through each other, and each having two full twists.

Getting back to our original Möbius strip, another ‘trick’ is to make your cut 1/3 from the one edge. If you keep cutting until reaching your original point, you end up with two loops – one new, thin Möbius strip (the center third of the original), which will be the same length as the original, and one thin loop with two twists and twice the length of the original.

In the same way that a Möbius strip (with 1 half-twist) when bisected, results in a new loop with 2 full twists, loops with more half-twists will lead to different end results. Generally speaking, a loop with N half-twists (where N is an odd number), when cut along its centre, becomes a loop with N+1 full twists.

These crazy, twisty loops are not only amusing things to cut up and play with – they have lots of useful practical applications. When designed in the shape of a Möbius strip, conveyer belts can be made to last longer, as wear and tear will be shared between the two sides. Similarly, typewriter ribbons can be more effective when shaped with a half-twist. Möbius strips, with their unique geometry, also occur in physics and chemistry, and even have applications in electric circuitry.

Amusing things indeed, and surely well deserving of their own special day!

Frederick Bowen and the fascinating ferns

Today we celebrate the birthday of one Frederick Orpen Bower, born 4 November 1855. Bower, an English botanist, was famous for his studies of the origins and evolution of primitive land plants such as ferns and mosses. In his research, published in books like Origin of a Land Flora (1908), Ferns (1923-28), and Primitive Land Plants (1935), Bower concluded that these plants had evolved from algal ancestors.

Ferns, the subject of much of Bower’s research, is a fascinating plant in many ways. Unlike mosses, ferns are vascular plants with stems, leaves and roots. Unlike other vascular plants, however, they reproduce via spores rather than flowers and seeds.

The shape and structure of young fern fronds can provide endless visual fascination.
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While we typically associate ferns with moist, shady areas, they can be found in a wide variety of habitats, from desert rocks to mountains to water bodies. They can prosper in marginal areas where many flowering plants fail to grow. This tenacity make certain fern species serious weeds, such as the Bracken Fern in Scotland, and the giant water fern, one of the world’s worst aquatic weeds.

From a biochemical point of view, ferns can be particularly useful in fixing nitrogen from the air into compounds usable by other plants, and for removing heavy metals from the soil.

Another beautiful young frond, appearing almost animal-like.
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Patterns and motives based on fern shapes are popular in traditional art and culture. In New Zealand, for example, the silver fern is a very prominent cultural symbol, featured often in traditional art. The leaf of the silver fern is also the proud emblem of many of the country’s top sporting teams such as All Blacks (rugby) and Silver Ferns (netball).

On a more esoteric level, ferns are a wonderful embodiment of mathematics in nature, with young fern fronds unrolling in stunning Fibonacci spirals. The patterns and structure of fern leaves can also be simulated by means of iterative mathematical functions.

Definitely a plant that fascinates on many levels. No wonder Frederick Bowen committed his life to studying these wonderful plants!

Having some fun on ‘Look for Circles Day’

Today, they say, is Look for Circles Day. The idea of the day, aimed mostly at entertaining the young ‘uns, is to see how many circles you can spot. We come across hundreds of circles each day, so in addition to the obvious ones, try to look for circles in unexpected places, and even look for implied circles (where objects occur, or are placed, in such a way that they form a circle).

A science lab can be an abundant source of circles.
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Liquid droplets splattering – a stunning symphony of circles and spheres.
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Look for Circles Day is a great opportunity to entertain kids of all ages with one of the most interesting shapes in nature, and to teach them some maths and geometry in the process. Here are some interesting circle facts:

  • A circle is an infinite set of points on a plane that are all the same distance from a specific, predefined point.
  • Of all shapes with a given perimeter, the circle has the largest area. Or to put it another way, circles have the minimum possible perimeter for a given area.
  • They are the only single-sided shape with an area.
  • A circle with an infinitely large radius is a straight line (there’s a hint to give you the upper hand when searching for circles!)
  • A circle can be split in two identical halves in an infinite number of ways, or stated more formally, a circle has an infinite number of lines of symmetry.
  • The circumference and perimeter of a circle are related through the mathematical constant pi, or π – a very interesting number in itself, as we discussed previously.
  • A solid circle is a wheel, and we all know what useful invention the wheel was!
  • Apparently, according to research done by the Max Planck Institute for Biological Cybernetics in Tübingen, when we have no way to navigate – for example in a thick fog, or a moonless night – we tend to walk in circles (literally).
  • There is a form of divination called ‘gyromancy’ where people are made to walk in a circle until they fall down from dizziness, and the location where they fell is then used to predict future events.

Yep, as I said – circles are amazing things… Happy circle spotting!

Get ready for the Molar Eclipse!

Today is October 23rd; 10-23.

If you’re involved in any way with the field of chemistry, 10 to the power 23 should ring a bell – Avogadro’s constant, 6.02 x 10^23, the number of particles in a ‘mole’ of a substance, is a basic quantity in chemistry.  To quote wikipedia:
“The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities (e.g., atoms, molecules, ions, electrons) as there are atoms in 12 grams of pure carbon-12, the isotope of carbon with atomic weight 12. This corresponds to a value of 6.02214179×10^23 elementary entities of the substance. It is one of the base units in the International System of Units, and has the unit symbol mol.”

In celebration of the above, Mole Day is celebrated on October 23rd, from 6:02 am to 6:02 pm. Generally the day involves activities that represent puns on ‘mole’ or ‘avogadro’. For 2012, the official theme for Mole Day is ‘Molar Eclipse’.

Mole Day was created to foster an interest in chemistry.
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So how do you celebrate this day? Well, you can go on a molercoaster ride, or make a meal with avogadro dip or guaca-mole sauce. Have a Rock ‘n’ Mole party. Whatever you do, involve as many of your friends as possible – the mole the merrier! 🙂

I’m sure readers of this blog will be able to come up with many funnier, punnier activities to engage in on Mole Day – any suggestions?

A celebration of the mind with Martin Gardner

Today we celebrate the birthday of Martin Gardner. Gardner, born in 1914, was a science writer specialising in the field of recreational mathematics, but also covering topics like magic, literature, scientific scepticism, philosophy and religion.

His most famous contribution to the popularisation of science and mathematics was the ‘Mathematical Games’ column he wrote for Scientific American for 25 year, between 1956 and 1981. Many of these columns have been collected and published as a series of books starting with ‘Mathematical Puzzles and Diversions’, first published in 1956.

So popular was Gardner (who passed away recently in 2010) that, in honour of his life and work, his birth date of October 21 has come to be known as the ‘Celebration of the Mind’. The Gathering for Gardner Foundation aims to use this day to “celebrate Martin’s life and work, and continue his pursuit of a playful and fun approach to Mathematics, Science, Art, Magic, Puzzles and all of his other interests and writings.” They encourage people to get together on the day to share mathematical or logic puzzles, paradoxes, illusions and magic tricks, or just in general engage in activities that gets the logical side of your brain buzzing.

One of the classic Gardner puzzles. Rearrange a triangle made up of six coins into a hexagon, by moving one coin at a time, with each move leaving every coin touching at least two others, as seen in the pictures.
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Examples of Gardner’s puzzles are readily available online. To immerse yourself in his world of puzzles, start with the classics on puzzles.com. There’s also an edition of the College Mathematics Journal dedicated to Martin Gardner available for free from the Mathematical Association of America.

As a teaser, here’s a few from the Gathering for Gardner website:

  • A woman either always answers truthfully, always answers falsely, or alternates true and false answers.  How, in two questions, each answered by yes or no, can you determine whether she is a truther, a liar, or an alternater?
  • You are in a room with no metal objects except for two iron rods. Only one of them is a magnet. How can you identify which one is a magnet?
  • Mr. Smith has two children.  At least one of them is a boy.  What is the probability that both children are boys?  Mr. Jones has two children.  The older child is a girl.  What is the probability that both children are girls?

Besides his interest in recreational mathematics, Gardner was an outspoken scientific sceptic with an uncompromising attitude towards pseudoscience. In his books he commented critically on a range of ‘fringe sciences’, from creationism to scientology to UFOs and the paranormal. This earned him many fans, but also many antagonists, particularly individuals operating in these fringe domains. While critical of conservative Christianity, Gardner considered himself a ‘fideistic deist’, believing in a god as creator, but critical of organized religion.

Gardner was also a leading authority on Lewis Carroll. He published ‘The Annotated Alice’, an annotated version of ‘Alice’s Adventures in Wonderland’ and ‘Through the Looking Glass’, where he discussed and explained the riddles, wordplay and literary references found in Carroll’s works. He also produced similar annotations of GK Chesterton’s works, ‘The Innocence Of Father Brown’ and ‘The Man Who Was Thursday’

Yet his most enduring contribution remains in the field of recreational mathematics and puzzles. It has famously been said that, through his writings on puzzles, tricks and paradoxes, he “turned thousands of children into mathematicians, and thousands of mathematicians into children”.

Math Storytelling Day, Einstein and a glass of milk

Today, my sources tell me, is Math Storytelling Day. One of several mathematically oriented holidays, the idea of this specific day is to focus on the anecdotal side of mathematics, to address mathematics in a manner that may be more acceptable to the ‘wordy types’ among us – the ones who prefer a good sentence to a good equation.

I was hoping to come up with an original story for this day, but sadly my muse failed to come to the party, so I will have to resort to sharing an existing mathematical anecdote, from our old friend Albert Einstein. OK, it’s only borderline maths, but what the heck…

A glass of milk – just the thing to explain the theory of relativity.
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Apparently, shortly after his appointment at Princeton, Einstein was invited to a tea in his honour. At the event, the excited hostess introduced the great man and asked if he could perhaps, in a few words, explain to the guests the theory of relativity.

Not missing a beat, he rose to his feet and shared the story of a walk he had with a blind friend. It was a warm day, so at one point Einstein said to his friend, “I could really do with a glass of milk!”

His blind friend asked, “I know what a glass is, but what is milk?”, to which Einstein replied, “Why, milk is a white fluid.”

“Now I know what fluid is,” the blind man responded, “but what is white?”

“Oh, white is the colour of a swan’s feathers.”

“Feathers, I know what they are, but what is a swan?”

“A swan is a bird with a crooked neck.”

“I know what a neck is, but what do you mean by crooked?”

Einstein realised the discussion could go on for a while, so instead he seized his blind friend’s arm, straightened it, and said “There, now your arm is straight.” He then bent his friend’s arm at the elbow, and said, “And now, your arm is crooked.”

To which his blind friend happily exclaimed, “Ah! Now I understand what milk is!”

At this point, Einstein politely smiled at his audience, and sat down.

Joining hands on Black Ribbon Day

Today is International Black Ribbon Day; also celebrated as the European Day of Remembrance for Victims of Stalinism and Nazism in Europe. While it is a day highlighting a dark part of history, more than anything else, today is a celebration of the human spirit, about unity and about how amazing things can be achieved by joining hands and standing together (quite literally, in this case).

Joining hands to overcome hardship (and to solve mathematical problems!).
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Black Ribbon Day originated in the 1980s, as a annual series of demonstrations, held on 23 August in various western countries to highlight crimes and human rights violations in the former Soviet Union. The date marks the anniversary of the signing of the Molotov-Ribbentrop pact between the Nazi and Soviet Communist regimes – an event described by President Jerzy Buzek of the European Parliament as “the collusion of the two worst forms of totalitarianism in the history of humanity.”

Starting with initial participation of western countries only, it spread to the Baltic states in 1987, and in 1989 culminated in a historic event known as the Baltic Way. The Baltic Way, also referred to as the Baltic Chain, the Chain of Freedom and the Singing Revolution, was a peaceful demonstration involving almost two million people joining hands to form a 600km long human chain across the three Baltic states (Estonian SSR, Latvian SSR, and Lithuanian SSR), to protest against continued Soviet occupation.

The Baltic Way was meant to highlight the Baltic states’ desire for independence and to show the solidarity between the 3 nations. It proved an effective, emotionally captivating event. Within 6 months of the protest, Lithuania became the first Republic of the Soviet Union to declare independence, with Estonia and Latvia following in 1991.

Now you may be wondering why I’m discussing International Black Ribbon Day and the Baltic Way on this blog. Well, besides it being an opportunity to celebrate the strength of the human spirit in overcoming adversity, what caught my attention was something small and (almost) unrelated that grew out of it – the Baltic Way Mathematical Contest.

This maths contest has been organised annually since 1990, in commemoration of the Baltic Way human chain demonstrations. It differs from most other international mathematical competitions in that it is a true team contest. Teams, consisting of 5 secondary school students each, are presented with 20 problems, and they have four and a half ours to collaboratively solve these.

Initial participation was limited to the three Baltic states, but the competition has grown to include all countries around the Baltic Sea. Germany participates with a northern regions team, and Russia with a team from St Petersburg. Iceland has a special invitation for being the first state to recognise the independence of the Baltic States, and guest countries (including Israel, Belarus, Belgium and South Africa) have been invited in particular years, at the discretion of the organisers.

From people joining hands to overcome political hardship to students teaming up to solve complex mathematical problems, today truly is a day to celebrate strength in unity.

Poet’s Day, mathematically speaking

Today is Poet’s Day, a day to celebrate the sensitive souls who, through the ages, shared their deepest thoughts through verse and rhyme. I have to admit to being more of a ‘prose person’ than a ‘poetry person’, but that by no means implies that I don’t have the greatest respect and admiration for a good poem – it’s simply not my very favourite literary form.

Of course there’s a close relation between poetry and mathematics – a subject that is close to my heart. It was Einstein who said: “Pure mathematics is, in its way, the poetry of logical ideas.”

Mathematics in general seem to play an important role in poetry. Not only is there mathematics in the structure and rhythm of poetry, but many poems have also been written that contain overt mathematical themes. In a 2010 article entitled Poetry Inspired by Mathematics, Sara Glaz from the University of Connecticut, discusses some examples of such poems. More examples can be found in an earlier article from 2006 by JoAnne Growney, Mathematics in Poetry. In the latter article, Growney elegantly states, “As mathematicians smile with delight at an elegant proof, others may be enchanted by the grace of a poem. An idea or an image expressed in just the right language–so that it could not be said better–is a treasure to which readers return.”

The wonderful Fibonacci number sequence not only pops out in nature, but now claims its place in the world of the poet as well.
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An interesting new poetic form which I’ve discovered while doing some background reading for today, is the so-called “Fibonacci poetry”, which is based on the Fibonacci number sequence. Fibonacci numbers are a sequence, starting with 0 and 1, where each subsequent number is the sum of the previous two, i.e. 0,1,1,2,3,5,8,13,21,34,…

Fibonacci numbers occur often in nature, as I’ve discussed in an earlier blog post.

In poetry, the number sequence can refer to the numbers of letters, syllables or words in successive lines of the poem. These poems, known as ‘Fibs’, are six lines long, typically starting with a single letter/syllable/word in the first line. They can, however, theoretically start with any number of letters/syllables/words in the Fibonacci sequence.

Even though this form, originally introduced by Gregory K in a blog post on the GottaBook blog, appears to still be more popular among mathematicians than among poets, it has managed to garner a mention in the New York Times Books section. Their example, based on syllables, neatly illustrate the concept:

Blogs 
spread 
gossip 
and rumor 
But how about a 
Rare, geeky form of poetry?

I like the idea, I really do – very cool indeed! So, without further ado, herewith my own humble Fib for the day:

Words
and
numbers
sequences
not just in nature
but warming the hearts of poets too.

(uhm, assuming ‘poets’ is a single syllable word, of course…)

Happy Poet’s Day, everyone!  And please do share some Fibs, if you’re that way inclined!

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!