Category: Mathematics

The last 484 feet!

Some milestones on the age of the solar system.

Forgive me, dear readers, but something light and simple for today.  I don’t mean in the sense of the content, far from it, just easy for me to put the post together as it is from a presentation that I gave a year ago.

Here’s a picture of our solar system.

Most of us are reasonably familiar with this visual concept of our solar system, but what of it’s age?  That’s much more difficult to embrace in a way that we can relate to.

So let’s use something to represent the age of our solar system, the distance from Phoenix to Payson.

In round terms, Payson is 80 miles North-East from Phoenix.  Put another way, that’s 422,400 feet!

So if those 80 miles represented the age of our solar system, what would be the significant milestones on this metaphorical journey?

Phoenix represents the start, the ‘start’ of our solar system some 4.54 billion years ago

It was 1,075,000,000 years before Blue-green algae appeared.  That is the equivalent of travelling 18.94 miles from Phoenix North-East along Highway 87.  Or looking back, those algae appeared some 3.465 billion years ago.

But on we travel, metaphorically an unimaginable 3,459,800,000 years after the arrival of Blue-green algae until the next milestone; the earliest hominids.  In terms of our Highway that’s a further 60.97 miles.  Again, looking back that was 5,200,000 years ago.

The sharp-eyed among you will see that 18.94 miles added to 60.97 miles is 79.91 miles.  Goodness that’s awfully close to the total distance of 80 miles between Phoenix and Payson!  In fact, the 0.09 miles to run is the equivalent of 484 feet!

So let’s look at those last 484 feet.

The first 465.20 feet represents the approximately 5 million years after the earliest hominids appeared before H. sapiens arrived, some 200,000 years ago.

The appearance of Homo sapiens brings us to just 18.6 feet from Payson.

But first, we travel 9.3 feet and see the arrival of dogs, generally regarded to have separated, in DNA terms, from the Grey Wolf 100,000 years ago.

And are you 60 years old?  You were born just 0.0669 inches or 7/100ths of an inch from Payson!  If my maths is correct (someone please check!) 0.0669 inches is about 34 times the thickness of the human hair!  That’s very close to Payson!

Don’t know about you but it puts the age of our solar system into a perspective one might be able to get one’s arms around.

On the scale used above, one inch represents 895.68 years, one foot the equivalent of 10,748.11 years and a mile represents 56,750,000 years.

Anybody want to hazard a guess as to the state of our planet in one further inch?

The difference an inch makes! 895.68 years!

OK, let me stay more or less on topic and just round things off.

EarthSky website seems to have some great items, including this one.

Ten things you may not know about the solar system

9 ) Pluto is smaller than the USA
The greatest distance across the contiguous United States is nearly 2,900 miles (from Northern California to Maine). By the best current estimates, Pluto is just over 1400 miles across, less than half the width of the U.S. Certainly in size it is much smaller than any major planet, perhaps making it a bit easier to understand why a few years ago it was “demoted” from full planet status. It is now known as a “dwarf planet.”

Go here for the full list of ten items.

Finally, just how far does it all go?

How far do the stars stretch out into space? And what’s beyond them? In modern times, we built giant telescopes that have allowed us to cast our gaze deep into the universe. Astronomers have been able to look back to near the time of its birth. They’ve reconstructed the course of cosmic history in astonishing detail.

From intensive computer modeling, and myriad close observations, they’ve uncovered important clues to its ongoing evolution. Many now conclude that what we can see, the stars and galaxies that stretch out to the limits of our vision, represent only a small fraction of all there is.

Does the universe go on forever? Where do we fit within it? And how would the great thinkers have wrapped their brains around the far-out ideas on today’s cutting edge?

For those who find infinity hard to grasp, even troubling, you’re not alone. It’s a concept that has long tormented even the best minds.

Over two thousand years ago, the Greek mathematician Pythagoras and his followers saw numerical relationships as the key to understanding the world around them.

But in their investigation of geometric shapes, they discovered that some important ratios could not be expressed in simple numbers.

Take the circumference of a circle to its diameter, called Pi.

Computer scientists recently calculated Pi to 5 trillion digits, confirming what the Greeks learned: there are no repeating patterns and no ending in sight.

The discovery of the so-called irrational numbers like Pi was so disturbing, legend has it, that one member of the Pythagorian cult, Hippassus, was drowned at sea for divulging their existence.

A century later, the philosopher Zeno brought infinity into the open with a series of paradoxes: situations that are true, but strongly counter-intuitive.

In this modern update of one of Zeno’s paradoxes, say you have arrived at an intersection. But you are only allowed to cross the street in increments of half the distance to the other side. So to cross this finite distance, you must take an infinite number of steps.

In math today, it’s a given that you can subdivide any length an infinite number of times, or find an infinity of points along a line.

What made the idea of infinity so troubling to the Greeks is that it clashed with their goal of using numbers to explain the workings of the real world.

To the philosopher Aristotle, a century after Zeno, infinity evoked the formless chaos from which the world was thought to have emerged: a primordial state with no natural laws or limits, devoid of all form and content.

But if the universe is finite, what would happen if a warrior traveled to the edge and tossed a spear? Where would it go?

It would not fly off on an infinite journey, Aristotle said. Rather, it would join the motion of the stars in a crystalline sphere that encircled the Earth. To preserve the idea of a limited universe, Aristotle would craft an historic distinction.

On the one hand, Aristotle pointed to the irrational numbers such as Pi. Each new calculation results in an additional digit, but the final, final number in the string can never be specified. So Aristotle called it “potentially” infinite.

Then there’s the “actually infinite,” like the total number of points or subdivisions along a line. It’s literally uncountable. Aristotle reserved the status of “actually infinite” for the so-called “prime mover” that created the world and is beyond our capacity to understand. This became the basis for what’s called the Cosmological, or First Cause, argument for the existence of God.

Think I need to lie down now!

Dogs and the mathematics of calculus.

A remarkable story about two very clever men and an equally clever dog!

Five days ago, I received an email from a Richard Hake quite out of the blue!  This is what it said,

Dear Paul Handover,

I’m taking the liberty of cc’ing this to Tim Pennings since he may be interested in your blog “Learning from Dogs.”
As founding author of the great blog “Learning from Dogs”, I thought you might be interested in the work of Tim Pennings.
Paraphrasing from the Hope College website:

*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*
Tim Pennings is a professor of mathematics at Hope College and owner of a famous Welsh Corgi dog, Elvis, who knows calculus. He has given over a hundred talks – including several speaking tours – based on his papers “Do Dogs Know Calculus?” [Pennings (2003)] and “Do Dogs Know Bifurcations?” [Minton & Pennings (2007)]. Articles about Elvis are easily found on Google and Youtube. For example:
1. “A Dog, a Ball, and Calculus” Ivars Peterson’s MathTrek,
2. “Calculating Dogs” Ivars Peterson’s MathTrek ,
3. “Dog Plays Fetch With Calculus” YouTube (see below)
4. “Elvis The Calculus Dog at Roanoke College” Vime0 Videos.
*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*

Regards,

Richard Hake, Emeritus Professor of Physics, Indiana University
Honorary Member, Curmudgeon Lodge of Deventer, The Netherlands
President, PEdants for Definitive Academic References which Recognize the Invention of the Internet (PEDARRII)

Well what fun!

Let me start with that YouTube video mentioned above:

Tim Pennings of Hope College in Madison, Wisconsin takes a look at the mathematics his dog Elvis uses to play fetch.

And here are a number of wonderful pictures of Elvis from which comes this one:

Elvis: Professor Tim Pennings' dog.

More may be learnt about Tim Pennings from here, from which I quote:

I am a professor of mathematics at Hope College. My areas of research and writing include dynamical systems (the shadowing property in particular), mathematical modeling, and the infinite. I have directed the Mathematics REU Site since 1995 and have mentored research students almost every year since 1990. A complete list of published papers, talks, and other professional activity is included in my vitae.

My interest in infinity stems from my intrigue with the rich stuff that lies in the confluence of mathematics, physics, philosophy, and theology. My paper, “Infinity and the Absolute: Insights into Our World, Our Faith, and Ourselves” is the backbone of my senior seminar course, Pondering the Big Questions. Several other math-theology papers, published in Perspectives include “A Life Lesson from Calculus” and “Haggai, Mathematical Dynamics, and the Nature of Good and Evil”.

I have a famous Welsh Corgi dog, Elvis, who knows calculus. Here are some pictures of us. We have given over a hundred talks – including several speaking tours – based on our papers “Do Dogs Know Calculus?” and “Do Dogs Know Bifurcations?” (written with Roland Minton) both published in the The College Mathematics Journal of the MAA. Articles about Elvis are easily found on Google and Youtube.

Finally, the author of the email, Richard Hake, is no slouch!  Here’s Richard’s Blog Hake’sEdStuff and information on his academic background.

Thanks, Richard, for getting in touch!

The only certainty is uncertainty.

The fascinating aspects of chaos.

I must immediately volunteer the fact that the thrust of this article is the result of a programme that we watched last Thursday night.  It was a programme originally featured on the UK BBC 4 channel in 2008.  Called High Anxieties, The Mathematics of Chaos, it is a fascinating examination into the way that mathematicians have fundamentally adjusted their views, from the certainty of Newtonian principles to the certain uncertainty offered by mathematicians such as Henri Poincare and Alexander Lyapunov.  The 60-minute documentary, directed by David Malone, is wonderfully interesting and much more relevant to the uncertainty surrounding all our lives than one might anticipate from any references to mathematicians!  It puts the collapse of Lehman Brothers, some 2 years ago, last Thursday, into an interesting perspective.

Here’s the first 9 minutes as offered on YouTube, introduced thus,

David Malone http://golemxiv-credo.blogspot.com author of The Debt Generation, directs and presents this film, It is the first part of a documentary first shown on BBC4 Television in the UK in September 2008. The film was first broadcast 2 days after the collapse of Lehman brothers at the start of the financial crisis. It looks at the discoveries in mathematics during the 20th Century which have challenged the view that the world is an essentially knowable and therefore controllable place. The film focuses on the economy and the environment and suggests that ideas about unpredictability, the butterfly effect and tipping points, stemming from mathematics, are part of what underlie some modern anxieties about the world we live in.

If this first part grabs your attention then finding the other 8 parts is easy on YouTube.  Alternatively, the complete set of videos is linked together as one film on Top Documentary Films, click  High Anxieties: The Mathematics of Chaos. where it is described as follows,

The documentary looks at the modern advances in mathematics and how they affect our understanding of physics, economics, environmental issues and human psychology.

The film looks at how developments in 20th Century mathematics have affected our view of the world, and particularly how the financial economy and earth’s environment are now seen as inherently unpredictable.

The film looks at the influence the work of Henri Poincare and Alexander Lyapunov had on later developments in mathematics. It includes interviews with David Ruelle, about chaos theory and turbulence, the economist Paul Ormerod about the unpredictability of economic systems, and James Lovelock the founder of Gaia theory about climate change and tipping points in the environment.

As we approach tipping points in both the economy and the climate, the film examines the mathematics we have been reluctant to face up to and asks if, even now, we would rather bury our heads in the sand rather than face harsh truths.

Very, very interesting and rather puts the pictures of the Petermann Glacier shown here into context.

The mystery of telepathy

Just a bit more science about that sixth sense.

Yesterday, I wrote about how science was coming up with some pretty strong evidence that humans do have the ability to communicate in a way that might be called ‘telepathic’.

If (and that’s a big ‘if’) I have any understanding of the science, I believe it has much to do with quantum physics.  So I thought it fun to take a small diversion in today’s Post and give you some material on this very strange world of the very, very small.

From A Lazyman’s Guide to Quantum Physics,

What is Quantum Physics?

That’s an easy one: it’s the science of things so small that the quantum nature of reality has an effect. Quantum means ‘discrete amount’ or ‘portion’. Max Planck discovered in 1900 that you couldn’t get smaller than a certain minimum amount of anything. This minimum amount is now called the Planck unit.

Why is it weird?

Niels Bohr, the father of the orthodox ‘Copenhagen Interpretation’ of quantum physics once said, “Anyone who is not shocked by quantum theory has not understood it“.

To understand the weirdness completely, you just need to know about three experiments: Light Bulb, Two Slits, Schroedinger’s Cat.

Two Slits

The simplest experiment to demonstrate quantum weirdness involves shining a light through two parallel slits and looking at the screen. It can be shown that a single photon (particle of light) can interfere with itself, as if it travelled through both slits at once.

Light Bulb

Imagine a light bulb filament gives out a photon, seemingly in a random direction. Erwin Schroedinger came up with a nine-letter-long equation that correctly predicts the chances of finding that photon at any given point. He envisaged a kind of wave, like a ripple from a pebble dropped into a pond, spreading out from the filament. Once you look at the photon, this ‘wavefunction’ collapses into the single point at which the photon really is.

Schroedinger’s Cat

In this experiment, we take your pet cat and put it in a box with a bottle of cyanide. We rig it up so that a detector looks at an isolated electron and determines whether it is ‘spin up’ or ‘spin down’ (it can have either characteristic, seemingly at random). If it is ‘spin up’, then the bottle is opened and the cat gets it. Ten minutes later we open the box and see if the cat is alive or dead. The question is: what state is the cat in between the detector being activated and you opening the box. Nobody has actually done this experiment (to my knowledge) but it does show up a paradox that arises in certain interpretations.

To conclude I will offer this quotation reputed to be from the great master himself, Albert Einstein,

The more success the quantum theory has, the sillier it looks.

Mandelbrot and fractals

Concluding article on the great Benoit Mandelbrot.

Yesterday, I wrote about Benoit Mandelbrot but wanted to save some additional information for today.

There’s a very comprehensive review of Benoit’s life on a website called NNDB.  In that review, it mentions his association with the IBM Thomas J. Watson Research Center where he worked for 32 years.  It was while working for IBM that he published the paper that established his credentials world-wide.  Taken from the IBM website is this extract,

The father of fractals, Dr. Benoit Mandelbrot, passed away from pancreatic cancer on October 16, 2010. He was 85.

Benoit, IBM Fellow Emeritus, joined the IBM Thomas J. Watson Research Center in 1958 where he worked for 32 years. His 1967 article published in Science, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, introduced the concept that a geometric shape can be split into pieces that are smaller copies of the whole. It wasn’t until 1975 that he defined the mathematical shapes as fractals.

Here is another website that has fractal images taken from the Mandelbrot set.  An example.

Just stunningly beautiful!

Finally, if you go to this website there is a slideshow of stunning images of fractals in honour of the great man.

Benoit Mandelbrot and the roughness of life.

There is so much about the lives of humans that is astoundingly beautiful.

Before I get started on this article, a few words about the Blog in general.  In recent times, the readership of Learning from Dogs has increased, frankly, to quite amazing levels.  Not really sure why but grateful, nonetheless.

Readers will recognize that articles written specifically about dogs are in the minority.  Even using dogs as a metaphor would still limit what could be published.  But as is written elsewhere on the Blog, ‘The underlying theme of Learning from Dogs is about truth, integrity, honesty and trust in every way.’  Dogs are integrous creatures; that’s all the example required.

We are at a point in the history of man where truth, integrity, honesty and trust are critically important (they have always been important but the economic and ecological pressures bearing down on us all make these values critical to mankind’s survival).  Thus the aspiration of Learning from Dogs is to offer insights on truth from as many perspectives as possible – and to make your experience as a reader sufficiently enjoyable that you will wish to return!

OK!  On to the topic for today!

There are many aspects of the world in which we live that are mysterious beyond our imagination.  Take the circle.  Practically everyone is aware that to calculate either the area or the circumference of a circle one needs to use a mathematical constant π or ‘pi’.  As a mathematician would put it,  π (sometimes written pi) is a mathematical constant whose value is the ratio of any circle’s circumference to its diameter; this is the same value as the ratio of a circle’s area to the square of its radius. π is approximately equal to 3.14159.

Note the word ‘approximately’!  Now read this,

(PhysOrg.com) — A computer scientist in France has broken all previous records for calculating Pi, using only a personal computer. The previous record was approximately 2.6 trillion digits, but the new record, set by Fabrice Bellard, now stands at almost 2.7 trillion decimal places.

Bellard, of Paris Telecom Tech, made and checked the calculation by running his own software algorithms for 131 days. The previous record calculation, set by Daisuke Takahashi at the University of Tsukuba in Japan in August 2009, took only 29 hours to complete, but used a super-computer costing millions of dollars, and running 2000 times faster than Bellard’s PC.

Full article is here.

Apart from the wonderful aspect of the need of a human being to go on determining the n’th value of π there is a deeper and more beautiful aspect (well to me there is!) and that is the acknowledgement that something as simple as, say, that round coin in your hand is an expression of the infinite.

Now to Benoit Mandelbrot, who died a little over six months ago, but in his lifetime also explored the wonder and magic of the infinite.

Here’s a video of Mandelbrot recorded in February 2010 in what would be his last year of his life on earth.

If you found that video fascinating then try this series of six videos presented by the one and only Arthur C. Clarke.

Benoit Mandelbrot died on the 14th October, 2010, a little over a month before his eighty-sixth birthday (born 20th November, 1924).  Here’s a nice tribute from the The New York Times.

Tomorrow, some more insights into the mysterious beauty of fractals.

Benoit Mandelbrot

Amazing man!