Archive for the ‘Mathematics’ Category
Modelling the future.
Can we trust the predictive output of computer modelling?
I would be the first to admit that this is not an area where I have anything more than general knowledge. However, what prompted me to think about this topic was a chance conversation with someone here in Payson. We were chatting over the phone and this person admitted to being less than fully convinced of the ’cause and effect’ of man’s influence on the global biosphere.
When I queried that, what was raised was the idea that all modelling algorithms used in climate change predictions must incorporate mathematical constants. I continued to listen as it was explained that, by definition, all constants were, to some degree, approximations. Take, for example, the obvious one of the constant π, that Wikipedia describes as: a mathematical constant that is the ratio of a circle’s circumference to its diameter. Pi, of course, would have to be rounded if it was to be used in any equation. Even taking it to thirty decimal places, as in 3.14159 26535 89793 23846 26433 83279, would mean rounding it to 3.14159 26535 89793 23846 26433 83280 (50288 being the 30th to 35th decimal places).
OK, so I must admit that I was leaning to the viewpoint that this person had a valid perspective. I then asked Martin Lack, he of Lack of Environment and a scientifically trained person, for his thoughts. The rest of this post is based on the information that Martin promptly sent me.
One of the links that Martin sent was to this post on the Skeptical Science blogsite. That post sets out the common skeptics view, namely:
Models are unreliable
“[Models] are full of fudge factors that are fitted to the existing climate, so the models more or less agree with the observed data. But there is no reason to believe that the same fudge factors would give the right behaviour in a world with different chemistry, for example in a world with increased CO2 in the atmosphere.” (Freeman Dyson)
The author of the Skeptical Science posting responds,
Climate models are mathematical representations of the interactions between the atmosphere, oceans, land surface, ice – and the sun. This is clearly a very complex task, so models are built to estimate trends rather than events. For example, a climate model can tell you it will be cold in winter, but it can’t tell you what the temperature will be on a specific day – that’s weather forecasting. Climate trends are weather, averaged out over time – usually 30 years. Trends are important because they eliminate – or “smooth out” – single events that may be extreme, but quite rare.
Climate models have to be tested to find out if they work. We can’t wait for 30 years to see if a model is any good or not; models are tested against the past, against what we know happened. If a model can correctly predict trends from a starting point somewhere in the past, we could expect it to predict with reasonable certainty what might happen in the future.
So all models are first tested in a process called Hindcasting. The models used to predict future global warming can accurately map past climate changes. If they get the past right, there is no reason to think their predictions would be wrong. Testing models against the existing instrumental record suggested CO2 must cause global warming, because the models could not simulate what had already happened unless the extra CO2 was added to the model. All other known forcings are adequate in explaining temperature variations prior to the rise in temperature over the last thirty years, while none of them are capable of explaining the rise in the past thirty years. CO2 does explain that rise, and explains it completely without any need for additional, as yet unknown forcings.
I strongly recommend you read the full article here. But I will republish this graph that, for me at least, is a ‘slam dunk’ in favour for modelling accuracy.
Sea level change. Tide gauge data are indicated in red and satellite data in blue. The grey band shows the projections of the IPCC Third Assessment report (Copenhagen Diagnosis 2009).
Not only does this show that the data is within the range of projections of the modelled output, more seriously the data is right at the top end of the model’s predictions. The article closes with this statement:
Climate models have already predicted many of the phenomena for which we now have empirical evidence. Climate models form a reliable guide to potential climate change.
There is a more detailed version of the above article available here. Do read that if you want to dig further down into this important topic. All I will do is to republish this,
There are two major questions in climate modeling – can they accurately reproduce the past (hindcasting) and can they successfully predict the future? To answer the first question, here is a summary of the IPCC model results of surface temperature from the 1800′s - both with and without man-made forcings. All the models are unable to predict recent warming without taking rising CO2 levels into account. Noone has created a general circulation model that can explain climate’s behaviour over the past century without CO2 warming. [my emphasis, Ed.]
Finally, back to Lack of Environment. On the 6th February, 2012, Martin wrote an essay Climate science in a nut fragment. Here’s how that essay closed:
Footnote:
If I were to attempt to go even further and summarise, in one single paragraph, why everyone on Earth should be concerned about ongoing anthropogenic climate disruption, it would read something like this:Concern over anthropogenic climate disruption (ACD) is not based on computer modelling; it is based on the study of palaeoclimatology. Computer modelling is based on physics we have understood for over 100 years and is used to predict what will happen to the atmosphere for a range of projections for CO2 reductions. As such, the range of predictions is due to uncertainty in those projections; and not uncertainties in climate science. Furthermore, when one goes back 20 years and chooses to look at the projection scenario that most-closely reflects what has since happened to emissions, one finds that the modelled prediction matches reality very closely indeed.
In his email, Martin included these bullet points.
- Concern over anthropogenic climate disruption (ACD) is not based on computer modelling.
- It is based on our understanding of atmospheric physics (and how the Earth regulates its temperature).
- Computer modelling is based on this physics (which we have understood for over 100 years).
- Models have been used to predict temperature and sea level rise for a range of projections for CO2 emissions.
- The wide range of predictions was due to uncertainty in those emissions projections not uncertainties in climate science.
- This can be demonstrated by looking at predictions made over 20 years ago in light of what actually happened to emissions.
- The model predictions for both temperature and sea level rise are very accurate (if not slightly under-estimating what has happened).
Sort of makes the point in spades! The sooner all human beings understand the truth of what’s happening to our planet, the sooner we can amend our behaviours. I’m going to pick up the theme of behaviours in tomorrow’s post on Learning from Dogs.
Finally, take a look at this graph and reflect! This will be the topic that I write about on Thursday.
Thinking outside the box
Strange theory reveals secrets of the universe, the logic of sycamore leaves and why even smart people struggle with new ideas.
A guest post from Pete Aleshire, Editor, Payson Roundup.
Introduction
The Payson Roundup is our local newspaper here in Payson, AZ. I first saw this article by Pete a couple of weeks ago and was just utterly engrossed by it. Not just the tantalising peek into a physics I know so little about but the beautiful prose. The latter is not surprising because as well as being editor of the paper, Pete also teaches the creative writing course at our local college. Jean and I had the benefit of attending the course, I guess about a year ago, and therefore can speak from experience.
So settle down and enjoy.
oooOOOooo
I finally got Drake Larson together with both sycamore leaves and Payson Mayor Kenny Evans. Moreover, I have been entrusted with a formula that may win me an invitation to Oslo if Drake gets a Nobel Prize.
But even if that don’t work out, I did get to eavesdrop on Drake and Evans. Quite the event, from my bemused point of view, since it shed light on dangerous delights of outside-the-box thinking and the Nature of the Universe.
But wait. You look confused.
Let me back up — and start somewhere closer to the beginning. Be patient with me — by the time we’re done, you’ll realize why God’s a math nerd, one surprising secret of Dark Energy, why farmers become original thinkers and what sycamore leaves tell us about the universe.
But first, I have to explain about Drake.
We grew up together, getting into (and mostly out of) various varieties of trouble. Very early on, I realized that he was much (much) smarter than me. This initially really irritated me, as I was previously inclined to vanity about my intelligence. Turns out, I love learning stuff other people have discovered, but Drake only gets truly excited when he has hold of a completely new idea that no one else can quite grasp. This prepared me, as it turns out, for meeting Kenny Evans — but that’s getting ahead of the story.
Drake and I grew up doing math homework together, before I wandered off into a career in newspapers. He got his degree in mathematics, turned down a job with the RAND Corporation and took up growing table grapes.
But he never quit picking at the lock of the universe.
Years ago when I was the science writer for the Oakland Tribune, he came to me all excited about a set of formulas he had. I did my best to follow the two pages of calculations, but all I can tell you is that they described instabilities of any sphere with uniform density. He predicted that when the Voyager spacecraft reached Jupiter, it would report inexplicable turbulence at a certain depth in the atmosphere. I ran his numbers past various top-level physicists and mathematicians who couldn’t find a flaw in his formulas — but concluded that it had to be wrong since it led to a violation of the keystone laws of conservation of mass and energy.
But I took note some months later when the Voyager spacecraft reported mysterious levels of turbulence deep within the atmosphere of Jupiter.
The years passed. Drake kept growing grapes, flowers, dates and vegetables — and working on his calculations. He wrote a book, “The Cults of Relativity,” in which he described a few of his theories, delighted in the conundrums of mathematics and pondered the curious resistance of even smart people to unconventional ideas.
We got together again recently. I took him down to Fossil Creek, all overhung with sycamores with the rustle of floppy, five-pointed leaves. Drake was his old self on our Fossil Creek tour as he tried to show me math’s beauty around us, although I was but a blind man clutching the tail of his mathematical elephant.
He had now connected his formulas to dark energy, a still hypothetical form of energy invoked by desperate cosmologists to explain the startling observation that the expansion of the universe is actually accelerating. To explain this seeming impossibility, they invented “dark energy” — which they figure pervades the universe and at certain densities creates a repulsive force stronger than the attractive force of gravity.
Anyhow, here’s the point: Drake was in awe that his mathematical wanderings offered a way to calculate dark energy’s cap within earth — it happened to be an “inside is now outside” inversion of Isaac Newton’s simplest integral. No. Please. Don’t ask me to explain that. But earth’s dark energy cannot exceed 17 pounds per square inch at a depth of about 1,500 miles. He’s been working with University of Southern California computer crunching guru professor Barry Boehm, and the University of California at Riverside geophysics professor Shawn Biehler on its implications. Among other things, it could explain the perplexing observation that major earthquakes increase the earth’s rotation rate.
No one knows how to measure such a quantity at present. Someday they will. If it turns out that 17 pounds per square inch is a relevant benchmark for earth’s dark energy, then this column will maybe win Drake the Nobel — and I’ll get to dress up and attend the ceremony.
So Drake and I spent the day wandering along the banks of Fossil Creek as he kept trying to come up with metaphors so I could grasp math’s secret within the beauty of Fossil Creek’s sycamore leaves. The well-designed sycamore leaves adhere to the Fibonacci sequence, a mysterious progression of numbers that crops up throughout nature — from the spiral of a nautilus shell to the layout of the ruins of Chaco Canyon.
So I figured I’d just write this — and get earth’s 17 PSI cap for dark energy out there in the time/date/ stamped world.
Oh, yeah: And about Kenny Evans.
So that night, I took Drake to the Payson council meeting. Turns out, Drake’s family was growing grapes in the Coachella Valley at the same time Evans was farming 10,000 acres in Yuma. They both managed to survive that tempestuous time when the United Farm Workers union organized agriculture workers.
I introduced them and listened as they recalled events and figured out whom they knew in common.
It was then that I decided to blame Drake for my faith in Evans’ ridiculous conviction that a university will build a campus here in this itty bitty tourist town — complete with a research center and convention hotel. No sensible small-town mayor would risk public ridicule while spending thousands of hours on such an outside-the-box notion … unless he’d learned to gamble on dreams and hard work during all those years as a farmer.
Evans’ notion is almost as silly as a farmer who calculates the amount of dark energy emanating from earth, while credentialed experts scratch their collective heads.
Still, I’m thinking maybe I’ll get a nice suit jacket — something I can wear to both the university’s groundbreaking and the ceremonies in Oslo.
Hey, never hurts to be prepared.
oooOOOooo
A big thank-you for the permission to republish this on Learning from Dogs. I have no doubt that many LfD readers enjoyed it as much as I did! Stay with me for tomorrow when the theme of thinking, innovation and craziness is explored a touch more.
The Higgs boson
Clarity of thought courtesy of The Economist
Like many people I had been aware of the hunt for this strange particle, the Higgs boson. Like many people as well, I suspect, I really didn’t comprehend what it was all about.
Then in The Economist print edition of the July 7th the newspaper’s primary story and leader were about the discovery of the Higgs announced on the 4th July. The leader, in particular, was both clear and compelling. I held my breath and asked for permission to republish that leader in Learning from Dogs.
Well the good people from the relevant department at The Economist promptly gave written permission for their leader to be available here for a period of one year. Thanks team!
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The Higgs boson
Science’s great leap forward
After decades of searching, physicists have solved one of the mysteries of the universe
Jul 7th 2012 | from the print edition
HISTORICAL events recede in importance with every passing decade. Crises, political and financial, can be seen for the blips on the path of progress that they usually are. Even the horrors of war acquire a patina of unreality. The laws of physics, though, are eternal and universal. Elucidating them is one of the triumphs of mankind. And this week has seen just such a triumphant elucidation.
On July 4th physicists working in Geneva at CERN, the world’s biggest particle-physics laboratory, announced that they had found the Higgs boson. Broadly, particle physics is to the universe what DNA is to life: the hidden principle underlying so much else. Like the uncovering of DNA’s structure by Francis Crick and James Watson in 1953, the discovery of the Higgs makes sense of what would otherwise be incomprehensible. Its significance is massive. Literally. Without the Higgs there would be no mass. And without mass, there would be no stars, no planets and no atoms. And certainly no human beings. Indeed, there would be no history. Massless particles are doomed by Einstein’s theory of relativity to travel at the speed of light. That means, for them, that the past, the present and the future are the same thing.
Deus et CERN
Such power to affect the whole universe has led some to dub the Higgs “the God particle”. That, it is not. It does not explain creation itself. But it is nevertheless the most fundamental discovery in physics for decades.
Unlike the structure of DNA, which came as a surprise, the Higgs is a long-expected guest. It was predicted in 1964 by Peter Higgs, a British physicist who was trying to fix a niggle in quantum theory, and independently, in various guises, by five other researchers. And if the Higgs—or something similar—did not exist, then a lot of what physicists think they know about the universe would be wrong.
Physics has two working models of reality. One is Einstein’s general relativity, which deals with space, time and gravity. This is an elegant assembly of interlocking equations that poured out of a single mind a century ago. The other, known as the Standard Model, deals with everything else more messily.
The Standard Model, a product of many minds, incorporates the three fundamental forces that are not gravity (electromagnetism, and the strong and weak nuclear forces), and also a menagerie of apparently indivisible particles: quarks, of which protons and neutrons, and thus atomic nuclei, are made; electrons that orbit those nuclei; and more rarefied beasts such as muons and neutrinos. Without the Higgs, the maths which holds this edifice together would disintegrate.
Finding the Higgs, though, made looking for needles in haystacks seem simple. The discovery eventually came about using the Large Hadron Collider (LHC), a machine at CERN that sends bunches of protons round a ring 27km in circumference, in opposite directions, at close to the speed of light, so that they collide head on. The faster the protons are moving, the more energy they have. When they collide, this energy is converted into other particles (Einstein’s E=mc2), which then decay into yet more particles. What these decay particles are depends on what was created in the original collision, but unfortunately there is no unique pattern that shouts “Higgs!” The search, therefore, has been for small deviations from what would be seen if there were no Higgs. That is one reason it took so long.
Another was that no one knew how much the Higgs would weigh, and therefore how fast the protons needed to be travelling to make it. Finding the Higgs was thus a question of looking at lots of different energy levels, and ruling each out in turn until the seekers found what they were looking for.
Queerer than we can suppose?
For physicists, the Higgs is merely the LHC’s aperitif. They hope the machine will now produce other particles—ones that the Standard Model does not predict, and which might account for some strange stuff called “dark matter”.
Astronomers know dark matter abounds in the universe, but cannot yet explain it. Both theory and observation suggest that “normal” matter (the atom-making particles described by the Standard Model) is only about 4% of the total stuff of creation. Almost three-quarters of the universe is something completely obscure, dubbed “dark energy”. The rest, 22% or so, is matter of some sort, but a sort that can be detected only from its gravity. It forms a giant lattice that permeates space and controls the position of galaxies made of visible matter (see article). It also stops those galaxies spinning themselves apart. Physicists hope that it is the product of one of the post-Standard Model theories they have dreamed up while waiting for the Higgs. Now, they will be able to find out.
For non-physicists, the importance of finding the Higgs belongs to the realm of understanding rather than utility. It adds to the sum of human knowledge—but it may never change lives as DNA or relativity have. Within 40 years, Einstein’s theories paved the way for the Manhattan Project and the scourge of nuclear weapons. The deciphering of DNA has led directly to many of the benefits of modern medicine and agriculture. The last really useful subatomic particle to be discovered, though, was the neutron in 1932. Particles found subsequently are too hard to make, and too short-lived to be useful.
This helps explain why, even at this moment of triumph, particle physics is a fragile endeavour. Gone are the days when physicists, having given politicians the atom bomb, strode confidently around the corridors of power. Today they are supplicants in a world where money is tight. The LHC, sustained by a consortium that was originally European but is now global, cost about $10 billion to build.
That is still a relatively small amount, though, to pay for knowing how things really work, and no form of science reaches deeper into reality than particle physics. As J.B.S. Haldane, a polymathic British scientist, once put it, the universe may be not only queerer than we suppose, but queerer than we can suppose. Yet given the chance, particle physicists will give it a run for its money.
Copyright © The Economist Newspaper Limited 2012. All rights reserved.
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Before signing off on this very important step forward for physics, here are a couple of footnotes.
First, here’s a video of the announcement that was widely shown on the 4th.
Secondly, the BBC News website had a really good piece on the 12th July written by their science correspondent, Quentin Cooper, called Higgs: What was left unsaid. Here’s a flavour taken from the early part of the article,
So that’s it, search over, Higgs boson found. Almost 50 years after physicist Peter Higgs first theorised it was out there, public elementary number one has finally been captured in the data from two detectors at the Large Hadron Collider at Cern. Case closed. Champagne popped. Boson nova danced.
If only. That handily simplified and heavily fictionalised telling of the tale has helped transform a spectacular scientific success story into one that is also global front page news. Without it the 4 July announcement might not have generated such a frenzy of coverage and so many claims about it being a historic milestone for our species. One particle physicist only half jokingly told me that in future the date may come to be celebrated as Higgs Day, rather than anything to do with American independence.
Don’t get me wrong. What has happened at Cern represents a magnificent accomplishment; big science at its biggest and boldest. And it’s fantastic that it has been perceived and received as being of such importance. It’s just that there is more to the story from the very beginning right through to the, probably false, ending.
For starters, as Peter Higgs himself acknowledges, he was just one of several scientists who came up with the mechanism which predicted the particle which bears his name, but the others rarely get a mention*. As to the finish – well, as small children are fond of saying, are we there yet? There is very strong evidence that the LHC teams have found a new elementary particle, but while this is exciting it is far less clear that what they’ve detected is the fabled Higgs. If it is, it seems curiously lighter than expected and more work is needed to explain away the discrepancy. If it’s not, then the experimentalists and theorists are going to be even busier trying to see if it can be shoehorned into the current Standard Model of particle physics. Either way, it’s not exactly conclusive.
Do take the simple step of clicking here and read the BBC piece in full.
Well done, Mr. Peter Higgs and all those very persistent scientists associated with the Large Hadron Collider; I suspect we haven’t heard the last of this!
And ‘thank you’ to The Economist.
One smart brain!
Talking physics with your dog!
Not as silly as one might think!
Back on the 25th April I ran a Post called Dogs and the Mathematics of Calculus that had been prompted by a lovely email from Richard Hake who is Emeritus Professor of Physics at Indiana University. (Now here’s a question for yours truly; what does it mean for a Professor to be an Emeritus Professor? Answers as comments please.)
It was very well received. Then just a few days ago Professor Hake, who admits to being a dedicated lurker of this blogsite, sent me another email with a number of fascinating links. So here goes with one of those links.
Talking to Your Dog About Physics
A conversation with Chad Orzel
So, why do you talk to your dog about physics?
Lots of reasons, but the main one is that I’m a physics professor. Talking about physics is what I do. Sooner or later I talk to everybody about physics.
I bet that’s a big hit at parties.
You might be surprised. I mean, sure, I get a lot of people making faces and saying how much they hated physics when they took it in college. But some of those same people turn right around and start asking interested questions about the subject.
OK, but why the dog?
Talking to the dog about physics is worthwhile because it can help me see how to explain physics to my human students. Humans all come at the subject with the same set of preconceptions about how the world works, and what “should” happen, and it can be very hard to shake those off. That’s a big barrier to understanding something like quantum physics.
Dogs look at the world in a very different way. To a dog, the world is a neverending source of wonder and amazement. You can walk your dog past the same rock every morning, and every morning, she’ll sniff that rock like she’s never sniffed it before. Dogs are surprised by things we take for granted, and they take in stride things that would leave us completely baffled.
Can you give an example?
Well, take the dog’s bowl, for example. Every now and then, we put scraps from dinner in the bowl when she’s not looking, and she’s become convinced that her bowl is magic– that tasty food just appears in it out of nowhere. She’ll wander over a couple of times a day, and look just to see if anything good has turned up, even when we haven’t been anywhere near the bowl in hours.
This puts her in a better position to understand quantum electrodynamics than many humans.
It does?
Sure. One of the most surprising features of QED, in Feynman’s formulation, is the idea of “virtual particles.” You have an electron that’s moving along, minding its own business, and every now and then, particle-antiparticle pairs just pop into existence for a very short time. They don’t stick around very long, but they have a real and measurable influence on the way electrons interact with each other, and with other particles.
You’re making this up, right?
No, not at all. One set of these interactions is described by a number called the “g-factor” of the electron, and this has been measured to something like fifteen decimal places, and the experimental measurement agrees perfectly with the theoretical prediction. If there weren’t electrons and positrons popping out of nowhere, there’s no way you could get that sort of agreement.
So, what’s this have to do with the dog?
Well, like I said, the dog is perfectly comfortable with the idea of stuff popping into existence out of nowhere. If a great big steak were to suddenly appear on your dining room table, you’d probably be a little perturbed. The dog, on the other hand, would feel it was nothing more than her due.
So she’s perfectly ok with the idea of virtual particles, unlike most humans, who tend to say things like “You’re making this up, right?” She was already convinced that there were bunnies made of cheese popping in and out of the backyard, and just regards QED as a solid theoretical justification for her beliefs.
And this helps humans, how, exactly?
Physics has a reputation as a difficult and unapproachable subject, especially in fields like quantum mechanics, where the predictions of the theory confound our human preconceptions. If you can put aside a few of your usual notions of how the world works, and think about how things look to a dog, some aspects of physics that seem absolutely impossible to accept become a lot more approachable.
Why does this matter, though? Isn’t this all stuff that you need a billion-dollar particle accelerator to see?
Actually, no. It’s a common misconception, but most of the really cool aspects of quantum mechanics that we talk about in the book are experiments that are done on a table-top scale. One of them, the “quantum eraser,” you can even do yourself with a laser pointer and a couple of pairs of polarized sunglasses.
OK, but what is it good for, in a practical sense?
Lots of things. It’s not an exaggeration to say that modern life as we know it would be impossible without an understanding of quantum phyiscs. You need to understand quantum ideas to build the lasers we use in modern telecommunications, and the transistors that are the basis of all modern electronics. The computer I’m typing this on wouldn’t exist without quantum physics.
And there are a whole host of future technologies that are based on quantum ideas. There are exotic applications like quantum computers that can do calculations that would be impossible with any normal computer, and quantum cryptography systems that allow us to make unbreakable codes. But even relatively mundane “green” technologies like more efficient light bulbs, batteries, and solar panels rely on quantum ideas to work.
Quantum physics is everywhere, and drives a huge amount of modern science and technology.
So that’s why people should teach quantum physics to their dogs?
Exactly. Also, it’s just about the coolest thing ever.
How to Teach Physics to Your Dog is published by Scribner can be ordered from Amazon.com, IndieBound,Barnes and Noble, and Powell’s.
How to Teach Relativity to Your Dog is published by Basic Books and can be ordered from Amazon, Barnes & Noble, Powell’s.
Talking or sleeping about physics?
OK, two thoughts to close this off. The first is to remind you of an early sentence that Chad Orzel wrote, “Talking to the dog about physics is worthwhile because it can help me see how to explain physics to my human students.” and to add that in my next life, I wouldn’t mind coming back as one of Chads dogs!
The second thought is that Chad’s talks with his dogs are pretty relaxed affairs, as the picture above bears out!
Thank you, Professor Hake!
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!
Paul Handover
