Monday, August 27, 2012

astro pictures for the day

image credit NASA/Mars Curiosity Rover

You get the feeling this Curiosity Mars Rover will give us lots to be excited about till the Pluto probe gets to Pluto!  Shoot, the spaceprobe around Vesta currently will probably get to Ceres before the Pluto encounter.  That's plenty of excitement.

---------------------------------crazy science/technology excitement extra for the day

If these people accomplish this, it will be just as affective imo as Feynman/Drexler's nanomanufacturing system.  Couple this with the chemical brain, and one can make almost anything!

Saturday, August 25, 2012

First man to set foot on the moon passes away

As the Wiki shows, Neil flew the X1b(as oppossed to the bell x-1 of Chuck Yeager, and the x-15 pictured above, then the Gemini and the Apollo 11 first human moon landing. The wiki has an embedded video of the first human step on the moon. 

The Apollo 11 landing was easily the most boring space mission after Neil stepped onto the moon. He also made the famous quote, "that's one small step for man, one giant leap for mankind."

I think that Neil died knowing that SpaceX has really gotten things going and most likely will open up the space frontier.

- It took the whole Saturn V rocket to get three men to the moon; it took half the lunar landing module to launch back up to lunar orbit.  Yes, the service module was needed to rocket back to the Earth; but, there is a striking difference in scale!  I mean, once a human presence is established in space, it's relatively easy to get around after that!  There's a wealth of materials and energy to tap that is easy to get to and use once in space.  Our problem is getting established out in space.

-------------------------just some of the crazy science/technology reported around the time of Neil Armstrong's passing away

This may not stop those who want quantum loop quantum/gravity, but it's going to make them think a lot harder!  Quantum loop gravity is a challenger to the String theory explanation of quantum gravity.

Nanotechnologists(dna nanotechnologists?) are excited about this; maybe they can speed up analyses of dna-nanotechnologies.  They've sped up dna-nanomanufacturing substantially, but the analyses was still hard.

Friday, August 24, 2012

youtube for the day/ panning inside and out of the Large Magellanic cloud's tarantula star factory nebula

I'll use this to point out something exciting nanotechnologists over the last couple of days,

Chemists have created a network database of all the chemical knowledge about organic chemistry right now; they can create any number of algorithms to solve whatever problems using all the known chemistry rules humanity has dug up over the last two hundred and fifty years!  That alone is a great accomplishment.  Nanotechnologists have been linking and posting about this as an automated engineering technology to probably make nano-mechanical parts.

Thursday, August 23, 2012

youtube for the day/ Planet Earth

About the only reason I want to show this video series is because it's the last pretty good science video I know about in the whole world.  That alone says something about the current human condition.  I figured I'd show it while I'm putting up all these other videos!

I suppose I could make some comment about this knowledge about Planet Earth.  I don't know who said it first, but it's been said that one of the most significant outcomes of the Apollo program and the space programs since the late 1950s starting with the Russians Sputnik is that of seeing our Earth directly for the first time. It's had some affect but perhaps not as much some would like!  The Hubble Space Telescope seems to me to have had more affect on Humanity than the space programs of various countries.

This Planet Earth video series also presents the scientific spirit; it shows some of the first accurate knowledge about the Planet Earth in human history.  Before a hundred years ago or so, nobody knew how old the Earth actually was.  Much can be said for the rest of human knowledge; for the first time, we have some answers to quite a bit of everything we see. 

------------------------------and I've managed to find some more crazy science/tech for the day!

I've heard of this as I've heard of much else;  i've always felt like I'm getting bread crumbs; but, I've also been satisfied with this bits here and there.  I figure, when it gets here, we'll all know.  But this blew me away!  It takes a little while to get into it, but this is an update that is hugh!  And, this talk was given back in march!  He seems to be hinting pretty well in my opinion that he can bootstrap to Feynman/Drexlerian nanomanufacturing.

Wednesday, August 22, 2012

thought for the day/ Mechanical Universe on thermodynamics

  When Physicists worked Newtonian mechanics, they didn't think they had to deal with temperature when working out force, energy, momentum vectors of masses.  Yet, when they wanted to keep the conservation of energy in the 1800s, they had to learn  and combine thermodynamics.

Today and the last so many decades, Chaos theorists wanted to say that mathematics role in science has been disproven.  They wanted to say that the particle physicists obsession with analyses and breaking things down to ever smaller levels of reality is not the way to go.  Well, this is true in some sense, but the chaos theorists wanted to show quantum chaos.  But, they've come to see that thermodynamics doesn't exist at the quantum level(yes, the physicists combine thermodynamics in the description of the big bang, but they're talking about the thermodynamcis of particles at a macroscopic level and not what happens when an electron absorbs a photon).  I bring this up because of my ideas(and Jacob Bronowski's) about certain ideas work in a network of ideas when nature is dammed up in certain ways.  When dammed up in other ways, other ideas take hold.  There is quantum chaos, but it's different from the chaos theory at our scale.

In this macro world with thermodynamics, some phenomenon arise - things like entropy.


are both related.

As the second video of this three says, thermodynamics suggesed a solution to the puzzle of time; but, Einstein's relativity theories and quantum mechanics absence of time suggests the problem of time is still a problem.  My concern is actually what thermodynamics entropy means for life.

While most biologists were concerned with molecular biology and genetics at various times . . . not to mention bacteriology, and zoology, there was a few that observed that if the second law is true(which had been established back in the 1800s; it was one of the major accomplishments of the 1800 science), then how come we have Earth brimming with life?  Life seems to be a counter example to the second law of thermodynamics.  Life certainly seems like a water wheel driving another water wheel to make water flow uphill instead of downhill.  As the second video shows, the efficiency gets better with the greater difference between the heat and cold loads.  How does life do it?  I'd suggest and so do many chaos theorists that chaos is at work in living beings.

A big part of the problem of human machines is they generally work as closed systems.  And really, this is the effort, and this is what thermodynamics disproves; that a closed system cannot run forever; energy is produced out of thin air.  But, in an open system, open to the flow of energy and matter, a given structure can be maintained and kept moving.  Kind of like keeping a pencil on its tip, life keeps patterns going by means of feedback loops.  There's two types of feedback loops - positive and negative.  Positive feedback is like sending the output back into the loop that generated that output signal.  This generaly results in an exponential growth and a blown up speaker.  Negative feedback loops are like a thermometer that one can set at a given temperature, and the electronic circuit either turns on or off depending on whether the temperature is higher than a given temperature or lower.  Notice, mammallian life generaly has an ability to keep the body at a certain livable temperature.

In chaos theory, unstable states are held stable by one or more negative feedback loops.  Chaos theory can keep heat engines at greater efficiencies. 

I and many chaos have this idea in the back of our heads that chaos in the brain is what allows us our free will thought, our creativity and the creativity of life.  Our flexibility that you don't see in general human industrial machines.

I've probably taken on more than I can chew at one time; but, I hope I've made the idea of chaos theory solving the second law of thermodynamics puzzle to how life prospers on earth . . . plausible.

I'd like to end by pointing out that chaos theory has been developed technologicaly.  They can use chaos, or they can go from a chaotic state to any number of the systems possible stable behaviors in any medium - chemical, mechanical, lasers, and electronic circuits.  I've seen it used in nuclear fusion experiments, and for the electric circuits in today's electron microscopes utilising the latest inverse refractive optics. Electron microscopes have always been the twentieth centuries underrated technology. 

Also, chaos theory does not mean the end of the mathematical description of nature.  Chaos theory is as much an abstraction as the number two.  A strange attractor, whether in chemical, mechanical, electronic systems, are abstractions also.

Tuesday, August 21, 2012

thought for the day/Mechanical Universe . . . Einstein's theories of Relativity


-I'd like to first comment about why there's no astronomy picture for the day.  I'm not sure what pictures I've got up.  I know I've got some repeats.  Unless there's some really new picture, I'm pretty much done right now, unfortunately.

The Mechanical Universe's account of Einstein's theories only covers his Special theory of Relavity.  In both cases, Special and General, mathematicians had a big role.  One could say that Einstein got the credit more because he derived the work of the mathematicians from axioms.  The Mechanical Universe episodes show most of this, except they don't note that it was a Minkowski who thought of spacetime unified.  Then, much later around 1910 to 1915, Einstein showed that mass bends spacetime. The videos explain the rest well enough.  In special relativity, speed is relative to the observer; in General Relativity, inertia and gravity are relative to the observer. 

For Physicists, the Michelson-Morley experiment was like dark energy today - a completely unexpected result.  Physicists thought that light being waves implied an aether.  The fact that they did follows my and Jacob Bronowski's points about defining a part of the universe generates 'infered units' as Jacob Bronowski calls them(see my third to first post about the nature and origin of mathematical knowledge).  Initially, one could argue that the Michelson-Morley experiments presents problems for Jacob Bronowski's theory of knowledge(and mine); but, the truth is that the Physicists failed to question assumptions.  And in mathematics, the reason why they do axiomatics and deductive proof of everything is  . . . at least one reason . . . is to question assumptions. 

For mathematicians, the relativity phenomenon destroyed their religion of group theory - at least for the most part.  Mathematicians, and perhaps Felix Klein specificaly, had started thinking group theory can unify all mathematics(geometry and algebra).  From late 1700s to early 1800s, mathematicians had come up with non-euclidean geometries.  Felix Klein had shown that group theory can describe and unify all these non-euclidean geometries. Riemanian geometry almost certainly took center stage by Einstein's time; well, that's not stricktly true.  Mathematicians had made connections between complex analyses, algebraic geometry, and Riemanian geometry long before.  But, that's another story.  Well, mathematicians had actualy thought of curved spacetime(both Riemann and I'm forgetting the name of another guy; i think he combined complex numbers with vectors . . . sorry, I can't remember his name). 

Getting back to the physicists, they were later to make the vacuum have properties kindof aether like with Paul Dirac's combining of special relativity and quantum mechanics - in his quantum electrodynamics. And of course, this leads to the effort to make a unified quantum theory of gravity.

Monday, August 20, 2012

astro picture for the day

Image Credit & Copyright: Bret Dahl

--------------------------------Mechanical Universe extra!

I've tried to straighten out the Mechanical Universe's presentation, but clearly I wasn't able to.  I've pretty much covered the quantum mechanics side of things.  I'm planning on covering the industrialism/thermodynamics which is kind of related to atoms; but, it's also somewhat related to the particle physics/cosmology side of things(the three quantum forces are generally thought to unite at very high temperatures; temperatures that mankind cannot duplicate; gravity is hoped to unite at temperatures that makes that unification energy seem kind of quaint).  But, I'm going to cover the Einstein Relativity theories second, and then backtrack to the thermodynamics later.

----------------------------crazy science/technology for the day

Nobody is quite sure where this will lead science or technology wise.

Sunday, August 19, 2012

My Arthur Koestler Sleepwalkers review

Mr Koestler begins trying to relate and show how science comes out of pre-science - namely religion and mythology.  He ends up making one of the best accounts of church and state - in terms of the whole Copernicus, Kepler, and Galileo stories and issues.
The book is pretty good, perhaps for its time.  Well, when was Ernst Cassirer and Suzanne K. Langer writing?  About his time and before.  But, well, that's an indication of how much knowledge is out there; it's hard to be objective.  All one can do is say where you're getting your knowledge and what knowledge you're using.  Here, Koestler excells; he's make your college compositon teachers proud!
The book begins with great poetic paragraphs, and I was like, "oh boy, here we go; a poetic account of the development of science; this could get . . . exciting!"  Only, he quickly turns his writing style to pure scientific account.
He starts out by suggesting that science and mathematics comes out of a mythological past, which I agree.  But, as I hinted at above, he clearly doesn't know about or think about the nature of abstraction and how that relates to science and mathematics(see Suzanne K Langer's "Introduction to Symbolic Logic" for the best account of the how abstraction works and its relation to symbolic logic; and that symbolic logic can be derived out of pure language.  See Ernst Cassirer and Jacob Bronowski for an further understanding of the nature of mythology and mathematics; Suzanne K Langer translated Cassirer's "Myth, Language, and Logic"; see Jacob Bronowski's 'Science and Human Values", and "Origin of Knowledge and Imagination" . . . and "Science, Magic, and Civilization"; all these are short but sweet books, and just as intellectually exciting as Koestler's book here!).  He also can't see like Jacob Bronowski for one how science relates to the human condition.  This is why I dock him one star.  This is where he goes wrong in his book - in terms of relations between faith and reason.
What got me to buy and read this book was a reference to it in John Stillwell's "Mathematics and It's History."  It seemed an odd reference.  I looked it up, read some reviews here at amazon mostly!  and thought, that sounds familier!  I went and looked up Carl Sagan's "Cosmos" references, and found it referenced!  The two major and most important chapters/episodes in Carl Sagan's "Cosmos" are three and seven; the chapters about Kepler and the Greek science.  Well, now I know where he got the majority of his material to make those chapters!  As for Plato and Christianity?  I'd reference the first post of my "Jacob Bronowski Scientific Humanism" blog!  Or, how about The Origins of Christianity & the Bible 
by Andrew D. Benson.  The pace of research in new testament and old testament studies is of course fast and hugh.  I'd recommend The Jesus Puzzle, Christ in Egypt.  Robert Eisenman's "James Brother of Jesus." Bottom line, Jesus Christ is a hellenistic sungod overwright for James the Just.
I'm going to go to his chapters about Copernicus, Kepler, and Galileo now.  I've never read a hundred pages about Copernicus, now I know why!  Copernicus's work is inaccurate!  Still, I've found one more example of how the vagueness gamers play their games.  For instance, they justified not burning Copernicus at the stake by saying his work is just a mathematical fiction; convenient, but not to be taken seriously.  Meanwhile, beleive in god, or you'll go to hell(and pay me money at church).  I mean, why has science had such a powerfull affect for atheism?  Because scientists have noted that they never need the god hypotheses to prove or calculate anything.  Try calculating 1+1 by saying god over and over again.  Sure, you could do it; but, the point is you don't need to do so to calculate anything mathematical. 
Anyways, Kepler . . . Kepler figured out the elliptic orbits before Galileo figured out that unequal masses fall in the vacuum at the same rate(and noted that objects thrown in the air follow an parabolic path), and before Newton/Liebniz figured out the calculus; it's one of the most amazing human accomplishments in all human history!  The human story is one of horrendous odds of dealing with rather unprofessional messing around society from Tycho Brahe to Castle owners.  Arthur Koeslter uncovers great details like he came up with his second law of equal areas before he figured out the elliptic orbits, and that Copernicus new of the eight minutes of arc discrepency in the orbit of Mars.  He traces how they went from obvious wrong ideas like epicycles, but then he tries to say how they couldn't possibly have taken them seriously. He contradicts himself, and due to his lack of knowledge about the nature of abstraction(see above), he doesn't know how to find and trace how constructive knowledge comes out of previous vague knowledge(like natural language).  While Kepler almost certainly believed in god, due to Arthur's great work, I thougth I detected a hint that even Kepler may have had moments of pause in his later years(as he did acknowledge that astrology maybe wrong).  It's curiose and makes for great reading(Arthur Koestler's book here) how the Greeks from Homer to some passages here and there in Plato(Plato did believe in god)  but, Kepler, Galileo struggled with science and what it could mean.
What Galileo's science meant for god and the bible is what got him into trouble. Koestler tries to say that Galileo's troubles came from being a jerk to Kepler and telling Jesuits they're a bunch of numbnuts; but, ultimately, when talking to some daughter or wife of some church guy and telling her that yes, the bible is wrong is what really set the wheels in motion.  Koestler mentions this episode as well. But, he tries to justify it all because Galileo was a jerk to Kepler. Once again, thanks Arthur for the great historical details of the scientific episodes; but, in the end, the conclusion is that the Christian church tried to put the flame of rational light out when things looked like they were getting out of control for them.
I actually got tired, and didn't read the last half of the Galileo trials, the Newton stuff(I've read Morris Kline's Calculus which teaches calculus by means of learning Newtonian mechancis and solving the kepler problem; i don't think I need to read this section), and his last philosophy of science chapter(see above comments of mine).
If you want the gory details about Copernicus, Kepler, and Galileo's life and scienc's struggle, great read this book!(just like reading the gory details about christianity; read Robert Eisenman's books!).  As for his account of mankinds path from pre-science to science, I'd recommend Morris Kline's "Mathematics in Western Civilization", E.T. Bell's "The Development of Mathematics", or James Burke's "Connections", and "The Day the Universe Changed." Point is that, either through knowledge being too big for even Koestler to read up and synthesize it all, or science and mathematics progressing, his accounts of that progress and evolution of consciousness and intelligence in this sector of the universe is now outdated.  I've tried, and will no doute continue to try(one reason why I read this book is to try again and get more details, which was great).  I would recommend my blog again because that is a big part of what I try to do there.

astro picture for the day

Credit & Copyright: Ken Crawford (Rancho Del Sol Observatory), Macedon Ranges Observatory

---------------------------------Mechanical Universe extra for the day!


I've already addressed these aspects of the physics and particle physics world.  I don't know why they felt like making two very similar episodes and side by side!  But, they did, and they made more than one extra 'review article' like episodes like episode 1.  I'll get around to posting those also.

However inadequate my post about quantum electrodynamics and chromodynamics, it's more than what "Mechanical Universe" does!  What more to say?  How about my theory of quantum gravity!

Quantum particles are perpetually jiggly things.  They also have an ability to disappear from one place and reappear on the other side of the wall; they're waves and particles at the same time.  There was a recent(about a year ago) report about experiment/observations of electrons that they are perfectly spherical(when not bound to an atom) within experimental error.  How do they manage that?!  Let me mention one more phenomenon about quantum particles which is rather famous with today's quantum computing efforts, quantum entanglement.  They showed experimentally I do believe around 1980 that if you entangle two electrons(quantum particles), shoot them at opposite directions, and then do something to one of the electrons, the other electron will feel that faster than the speed of light.  How does that work!?

My solution is higher dimensional spaces going in and out of lower dimensional spaces.  The famous example of this is the flatland story of the 1800s.  I recall the authors name is Abbot; i havn't read the book; i need to; i get the info from any number of books.  Anyways, the idea is that take a sphere and pull it through an euclidean plane; at first, you see a point(tangent contact), then circles, first small then ever growing larger in diameter circles; then, the circles diameters get smaller as the sphere goes through the plane till it goes to a point and disappears all over again.  Sound familiar?  Yes, quantum particles! 

How does this explain the quantum entanglement experiment?  And how quantum particles can seemingling walk through walls?  Quantum particles are not just spheres going through planes, but much more complex higher dimensional shapes going through lower dimensional spaces.  Complex shapes that as they go through lower dimensional spaces, their shadows show up on one side, maybe predominantly, but, then they disappear and reappear somewhere else, maybe on the other side of the wall.  The quantum entanglment experiment?  By quantum entangling, a more complex higher dimensional shape has been created, and shooting them off just stretches it out; doing anything to one side obviously makes a reaction to the other side.

So much for quantum mechanics.  But, what about gravity?  I look to inflationary theory.  The inflationary period occurs at the planck time scales . . . it seems to be like a sphere going through a plane; it starts out as a point, then ever larger circles, and then smaller circles.  Well, in the big bang, the circles, actually in the big bang, it would be spheres, get larger and larger, and then they stay larger.  In fact today, we feel like there's an accelleration.  I've suggested this is just an inflationary aftershock.  But, it seems to me the inflationary theory seems like a higher dimentional shape going through a lower dimenstional space.  We can see easily how this higher dimensional interpretation of inflationary theory and quantum particles as I've described above are analogous, but how does gravity work here?  Well, the higher dimensional shape going through a lower dimensional space is like pulling apart a rubber band(or, one could even say like in Einstein's general theory of relativity, a mass bands spacetime). 

------------------------------------crazy science/technology extra for the day!

First I'd like to mention something similar to something I just found.  In "the Turing Option", probably the best hard sci-fi book next to "the Daimond Age" I've yet found, at the end of the book, they talk about this one quantum one way thermos bottle like material.  They mention how it would allow beer cans to stay cold for years, and one can bring down the cold from the north and south poles and use superconductivity more efficiently.  I've tried to see if anybody has worked on it; they have; but, it appears to be something that maybe only quantum computers can compute.  But, now to something I found just now.

This work can reduce energy loss in electronic devices considerably.  If anything, It allowed me to mention the above wild idea from a mere sci-fi book!

Saturday, August 18, 2012

astro picture for the day

Credit: NASA and the Hubble Heritage Team (AURA/STScI)

I found it!  This is not the Trantulla nebula of the large magellanic cloud(one our galaxies irregular satellite galaxy); this one is even bigger; it has to be; it's in a galaxy three million light years away!  This is a Hubble picture that I think probably will be forgotten(already has).

-----------------------------------Mechanical Universe extra for the day!

Isaac Newton's derivation of Kepler's laws from his inverse square law and his three fundamental laws of physics was one of the first such great syntheses in mathematical science.  Another would be quantum mechanics deriving all of the chemistry properties of the previous hundred years!  Another would be General Relavity making Newtonian mechanics a low speed special case.  Maxwell's deriving of previous laws of electricity and magnetism.  I've seen but would have to review that they've done such vast generalisations in thermodynamics; i think it was combining Newton's dynamical laws with thermodynamics by a Pierre Dunham. 

When new theories take in old theories as general cases, they are able to establish the old facts much more directly.  All theories are established indirectly at first(with the discovery of the sun centered solar system being the classic example), and then when Isaac Newton's come around to derive the old theorems by means of vaster generalisations, well, the old stuff is made much more systematic. 

-------------------------------------wild science/technology find for the day

from qubits to  . . . drumroll . . . superqubits!  Supersymmetric generalisation of qubits!

Friday, August 17, 2012

astro picture for the day

Credit: Hubble Heritage Team (AURA / STScI), Y.-H. Chu (UIUC) et al., ESA, NASA

-------------------------------Mechanical Universe extra for the day-------------------------------

The Heisenberg UnCertainty principle is more famous for the unpredictability it brings to science and the cosmos.  Quantum psychologists everywhere try to turn scientific method upside down.

The true significance of the Uncertainty principle is that it defines the quantum forces - electromagnetism, the strong and weak nuclear forces.  Particles can come out of the vacuum so long as they disappear back into non-existence within the uncertainty principle. The more massive the particles, the faster and shorter distance those particles can be allowed to go before it has to disappear within the uncertainty principle.  If these particles hit or are absorbed by other particles then they are exchange particle - force carriers.   The weak and strong nuclear force are short range because their exchange particles are massive, while the electromagnetic force can go the length of the universe because it's exchange particle is massless(the photon).

This whole understanding of the uncertainty principle came from Paul Dirac's unification of special relativity with shroedinger quantum mechanics.  Is so doing, he predicted anti-matter.  Anti-matter was shortly observationaly confirmed.  But, the theory had certain problems of infinity - self energy infinities.  This was the hardest physics problem of its time; it led to the search for unification theories.  It turned out that by unifying the fundamental quantum forces, the infinities could be cancelled out; this is called renormalisation.  Some say the problem has never been completely solved.  But, one thing is for sure, the three quantum forces led to electro-weak unification and then 'gut' theories(grand unification theories). 

Murry Gell-Mann solved the strong nuclear force by means of quarks.  Quarks are subatomic particles that come in pairs.  They can never be seen individually; if you put in enough energy to separate them, they just form another quark(by means of e=mc^2).  But, Feynman and experimentalists figured out that particle accellerators can act as a giant microscope that can slow down the process(just like in special relativity where time can be slowed down by getting close to teh speed of light; particles that normally decay in less than nanoseconds can be made to last longer, and so particle physicists can study them, or even use them to collide with other particles before they decay); the Stanford linear particle accellerator was able to view quarks without tearing them out of their quark couples and triplets.  Unification theorists soon found that by getting the number of quarks to match up with electrons, muons, and I feel like I'm forgetting another electron like particle, then they could solve the renormalisation problems for the strong nuclear force.  This shows the significance of the 1990s discovery of the Top quark; it was the last quark needed to confirm the standard model.  Well, the Higgs was next; but, the Top quark pointed particle physicists on where approximatelly the Higgs particle was.  The Higgs particle of course has recently been discovered.  The Higgs has a role to play in the Inflationary generalisation of the big bang theory.  One could say, the big bang was a highly improbably quantum fluctuation of the uncertainty principle.  But, we have plenty of evidence such as the Cobe and Wmap satellite data on the cosmic background radiation that the inflationary theory did happen.  And, of course, the Higgs has been discovered which further confirms the inflationary theory.  We've clearly come a long way since J.J. Thompson's discovery of the electron and even Maxwell's great unification of electric and magnetic fields.  But, we've got even bigger game - quantum gravity.

We know that with black holes and the big bang, gravity and the quantum forces need to be unified; but, gravity is not a quantum force defined by the heisenberg uncertainty principle.  It's a big problem!  We've also got other puzzles to solve - dark matter and dark energy(which I've often thought is just an inflationary aftershock).  The LHC that discovered the Higgs may be able to solve those problems as well.

I've actually gotten ahead of the story of quantum and Einstein's relativity theories over the last hundred now plus years.  Einstein's General theory of relativity led to the Big bang theory before Edwin Hubble deduced it by the motions of galaxies.  But, particle physics had also been hinting at a big bang theory of the universe.  Quantum physicists had discovered the mechanism that powers the stars - nuclear fussion.  They had soon figured out more or less how supernova work.  They learned how the different atoms were generated in the cosmos; they learned that we and our planet earth is stardust generated in supernova. But, they found that the proportions of helium and hydrogen in the cosmos is not accounted for by the supernova.  This 70-30 proportion seems to hint at a big bang a long time ago.  They also predicted the cosmic radiation background now used to study the big bang(and confirm the inflationary generalisation of the big bang universe).  This was discovered in the 1960s and established the big bang theory over other theories like the steady state theory.  But, particle physics soon hinted at the big bang theory in other ways in the 1970s.

In the 1970s, the unification of forces to solve the renormalisation problems seemed to occur at super hot temperatures.  At these temperatures, the three forces melt into one another, and at temperatures that dwarf those temperatures, it's hoped that gravity will also meld with the three quantum forces.  Where else do these super hot temperatures occur?  In black holes and the big bang.

With Isaac Newton's view of the universe as so many billiard balls richecheting off one another leading to Laplace's saying, "get the initial conditions established precisely, one can predict both all the past and the future" led to the idea of reductionism science.  In truth, if you study the nature of mathematics(which they did not), you see abstraction(common forms of many different structures), and the structural relations, one sees mathematics as making connections, unifications.  James Burke tries to show connections between all historic forces and technologies; i've shown that some of them are interesting, other connections are a little more spurrious.  But, mathematics is the real unified theory or everything(and so does Jacob Bronowski).  These are the realconnections.  The supernatural religious will say that science cuts us off from the universe, when, the truth is they're saying that spirit above matter, and the cosmos elements are different from earth elements because that's where god must reside(because he clearly doens't live on earth) is the real divisive split.  It was Isaac Newton, mathematicians, and mathematical science that has made the real connections between the cosmos and humanity.

---------------------------------dna nanotech news for the day------------------------------------------

A book has been encoded into dna.  The only problem is reading it out.

Thursday, August 16, 2012

astro picture for the day

Credit: ESO

---------------------------------extra goodie for the day------------------------------------------------------

The electromagnetic nature of light was certainly a great surprise; but, that discovery was just a tip of an iceberg.  The electric nature of atoms was to lead to much; the unification theories of the universe; the nature of the stars; that we are stardust, and eventualy, the atom seems to be like a snowflake; the clues to the origin of our universe is imprinted in atoms.

-----------------------------------quote for the day----------------------------------------------------------

"the roads that lead man to knowledge are as wondrous as that knowledge itself." - Johannes Kepler

---------------------------------nanotechnology and quantum computing news for the day-----------

D-Wave has been making special purpose quantum computers for a few years now(with some controversy on whether they're machines actually work . . . just like when Galileo was spotting moons going around Jupiter with his little telescope?).  Well they've just succeeded in calculating some protein folding with a quantum simulation machine.  This used 81 qubits; they have a 500 plus quibit machine going into commercialisation right now and plan on building a two thousand plus qubit machine in a few years.  But, with this demonstration, we're already in the quantum computing age!  Maybe we will just compute a primitive protein nanomachine(when the two thousand plus qubit machine gets built).  And, quantum computer breakthroughs occur all the time(today, there was a recent announcement of reading out qubits of a single atom that lasts for seconds(as oppossed to less than a nanosecond). 

Wednesday, August 15, 2012

astro picture for the day

Credit: Y. Nazé (Univ. Liège), Y.-H. Chu (Univ. Illinois), ESA, NASA

-------------------------------Mechanical Universe extra goodie for the day----------------------

In the 1800s, astronomers were told they couldn't tell what atoms were on stars and the sun, only for spectroscopists to find the fingerprint of the atoms.  The spectroscope was essentially started by Isaac Newton but needed some microscope to really see the spectral lines.

Later of course, philosophers thought that atoms weren't real or nobody could learn about them if they were true.  Then came the electron. 

This Mechanical Universe video shows how theory kind of dictated what should be there; if Millikan couldn't find the right numbers, he realized he needed to do the experiment better; the data simply wasn't good enough.

As the electron and the atom was becoming discovered, they never thought, "Oh, we could make nanomanufacturing with them."  It took generations and Richard Feynman to first think of it; and he was just joking around.  Anyways, I've gotten some recent progress that I'm not too sure they want said out loud; i won't tell you who, or where this work was done; but, I'll give a picture, and tell you what it is.

This is a single atom and electron transistor made by a scanning tunneling microscope lithografic process.  Word out is this is just some of what's been done.  They've made precise atomic structures to make precise quantum computing. Another thing said is that it would be nice if they could make microscopic stms; that's a little disappointing; i've been waiting for some news that they've done so.  Still, the above and apparently other nanotechnologies were made by an automated ability to make arbitrary precise atomic structures.

Tuesday, August 14, 2012

astro picture for the day

Credit & Copyright: Jon Christensen

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"The roads by which men arrive at their insights into celestial matters seems to me almost as worthy of wonder as those matters in themselves." - Kepler

-----------------------------------youtube for the day/Mechanical Universe and Beyond/Maxwell's equations

- I'd like to note first that I've seen all the mechanical universe videos on youtube before, only to see them all taken down; so, I hope if you're interested that you take advantage; who knows when they're taken down again!  These can be bought(which I did).

- In the history of physics, electricity/magnetism has an interesting history.  Isaac Newton pointed the way in his Principia; he mentions at the end that electricity/magnetism seems to be the next phenomonen needing to be mathematized.  But, nobody really took up his suggestion.  Instead, mathematicians like Laplace felt that astronomy was the ultimate science(in some ways, it is; but, until the twentieth century, we couldn't understand that we need to understand both the large and the small).  Mathematicians extensivelly extended analyses(the mathematicians word for the calculus) and went about applying it to astronomy(and mechanical concerns like vibrations of strings).  I've already pointed this out in the history of the discovery of Neptune and Laplace's five volume generalisation of Newton's Principia.

While the digression to astronomy may have halted the study of electricity, magnetism, and chemistry, the mathematics of the heavens did come back in quantum mechanics; but, that's a later story!  And, experimentalists did work on electricity and magnetism; but, untill, Maxwell put mathematics to it all, we never saw the nature of light!

Really, in some ways, I've messed up the history a little bit.  Newton's mechanics was the first unified account of everything.  A Romer around late 1600s noticed some inconsistent results of applying Newtonian mechanics to the orbits of the motions of the Galilean satellites of Jupiter.  Instead of announcing that Newtonian mechanics is wrong, he deduced the finite speed of light!  Despite the digression of sciences study of electricity and magnetism, James Clark Maxwell applied mathematics to the experimental discoveries of electricity and magneticsm and ended up predicting and calculating the finite speed of light!(there's some easy mathematics that one can do; but, one must understand that many experimentalists had to do much theory and experiment to get those constants). 

James Clark Maxwell's equations of electomagnetism and the realization that light is electromagnetic waves was the greatest achievement in science since Isaac Newton's Principia. Maxwell's electromagnetism would lead to problems which would create quantum mechanics and Special/General relativity. 

When Isaac Newton's Principia came about, philosophers would rewright all of the philosophy of ethics, knowledge.  They'd come up with a philosophy of reductionism which has haunted the philosophy of mathematics and science ever since.  Many would try to bring down all of science.  When Maxwell's equations led to quantum mechanics and Einstein's work, philosophers of science, mathematics had a hard time understanding it.  Mathematicians had a clue for a long time in the non-euclidean geometries, abstract algebras; but, this was inaccessable to philosophers; mathematicians didn't make much of it other than develop the mathematics. Well there were a few like Bertrand Russel who wanted to derive all of mathematics from logic alone; but, Kurt Godel's 1931 theorems about the incompleteness of finite sets of axioms gave them pause. In my opinion, Jacob Bronowski weighed it all out; seems that the whole world including the mathematicians have forgotten Jacob Bronowski.  It seems that the implications of the human species being defined and distinguished from all other life by it's ability and need to figure out the universe is too much for the latest generation.

that's probably enough for now!

Monday, August 13, 2012

astro picture for the day

Image Credit: Data Collection: Hubble Legacy Archive; Processing: Danny Lee Russell

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Looks like John Romer's "Seven Wonders" documentary has been put up by somebody; i've already posted some thoughts about this; but, at the time, there was just a little piece of the documentary posted on youtube.

Sunday, August 12, 2012

astro picture for the day

Image Credit: NASA/JPL-Caltech/MSSS

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Nasa/Jpl's Mars Curiosity rover has generated lots of excitement, and all it has done so far is land! Mars has held mankind's fascination for a while.  For Kepler, Mars presented a problem.  Untill, he mapped out the actual shape of Mars, he couldn't figure out what the problem was!

What's perhaps more remarkable about Kepler's mathematical understanding of the planets of the solar system is that he did it before Galileo figured out inertia, and that different massive objects have the same rate of fall in a vacuum.  These were the problems that stopped people from working on the heliocentric hypotheses.

Further, most people have heard of Isaac Newton's inverse square law; that mathematics is actually pretty easy when you check it out.  What most people don't realize is that Isaac Newton derived Kepler's laws from his three laws and his inverse square law. Newton's syntheses of Galileo and Kepler's findings(remarkable in themselves) requiered more mathematics; he had to come up with a theorem about how one doesn't have to calculate all the vectors of a mass - just know the center of mass. Newton went on to of course calculate the tides, the precession of the equinoxes, predict Haley's comet, the flattening of the Earth at the poles.  He also started the work to be able to derive the likely substances of the planets just by knowing the mass, rotation rate and speed around the sun.

The Greeks mathematics and physics was of course left to collect dust on the shelf(although, mathematicians still find it fun to relate the ancient mathematics to modern mathematics).  Isaac Newton's synthesis left the old agricultural civilization.  They were dependend on the calendar(not that are not; but, now clocks take precedence); they made their mythology(the science of the day) according to the zodiac and the precession of the equinoxes.  There's the twelve constellations always personified from the Egyptians to whoever made the Gospels.  If you'll notice Taurus the Bull, aries the ram, and Pisces the fish, you'll see that the Judeo-Christian religion seems to be about being at the right place at the right time; to kind of wright themselves into the history books.  The Minoans were full of Bulls; the Jews were into Rams horns, and Jesus Christ is always represented by the fish of the Pisces constellation.  The passover is about the transition from the pole star being in a given zodiacal constellation, taurus, aries, and pisces.  Well, it follows the pattern. Anyways, these concerns have for the most part been swept away.  Today, we see some people hoping that Maya myths of 2012 are true to sweep away this troubled world and put themselves into power.  No doubt, christians everywhere are waiting for the second coming at the coming passover(around 2100A.D; the passover, or the precession of the equinoxes from one constellation to another goes every 2100 years or so).  Many people don't know or care or believe this(I of course don't; but, i do believe that people believed in this mythology back then and tried to position themselves to it).

Newton's every action gives and equal and opposite reaction led to the rocket; i of course, hope that one day humanity expands out to space; out there, mankind will further grow up.

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I tried to explain awhile back how if civilization fell today, most people would die.  They don't know how to live off the land; they're living two many steps past 'living off the land' such as all life.  Let's say, they're that dependent on science and technology.  The few survivors will not be able to teach others how to start civilization again; All the stuff that got in the way of people thinking about science and mathematics, wild animals, weather, and others would keep people from learning everything before the next generation. they will have to start all over again.  I pointed out how the precision of the machine tools comes from astronomy(the clockwork precision of circular orbits).  So, they'd have to make stonehenges again; basically, they'd have to start all over again back to the stone age. 

The Kepler/Tycho Brahe events highlights our delicate dependence on science and technology in another way; people grew up with non-science beliefs; too leap to science thought required questioning those beliefs - something those beliefs do not want people to do. This is what I meant to point when I first posted Carl Sagan's episode 3 about Johannes Kepler(of his cosmos book/videos). 

Saturday, August 11, 2012

astro picture for the day

Credit & Copyright: Jean-Charles Cuillandre (CFHT), Hawaiian Starlight, CFHT

------------------------------quote for the day-----------------------------------------------------

"Free in mind must he be who desires to have understanding" - George Joachim Rheticus

Rheticus as he liked to call himself(because he learned Latin; he changed his name) persuaded Copernicus to publish his book about the heliocentric theory of the solar system.  This is all the more curious since it appears that the Copernicus got the Heliocentric idea from various people in Padua while he was learning mathematics and astronomy there.   Those people were too close to rome to publish.  As for Copernicus, he may have had misgivings about it himself; he may have been Catholic and didn't want to shake up the faith.  Rheticus was protestant and was willing to shake up a few people's faith.

Rheticus appears to not have been the most rational person around either.  He made an astrological epistle showing how the new heliocentric theory predicts the fall of the Roman and Arab spain empires and the age of the universe to 6000 years.

As for the Catholic Roman Vatican, they were initially willing to embrace the theory; but, that was only because they considered it 'just a theory.'  If it were to prove real, then they'd have a problem. 

-more interesting stuff about Rheticus; he noticed there was a problem with the orbit of Mars.  Johanes Kepler even knew that Rheticus noticed this before him. This is interesting.  Generally speaking, a scientists is suppose to throw out a theory if just one data point disproves it.  But, Rheticus and Copernicus chose not to throw out the Heliocentric hypothesis.

Thursday, August 9, 2012

quote for the day

" . . . Men imagine gods to be born, and to have clothes  and voices and shapes like theirs . . . . Yea, the gods of the Ethiopians are black and flat nosed, the gods of the Thracians are red-haired and blue-eyed . . . . Yea, if oxen and horses and lions had hands, and could shape with their hands images as man do, horses would fashion their gods as horses, and oxen like oxen" - Xenophanes

Wednesday, August 8, 2012

astro picture for the day

Copyright: Davide De Martin. skyfactory

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I can't seem to find a youtube; so, I'm linking directly to physorg article with a video of a galaxy survey(well worth viewing!).

The above is an image of quantum entanglement. Quantum computer researchers have been working to make quantum computers by making and manipulating quantum entanglement; but, these researchers are showing the relations between the quantum world and the classical may allow an easier way!

-Quantum computing might be easier, is Feynman/Drexler nanomanufacturing easy?  The obvious solution from today's(or yesterdays) technological base to feynman/drexlerian nanomanufacturing is to use stms to build smaller stms; but, that has proven to be a long road.  Eric Drexler has tried throughout his life really to find a shortcut; is there a shortcut(whether easy or hard) to get from today's technological base to nanomanufacturing?  So far, not really; or, the shortcut isn't exactly an easy one.   Could this result lead to an easy path towards nanomanufacturing?

Dna-nanomanufacturing has been shown now to be able to make arbitrary shaped nanomechanical gold parts with probably generalizations to other metals at least.  Or, this ability could be used to design specific catalysts which can do discrete chemical reactions(building nanomechanical parts). 

Dna-nanomanufacturing has been shown to be able to self organise the placement of arbitrary nanomechanical parts already.  So, if it can make them as above and self-organise them, then, we may start to see some nanomanufacturing from dna-nanotechnology.  Dna nanomanufacturing may yet provide an easy shortcut to feynman/drexlerian nanomanufacturing.  In other words, we might be able to get nanomanufacturing within our lifetimes!

Tuesday, August 7, 2012

astro picture for the day/thought for the day/ the mathematics of the circle

credit Leonard Mercer

---------------------------------Mechanical Universe extra----------------------------------------------


New account of the ancient mathematics of the circle -

The mathematics of Thales was interesting because of the way the theorems related to one another, in my opinion.  Similarly, for Babylonian geometric algebra and the Pythagorean theorem(which I've yet to write up!). Similarly, the mathematical story of the circle is interesting in the mathematics that is generated.  Sometimes a wrong path still makes for interesting mathematics, and then the mathematics along the correct road is holy!

The earliest mathematical knowledge of the circle before the Greeks was the Babylonians.  The neat thing they did was inscribe a hexagon in a circle.  It's the easiest polygon construction one can do!  Use the circles radius, and make equivalent lines starting from one point of the circle where a given radius meets the circle, let the other point of the equivalent line meet the circle, and the next line starts from that endpoint, all the way around the inside of the circle, and you get a six sided figure. So, the Circumference of the hexagon is 6, and the diameter is 2, so, circumference is approximated as 3.

The Babylonians hit on the insight that different size circles have the same ratio of diameter to circumference. C/D="pi", or C=(pi)D.  Either way, "Pi" come out of know-where, almost Jacob Bronowski 'inferred units like'(see my nature and origin of mathematical knowledge; third post of this blog!).  It also hides behind some mysterious symbolism much as the square root of two! But, that's getting ahead of the story.

In a connection between the mathematics of Thales and that of Pi, Thales most exciting theorem, "angles in a semi-circle are right angles" is used to by a Hippocrates to solve the 'quadrature of the lune'.

Hippocrates is also credited, by the Greeks, to having composed an 'Elements' like Euclid did hundreds of years later. Thales either discovered deductive proof, or as some evidence was found and communicated by Van Der Waerden, in his "Geometry and Algebra in Ancient Civilizations" book, Thales learned from a grapevine coming all the way from India!  Then, Thales taught Pythagoras, and the Greek mathematical awakening was lit.  The famous example of deductive reasoning is Plato's syllogism "All men are mortal . . . Socrates is a man . . . therefore, Socrates must die". The Deduction is two assumptions(in this case), and then a third conclusion that follows 'logically'!  The Greeks, apparently, saw that one could perhaps try to prove everything deductively from a finite set of axioms/postualtes.  The two assumptions mentions in the above Plato syllogism are like these axioms for that deduction. Hippocrates appears, as far as anyone can tell, to be the first to postulate all the results of Greek mathematics up to his lifetime; essentially, he codified all of Pythagorean mathematics!  The result is lost. In fact, the only original Greek mathematics that has come down to us is a result of Hippocrates - the Quadrature of the Lune. This was about 440 B.C. This was a major result at the time of Greek mathematics. It was also before Athenian Greek history.

The Greeks at this time hoped to tame the circle by squaring it. In fact, the Greeks hoped to square everything!  Because the calculation of the square is easy, if they could turn everything into a square, then, they can solve everything in the universe! The ancient mathematics of the circle is kind of hard to put in a straight line. Because there's connections between the mathematics of squaring everthing and the irrationality of the square root of two, I'm going to back up and present that mathematics, and then get back to showing Hippocrates "Quadrature of the Lune."

- One of the greatest Greek theorems before Hippocrates 'Quadrature of the Circle" is Pythagorases theorem about right triangles.  The right angles triangle has a right angle in it which generates a hypotenuse diagonal side.  A right angle is one that is a 90 degrees angle. The Pythagoras is credited for proving this deductively. The Babylonians may have proved a linear case of this.

Start with a square, draw two diagonals inside the square, generating four equal triangles.  On one side of the square, make a triangle on top of the square equal to one of those triangles.  Draw two square on the sides of that triangle.  They will both be equal, because the sides of the triangle are equal.  Make a diagonal inside those two smaller squares(this figure should look kind of funny . . . !).  So, you have four triangles between the two smaller squares; that's the same number of triangles in the larger square.  The triangle on top of the square now has two squares on it's right angle sides that sum to the same area as the square on its hypotenuse!  This is about as easy a proof of an interesting mathematical fact as you can get!

The above is a linear case of the Pythagorean theorem.  Pythagoras(or the Pythagoreans) solved a non-linear case.  And more or less showed this theorem is true 'in general.' Pythagoras almost certainly got this also from the Babylonians. Start with a square, and arrange around the inside four right angled triangles.  At this stage, you have a square on the inside, which is also 'on' the hypotenuse of each right angles triangle inside.  Now, mathematicians of the Babylonians and Pythagoras, rearrange the triangles into two rectangles.  I put one rectangle, horizontally, on the upper right hand side . . . and the other, vertically on the lower right hand side.  Now, one should probably put an imaginary line from the top rectangles left lower vertex. . . the line should extend from the bottom side of the upper rectangle to the other side of the square.  Now, we have a small square in the upper left side, on the small side of the rectangle above.  Below, there's a medium sized square on the other right angled side of the lower rectangle.

- Now, for Euclid's famous proof of the Pythagorean theorem, in his then and now monumental axiomiatic unification of all Greek mathematics up to his time.  Euclid lived around 300 B.C. and he did his work in the new city and library of Alexandria.  Alexander the Great has been dead for a little while. But, the Greeks seemed to be in charge of the whole known world.  This was before Alexander the Great's sons started waring with one another, and then the Romans went around cleaning them up from behind.   But, anyways!

Before I get to Euclid's theorem, I need to explain all kind of mathematics that needed to be solved before Euclid could see a new more general proof than even the Pythagoreans(the Babylonians).

The simple geometric proof of the area of a triangle is to take a rectangle, or even a square, and divide it in half by a diagonal.  The area of the rectangle is of course base times height.  Now, we see an extraordinary formula comes out quite simply - the area of a triangle is 1/2base 'x' height. But, triangles can come in a variety of sizes and shapes.  Scalene, acute triangles . . . basically, one vertex can extend outside of the base, or well inside. What about these cases?  Or, we could say, these are non-linear cases; how to prove them?  Euclid breakes it down into every possible case in book one of his Elements - from about 1-35 to 1-41(the Pythagorean theorem is famous 1-47; proved very early in Euclid's elements; what other wonders are in twelve other books?  It's noted by mathematicians that Euclid chooses his parallel axiom wisely, and doesn't use it till like book two.  Euclid's proof of the Pythagorean theorem is considered a great theorem for this and other reasons).

I won't get into every case; but, what he's doing is showing that when variously, triangles or squares are in the same base and parallels, they have the same areas. You could have a figure like a parallelogram, where there's a small base side, and a longer top side, and hence the two end sides are angled outward.  It looks likea bit of a bowl, but with straight lines.  Drawing straight parallel lines from the vertices of the bottom base to the opposite end sides generates two overlapping parallelograms.  Then through checking off various properties of opposite angles are equal, and similar parts, one sees these paralleograms are of same area. What the Greeks found is that, no matter how skewed the angle of the triangles, the area is the same!  Euclid used this property to make a very prove the Pythagorean theorem in general!

I've generally tried not to put in a picture in this post because they get taken down; but, I can't help putting in a picture of Euclid's famous proof of the Pythagorean theorem,

image credit, The British Library Board

Well, this is a 1200 A.D. Arabic copy of a page of Euclid's Elements, showing Euclid's proof of the Pythagorean theorem.  Here, you only see one set of lines relating the rectangle area on the right side to the small square on top.  But, the basic idea, and point of it all is as described above.

The Pythagorean theorem makes all kinds of things possible, from making spirals, to number theory of Pythagorean triples, to trigonometry, and differential geometry of Einstein's spacetime.  It also leads mankind being able to deal with more non-linear cases of the world - irrational numbers.

Getting back to the linear case of the Pythagorean theorem actually!  Where both right angled sides equals one . . . if you make one of those sides base, and both sides equals one . . . then let the hypotenuse fall down on a side, the hypotenuse side extends past whichever base side you choose to a point, that turns out to be an irrational number!

Here's an account of the Pythagorean proof of the irrationality of the square root of 2 that I made elsewhere . . . well, a little more explanation. The algebraic formulation of the Pythagorean theorem is A^2+B^2=C^2 (the "^" means exponent here; I can only hope you know what an exponent means! If not, google it!)  In terms of our linear case, with the hypotenuse allowed to fall down on a number line, generating some mystery number, we get 1^2=1^2=C^2, or 1+1=2, and 2=C^2; now, in algebra, to get rid of a exponential, you make a radical sign to both sides . . . the radical sign and the "^2" on the right hand side disappears to just end up with C, and on the other side, you get radical 2, or "what's the square root of 2?"(once again, if you don't know what I'm talking about, just google, square root of 2, or radical sign, or something like that!)

"there's a little theorem of Pythagoras that proves irrational numbers . . . exist! (square root of; or sr)2=a/b(for a rational number, because the Pythagoreans thought all numbers should be these nice little numbers at the most!).  Now, in algebra, there's this rule that you can take the radical off a root of a number by squaring it(if this is a root of 2), and squaring the other side, so we get 2=(a^2)/(b^2). Now, some algebraic switcheroo, gives 2(b^2)=a^2.  So, a is even, therefore, we can bring in a=2b; so, we substitute this into a^2=2(b^2), so (2b)^2, gives 4b^2=2(b^2) . . . dividing out 2, gives 2b^2!(not the factorial exclamation mark!).  We can go the other way, which means b is also even, which means a/b are reducible further . . . which contradicts that a/b is not irreducible; hence, irrational number(in this case the diagonal of a right triangle, which involves the Pythagorean theorem by the way)  . . . drumroll . . . exist!|

I don't really want to get into the deductive proof niceties of this proof here.  Something I should have said somewhere above when I mentioned deductive proof is . . . Human language is radically different from all other life on Earth.  Whether discovered by the Indians(seems to be the case actually!), or the Greeks(they seem to be the first to really take it seriously), deductive proof is as radical an advancement of language, and human reason, and natural human language is over animal language . . . perhaps in my opinion.  Here, with the deductive proof of a new number, Human's have unlocked a secret of nature, and gone to a new stage of intelligence(well, the Greeks from Thales on did anyways)  Deductive proof unifies, things, it allows one to look behind what our senses immediately tells us(questioning of assumptions), kind of like how our senses say the Earth is flat, and mathematical science allows us to deduce that the Planets goes around the Sun, and not the sun around the Earth. Deductive proof solved problems by showing the balance between two much and two little. It tunes to the exact solution. It shows how things fits exactly. I couldn't stress enough the landmark irrational numbers and the Greek/Pythagorean deduction of it is for mankind. It's like having a spacefship, and parking it in your garage; or, at least it was for these Greeks, who barely wore any clothing at all. It's like some god giving Perecles a sword of some unknown metal never heard of before in "Clash of the Titans(see the original 1981 movie!).  Moving on . . . back to Hippocrates 'Quadrature of the Lune' . . . we'll relive the irrational numbers a bit later on though!

- After the Pythagoreans, the next major mathematical milestone was Hippocrates Quadrature of the Lune. The lune is like the crescent phase of the moon.  Hippocrates uses 1) the Pythagorean theorem, 2) angles in a semi-circle are right angles, and 3! the areas of two circles, are to one another as the squares on their diameters(Proof is in Euclid's "Elements", book XII-2.

Proof is on a semi-circle, split it in half from the diameters half line . . . on one half(usually on the left side!), draw a straight line from the top of the perpendicular line from the half point of the diameter, to the edge of the diameter on the left side. That should generate a triangle on the left side.  On the diameter of triangle, draw a semi-circle. - the angle in the semi-circle is of course, a right angle, so apply the Pythagorean theorem.  Well, the legs of the triangle inside the semi circle are of course 1 each, well area of large is 2, while area of small is one, and because of the Euclid XII-2 above, we get 1/2.  (AC)^2/2(AC)^2, and the AC's cancel out, leaving 1/2.  The bottom is the larger semi-circle, so it's 2 of the semi-circle built on the left side, so it's 1/2, by a theorem that hasn't been proved yet. But, this is how Hippocrates did it! 

Hippocrates thought he had proved the area of two circles are as their squares; but, he didn't; he couldn't have.  He also thought he had squared the circle.  But, he didn't.  The crescent is not exactly even squaring one quadrant of the circle!  It's something built on top!  But, he tried, he did some heptagon thing, which didn't work. This effort was a dead end; but, mathematicians couldn't prove it till like the 1800s when Lindemann proved that Pi is a transcendental number(won't explain here; well, kind of like a curve that intersects a line in the Cartesian plane off to infinity; if you have an algebraic . . . which isn't accurate statement . . . equation that intersects the line off to infinity . . . that's a transcendental number; at that point, you should realize, the Greeks were messing with something far more mathematically scarry than even irrational numbers!).  But, I'm going to describe some more mathematics that came out of Hippocrates and the Greeks efforts to square . . . everything, and then get into Archimedes.

Because I'm not really finished with Hippocrates effort to square the lune!  Hippocrates showed that the lune on the left side is equivalent to the triangle on the left side. Subtract the left quadrant of the large semi-circle from the small semi-circle built onto the left side; the remainder is the lune(a crescent shape). That cresent in eual to the triangle generated earlier. How is that squared?  The Greeks came up with a way to square the triangle.

A triangle is 1/2b'x'h, and a rectangle is of course b'x'h, (I'm using x to mean multiplication; should have explained above!). So, if we make the small side equal to half the height of the triangle, and the base of the rectangle equal to the base of the triangle, we make a rectangle equal in area to the triangle. Then, from there, we make a square equal in area to the rectangle.  The squaring of the rectangle involves the Pythagorean theorem.  The rectangle long side base is estended to the length of its short side.  A square is made where one side is equal to twice the length of the short side of the rectangle. A semi-circle is used to make an intersection point. A triangle is formed, and the Pythagorean theorem is used to equate squares on sides. Some of the geometric algebra looks like EF= FG-EG, BE= BG+GE, these are a-b and a+b respectively.  If you take the Pythagorean theorem, you have A^2= B^2 + C^2(I did Pythagorean theorem algebraically wrong above; but, It probably doesn't totally ruin what's said above, fortuitiously), then rearrange the Pythagorean theorem algebraically, to A^2 - B^2 = C^2.  Now, more algebra turns the left side into (a-b)(a+b)!  Base times Height!

Hipprates once again, proved his theorem, most likely without having proved the different circles are to one another as the squares on their diameters.  Remarkable enough, but lets describe some of which is proved in Euclid's elements, hundred plus years after Hippocrates time.

The theorem was most likely observed by the Babylonians, and proved and placed in Euclid's "Elements" XII-2. It relies on two theorems in Euclid's Elements book X-1 and 2. All these proofs are long, and, like describing the proofs above, I'm just going to give the basic ideas of them here. It's like in infinit series, were you have a point at say 1, and you make one point 1/2, the next point 1/4th, and you make half of each succeeding interval that gets closer to the point you want, this is more or less what's done.  Actually, there' s a remarkable relation to Euclid's algorithm(subtractive version and not division; there's two versions).  You take a smaller part from a larger part, but the smaller part is just more than half the size of the larger, and you repeat the process. X-1 sets up two of these processes and shows them to be equal!

XII-1 is a polygon version of XII-2(circles are to one another as the squares on their diameters).  Really, as far as I can tell, all that really happens is that the theorem shows that the squares never exceed certain bounds. XII- 18 is the sphere's version!  Archimedes uses this as well.  Here, the sphere are to one another as the triplicate ration of their diameters.

Archimedes certainly didn't know you can't square the circle; but, he knew nobody had succeeded.  Archimedes by the way got his initial schooling at the Library of Alexandria.  He was of nobility.  His father was King of Syracuse, on the island of Sicily. I once lived there and more or less saw various things of Syracuse. Sicily has all kinds of ruins going back to pre-Greek and Roman times actually.  But, Archimedes lived in a time when the Roman's were consolidating power.  They had just defeated the Cartheginians(what was left of the Phoenicians/Minoans.  They weren't in the mood for a wise-cracker like Archimedes. Getting back to the mathematics.

Archimedes proved that one can triangulate the circle.  He made one side of the triangle equal to the radius of the circle, and the other the circumference of the circle. The areas of the circle equal to the hypotenuse; this last part should suggest he's relating different dimension quantities! But, lets back up to his proving the area of a polygon = 1/2 hQ

Imagine any polygon, send out diameters to each vertex; triangulating the entire polygon. So, for each triangle, 1/2b'x'h+ 1/2b'x'h . . . ; then factor out the b'x'h, and you get 1/2hQ  Seems obvious enough!

Getting back to the triangulation of the circle . . .  area of triangle is T=1/2rC!  (C is of course circumference as one of the triangles dimensional quantities to be multiplied). . . . he makes two cases, 1-A> T.  So, 1-A> T would leave an excess.  But, one can imagine inscribing a square in the circle, doubling it's sides till it overcomes that excess boundary.  So, contradiction. I'm giving a rough description here! But, as always, I'm giving the gist of it.  Archimedes then does this in reverse, <, and gets another contradiction;  What he's done is a double proof by contradiction!

We have the area of a triangle which is not < or >, but must equal it by deductive proof!  Hence, A='Pi'r^2(by the polygonal theorem above!). Or 1/2rC, and plugging in 2'pi'r(circumference, 1/2r(2'pi'r), gives 'pi'r^2

Here, Archimedes sees that he needs to calculate Pi. His plan is to inscribe/circumscribe polygons in/on a circle.  This was a plan started by Eudoxus, called method of exhaustion. Eudoxus found the fourth term of proportionality, found in Euclid's Elements book VI I do believe.  This is odd, because it's known that book V of Euclid's Elements, all about advanced proportions, is composed by Eudoxus.  Eudoxus proves things like A+B/c= D+E/f, and A+B+C.../A'+B'+C'...  I mention some more about the genius of Eudoxus in my accounts about Greek Astronomy.  But, if you look at inscribing/circumscribing polygons in circles, you'll find that each type of polygon takes a new insight/proof/construction to inscribe and circumscribe around circles!  Archimedes finds a proof that allows him to circumvent having to to go through all these different specialized constructions - Euclid's Elements VI-3.  I'll describe it later; first, I want to describe some of these constructions!

- To find the center of a circle.  Draw a line at your convenience through a circle.  Then, bisect it; draw a perpendicular through that bisect point.  Then find the bisection of that perpendicular line; that's the center of the circle!  The proof is just checkoffs of conditions for it to be so!

I'm just going to describe some inscribing of polygons and not the circumscribing to make this whole writeup a little bit shorter! To inscribe a triangle. Draw a tangeant line to the circle to inscribe the polygon/triangle. Make an angle on one side of the tangeant angle equal to one side of the triangle to inscribe.  Then on the other side of the tangeat line, make an angle equal to another angle of the triangle. Then join sides of the angles inside the circle!

A square is the easiest! Draw to lines through the center of the circle, and each perpendicular to each other.  Join lines to the the points of intersection between the circle and these lines!

Now, we get the exciting polygon - the Pentagon.  This was solved by non other than the Pythagoreans. We inscribe an isosceles triangle like the triangle inscribing method above. Then, make bisection lines from each vertex of that triangle, draw the lines to the other side of the circle. Connect dots by drawing lines from those opposite points to the other points on the circle. Two other iscosceles triangles are generated, by the way. There's actually another way, involving decomposing into triangles.

As the Pythagoreans found out, the Pentagon needed to use certain scaline triangles to decompose it while the other polygons used the same triangles. The Pentagon was a special polygon in that sense. This ability to make a Pentagon was essential to make the Dodecahedron and Icosahedron(the two hardest, and the only really hard ones). Eudoxus also found a proportions method to making the Pentagon.

But, Archimedes actually found a shortcut around having to go through all these polygons and more(he got up to 96 sides; the pentagon in only 5).  Euclid's Elements theorem VI-3

If an angle of a triangle be bisected and a straight line cutting the angle cuts the base also, the base will have the same ratio the remaining sides of the triangle.  The proof takes a parallel line outside of the triangle, and so uses properties of parallelism as usuall.  It basically starts equating angles, even using a few syllogisms, like this angle is equal to this angle, so that first angle is equal to this third angle. So, if you actually check say Sir Thomas Heath's collected works of Archimedes, and see the proof, you'll see a diagram of a quadrant of a circle, with rays coming out of the center point to the circle on the left, and they are all bisecting each other.  Which is all great, except, the first one uses a square root of three . . .

Back to the proof of the square root of two.  One could do a similar proof for the square root of 3!  Thaetetius and a Theodorus did this way back in Athenian times, and went as far as square root of 17.  They then did a rather elaborate theory of incommensurability.  But, really, we're getting out of subject for this writeup. Archimedes has to make an approximation to this square root of 3.  And here, scholars are either mixed, or they just say, they don't know. Nowhere, not Sir Thomas Heath, not Van Der Waerden, not William Dunham, do I ever see anyone actually work this out.  They just say, there's two educated guesses of how Archimedes made approximations to this, and three other mystery numbers like this.  One way is what's called side and diagonal numbers; modern mathematicians recognize this as a pellian equation of number theory. I quote a passage of Theon of Smyrna, who's actually living around 100 A.D. to give us this account and idea of what Archimedes might have done,

"As the source of all numbers, unity is potentially a side as well as a diagonal. Now let two unts be taken, one lateral unit and one diagonal unit; then a new side  is formed  by adding the diagonal unit to the lateral unit, and a new diagonal by adding twice the lateral unit to the diagonal unit."

Let's look at the mathematics of this Pythagorean mathematics.  Yes, this is yet another piece of Pythagoreanism.  This time, they're pretty sure some Pythagorean, and not Pythagoras did this. Pythagoras set up a school of mathematicians. And, they did mathematics.  It was probably the first of its kind. A pellian equation is kind of a Pythagorean equation, but like this, d^2=2a^2+/-1

The algebraic generalisations of the Theon description is A(subscript n+1)=A(subscript n)+D(subscript n) , D(subscript n+1)=2a(subscript n)+d(subscript n)  .  Van Der Waerden gives a educatd guess at a mathematical way the Pythagoreans could have come to this,

Perform an Euclidean algorithm on a triangle.  Take a side of a right triangle and subtract it from the hypotenuse. Then, subtract b-a from that smaller side, and so on.  From b'=a'-a' , and a'=b-a, we get b'+a'=a (see the algebra from b'=a'-a'?  Add +a to both sides, the other one leaves me scratching my head a little bit actually!, and b= 2a'+b' from  I see how the 2a' comes from a'= -a, but somehow he fills in b on the 2a' side.  There's a mathematical induction performed by the Pythagoreans which seems a bit more solid.

d^2=2a^2+/-1 is proved geometrically in Euclids Elements II-11.  The diagram is elaborate, so, I'm going to give a briefer 'model' of what's going on there.  Take a straight line, mark acbd, spaced out from left to right.  AB is bisected by C, and produced to D, then (AD)^2 + (DB)^2 = 2(AC)^2 + 2(CD)^2.  A reexpresing by means of AC=CB=x, BD=y, leads to (2x+y)^2 + Y^2 = 2x^2 + 2(x+y)^2.  If you subtract y^2= 2x^2+/-1 from that last equation, you get (2x+y)^2= 2(x+y)^2 +/-1.  This I kind of like replacing x by (x+1) in the mathematical induction process. The Pythagoreans had essentially proved their side and diagonal method.  And so, now, we can get on to Archimedes calculation of Pi!

All the above is the mathematical history of Pi.  Today, we solve it by calculus methods, and some infinit series methods(Vieta has a remarkable trigonometry method). I actually stop here, because, I found that I could not find his first mystery number by the side and diagonal numbers! Archimedes has a 265/153 first approximation to square root of 3.  He goes on to far larger fractions.  I actually got the 265; but, for some reason, I couldn't get the 153!  He uses the Pythagorean theorem, and some of the advanced proportions of Eudoxus to twist and turn numbers as he goes.

I'd like to finish by pointing out that Archimedes goes on to suggest a "cattle of the sungod" problem.  Which turns out to be a pellian equation, and even systems of equations.  Here's the pellian equation. U^2-(609)(7766)v^2=1.  Well, here's a wiki --> Archimedes' cattle problem

- I can't help pointing out more about Archimedes. Everyone knows he came up with the Archimedes screw; there's suggestions he made burning mirror to set ships on fire; but, I'm sure all they did was get in the eyes of the Roman soldiers.  They also say he had a mechanical arm that would lift the Roman's ships out of the sea and drop it.  That sound's remarkably impractical(would the Roman's allow the Syracusans to hook up the contraption? Mathematically, he did more. He found center points of triangles and ellipses and such. He used Euclid's Elements propostion XII-18 to find areas and volumes of Sphere's like he did for the circle!  There's a Heron's area of a triangle formula.  A general equation for finding triangular area other than 1/2 b'x'h. It was attributed to Archimedes by the Spanish Arabs. He found the area of a parabola.  He used spirals to solve trisection of an angle, he solved third degree equations by means of intersection of conics(today, this wouldn't seem so hard; but, he's doing things by geometric algebra).  Getting back to physical science, he found Buoyancy.  He helped his King find fraudulent gold by means of density.  Well, he supposedly discovered it while bathing, and got out, ran around naked, screaming Eureka!  Well, I'm sure back then, people wouldn't have been that disturbed by some guy running around naked in an excited state!