Monday, July 30, 2012
Demons in the History of Science
Part one of two: Laplace’s Demon
Some might say that the modern day physicists have it easy; they can appeal to the public with their stories of eleven-dimensional universes, time travel, and stories of a quantum world that is stranger than fiction. But the basis of such appeal remains the same as the appeal for pursuing science always was and will be: a greater understanding of the environment, ourselves, and knowledge itself.
Just like Schrödinger’s cat, a popular thought experiment by famous physicist Erwin Schrödinger, Laplace’s Demon and Maxwell’s Demon are two other thought-experiments in scientific thinking which are important for what they reveal about our understanding of the universe. It may only interest you to learn of these thought-experiments for the sake of reinforcing the philosophical relevance and beauty that science has always sought to provide.
Jim-Al Khalili, author of Quantum: A Guide for the Perplexed, affirms that fate as a scientific idea was disproved three-quarters of a century ago, referring to the discoveries of quantum mechanics as proof, of course. But what does he mean when he says this? Prior to such discoveries, it may have been okay to argue for a deterministic universe, meaning that scientists could still consider the idea of a world in which one specific input must result in one specific output and thus the sum all these actions and their consequences could help “determine” the overall outcome, or fate, of such a world.
Pierre-Simon Laplace, born on March 23, 1794, was a French mathematician and astronomer whose work largely founded the statistical interpretation of probability known as Bayesian Probability. He lived in a world before Heisenberg’s Uncertainty Principle and Chaos Theory and thus he was allowed to imagine such a deterministic universe:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
Laplace, A Philosophical Essay on Probabilities

Laplace thought about what it would be like if it were possible to know the positions, masses, and velocities of all the atoms in existence and hypothesized a being, later known as Laplace’s Demon, which would be able to know such information and such calculate all future events. 
With our knowledge of physics, The Heisenberg Uncertainty Principle and Chaos Theory, such a being could not exist because such information about atoms cannot be observed with enough precision to calculate and predict future events. (By the way, “enough” precision means infinite precision!) This might be good news for those who believe in free will as its concept would not be permitted in a deterministic universe governed by Laplace’s demon. 
Interestingly enough, The Heisenberg Uncertainty Principle and Chaos Theory are not the only restrictive challenges that scientists have faced in trying to understand the properties and bounds of our universe. The Second Law of Thermodynamics is also of concern to scientists and philosophers alike, as we will learn with the birth of another mind-boggling demon.

Demons in the History of Science

Part one of two: Laplace’s Demon

Some might say that the modern day physicists have it easy; they can appeal to the public with their stories of eleven-dimensional universes, time travel, and stories of a quantum world that is stranger than fiction. But the basis of such appeal remains the same as the appeal for pursuing science always was and will be: a greater understanding of the environment, ourselves, and knowledge itself.

Just like Schrödinger’s cat, a popular thought experiment by famous physicist Erwin Schrödinger, Laplace’s Demon and Maxwell’s Demon are two other thought-experiments in scientific thinking which are important for what they reveal about our understanding of the universe. It may only interest you to learn of these thought-experiments for the sake of reinforcing the philosophical relevance and beauty that science has always sought to provide.

Jim-Al Khalili, author of Quantum: A Guide for the Perplexed, affirms that fate as a scientific idea was disproved three-quarters of a century ago, referring to the discoveries of quantum mechanics as proof, of course. But what does he mean when he says this? Prior to such discoveries, it may have been okay to argue for a deterministic universe, meaning that scientists could still consider the idea of a world in which one specific input must result in one specific output and thus the sum all these actions and their consequences could help “determine” the overall outcome, or fate, of such a world.

Pierre-Simon Laplace, born on March 23, 1794, was a French mathematician and astronomer whose work largely founded the statistical interpretation of probability known as Bayesian Probability. He lived in a world before Heisenberg’s Uncertainty Principle and Chaos Theory and thus he was allowed to imagine such a deterministic universe:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

Laplace, A Philosophical Essay on Probabilities

Laplace thought about what it would be like if it were possible to know the positions, masses, and velocities of all the atoms in existence and hypothesized a being, later known as Laplace’s Demon, which would be able to know such information and such calculate all future events. 

With our knowledge of physics, The Heisenberg Uncertainty Principle and Chaos Theory, such a being could not exist because such information about atoms cannot be observed with enough precision to calculate and predict future events. (By the way, “enough” precision means infinite precision!) This might be good news for those who believe in free will as its concept would not be permitted in a deterministic universe governed by Laplace’s demon. 

Interestingly enough, The Heisenberg Uncertainty Principle and Chaos Theory are not the only restrictive challenges that scientists have faced in trying to understand the properties and bounds of our universe. The Second Law of Thermodynamics is also of concern to scientists and philosophers alike, as we will learn with the birth of another mind-boggling demon.

Thursday, July 5, 2012

Two days ago, the CERN team announced that they had found a new particle whose properties are consistent with the long sought-after Higgs Boson’s. Whether or not it is the elusive boson however, is still to be determined by further research. To read more about the event, follow this link to the new BBC article.

If you have no clue what this is about, the above video is a quick and nice introduction to the Higgs Boson submitted by one of our followers, the lovely oh-yeah-and-what. Thanks for the awesome submission!
SIWS loves feedback from followers and we’ll do our best to respond. If you have any questions, ideas, or concerns, feel free to drop us a message, email us at sayitwithscience@gmail.com or like and post on our Facebook page. You can even make a submission post and we might publish it and credit you, like we did with this one!
Take care and happy science-ing! 

Wednesday, October 12, 2011
Hypercubes
What is a hypercube (also referred to as a tesseract) you say! Well, let’s start with what you know already. We know what a cube is, it’s a box! But how else could you describe a cube? A cube is 3 dimensional. Its 2 dimensional cousin is a square. 
A hypercube is just to a cube what a cube is to a square. A hypercube is 4 dimensional! (Actually— to clarify, hypercubes can refer to cubes of all dimensions. “Normal” cubes are 3 dimensional, squares are 2 dimensional “cubes, etc. This is because a hypercube is an n-dimensional figure whose edges are aligned in each of the space’s dimensions, perpendicular to each other and of the same length. A tesseract is specifically a 4-d cube). 

[source]
Another way to think about this can be found here:

Start with a point. Make a copy of the point, and move it some distance away. Connect these points. We now have a segment. Make a copy of the segment, and move it away from the first segment in a new (orthogonal) direction. Connect corresponding points. We now have an ordinary square. Make a copy of the square, and move it in a new (orthogonal) direction. Connect corresponding points. We now have a cube. Make a copy and move it in a new (orthogonal, fourth) direction. Connect corresponding points. This is the tesseract.

If a tesseract were to enter our world, we would only see it in our three dimensions, meaning we would see forms of a cube doing funny things and spinning on its axes. This would be referred to as a cross-section of the tesseract. Similarly, if we as 3-dimensional bodies were to enter a 2-dimensional world, its 2-dimension citizens would “observe” us as 2-dimensional cross objects as well! It would only be possible for them to see cross-sections of us.
Why would this be significant? Generally, in math, we work with multiple dimensions very often. While it may seem as though a mathematican must then work with 3 dimensions often, it is not necessarily true. The mathematician deals with these dimensions only mathematically. These dimensions do not have a value because they do not correspond to anything in reality; 3 dimensions are nothing ordinary nor special. 
Yet, through modern mathematics and physics, researchers consider the existence of other (spatial) dimensions.  What might be an example of such a theory? String theory is a model of the universe which supposes there may be many more than the usual 4 spacetime dimensions (3 for space, 1 for time). Perhaps understanding these dimensions, though seemingly impossible to visualize, will come in hand. 
Carl Sagan also explains what a tesseract is. 
Image: Peter Forakis, Hyper-Cube, 1967, Walker Art Center, Minneapolis

Hypercubes

What is a hypercube (also referred to as a tesseract) you say! Well, let’s start with what you know already. We know what a cube is, it’s a box! But how else could you describe a cube? A cube is 3 dimensional. Its 2 dimensional cousin is a square. 

A hypercube is just to a cube what a cube is to a square. A hypercube is 4 dimensional! (Actually— to clarify, hypercubes can refer to cubes of all dimensions. “Normal” cubes are 3 dimensional, squares are 2 dimensional “cubes, etc. This is because a hypercube is an n-dimensional figure whose edges are aligned in each of the space’s dimensions, perpendicular to each other and of the same length. A tesseract is specifically a 4-d cube). 

[source]

Another way to think about this can be found here:

Start with a point. Make a copy of the point, and move it some distance away. Connect these points. We now have a segment. Make a copy of the segment, and move it away from the first segment in a new (orthogonal) direction. Connect corresponding points. We now have an ordinary square. Make a copy of the square, and move it in a new (orthogonal) direction. Connect corresponding points. We now have a cube. Make a copy and move it in a new (orthogonal, fourth) direction. Connect corresponding points. This is the tesseract.

If a tesseract were to enter our world, we would only see it in our three dimensions, meaning we would see forms of a cube doing funny things and spinning on its axes. This would be referred to as a cross-section of the tesseract. Similarly, if we as 3-dimensional bodies were to enter a 2-dimensional world, its 2-dimension citizens would “observe” us as 2-dimensional cross objects as well! It would only be possible for them to see cross-sections of us.

Why would this be significant? Generally, in math, we work with multiple dimensions very often. While it may seem as though a mathematican must then work with 3 dimensions often, it is not necessarily true. The mathematician deals with these dimensions only mathematically. These dimensions do not have a value because they do not correspond to anything in reality; 3 dimensions are nothing ordinary nor special. 

Yet, through modern mathematics and physics, researchers consider the existence of other (spatial) dimensions.  What might be an example of such a theory? String theory is a model of the universe which supposes there may be many more than the usual 4 spacetime dimensions (3 for space, 1 for time). Perhaps understanding these dimensions, though seemingly impossible to visualize, will come in hand. 

Carl Sagan also explains what a tesseract is

Image: Peter Forakis, Hyper-Cube, 1967, Walker Art Center, Minneapolis

Friday, August 26, 2011
[Image source]
What exactly is “redshift”? 
Redshift is defined as: 

a shift toward longer wavelengths of the spectral lines emitted by a celestial object  that is caused by the object moving away from the earth.

If you can understand that, great! But for those of us who cannot, consider the celestial bodies which make up our night sky. Did you think they were still, adamant, everlasting constants? They may seem to stick around forever, but…
Boy, you were wrong. I’ll have you know that stars are born and, at some point, they die. They move, they change. Have you heard about variable stars? Stars undergo changes, sometimes in their luminosity. (We are, indeed, made of the same stuff as stars).
So, stars move. All celestial bodies do, actually. You might have heard about some mysterious, elusive thing called dark energy. Dark energy is thought to be the force that causes the universe to expand at a growing rate. If it is proven to exist, dark energy will be able to explain why redshift occurs.
Maybe you can understand redshift by studying a visual:

[Image source]
These are spectral lines from an object. What do you notice is different in the unshifted, “normal” emission lines from the redshifted and blueshifted lines?
The redshifted line is observed as if everything is “shifted” a bit to the right— towards the red end of the spectrum; whereas the blueshifted line is moved to the left towards the bluer end of the spectrum.
Imagine if you were standing here on earth and some many lightyears away, a hypothetical “alien” was standing on their planet. With this image in mind, consider a galaxy in between the two of you that is moving towards the alien. You would then observe redshift (stretched out wavelength) and the alien would observe blueshift (shortened wavelength). 
Here, Symmetry Magazine explains redshift in their “Explain it in 60 seconds” series. 
A simple, everyday example of this concept can be observed if you stand in front of a road. As a car (one without a silencer) drives by, the pitch you observe changes. This is known as the Doppler effect. Watch this quick youtube video titled “Example of Dopper Shift using car horn”: 
(You may not be able to view it from the dashboard, only by opening this post on the actual blog page. You can watch the video by clicking this link). 

Notice how as the car drives past the camera man, the sound changes drastically.
Understanding redshift is important to scientists, especially astronomers and astrophysicists. They must account for this observable difference to make the right conclusions. Redshift is one the concepts which helped scientists determine that celestial bodies are actually moving further away from us at an accelerating rate.

[Image source]

What exactly is “redshift”? 

Redshift is defined as: 

a shift toward longer wavelengths of the spectral lines emitted by a celestial object  that is caused by the object moving away from the earth.

If you can understand that, great! But for those of us who cannot, consider the celestial bodies which make up our night sky. Did you think they were still, adamant, everlasting constants? They may seem to stick around forever, but…

Boy, you were wrong. I’ll have you know that stars are born and, at some point, they die. They move, they change. Have you heard about variable stars? Stars undergo changes, sometimes in their luminosity. (We are, indeed, made of the same stuff as stars).

So, stars move. All celestial bodies do, actually. You might have heard about some mysterious, elusive thing called dark energy. Dark energy is thought to be the force that causes the universe to expand at a growing rate. If it is proven to exist, dark energy will be able to explain why redshift occurs.

Maybe you can understand redshift by studying a visual:

[Image source]

These are spectral lines from an object. What do you notice is different in the unshifted, “normal” emission lines from the redshifted and blueshifted lines?

The redshifted line is observed as if everything is “shifted” a bit to the right— towards the red end of the spectrum; whereas the blueshifted line is moved to the left towards the bluer end of the spectrum.

Imagine if you were standing here on earth and some many lightyears away, a hypothetical “alien” was standing on their planet. With this image in mind, consider a galaxy in between the two of you that is moving towards the alien. You would then observe redshift (stretched out wavelength) and the alien would observe blueshift (shortened wavelength). 

Here, Symmetry Magazine explains redshift in their “Explain it in 60 seconds” series. 

A simple, everyday example of this concept can be observed if you stand in front of a road. As a car (one without a silencer) drives by, the pitch you observe changes. This is known as the Doppler effect. Watch this quick youtube video titled “Example of Dopper Shift using car horn”: 

(You may not be able to view it from the dashboard, only by opening this post on the actual blog page. You can watch the video by clicking this link). 

Notice how as the car drives past the camera man, the sound changes drastically.

Understanding redshift is important to scientists, especially astronomers and astrophysicists. They must account for this observable difference to make the right conclusions. Redshift is one the concepts which helped scientists determine that celestial bodies are actually moving further away from us at an accelerating rate.

Tuesday, August 2, 2011
Number Harmony
It is easy to recognize octaves because the frequency of an octave above a certain pitch is exactly twice the frequency of that pitch. Octaves harmonize so well that they almost sound identical, so we call these notes by the same name: an octave above or below middle C is  another C; an octave above or below concert A, 440 Hz, is another A (880  or 220 Hz). Mathematically, if a certain note H has frequency f then a note with frequency 2nf, where n is an integer, is n octaves above H (if n is negative, it is a positive power of 1/2 and represents |n| octaves below H).
Not alone in their ability to harmonize well, octaves are joined by all the intervals that make up a major or minor scale (in the Western music system), notably including perfect fifths (fifth note of a scale, 3/2 times the frequency of the starting note) and major or minor thirds (third note of a scale, respectively 5/4 or 6/5 times the frequency of the starting note). All these frequencies are ratios of relatively small whole numbers - this contributes to the harmony of the notes, just like the ratio 2/1 does for octaves. The simpler the frequency ratio, the higher the quality of harmony achieved by an interval when played out loud. The only requirement is for  the ratio to be a (positive) rational number, able to be written with whole numbers for the numerator and denominator.
However, suppose you tuned a piano perfectly according to one of the scales. Then you can play that scale and it would be perfectly in tune - but the harmony of all the other scales get thrown off! For example, E is both the third note of a C major scale and the second note of a D major scale. By tuning the piano to the C major scale, you guarantee that an E has frequency 5/4 times the frequency of a C (C to E is a major third). In a perfect C scale, D has frequency 9/8 times that of a C. Call these frequencies fC, fD, and fE.
fE = (5/4)fCfD = (9/8)fC⇒ fE = (5/4)(8/9)fD = (10/9)fD
This is still a relatively simple rational number ratio, but it’s the wrong ratio. In a perfect D major scale, E has frequency 9/8 that of D. The relative error when tuning to C is
|10/9 - 9/8| = |80-81|/72 = 1/72.
In the first days of the harpsichord and piano (keyboard instruments), tuners chose one scale to tune to, sacrificing the harmony of the other scales. Interestingly, some of the music from that era took that into account; on one hand some scales were considered “sweeter” than others based on common tuning practices, and on the other some songs were purposely written in one of the sour-sounding scales for their dissonant harmonies.
Today’s most common tuning, or temperament, is called equal temperament. Each scale sounds equally good (or equally bad, depending on your tolerance of imperfection), and the only interval which is perfectly preserved is octaves. Since, in the Western music system, there are 12 semitones from octave to octave (12 white and black keys from a note to an octave above the note), each of those keys is assigned the frequency of exactly the twelfth root of 2 times the key preceding it. What’s great about that, of course, is that this is a completely egalitarian system: no scale is sweeter or sourer sounding that any other. Yet the cost is the complete destruction of the rational number harmonies: the twelfth root of 2 is as irrational as they come, and could never in any number theorist’s wildest dreams be written as a ratio of whole numbers.
—
Further reading:”Why you’ve never really heard the Moonlight Sonata,” Jan Swafford, Slate Magazine

Number Harmony

It is easy to recognize octaves because the frequency of an octave above a certain pitch is exactly twice the frequency of that pitch. Octaves harmonize so well that they almost sound identical, so we call these notes by the same name: an octave above or below middle C is another C; an octave above or below concert A, 440 Hz, is another A (880 or 220 Hz). Mathematically, if a certain note H has frequency f then a note with frequency 2nf, where n is an integer, is n octaves above H (if n is negative, it is a positive power of 1/2 and represents |n| octaves below H).

Not alone in their ability to harmonize well, octaves are joined by all the intervals that make up a major or minor scale (in the Western music system), notably including perfect fifths (fifth note of a scale, 3/2 times the frequency of the starting note) and major or minor thirds (third note of a scale, respectively 5/4 or 6/5 times the frequency of the starting note). All these frequencies are ratios of relatively small whole numbers - this contributes to the harmony of the notes, just like the ratio 2/1 does for octaves. The simpler the frequency ratio, the higher the quality of harmony achieved by an interval when played out loud. The only requirement is for  the ratio to be a (positive) rational number, able to be written with whole numbers for the numerator and denominator.

However, suppose you tuned a piano perfectly according to one of the scales. Then you can play that scale and it would be perfectly in tune - but the harmony of all the other scales get thrown off! For example, E is both the third note of a C major scale and the second note of a D major scale. By tuning the piano to the C major scale, you guarantee that an E has frequency 5/4 times the frequency of a C (C to E is a major third). In a perfect C scale, D has frequency 9/8 times that of a C. Call these frequencies fC, fD, and fE.

fE = (5/4)fC
fD = (9/8)fC
fE = (5/4)(8/9)fD = (10/9)fD

This is still a relatively simple rational number ratio, but it’s the wrong ratio. In a perfect D major scale, E has frequency 9/8 that of D. The relative error when tuning to C is

|10/9 - 9/8| = |80-81|/72 = 1/72.

In the first days of the harpsichord and piano (keyboard instruments), tuners chose one scale to tune to, sacrificing the harmony of the other scales. Interestingly, some of the music from that era took that into account; on one hand some scales were considered “sweeter” than others based on common tuning practices, and on the other some songs were purposely written in one of the sour-sounding scales for their dissonant harmonies.

Today’s most common tuning, or temperament, is called equal temperament. Each scale sounds equally good (or equally bad, depending on your tolerance of imperfection), and the only interval which is perfectly preserved is octaves. Since, in the Western music system, there are 12 semitones from octave to octave (12 white and black keys from a note to an octave above the note), each of those keys is assigned the frequency of exactly the twelfth root of 2 times the key preceding it. What’s great about that, of course, is that this is a completely egalitarian system: no scale is sweeter or sourer sounding that any other. Yet the cost is the complete destruction of the rational number harmonies: the twelfth root of 2 is as irrational as they come, and could never in any number theorist’s wildest dreams be written as a ratio of whole numbers.

Further reading:
Why you’ve never really heard the Moonlight Sonata,” Jan Swafford, Slate Magazine

Friday, July 29, 2011
The Neutron Star
When a star runs out of fuel it eventually goes through a gravitational collapse. There are several possible outcomes, and three come about by simply taking into account only the mass of the star that has just collapsed. If the star is less than 1.5 solar masses, then a white dwarf is formed. Yet if its mass is larger than 5 solar masses, it will create a black hole. What about if the star’s mass was in between those two points? Well, that’s when a neutron star is formed. Due to the inward collapse of such a star, electrons combine with protons to form neutrons – thus giving the resultant celestial body the name “neutron star”.
Neutron stars characteristically have extremely high densities. Anything falling into a neutron star is super-accelerated by gravity (which is 100 billion times stronger than what we experience on earth). “If you dropped a marshmallow onto a neutron star, it would have the energy of an atomic bomb,” says Chip Meegan from the Marshal Space Flight Center of NASA. Neutron stars also have insanely strong magnetic fields, approximately 2 x 1011 times those of Earth. These stars are usually very hot. The degeneracy pressure due to the Pauli exclusion principle (no two neutrons or any other fermions can occupy the same place and quantum state simultaneously) ensures the neutron star’s stability and prevents it from collapse. (The only situation in which a neutron star would collapse into a black hole is if it is gradually absorbing matter from an accompanying binary star.)
Pulsars!
Very simply, pulsars are rotating neutron stars. Because they conserve the angular momentum of the stars from which they were formed, pulsars can spin at rates over 700 times revolutions per second. They stream jets of highly energetic particles (which have speeds close to that of light) out from their magnetic poles, producing extremely powerful beams of radiation. This combined with their rotational movement allows them to appear as if they are pulsing when they are observed. The rotational and magnetic axes of these stars are misaligned, which causes the beam of light from the jets to “sweep” around as the pulsar rotates, giving rise to the lighthouse effect.

The Neutron Star

When a star runs out of fuel it eventually goes through a gravitational collapse. There are several possible outcomes, and three come about by simply taking into account only the mass of the star that has just collapsed. If the star is less than 1.5 solar masses, then a white dwarf is formed. Yet if its mass is larger than 5 solar masses, it will create a black hole. What about if the star’s mass was in between those two points? Well, that’s when a neutron star is formed. Due to the inward collapse of such a star, electrons combine with protons to form neutrons – thus giving the resultant celestial body the name “neutron star”.

Neutron stars characteristically have extremely high densities. Anything falling into a neutron star is super-accelerated by gravity (which is 100 billion times stronger than what we experience on earth). “If you dropped a marshmallow onto a neutron star, it would have the energy of an atomic bomb,” says Chip Meegan from the Marshal Space Flight Center of NASA. Neutron stars also have insanely strong magnetic fields, approximately 2 x 1011 times those of Earth. These stars are usually very hot. The degeneracy pressure due to the Pauli exclusion principle (no two neutrons or any other fermions can occupy the same place and quantum state simultaneously) ensures the neutron star’s stability and prevents it from collapse. (The only situation in which a neutron star would collapse into a black hole is if it is gradually absorbing matter from an accompanying binary star.)

Pulsars!

Very simply, pulsars are rotating neutron stars. Because they conserve the angular momentum of the stars from which they were formed, pulsars can spin at rates over 700 times revolutions per second. They stream jets of highly energetic particles (which have speeds close to that of light) out from their magnetic poles, producing extremely powerful beams of radiation. This combined with their rotational movement allows them to appear as if they are pulsing when they are observed. The rotational and magnetic axes of these stars are misaligned, which causes the beam of light from the jets to “sweep” around as the pulsar rotates, giving rise to the lighthouse effect.