After explaining to someone the multidimensional Calabi-Yau manifolds, which are curled up so small that they are below the observable length (Planck length), he remarked something quite insightful:
"The Planck length is like the pixel of the universe."
How far can we "zoom in" on physical things in the universe? Is there some stopping point, where one can't zoom in anymore, because he or she is at the irreducible building-block of the universe? Or can one keep zooming in further and further, finding more and more complexity, as if he or she were zooming in on a Mandlebrot fractal?
Wednesday, June 29, 2011
Thursday, February 17, 2011
Mutations vs. Material Defects
DNA mutations have a striking similarity to defects in engineering materials. There are three main classes of both DNA mutations and defects in engineering materials (the first term is how it is referenced in biology, the second term is how it is referenced in materials science):
In DNA, point mutations usually substitute one base pair for another, such as an adenine for a guanine. In engineering materials, point defects involve substitutions of one cation for another, or one anion for another, or a vacancy*. In DNA, insertions or deletions of nucleotides change the length of the strand, and thus shift everything over; in engineering materials, line defects (dislocations) consist of a sheet of molecules being inserted into a crystal lattice (edge dislocation), which disrupts the structure of the crystal lattice and makes it effectively longer or shorter and shifts things. In DNA, structural rearrangements reshuffle the DNA sequence and effect the DNA strand on a more macroscopic scale; in engineering materials, volume defects include more macroscopic things that disrupt the structure of the material, such as twinning.
*Note that this vacancy does not disrupt the crystal structure of the material, just as a point mutation does not change the length of the DNA strand.
- singe-base-pair changes (point mutations) - point defects
- insertions or deletions of nucleotides - line defects (dislocations)
- structural rearrangements - volume defects
In DNA, point mutations usually substitute one base pair for another, such as an adenine for a guanine. In engineering materials, point defects involve substitutions of one cation for another, or one anion for another, or a vacancy*. In DNA, insertions or deletions of nucleotides change the length of the strand, and thus shift everything over; in engineering materials, line defects (dislocations) consist of a sheet of molecules being inserted into a crystal lattice (edge dislocation), which disrupts the structure of the crystal lattice and makes it effectively longer or shorter and shifts things. In DNA, structural rearrangements reshuffle the DNA sequence and effect the DNA strand on a more macroscopic scale; in engineering materials, volume defects include more macroscopic things that disrupt the structure of the material, such as twinning.
*Note that this vacancy does not disrupt the crystal structure of the material, just as a point mutation does not change the length of the DNA strand.
Friday, July 9, 2010
Einstein Rings vs. Newton Rings
Newton Rings are an interference phenomenon that occur when a curved lens and a flat lens are placed on one another. The equation governing this phenomenon follows:
where r_N is the radius of Nth Newton's bright ring, N is the bright ring number, R is the radius of curvature of the lens the light is passing through, and lambda is the wavelength of the light passing through the glass.
An Einstein ring is a sort of gravitational lensing that occurs around stars. It is governed by the following equation:
For a long time, I was entertaining the thought that these two phenomena were somehow linked. The physical mechanisms that cause each are very different, but both include some sort of lenses or lensing (Newton's rings require curved and flat glass lenses, Einstein's rings require gravitational lensing). However, the equations seem very different; the Newton's ring equation depends on lambda (the wavelength of the light), while the Einstein's ring equation doesn't. Also, Newton's rings refers to a set of concentric rings produced by one lens set, while Einstein's rings refer to a single ring or fragments of a ring around one light-emitting body in space.
So, the word for now is that Einstein's and Newton's rings are not linked... but we shall see -- more research is to come.
where r_N is the radius of Nth Newton's bright ring, N is the bright ring number, R is the radius of curvature of the lens the light is passing through, and lambda is the wavelength of the light passing through the glass.
An Einstein ring is a sort of gravitational lensing that occurs around stars. It is governed by the following equation:
For a long time, I was entertaining the thought that these two phenomena were somehow linked. The physical mechanisms that cause each are very different, but both include some sort of lenses or lensing (Newton's rings require curved and flat glass lenses, Einstein's rings require gravitational lensing). However, the equations seem very different; the Newton's ring equation depends on lambda (the wavelength of the light), while the Einstein's ring equation doesn't. Also, Newton's rings refers to a set of concentric rings produced by one lens set, while Einstein's rings refer to a single ring or fragments of a ring around one light-emitting body in space.
So, the word for now is that Einstein's and Newton's rings are not linked... but we shall see -- more research is to come.
Tuesday, June 22, 2010
Hydraulic Jumps, Breaking Waves, and Waterfalls
Upon studying open-channel fluid flow, I have observed that hydraulic jumps, breaking waves, and waterfalls all seem to be the same phenomenon.
DEFINITIONS
Let us first define each phenomenon as it is conventionally understood:
A hydraulic jump is a discrepancy in water depths. When water in a stream flows over a rock, for example, the water may reach a very high speed due to the decrease in the height of the water. (This is due to the continuity equation, (initial velocity)(initial height) = (final velocity)(final height).) If the speed increases greatly, it may exceed the speed that waves propagate at; such a speed is called "super-critical" flow. In this case, a sudden jump in the water's height will occur so that the speed of the water is reduced to sub-critical speed.
A breaking wave is a wave where the back of the wave catches up to the front and tumbles over, causing the wave to break. Once the wave has been broken for some time, it looks like it does in the photo. I will refer to such a wave as a "developed breaking wave".
A waterfall is an amount of water that falls over the top of a ledge of some sort, falling into a body of water below.
PART I: VISUAL SIMILARITY
Now, let us study photographic images of these three phenomena. Pay attention to the regions outlined in pink - they look very similar to each other.
The Hydraulic Jump
Photo Credit: Phy Schlafly
The Developed Breaking Wave
Photo Credit: http://s0.geograph.org.uk/photos/13/87/138779_671fe88b.jpg
The Waterfall
Photo Credit: http://www.tripadvisor.com/ReviewPhotos-g147313-r5396055-Negril_Jamaica.html
In each photograph, there is a cusp of whitish foam that appears right where a change in the depth of the water occurs. In the case of the hydraulic jump and waterfall, this cusp of foam is preceded by a region of water that is higher velocity and smaller depth (we'll call this Region 1). In the developed breaking wave, if you observe the wave in a control volume such that the cusp of whitish foam is stationary, this is also true. After the cusp, there is a region of lower-velocity, larger-depth water (we'll call this Region 2). So, the hydraulic jump, developed breaking wave, and waterfall are all similar in that there is a high-velocity, small-depth region of water, then a cusp of whitish foam, and then a low-velocity, large-depth region of water.
PART II: SCHEMATIC SIMILARITY
In Fluid Mechanics, Volume 10 by Pijush K. Kundu and Ira M. Cohen, the following diagram appears, illustrating three different hydraulic jumps:
The first hydraulic jump appears to be the same as a waterfall (take note, however, that it is not quite like a waterfall because a waterfall's Region 1 is airborne and does not touch the bottom.) The second hydraulic jump pictured is a standard, stationary hydraulic jump. The third hydraulic jump is moving and appears to be the same as a developed breaking wave.
PART III: MATHEMATICAL SIMILARITY
So, these three phenomena seem to be very similar. How could we prove that they are the same? Well, the hydraulic jump is governed by a particular equation. If this equation holds true for developed breaking waves and waterfalls as well, that would prove that they are all the same phenomena.
The equation is:
Where
We will now try to apply this hydraulic jump equation to developed breaking waves and waterfalls.
In a hydraulic jump, the cusp of white foam that we mentioned earlier occurs just where the speed of the water becomes exactly critical: that is, where the speed of the water in Region I reaches sqrt(gY_1).
THIS SECTION IS NOT FINISHED
PART IV: CONCLUSION
Our conclusion is that developed breaking waves are hydraulic jumps in series:
THIS SECTION IS NOT FINISHED
Our conclusion is that waterfalls are hydraulic jumps only if the water is not airborne:
THIS SECTION IS NOT FINISHED
DEFINITIONS
Let us first define each phenomenon as it is conventionally understood:
A hydraulic jump is a discrepancy in water depths. When water in a stream flows over a rock, for example, the water may reach a very high speed due to the decrease in the height of the water. (This is due to the continuity equation, (initial velocity)(initial height) = (final velocity)(final height).) If the speed increases greatly, it may exceed the speed that waves propagate at; such a speed is called "super-critical" flow. In this case, a sudden jump in the water's height will occur so that the speed of the water is reduced to sub-critical speed.
A breaking wave is a wave where the back of the wave catches up to the front and tumbles over, causing the wave to break. Once the wave has been broken for some time, it looks like it does in the photo. I will refer to such a wave as a "developed breaking wave".
A waterfall is an amount of water that falls over the top of a ledge of some sort, falling into a body of water below.
PART I: VISUAL SIMILARITY
Now, let us study photographic images of these three phenomena. Pay attention to the regions outlined in pink - they look very similar to each other.
The Hydraulic Jump
Photo Credit: Phy Schlafly
The Developed Breaking Wave
Photo Credit: http://s0.geograph.org.uk/photos/13/87/138779_671fe88b.jpg
The Waterfall
Photo Credit: http://www.tripadvisor.com/ReviewPhotos-g147313-r5396055-Negril_Jamaica.html
In each photograph, there is a cusp of whitish foam that appears right where a change in the depth of the water occurs. In the case of the hydraulic jump and waterfall, this cusp of foam is preceded by a region of water that is higher velocity and smaller depth (we'll call this Region 1). In the developed breaking wave, if you observe the wave in a control volume such that the cusp of whitish foam is stationary, this is also true. After the cusp, there is a region of lower-velocity, larger-depth water (we'll call this Region 2). So, the hydraulic jump, developed breaking wave, and waterfall are all similar in that there is a high-velocity, small-depth region of water, then a cusp of whitish foam, and then a low-velocity, large-depth region of water.
PART II: SCHEMATIC SIMILARITY
In Fluid Mechanics, Volume 10 by Pijush K. Kundu and Ira M. Cohen, the following diagram appears, illustrating three different hydraulic jumps:
The first hydraulic jump appears to be the same as a waterfall (take note, however, that it is not quite like a waterfall because a waterfall's Region 1 is airborne and does not touch the bottom.) The second hydraulic jump pictured is a standard, stationary hydraulic jump. The third hydraulic jump is moving and appears to be the same as a developed breaking wave.
PART III: MATHEMATICAL SIMILARITY
So, these three phenomena seem to be very similar. How could we prove that they are the same? Well, the hydraulic jump is governed by a particular equation. If this equation holds true for developed breaking waves and waterfalls as well, that would prove that they are all the same phenomena.
The equation is:
Where
We will now try to apply this hydraulic jump equation to developed breaking waves and waterfalls.
In a hydraulic jump, the cusp of white foam that we mentioned earlier occurs just where the speed of the water becomes exactly critical: that is, where the speed of the water in Region I reaches sqrt(gY_1).
THIS SECTION IS NOT FINISHED
PART IV: CONCLUSION
Our conclusion is that developed breaking waves are hydraulic jumps in series:
THIS SECTION IS NOT FINISHED
Our conclusion is that waterfalls are hydraulic jumps only if the water is not airborne:
THIS SECTION IS NOT FINISHED
The Fourth Dimension
I have always been severely disturbed by the fact that time is generally accepted as the fourth dimension.
Time, unlike the x, y, and z directions, cannot be assigned a vector. David Z Albert's book, Quantum Mechanics and Experience, states: "... the dimension of a space is equal to the number of mutually perpendicular directions in which vectors within a space can point" (Albert, 21). Time, however, cannot be assigned a vector, let alone a perpendicular one. Thus, it cannot define a perpendicular direction, and cannot be counted as a dimension.
More on this argument to come.
Time, unlike the x, y, and z directions, cannot be assigned a vector. David Z Albert's book, Quantum Mechanics and Experience, states: "... the dimension of a space is equal to the number of mutually perpendicular directions in which vectors within a space can point" (Albert, 21). Time, however, cannot be assigned a vector, let alone a perpendicular one. Thus, it cannot define a perpendicular direction, and cannot be counted as a dimension.
More on this argument to come.
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