tag:blogger.com,1999:blog-89207959897604283892019-03-23T01:42:45.319-07:00Math MathKen Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comBlogger230125tag:blogger.com,1999:blog-8920795989760428389.post-28526997293755608482019-02-21T11:27:00.000-08:002019-02-22T07:37:06.277-08:00F=m*a derived from scratch<font size="4"> <b>F=m*a derived from scratch</b><br><br> F=m*a is probably the most famous equation in physics. If a force F acts on a mass m to produce an acceleration a then F=m*a.<br><br> Or another way to look at this. If you find a mass m undergoing an acceleration a then look around. You'll find a force F and F=m*a.<br><br> But there's yet another way to look at this. If flux is the number of force lines through a space then the rate of change of flux is a force. This is certainly true in electromagnetism where it's called Faraday's Law. <br><br> Faraday's Law - When the magnetic flux linking a circuit changes, a force is induced in the circuit proportional to the rate of change of the flux linkage.<br><br> Let's assume this is also true for gravity..<br><br> Gravitational equivalent - When the gravitational flux linking a circuit changes, a force is induced in the circuit proportional to the rate of change of the flux linkage.<br><br> The number of lines of force is determined my m. And the rate of change is determined by a. So F=m*a<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-89391946316812714032019-01-13T11:50:00.000-08:002019-01-21T05:52:26.632-08:00Physics of Electrical Generators<font size="4"> <b>Physics of Electrical Generators</b><br><br> The physics is simple...<br><br> First, use the "lines of force" model for a magnetic field. The lines give the direction and intensity of the field. Intensity is just the number of lines per unit area. <br><br> Now take a close loop of conductor.. which of course in the real world will contain a load such as an LED lamp etc.<br><br> Now *change* the flux of magnetic field through this loop. The flux is just the number of magnetic lines of force that link the loop. The key word here is *change*.<br><br> A change will generate a current in the loop. And the size of the current depends on the *rate of change* of the flux. Change the flux faster and you get more current.<br><br> How can you change the flux through the loop? Several options..<br><br> 1) Change the strength of the magnetic field.<br>2) Change the size of the loop.<br>3) Increase the number of loops.<br>4) Change the orientation of the loop.<br>5) Move the loop in and out of the magnetic field.<br><br> That's it. This is the basis of all conventional electrical generators. <br><br> The trick is to come up with a design that changes the flux as fast as possible. The faster the change the more current you generate. <br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-18370342484012700632018-11-28T11:33:00.001-08:002018-12-01T06:20:18.519-08:00Spooky Sets<font size="4"> <b>Spooky Sets</b><br><br> Math deals with sets. And finite sets have an integer number of members.<br><br> But why?<br><br> Math is all about extending definitions and concepts. So why does the number of members of a set need to be an integer?<br><br> What about a rational number, or an irrational number. Or if you want very spooky sets what about a complex number!<br><br> How could this extension be done? I don't know.<br><br> And if x is the number of members would we still have the number of subsets = 2^x<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-31301858191890780192018-11-28T07:42:00.000-08:002018-11-29T08:18:25.083-08:00Shannon Entropy Explained - Fast<font size="4"> <b>Shannon Entropy Explained - Fast</b><br><br> Suppose you have a system that has n discrete states, where n is an integer.<br><br> And suppose the probability of finding the system in a given state is the same for all states.<br><br> Then is you solve n=2^x for the variable x you've just found the Shannon Entropy of the system.<br><br> Shannon Entropy is also known as Information Entropy.<br><br> For example: Consider a 6 sided dice. It has 6 states (meaning that when thrown it can land in 6 different ways), so solving 6=2^x gives x=2.5849.. and that's the Shannon Entropy of your dice.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-47214176527734106082018-11-18T14:46:00.003-08:002018-11-20T14:21:37.905-08:00Stephen Hawking Black Hole Formula<font size="4"> <b>Stephen Hawking Black Hole Formula</b><br><br> Hawking applied Quantum Mechanics to Black Holes in order to derive his famous formula.. <br><br> <div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-LxsCivkVoLE/W_HrF_W4ImI/AAAAAAAAIXE/v-eWH3KRTaMu7ryAEYnrShAwHmR-TvvrQCLcBGAs/s1600/hawkins-formula.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-LxsCivkVoLE/W_HrF_W4ImI/AAAAAAAAIXE/v-eWH3KRTaMu7ryAEYnrShAwHmR-TvvrQCLcBGAs/s320/hawkins-formula.png" width="320" height="167" data-original-width="311" data-original-height="162" /></a></div><br> S is Entropy and A is the area of the event horizon. All the other things in the formula are constants of nature.<br><br> So the formula says the entropy of a Black Hole is directly proportional to the area of its event horizon.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-31295484229463684212018-08-28T07:31:00.001-07:002018-11-15T07:47:58.287-08:00The classification of integers by missing primes<font size="4"> <b>The classification of integers by missing primes</b><br><br> If p1,p2,p3,.. is the sequence of prime numbers then any positive integer can be written:<br><br> n=pi^a1*p2^a2*p3^a3...pq^aq<br><br> where q is a positive integer and ai are integers greater than or equal to zero.<br><br> Of course, if ai=0 then pi^ai=1 and we say that the prime pi is "missing" from n.<br><br> I wonder if it would be instructive to classify integers by their missing primes?<br>As a special case: if n is prime then it has the maximum possible number of missing primes. <br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-44947614011735343832018-07-23T08:11:00.001-07:002018-07-31T17:04:07.567-07:00The Human Retina - Explained in 90 Seconds<font size="4"> <b>The Human Retina - Explained in 90 Seconds</b><br><br> The "rod" cells in the retina can detect a single photon. These cells are responsible for peripheral vision and low light vision. They provide black/white/grey vision and cannot detect color.<br><br> Special proteins in the cells can absorb an individual photon and as a result change the cell's membrane potential enough to trigger the cell. <br><br> The retina contains about 100 million rod cells.<br><br> The "cone" cells in the retina provide color vision. The retina has about 6 million cone cells and almost all are concentrated in a tiny "dimple" on the retina called the macula. It's only a few mm wide. This minuscule structure gives us our central field vision and our color vision!!!!!<br><br> Cone cells can detect frequency of light and there are 3 types of cones, each sensitive to a different frequency range. The "B" cones are most sensitive to blue, the "G" cones are most sensitive to green and the "R" cones are most sensitive to red. So our color vision is called "tricolor vision". By comparing outputs from the 3 cell types the brain "mixes" the 3 color signals to detect any color!<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-15086550092195673042018-07-18T05:11:00.002-07:002019-02-12T06:04:41.355-08:00Are Extra Dimensions a Dead End?<font size="4"> <b>Are Extra Dimensions a Dead End?</b><br><br> Extra Dimensions are much talked about. They are central to String Theory. The LHC is looking for them (with no success so far). <br><br> Now some think Gravitational Waves may be the key to finding them. <br><br> But suppose they did not exist for an incredibly simple reason: at very short distances the concept of dimension itself simply breaks down and gets replaced by something very different. <br><br> So we're looking for something that does not exist - literally. <br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-57315066936744344062018-07-13T13:57:00.002-07:002018-07-21T05:41:43.946-07:00Optical Computers - When Photons Replace Electrons<font size="4"> <b>Optical Computers - When Photons Replace Electrons</b><br><br> The electron totally dominates our current computing technology - with billions of electronic logic gates packed onto silicon chips. So if you want to build an optical computer the first thing you need to do is develop a photon logic gate.<br><br> Now a research group says they have done just that.. <a style="text-decoration:underline" href="http://science.sciencemag.org/content/361/6397/57">photon gates</a><br><br> Photons already dominate long haul data transmission (optical fiber). But a photon computer would be truly revolutionary!!!!!!<br><br> And let's not forget Quantum Mechanics. Photons are spin 1 bosons, while electrons are spin 1/2 fermions. Meaning photons play by a whole different set of rules than electrons.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-80451450750992265292018-07-11T10:34:00.002-07:002018-11-17T11:57:51.477-08:00The Relationship Between Mathematics & Physics<font size="4"> <b>The Relationship Between Mathematics & Physics</b><br><br> Understanding the relationship can eliminate much confusion.<br><br> In mathematics you are free to construct anything you like as long as it's consistent. So for example, in mathematics n-dimensional spaces really exist. They are well defined mathematical objects. <br><br> Physics uses mathematical objects to model reality - with varying levels of success. The fact that physics may use a mathematical object to model reality does not mean that object now exists outside of mathematics. It doesn't. It simply means the object is useful in modeling reality. <br><br> So for example, do n-dimensional spaces "really" exist? The question is meaningless. They exist in mathematics, and they are useful in modeling reality. But to ask if they "really" exist is a meaningless question. <br><br> Heck, it's even conceivable that something other that mathematics will ultimately be better at modeling reality. Very unlikely, but conceivable.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-15297548369535697792018-06-24T12:06:00.001-07:002018-07-11T10:31:54.758-07:00Building 3D Space<font size="4"> <b>Building 3D Space</b><br><br> Suppose I have a collection of identical balls. Each has 6 connectors on its surface - 2 red, 2 green and 2 blue. <br><br> Then I have cables that come in 3 varieties, red, green and blue. Red cables can only plug into red connectors, green cables can only plug into green connectors and blue cables can only plug into blue connectors. <br><br> Now I specify a simple constraint: ever ball must have all its connectors filled. <br><br> Bingo - the system is a 3D system. Or is it?<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-64642590676903214092018-06-22T11:16:00.004-07:002019-03-22T08:21:03.225-07:00Spin Inversion Symmetry (SIS)<font size="4"> <b>Spin Inversion Symmetry (SIS)</b><br><br> SIS says every elementary particle in the Standard Model (except for the Higgs) of spin s has a partner of spin 1/s. It also applies to the hypothetical Graviton.<br><br> I have no theoretical justification for SIS, but I do have a rather amusing story of how I came up with it. <br><br> I was imagining how you might model a spin 1/2 elementary particle with an everyday object. After a bit of thought I decided on a Mobius strip (strip of paper joined after applying 1 half turn). I figured 1 half turn = spin 1/2. <br><br> Then I made a strip with 4 half turns and discovered something rather elegant.. it will naturally "flip" into a double thickness Mobius strip. <br><br> In other words a strip with 4 half turns (spin 2) naturally flips into a strip with 1 half turn (spin 1/2). <br><br> That's a Boson to Fermion transition. Think of space time (a Boson spin 2 structure) "condensing" into spin 1/2 Fermions.<br><br> It explains why every elementary matter particle has spin 1/2. They have no choice!<br><br> So there you have it - a desktop version of SIS. All you need is paper, scissors, a touch of glue, and a vivid imagination!<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-11756977871094981102018-06-19T07:36:00.003-07:002018-06-19T07:43:58.481-07:00Metrics in Physics - An Unexplored Resource?<font size="4"> <b>Metrics in Physics - An Unexplored Resource?</b><br><br> We could probably do a lot in Physics by simply "tweeking" the metric. <br><br> For example: Special Relativity uses the familiar metric x^2+y^2+z^2-t^2. But this is just one in an infinite family of metrics.. x^q+y^q+z^q-t^q where q=1,2,3,4,... And there are many other metrics. <br><br> Even General Relativity, THE metric theory, modifies the metric using coefficients but still retains q=2 and never even considers changing that!<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-2312315679333436992018-06-16T13:38:00.003-07:002018-11-28T10:59:30.391-08:00The Holographic Principle - An Interesting Example<font size="4"> <b>The Holographic Principle - An Interesting Example</b><br><br> The Holographic Principle talks about regions of space where the surface area is just as important as the volume. So I decided to try and construct a specific example.<br><br> I did. And the result is both simple and elegant.<br><br> Consider objects that have their surface area equal to their volume.<br><br> For a circle of radius r in 2D we have surface area (circumference)= 2*pi*r and volume (area)=pi*r^2. Equating these two gives r=2. Which is the dimension of the space!<br><br> Doing the same for a sphere in 3D we get r=3. Which is the dimension of the space!<br><br> Is this always true? Yes!<br><br> Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)<br><br> Then the surface of a sphere of radius r is the set of points with:<br>x1^2+x2^2+x3^2+....+xn^2=r^2<br><br> and the volume consists of all the points with:<br>x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2<br><br> The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)<br><br> So equating them gives r=n. <br><br> Note that the function c(n) cancels out so its value is not needed, but c(n)=(pi^(n/2))/gamma(1+n/2). Where gamma is the Euler Gamma Function.<br><br> So we can say in general..<br><br> "In n-dimensional Euclidean space a sphere with surface area=volume has radius n"<br><br> The radius of the sphere is equal to the dimension of the space!!!! <br><br> This result is related to the Holographic Principle because it equates surface area and volume.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-12651463001175407552018-06-15T05:03:00.000-07:002018-06-15T06:34:11.906-07:00CERN - LHC to Get Major Upgrade<font size="4"> <b>CERN - LHC to Get Major Upgrade</b><br><br> CERN has officially begun work on a major upgrade to the Large Hadron Collider (LHC) to boost luminosity. <br><br> When the upgrade is complete in 2026 the LHC be able to collect data at almost 10X its current rate.<br><br> CERN says..<br><br> "The secret to increasing the collision rate is to squeeze the particle beam at the interaction points so that the probability of proton-proton collisions increases. To achieve this, the HL-LHC requires about 130 new magnets, in particular 24 new superconducting focusing quadrupoles to focus the beam and four superconducting dipoles. Both the quadrupoles and dipoles reach a field of about 11.5 tesla, as compared to the 8.3 tesla dipoles currently in use in the LHC. Sixteen brand-new “crab cavities” will also be installed to maximise the overlap of the proton bunches at the collision points. Their function is to tilt the bunches so that they appear to move sideways – just like a crab."<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-56728430660367351392018-06-13T05:35:00.004-07:002018-06-14T05:25:37.752-07:00A Prime Number Conjecture in Binary<font size="4"> <b>A Prime Number Conjecture in Binary</b><br><br> Take any prime number p and write it in binary format. <br><br> From this prime now generate a series of numbers using this simple algorithm:<br><br> You can insert a single digit (0 or 1) anywhere you wish. (But you can't insert a 0 at the beginning because that just gives p unchanged.)<br><br> If p is m digits long in binary you will generate 2*m+1 new binary numbers. (btw: some may are duplicates.)<br><br> CONJECTURE: At least one of these new numbers is prime.<br><br> Let me illustrate the process with a simple example. <br>Take the prime 5, which in binary is 101. It has 3 digits, so we know the algorithm will generate 2*3+1=7 numbers. Here they are (notice the duplicates)..<br><br> 1010=10<br>1011=11<br>1011=11<br>1001=9<br>1101=13<br>1001=9<br>1101=13<br><br> So in this case we've generated two new prime numbers, 11 and 13.<br><br> A note on duplicates: <br>I think you will always get m duplicates. So if these are removed the algorithm generates just m+1 numbers and the conjecture says at least one of these is prime.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-79857861257529995332018-06-12T06:45:00.002-07:002018-06-13T11:27:46.033-07:00Could General Relativity Fail Completely?<font size="4"> <b>Could General Relativity Fail Completely?</b><br><br> General Relativity sets gravity apart from all other fundamental interactions by claiming it's simply curvature of spacetime. It's not even a force!<br><br> This is why GR can't be reconciled with Quantum Mechanics. <br><br> So here's a strange thought. What if spacetime simply ceased to exist (as we currently know it) in certain circumstances? <br>Then General Relativity would fail completely. <br><br> What might these "certain circumstances" be? <br>I don't know. But a great place to start is the interior of a spinning black hole.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-6026075946388092672018-06-10T12:36:00.001-07:002018-06-21T07:42:17.860-07:00A Special Sphere in n-Dimensional Space<font size="4"> <b>A Special Sphere in n-Dimensional Space</b><br><br> It's really interesting to consider objects that have their surface area equal to their volume.<br><br> For a circle of radius r in 2D we have surface area (circumference)= 2*pi*r and volume (area)=pi*r^2.<br><br> Equating these two gives r=2. Which is the dimension of the space!<br><br> Doing the same for a sphere in 3D we get r=3. Which is the dimension of the space!<br><br> Is this always true? Yes.<br><br> Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)<br><br> Then the surface of a sphere of radius r is the set of points with:<br>x1^2+x2^2+x3^2+....+xn^2=r^2<br><br> and the volume consists of all the points with:<br>x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2<br><br> The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)<br><br> So equating them gives r=n. <br><br> Note that the function c(n) cancels out so its value is not needed, but c(n)=(pi^(n/2))/gamma(1+n/2). Where gamma is the Euler Gamma Function.<br><br> So we can say in general..<br><br> "In n-dimensional Euclidean space a sphere with surface area=volume has radius n"<br><br> The radius of the sphere is equal to the dimension of the space! I think this is a neat result because it relates the dimension of the space to a certain class of spheres within it. It may also be related to the Holographic Principle because it equates surface area and volume.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-31108263964529758182018-06-10T08:51:00.001-07:002018-06-11T05:39:13.993-07:00Making Prime Numbers - A Conjecture<font size="4"> <b>Making Prime Numbers - A Conjecture</b><br><br> Take any prime number. You are allowed to insert a single digit (i.e. 0-9) anywhere in the number, including the beginning and end (but we don't count adding a 0 at the beginning of the number because that just gives the original number.)<br><br> Conjecture: This process will always generate at least one new prime number.<br><br> Doing this in binary (base 2) means the only digits I can insert are 0 and 1 and it makes the process very simple. The conjecture of course remains the same.<br><br> If you write a program to test this conjecture I would love to hear about your results.<br><br> <b>Examples in base 10</b><br>503 is prime. I decide to insert 2 before the 3 to get 5023, which is prime. <br>20129 is prime. I decide to insert 5 at the beginning to get 520129, which is prime.<br><br> <b>Example in binary</b><br>Take the number 3 in binary (11), so the process generates 3 new numbers..<br><br> 111=7<br>110=6<br>101=5<br><br> And in this case it has generated two new primes, 5 and 7.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-16653293482238696792018-06-08T13:02:00.002-07:002018-06-16T13:33:11.276-07:00Sphere in n-Dimensional Space<font size="4"> <b>Sphere in n-Dimensional Space</b><br><br> Here's the elegant result clearly explained. <br><br> Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)<br><br> Then the surface of a sphere of radius r is the set of points with:<br>x1^2+x2^2+x3^2+....+xn^2=r^2<br><br> and the volume consists of all the points with:<br>x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2<br><br> Then the Theorem is simply this..<br><br> "In n-dimensional Euclidean space a sphere with surface area=volume has radius n"<br><br> The radius of the sphere is equal to the dimension of the space!!!! I think this is a neat result because it relates the dimension of the space to a certain class of spheres within it. It also has a "holographic" aspect because it equates surface area and volume.<br><br> If you're interested in the proof it goes like this..<br><br> The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)<br><br> So equating them gives r=n. <br><br> Note that the function c(n) cancels out so its value is not needed, but c(n)=(pi^(n/2))/gamma(1+n/2). Where gamma is the Euler Gamma Function.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-90306769118018000872018-05-31T11:56:00.003-07:002018-06-15T12:24:22.550-07:00The Day Einstein Got Lucky<font size="4"> <b>The Day Einstein Got Lucky</b><br><br> When physicists describe a moving object they measure its position at different times, and obviously these positions have to be given relative to some reference points. These reference points are called a reference frame. <br><br> Special Relativity showed that the Laws of Physics are independent of the choice of reference frames - when the reference frames are moving with constant velocity. <br><br> It was a huge success. But how could Einstein generalize it? <br><br> Easy.. just do the same thing for reference frames that are accelerating. Makes total sense. <br><br> So Albert went about the usual business that had been so successful before. But then he suddenly realized something. <br><br> On a local basis acceleration is indistinguishable from gravity. Bingo. Albert could do his usual thing with reference frames and describe gravity at the same time.<br><br> A huge bonus!!!!<br><br> So he produced General Relativity. And it was a massive success. That is, until it met Quantum Mechanics. The two did not get along. And still don't.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-55503140915691786402018-05-30T11:47:00.002-07:002018-05-30T11:47:29.565-07:00Was Einstein Toying With Us?<font size="4"> <b>Was Einstein Toying With Us?</b><br><br> First we invent the idea of a coordinate system. That's very helpful. <br><br> Then we frame all our physics in terms of it. <br><br> But we know that the laws of physics must be covariant - i.e. independent of our choice of coordinate system. <br><br> General Relativity showed that, and that alone would have been a very nice achievement. <br><br> But no, Einstein wove into it a description of gravity - placing gravity in a totally unique position. <br><br> No wonder we can't unify gravity with the rest of physics!!!!!!<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-44463022311426442142018-05-28T11:53:00.001-07:002018-11-17T11:47:48.973-08:00The Fluid Dynamics of Spacetime<font size="4"> <b>The Fluid Dynamics of Spacetime</b><br><br> The unification of the two great theories of the 20th Century - Quantum Mechanic (QM) and General Relativity (GR) - still seems far away.<br><br> Why?<br><br> I think both theories will require major modification before they can be united.<br><br> The modification will be in how they treat spacetime.<br><br> In QM spacetime is a stage on which the action occurs. In GR spacetime is the action!<br><br> That's a massive difference.<br><br> Think of a fluid - at the macro level an excellent description is fluid dynamics - the Navier-Stokes Equations. But at the micro level molecules rule.<br><br> So, the EFE (Einstein Field Equations of General Relativity) are the "Navier-Stokes Equations" of spacetime - an excellent description at the macro level.<br><br> But at the micro level of spacetime we have what? Nobody knows. Our theories are not there yet.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-81059780565920985322018-05-23T11:37:00.002-07:002018-06-07T06:00:13.286-07:00Fermat's Last Theorem - The Search for a Generalization<font size="4"> <b>Fermat's Last Theorem - The Search for a Generalization</b><br><br> One elegant approach is to write the theorem in the language of metric spaces..<br><br> Let (x1,x2,..,xn) be a point in a n-dimensional vector space.<br><br> Then r^q=|x1|^q+|x2|^q+...+|xn|^q defines a series of metrics for q=1,2,3,.. where r is the distance of the point from the origin and |xi| is the absolute value of xi.<br><br> This allows a generalized version of Fermat's Last Theorem to be written as follows..<br><br> "An integer point (x1,x2,..,xn) is never an integer distance (r) from the origin when q>n"<br><br> It's interesting to note that the generalization holds when the metric parameter (q) exceeds the dimensionality of the space (n).<br><br> The special case of n=2 was proved in 1994 by Andrew Wiles. It was an amazing achievement because mathematicians had been trying to prove it since Fermat first suggested it in 1637.<br><br> But is this generalized version true? <br><br> At first I though it was, but one of my readers pointed out a counter example. And one counter example is all you need to disprove a conjecture! However, I am now investigating ways the conjecture may be modified (or even generalized further) so it still holds.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-36220081262286282872018-05-21T10:36:00.005-07:002018-11-17T11:58:57.701-08:00When Gravity Fails<font size="4"> <b>When Gravity Fails</b><br><br> Einstein's General Theory of Relativity is now over 100 years old and still remains our best description of gravity. <br><br> But the elegant Tensor form of Einstein's Field Equations (EFE) masks a critical property - they are basically differential equations of derivatives with respect to spacetime i.e. with respect to the x,y,z and t coordinates. <br><br> But, as any mathematician will tell you, derivatives are only valid for "very smooth" functions. Meaning if spacetime becomes "granular" EFE will not apply - for the simple reason that the derivatives are no longer defined. <br><br> At some point this granularity of spacetime will become apparent - the center of a rotating black hole could be an example. <br><br> So, it's not that you can't solve the EFE in this situation - it's that you can't even write them down!<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbott PhDhttps://plus.google.com/101487824185426724709noreply@blogger.com