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Learn Algebra in 5 Minutes

Learn Algebra in 5 Minutes

Consider this problem, "what number, when added to 5, gives the result 20".

Instead of a sentence, this problem can be written much shorter and clearer as an equation, like this..

5+x=20

where x denotes the number we are trying to find.

Of course, we could also write it as x+5=20 and this is exactly the same equation. Or we could write 20=x+5 which is of course the same thing.

If we manage to find x we say that we've "solved" the equation. Can we solve this equation? Well, we could guess a few numbers for x and try them out. Does x=9 work? Let's see, 5+9=14, so x=9 is not a solution. After a few tries we get the solution, which is x=15.

Guessing a solution is perfectly fine, but it's very time consuming, especially for more complex equations. Of course, we could program a high speed computer to guess solutions and try them out ultra fast until we finally hit on the right solution. And for some very tough equations this is indeed the method used. But this method has one huge flaw.. if it fails to find a solution it does not mean the equation has no solution. That's because even the fastest computer can only make a limited number of tries.. and the actual solution may be something we never get around to trying.

So, coming back to our equation 5+x=20 we should ask if there is a foolproof method that's guaranteed to find the solution. The answer is yes, and it's all about the = sign. Once you truly understand this simple sign solving the equation is easy.

So what does this sign really mean? It means the "object" on the left of the sign is the same exact object as that on the right. They are the same thing.. exactly the same thing. They are the same exact mathematical object but just written in different ways. So there's really only one object!

OK, so our equation says that 5+x is exactly the same object as 20. So, if I do something to 5+x and then I do the same thing to 20 the results will still be equal. Cool. So lets subtract 5 from 5+x to get the result x. Now do the same exact thing to the other side, I'll subtract 5 from 20 to get the result 15. But these two results must be the same, so I can write them as equal to each other, that is x=15.

Bingo, we've solved the equation without any guessing!

Also, I'm not sure if you noticed this, but we just did some basic algebra. Don't let algebra intimidate you, it's just the art of manipulating equations until you get what you want!

Let's look at a slightly more complicated example..

3*x+2=17

Let's use the same method. First subtract 2 from both sides..

3*x=17-2=15

Now divide both sides by 3..

x=15/3=5

And that's the solution!

Here's another method I use. It's exactly the same technique as above, but it's faster and easier to handle. Or at least I think so, and I've used it over the years to do massive amounts of algebra!

First move the 2 over to the other side. It was adding, so when it moves over it subtracts, like this..

3*x=17-2=15

Now move the 3 over. It was multiplying, so when it moves over it divides, like this..

x=15/3=5

This technique is quite general and can be used for any equation. But notice that the order in which you do things is important. For example, you need to get the 2 over to the other side before you can handle the 3.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Contactless Cards change the way you pay

Contactless Cards change the way you pay

If your credit/debit card has the NFC symbol then it's a contactless card.

So you can just tap to pay. No swiping. No inserting in a chip reader. Just tap and go. It's super fast.

Of course, the merchant needs an NFC reader. Many do but they don't advertise the fact.

For example, I tapped to pay for a croissant at a bakery in NYC and then for my medications at CVS.

Oh.. I almost forgot. NFC is Near Field Communication. It's a communication protocol with a range of just a few inches. Perfect for tap and pay.

Amazon Shop Now!

Content written and posted by Ken Abbott abbottsystems@gmail.com
Disclosure: As an Amazon Affiliate I earn a small commission when people click on the above links and buy something.

A Wind Turbine for your Home

A Wind Turbine for your Home

First there are commercial wind turbines. These are giant machines, the things you see rotating in the distance on hillsides.

Vestas has a worldwide installed wind turbine capacity of about 590GW. Siemens has about 100GW of wind turbine capacity in operation. GE rounds out the top three manufacturers with a total installed capacity of 62GW.

A GW (gigawatt) is a billion watts. It's enough to power about 725,000 homes.

But what about a wind turbine for your home?

No problem, try a VAWT (vertical axis wind turbine). These things are around 2 feet in diameter and several feet high. Their capacity ranges from 100 watts to about 5 KW. And prices start at just a few hundred dollars.

Could one of these power your entire house? No, but it could help. The average house needs about 5KW. So if one doesn't do it buy two or three. The wind is free and it doesn't care how many VAWTs you deploy.

And unlike the sun, the wind is in action overnight. So just connect your VAWT to a high-tech storage battery. Then you have free wind energy even when the wind doesn't blow!

Amazon Shop Now!

Content written and posted by Ken Abbott abbottsystems@gmail.com
Disclosure: As an Amazon Affiliate I earn a small commission when people click on the above links and buy something.

Prime Numbers Explained a Different Way

Prime Numbers Explained a Different Way

Let's consider the positive integers greater than 1, that is 2,3,4,5,..

Suppose we are given the first integer and asked to make all other integers using only the multiply operation.

We soon run into problems because 2*2=4 and we have no way to make 3.

OK, we just add 3 to our set of given numbers g, so now g={2,3}

Can we make 4? Yes, 2*2=4

Can we make 5? No, all our tries fail, so we add 5 to our set of given number g={2,3,5}

Can we make 6? Yes, 2*3=6

Can we make 7? No, all our tries fail, so we add 7 to our given numbers g={2,3,5,7}

Can we make 8? Yes, 2*2*2=8

Can we make 9? Yes, 3*3=9

Can we make 10? Yes, 2*5=10

Can we make 11? No, so we add it to the set g={2,3,5,7,11}

Can we make 12? Yes, 2*2*3=12

Can we make 13? No, so add it to the set g={2,3,5,7,11,13}

Can we make 14? Yes, 2*7=14

Can we make 15? Yes, 3*5=15

Can we make 16? Yes, 2*2*2*2=16

What is the set g that we are generating by this process? It's the set of prime numbers! This is simply another way to explain prime numbers.

It's a nice demonstration because it shows how prime numbers generate all numbers using only the multiply operation. You can also see that as g gets bigger we can obviously make more numbers from it, so prime numbers become less and less frequent.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Australian Fires - The Terrible Numbers

Australian Fires - The Terrible Numbers

One billion animals burned.

Fires are top of the news again today, and finally they are mentioning wildlife. Why is the government not setting up animal rescue centers? Much of the wildlife is unique to Australia. The government should be setting up major centers to preserve species. There seems to be no plan.

Don't fight the fires, that's a totally lost cause, save the wildlife.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Wind Turbines - From Hillside to Home

Wind Turbines - From Hillside to Home

Vestas has a worldwide installed wind turbine capacity of about 590GW.

Siemens has about 100GW of wind turbine capacity in operation.

GE rounds out the top three manufacturers with a total installed capacity of 62GW.

A GW (gigawatt) is a billion watts. It's enough to power about 725,000 homes.

These are giant machines, the things you see rotating in the distance on hillsides. What if you want something for your home?

No problem, try a VAWT (vertical axis wind turbine). These things are around 2 feet in diameter and several feet high. Their capacity ranges from 100 watts to about 5 KW. And prices start at just a few hundred dollars.

Could one of these power your entire house? No, but it could help. The average house needs about 5KW. So if one doesn't do it buy two or three. The wind is free and it doesn't care how many VAWTs you deploy.

And unlike the sun, the wind is in action overnight. So just connect your VAWT to a high-tech storage battery. Then you have free wind energy even when the wind doesn't blow!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Math of Building Wealth

The Math of Building Wealth

A friend of mine is an expert on advanced life insurance policies as a method of building wealth.

I was a bit skeptical so I asked him for numbers based on investing just $1,000 a month starting at age 25.

Here's what he said..

"Starting at age 25, if you were to deposit $1,000 every month, you would start immediately with a $1,060,000 death benefit, and at age 65 you would have roughly $1,200,000 in cash. Your death benefit would have grown to roughly $2,400,000 and you would no longer be required to pay premiums. The policy would continue to grow for the rest of your life, both in cash and death benefit, and with tax benefits."

If you need more details just email Les Himel at lnhimel@gmail.com

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Problem with 2020

The Problem with 2020

No, not the year itself, but how you write the year on important documents such as checks, invoices and contracts.

The year 2020 provides scam artists a unique opportunity to forge documents. How?

For example, if you write a check and date it 1/3/20 a scammer could easily change the date to 2017 and then try to cash it as an "old" check. Or a scammer could make an invoice look "old" and unpaid.

The solution is easy. Write the year in full 2020.

Personally I think the year 1/1/2020 - 12/31/2020 will be a great year. But the year 1/1/20 - 12/31/20 not so much.

Oh, and a quick note for my international readers. I'm writing dates in "American", that's month/day/year and not the day/month/year that you use. But the scam avoidance is still the same!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Physics and the Vacuum

Physics and the Vacuum

We live in a causal universe - meaning there’s a maximum finite speed for causality. It’s a fundamental constant of the universe and it’s the same for all observers. It’s by no means obvious that light should travel at this speed. But it does.

The actual value for the speed of causality is determined by the structure of the “vacuum”. The thing we call the vacuum is complex, with lots happening deep down at the quantum level. Plowing through it is hard work, even for light and gravity waves. And far harder for matter particles.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Boeing 737 Max Saga

Boeing 737 Max Saga

Here's my explanation of this terrible tale.

MCAS is a software patch in the 737 Max flight control computer that tries to compensate for a dangerous aerodynamic instability. The instability came about due to engine choice.

Boeing used the LEAP-1B engine on the 737 Max. It's a huge diameter bypass engine and it pushed the center of gravity of the aircraft into strange territory.

Hence the aerodynamic instability. Hence the MCAS software which put pilots into a terrible situation trying figure out what was happening.

In two cases they could not, and the planes oscillated under MCAS control until they fell out of the sky.

Many people died.

Only then was the decision made to ground the plane.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Climate Change and Sea Level

Climate Change and Sea Level

The Greenland ice sheet is massive. If it melted sea levels would rise about 23 feet.

But the Antarctic ice sheet is way bigger. If that sheet melted sea levels would rise about 200 feet and wipe out almost every major city on earth.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Twin Primes Observation

Twin Primes Observation

About the distribution of Twin Primes.

Let p1 be the first prime of a twin prime pair and let p2 be the first prime of the next consecutive twin prime pair. Then, I'm quite surprised how small p2-p1 stays. And when it does increase it will suddenly drop back to a very small number such as 12.

Anybody have any input on this?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Einstein's Amazing Luck

Einstein's Amazing Luck

Was it amazing luck or was it amazing insight? Whichever it was, at some point Einstein realized..

At a given point in space it’s impossible to distinguish between the effects of gravity and the effects of acceleration.

So, when he was trying to generalize Special Relativity to accelerating frames of reference he got an incredible bonus.. gravity suddenly appeared.. and General Relativity became a theory of gravity.

The rest is history.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Simple and Powerful Diet Secret

Simple and Powerful Diet Secret

Read the nutritional label on the package. Or Google the food to read its nutritional label online. Just Google "nutritional label for xxx" where xxx is the name of the food, examples: beans, steak, donuts etc.

Then..

Don't worry too much about calories!
Look for high protein. The higher the better.
Look for high fiber. The higher the better.
Look for low sugar. The lower the better. Zero if possible.
Look for low saturated fat. The lower the better. Zero if possible.
Simple.

Content written and posted by Ken Abbott abbottsystems@gmail.com

My Favorite Amazon Deals

My Favorite Amazon Deals

Here are a few of my favorite Amazon deals..

AMAZON Business Account Quantity discounts and more.
CELL PHONES Huge Savings & Huge Selection.
AMAZON Echo The amazing Alexa voice interface.
AMAZON Echo Studio Alexa for Music Lovers.

Content written and posted by Ken Abbott abbottsystems@gmail.com
Disclosure: As an Amazon Affiliate I earn a small commission when people click these links and buy something.

CERN LHC Explained

CERN LHC Explained

How did the CERN LHC (Large Hadron Collider) get its name?

Well, it's a large machine, about 27 kilometers in circumference and it collides protons. So it's a large collider. But what about the Hadron in the name? It turns out the proton belongs to a class of particles called Hadrons. The Proton is a Hadron. So we have a Large Hadron Collider.

Why collide protons anyway?

The LHC accelerates two counter-rotating beams of protons. The beam pipes are about 27 kilometers in circumference, and the protons are steered around the machine by powerful superconducting magnets. The two beams intersect at 4 points and this is where the protons collide head-on. The "debris" from the collision is measured to learn about the internal structure of the proton.

It turns out the proton is a complex object. Current theories describe it as composed of three quarks bound together by an incredibly strong force field mediated by particles called gluons.

So colliding protons may result in their constituent quarks colliding and yielding information about quarks. The gluons could also collide and yield more information about the force field within the proton.

The LHC is currently shut down for a major upgrade. But when it's operating this dashboard lets you see the LHC machine status in realtime. Check machine energy, ramps, beam status, machine tests, detector status and more. Plus, see messages from machine operators.

Watch LHC operations in realtime

It doesn't matter if you're a physicist or simply an interested consumer.. the mystery of the proton is this.. why does nature pack such amazing complexity into such an incredibly small object?

Some interesting LHC stats:

Proton are bunched in the LHC beam pipes.
There are 2808 bunches in each beam pipe.
Each bunch is about 30 cm long and contains approx 10^11 protons.
After full acceleration the protons are traveling at 0.999999991 the speed of light.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Amazon Echo and the Elderly

Amazon Echo and the Elderly

My mother-in-law is 94 and she lives alone. Obviously there is much she cannot do. But I setup Amazon Echo for her. And she uses it constantly for..

-The weather
-The time
-Switching lights on and off (and dimming them)
-Switching the TV on and off (and changing volume)
-Placing phone calls
-Playing her favorite music
-Getting flight status information

Amazon Echo is an amazing tool for the elderly. I wish this got more publicity.

Get Amazon Echo. Amazon Echo

TIP: If you're a music lover get the version of the Echo with high fidelity speakers. Amazon Echo Plus for Music Lovers

Content written and posted by Ken Abbott abbottsystems@gmail.com

Speed of Light

Speed of Light

If you believe we live in a causal universe then there must be an upper (finite) limit on the speed of causality. Otherwise the universe would not be causal.

It is by no means obvious that this speed also equals the speed of light. But apparently it does.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Plot Functions with Google Search

Plot Functions with Google Search

The Google Plot Function is powerful, fast and free. It's a great way to teach yourself functions, or to make math homework easier.

To plot a function just type your request into Google. The format of the request is plot {function}, for example..

To plot the function f(x)=x^3 just enter plot x^3 into Google.

Of course, it can handle more complex functions, for example try entering plot 1/(1+x^2)

Notice the symbols used by the Google Plot function: x^n means raise x to the power n and * is used to denote multiplication.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Gigabit vs Gigabyte

Gigabit vs Gigabyte

The prefix "giga" mean a billion. So a Gigabit is 1 billion bits and a Gigabyte is 1 billion bytes.

There are 8 bits in a byte. So a Gigabyte is 8x bigger than a Gigabit.

A Gigabit internet connection is one that can transfer data at the rate of 1 Gigabit per second. A Gigabyte internet connection is 8x faster.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Artificial Gravity

Artificial Gravity

Artificial Gravity is a force that mimics gravity.

You live on a space station. It's the shape of a giant donut.

Everything is great, but you and the crew are getting sick of floating around in zero gravity.

What you need is to generate some "artificial gravity". But how?

That's easy. Just fire a small booster rocket and spin the space station. You'll be drawn the outside wall of the station which will become your floor. So you can stroll around as usual. And if you drop an object it will fall to the floor.

So now you have "artificial gravity".

And if you spin the station at the correct rate you can match the gravity on Earth.

Spinning is a form of acceleration. But why is that important?

Because one of the deepest physics insights was Einstein’s observation that, for a small volume of space, the effects of acceleration cannot be distinguished from the effects of gravity. So your “artificial gravity” is not so artificial after all.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Giant Magellan Telescope

Giant Magellan Telescope

We're going to get a new view of the universe.

The Giant Magellan Telescope is currently under construction. It will be 10X more powerful than the Hubble Space Telescope. It will be installed on a mountain top in Chile and begin operation in 2027.

It's a multi-mirror system with 7 primary mirrors and topology adaptive secondary mirrors to eliminate atmospheric distortion.

Giant Magellan Telescope

Content written and posted by Ken Abbott abbottsystems@gmail.com

Troubleshooting a Gigabit Internet Connection

Troubleshooting a Gigabit Internet Connection

For the longest time I've been using a Verizon Fios 100 Megabit internet connection.

But Verizon now offers an amazing 1 Gigabit connection. So I upgraded. But when I ran the Verizon speed test I only saw a minor improvement.

Then it occurred to me. WiFi was the bottleneck. So I hardwired my computer to the router and instantly got the full gigabit 1000/1000 speed. Crazy fast.

One Gigabit is a truly superb service from Verizon. But this connection is so fast many WiFi routers become a bottleneck. You'll only see the full speed if you hardwire your computer, or if you get a new generation WiFi router that supports gigabit connections.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Shannon Entropy Explained

Shannon Entropy Explained

Shannon Entropy (also called Information Entropy) is a concept used in physics and information theory. Here's the scoop..

Suppose you have a system with n states i.e. whenever you make an observation of the system you find it's always in one of the n possible states.

Now make a large number of observations of the system, then use them to get the probability pi that if you make an observation the system is in state i. So for every state of the system you have a probability pi.

Now construct this crazy sum = p1*log(p1) + p2*log(p2) +... + pn*log(pn) where the sum is over all the states of the system.

If the log is base 2 then (-1)*sum is called the "information entropy" of the system.

Note that "information entropy" applies to a complete system, not individual states of a system.

Here's a simple example..

My system is a penny and a table.
I define the system to have 2 states.. penny lying stationary on the table with heads up or with tails up.

My experiment is to throw the penny and then observe which state results.

I throw the penny many times and make notes. It lands heads up 1% of the time and tails up 99% of the time (it's a biased penny).

The crazy sum is 0.01*log(0.01) + 0.99*log(0.99) = 0.01*(-6.643856) + 0.99*(-0.0145) = -0.08079356

So the information entropy of the system is (-1)*(-0.08079356) = 0.08079356

Content written and posted by Ken Abbott abbottsystems@gmail.com

Hydrogen Cars Explained

Hydrogen Cars Explained

Hydrogen cars are very different from all other cars (gasoline, battery, hybrid). Here's how they work..

A hydrogen car is a 100% electric car, but instead of a battery it uses hydrogen fuel cells. These are simple, feed them hydrogen gas and they generate electricity. They are like a battery that never needs recharging. There is no combustion, nothing burns, hydrogen gas is simply fed to the fuel cells to produce electricity.

So where do we get the hydrogen gas? The cars carry high pressure tanks of hydrogen gas to supply their fuel cells. The tanks get refilled at a hydrogen gas station, and the filling process is very similar to regular gasoline filling. The hydrogen gas station produces its hydrogen on the spot. How? By electrolysis of water.

Yes, the raw material to generate hydrogen gas is water!

So, the hydrogen gas station makes hydrogen from water. Cars refill their hydrogen gas tanks. The hydrogen goes through the car's fuel cells and generates electricity to drive the car. And the exhaust? The car's exhaust is water vapor. There is zero pollution!

The whole thing is a water to water cycle!

The range of a hydrogen car is about the same as a gasoline car, and the refueling time is about the same.

Also, hydrogen is the most abundant element in the universe. So I doubt we'll run out!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Optical Computers Explained

Optical Computers Explained

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.

Now a research group says they have done just that.. photon gates

Photons already dominate long haul data transmission (optical fiber), but a photon computer would be truly revolutionary!

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.

Content written and posted by Ken Abbott abbottsystems@gmail.com

A Family of Gravitons?

A Family of Gravitons?

I would like to make a conjecture and call it "Spin Inversion Symmetry". It says: Every elementary particle in the Standard Model (except for the Higgs) of spin s has a partner of spin 1/s.

So the fermions (spin 1/2) in the Standard Model would have spin 2 partners and the bosons (spin 1) in the Standard Model would be their own partners.

Of course, for this to work we need to have at least one spin 2 particle. Maybe many more. A Family of Gravitons?

Content written and posted by Ken Abbott abbottsystems@gmail.com

What is time?

What is time?

I have a system which can exist in just 2 states and it transitions between them constantly. I'll call this system a "clock" and one transition a "tick".

Now I have another system that I'm trying to study. It makes just one big transition.

So while the system makes one big transition how many ticks does my clock make?

The answer is called time.

All I'm doing is comparing two systems, the one I'm studying and a reference system.

If we take this as a definition of time then such things as "time travel" are difficult to explain. Actually, difficult to conceive. Actually, non existent.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Entanglement of Socks

Entanglement of Socks

We're sitting at a table. On the table are two brown paper bags, a black sock and a white sock.

We both close our eyes, grab a sock, put it in a paper bag and close the bag.

Then we open our eyes. Neither one of us has any idea which sock is in our bag.

Then you say to me..

"Tell me exactly which sock is in my bag but without opening the bag."

My first reaction is that's impossible. So I say the content of your bag is a supposition of a black and white sock. Which is just a fancy way of saying I have no clue.

But then it occurs to me..

I can find out what's in your bag by opening my bag.

I open my bag, find a white sock, and tell you your bag contains a black sock.

No observation of any kind has been made on your bag. It's still sitting on the table unopened. But now we both know the content.

What this is saying about the nature of information?

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Speed of Light

The Speed of Light

You could say that Special Relativity says all speeds are relative.. except for one.. the speed of light. It’s the same for every observer no matter how they are moving and no matter how the source of light is moving. Relativity did not produce this result, Einstein fed it in as a postulate. And the rest is history.

Let's explain it plainly.

You're standing by the railway tracks. I come speeding past in my new engine going at 0.9c. I switch on the headlights and you measure the speed of light from them. It should be 0.9c+c=1.9c. Right? Wrong. The result you get is still c.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Me and Otto Frisch

Me and Otto Frisch

Otto Frisch was one of the great physicists of the 20th Century. I was not. But we did develop a certain relationship.

It was 1971 and I was a PhD physics student at the Cavendish Lab on Free School Lane at the University of Cambridge.

There were certain rituals. In mid morning everyone met for coffee. And in mid afternoon for tea.

The tea was my problem.

After everyone was seated and chatting a guy came in, got his tea cup with no saucer, and got a slew of McVities Digestive Biscuits with no plate.

Then he sat next to me, slurped his tea and broke his biscuits so they sprayed crumbs far and wide. And all over me.

After a few of these incidents I told my fellow PhD student that if this happens again I'll call the guy out.

He totally encouraged me with a wicked gleam in his eye.

It happened again.

So I turned to the guy and said..

"Who the hell do you think you are?"

He replied very politely..

"Oh, I am Otto Frisch. I built the Atomic Bomb on the Manhattan Project with my good friend Enrico Fermi. And may I please ask who you are?"

The whole room was deadly silent and looking in my direction. So I didn't mess around. I told him I was an idiot PhD Physics research student with no clue.



Content written and posted by Ken Abbott abbottsystems@gmail.com

Special Relativity Explained - In 5 Minutes

Special Relativity Explained - In 5 Minutes

Albert and myself are standing by the railway tracks. He just came over to show me his new high power engine.

It looks impressive.

But then Albert pulls out a ruler. He asks me to measure it.

I have a device that can accurately measure the length of any object, even if the object is moving.

I measure the length of Albert's ruler. It's exactly 1 meter.

Then Albert bolts the ruler to the side of his train. He backs up a few miles and accelerates like crazy so he passes me at high velocity.

My device measures the length of the ruler as it passes.

Albert stops his train, walks over to me and asks for the result.

To my surprise I have to report the length was a bit less than 1 meter. Has the ruler shrunk? I measure it again while it's stationary and it's exactly 1 meter. So the ruler has not shrunk.

We both look at each other and ask if the length I measure could be dependent on the velocity of the train.

That's easy to test. Albert backs up his train a greater distance, accelerates more and passes me at a higher velocity.

Sure enough, the length I measure is even shorter.

We do the experiment for higher and higher velocities and I get shorter and shorter lengths for the ruler. But I notice something else strange. The higher the velocity gets the harder it becomes to increase. Meaning, Albert has to accelerate harder and harder just to get a marginal increase in velocity. It's like the velocity has some kind of maximum. I call this maximum vmax.

I then analyze the data and find a formula that fits all the experimental results. Here it is..

x^2+y^2=1

x=rv/r0 and y=v/vmax
rv is the length of the ruler when the train has velocity v, r0 is the length of the ruler when the train has velocity 0 i.e. it's stationary, and vmax is the maximum velocity.

It's a neat formula because when I plot the data in terms of x and y I get a circle of radius 1.

Albert then asks the obvious question - what would the length of the ruler be if he measured it on the train?

That's also easy to test. We load my measuring device on the train. Albert accelerates to a high velocity, then zips past me as he measures the length of the ruler.

He stops the train and gives me the result. The length he measured was exactly 1 meter.

At first we think this is crazy, but then we realize it makes sense. When Albert did his measurement the ruler was not moving for him. It was totally stationary, so it should be exactly 1 meter. And it was.

What about the number vmax? When I plot all the train velocities I see them approaching a maximum. They never reach the maximum but they get closer and closer. So from my data I can see what the maximum must be. It turns out to be a very famous number, vmax=c the velocity of light!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Proton and Borromean Rings

The Proton and Borromean Rings

Here's a nice piece of topology.

Notice that no two rings are linked, yet all 3 rings are linked.

You could use this a a model for the proton - the rings representing the 3 quarks that make up a proton and "ring linkage" representing the color force.

Then we see that the color force is very unusual. It's a 3-force, meaning it only appears when you have 3 objects, the force between any 2 objects being zero.

This model even shows the famous "asymptotic freedom" and "confinement" of bound quarks. When the rings are very close together you can move them around as if they were free (asymptotic freedom), but when you try to pull them apart a strong force appears to stop you (confinement).

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Mathematics of Lost Money

The Mathematics of Lost Money

The United States is holding about $58 BILLION in lost money. Check if any is yours and claim it. I've claimed several times and got money!

Select a State below to go to the website of the State Comptroller. Then do a search for money.



Amazon does not sell lost money - But they sell almost everything else.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The story of Georg Cantor

The story of Georg Cantor

The prefix "trans" means "beyond". So a transfinite number is one that's beyond the finite. There's only one and that's infinity, right? Wrong. It turns out there are many transfinite numbers. Infinity is just a general concept, and the real mathematics is the study of transfinite numbers.

This work is due to Georg Cantor, who showed that there are many transfinite numbers, and some are bigger than others! He even developed an arithmetic for working with transfinite numbers. He denoted them by the Hebrew letter "aleph".

His work stands as one of the most elegant pieces of mathematics ever.

So what did Cantor do?

He formalized counting. He started with the integers {1,2,3,...} and asked what other sets could be placed in 1-to-1 correspondence with the integers. Instead of just saying there are an infinite amount of integers he denoted the number of integers by aleph0 and developed an arithmetic that in many ways treated aleph0 as a regular number. But he went further..

He showed that the rational numbers (fractions) could also be placed in 1-to-1 correspondence with the integers. So counterintuitively, there are only as many rational numbers as there are integers. Not more!

But when it comes to irrational numbers, there are many more. He called this number aleph1 and he showed that it was different and bigger than aleph0. He proved that the number of subsets of the set of integers {1,2,3,...} is also aleph1 and he produced this amazing result..

aleph1=2^aleph0

He even asked if there was an aleph number between aleph0 and aleph1.

During his lifetime Cantor was ridiculed, not by the general public, but by his fellow mathematicians. Today his work is regarded as brilliant and is taught as part of the standard university mathematics curriculum.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Physics and the Number 3

Physics and the Number 3

It seems that Physics likes the number 3 at all levels, from macroscopic down to the quantum level. Consider this..

1. There are 3 macroscopic spacial dimensions.
You can think of this as 3 families each containing 2 members..

Up/Down
Left/Right
Forward/Backward/

2. Quarks consist of 3 families each containing 2 members..

Up/Down
Charm/Strange
Top/Bottom

3. Quarks are bound together to form composite particles (such as protons) by an incredibly strong force known as the color force. Gluons are the particles that mediate this force. For this model to work quarks must have a color charge - and it comes in 3 families each containing 2 members..

Red/Anti-Red
Green/Anti-Green
Blue/Anti-Blue

4. Leptons consist of 3 families each containing 2 members..

Electron/Electron Neutrino
Muon/Muon Neutrino
Tau/Tau Neutrino

I guess nature loves 3. But why?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Gravity and The Standard Model

Gravity and The Standard Model

Spin Inversion Symmetry is a very simple conjecture, it says: Every elementary particle in the Standard Model (except the Higgs) of spin s has a partner of spin 1/s.

So the bosons (spin 1) in the Standard Model are their own partner, but the fermions (spin 1/2) in the Standard Model have spin 2 partners.

Of course, for this to work we need at least one spin 2 particle. Maybe more. So gravity appears automatically.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Why is light so slow?

Why is light so slow?

It takes light approx. 100,000 years to cross our Galaxy. Why not 2 minutes?

I mean, there are about 100 billion galaxies in the Universe. Light needs to pick up the pace.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Spin Inversion Symmetry - Predicting the Graviton

Spin Inversion Symmetry - Predicting the Graviton

I would like to make a conjecture and call it Spin Inversion Symmetry. It says: Every elementary particle in the Standard Model (except the Higgs) of spin s has a partner of spin 1/s.

So the fermions (spin 1/2) in the Standard Model would have spin 2 partners and the bosons (spin 1) in the Standard Model would be their own partner.

Of course, for this to work we need to have at least one spin 2 particle. Maybe many more. So gravity appears automatically.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Alexa Does Math (And Way More)

Alexa Does Math (And Way More)

You must try the Amazon Echo.

You can ask Alexa almost anything. It's kinda scary.

Just buy an Amazon Echo. It's a one time purchase - no subscription. Then install and ask away.

- Ask for weather forecasts.
- Ask for anything on the internet.
- Ask to place a phone call.
- Ask for flight status.
- Ask to play music. If you have Amazon Prime Alexa will play 2 million songs for free!

But I had to try math, "Alexa, is 1949 a prime number?" She got it right. And fast.

I should also mention this. My mother-in-law is 94 and she lives alone. Obviously there is much she cannot do. But I setup Alexa for her. And she uses it constantly for..

-The weather
-The time
-Switching lights on and off (and dimming them)
-Switching the TV on and off (and changing volume)
-Placing phone calls
-Playing her favorite music
-Getting flight status information

Alexa is an amazing tool to the elderly. I wish this got more publicity.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Can Gravity Produce Elementary Particles?

Can Gravity Produce Elementary Particles?

First I postulated Spin Inversion Symmetry (SIS). It 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.

Then I wanted to 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.

Then I made a strip with 4 half turns and discovered something elegant.. it will naturally "flip" into a double thickness Mobius strip.

In other words a strip with 4 half turns (spin 2) naturally flips into a strip with 1 half turn (spin 1/2).

That's a Boson to Fermion transition. Think of gravity (a Boson spin 2 structure) "condensing" into spin 1/2 Fermions.

And note that all Fermions in the Standard Model are spin 1/2.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Area of a Circle - Simple Derivation

Area of a Circle - Simple Derivation

There are several ways to derive the formula for the area of a circle. This is my favorite because it's so easy. I call it the "pizza method".

Consider a regular convex polygon with n sides each of length b. It has a "radius" r which is the perpendicular from the center of the polygon to the center of a side.

So the area of one triangular "pizza slice" of the polygon is b*r/2 and thus the total area of the polygon is n*b*r/2

But n*b is the length of the periphery of the polygon, let's call this c for "circumference". So the area of the polygon is c*r/2

So we've found the area of any regular convex polygon, no matter how many sides it has.

But a circle is just a regular convex polygon with an infinite number of sides. Our formula is still valid. So the area of a circle is c*r/2

But by definition pi=c/(2*r) which means c=2*pi*r and so we get the famous formula for the area of a circle pi*r^2

Even though this formula is famous I prefer c*r/2 because it works for a circle and also for any regular convex polygon. So the next time somebody asks you for the area of a circle confuse them and say c*r/2

Content written and posted by Ken Abbott abbottsystems@gmail.com

Elementary Particles & Paper Strips

Elementary Particles & Paper Strips

I was wondering if a spin 1/2 elementary particle could be modeled 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.

Then I made a strip with 4 half turns and discovered something elegant.. it will naturally "flip" into a double thickness Mobius strip.

In other words a strip with 4 half turns (spin 2) naturally flips into a strip with 1 half turn (spin 1/2).

That's a Boson to Fermion transition. Think of space time (a Boson spin 2 structure) "condensing" into spin 1/2 Fermions.

It explains why every elementary matter particle has spin 1/2. They have no choice!

This led me to conjecture Spin Inversion Symmetry. It says every elementary particle in the Standard Model (except for the Higgs) of spin s has a partner of spin 1/s.

So the spin 1 elementary particles in the Standard Model are their own partner, but the spin 1/2 elementary particles in the Standard Model have spin 2 partners.

This implies there is at least one spin 2 particle (graviton) and maybe more. Maybe a family of Gravitons!

So there you have it - the quantum mechanics of paper strips, and all you need is paper, scissors, a touch of glue, and a vivid imagination!

Content written and posted by Ken Abbott abbottsystems@gmail.com

F=ma derived from scratch

F=maa derived from scratch

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.

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.

But there's yet another way to look at this. If flux is the number of force lines through space then the rate of change of flux is a force. This is certainly true in electromagnetism where it's called Faraday's Law.

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.

Let's assume this is also true for gravity..

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.

The number of lines of force is determined my m. And the rate of change is determined by a. So F=m*a

Content written and posted by Ken Abbott abbottsystems@gmail.com

Physics of Electrical Generators

Physics of Electrical Generators

The physics is simple...

First, use the "lines of force" model for a magnetic field. The lines give the direction of the field and the intensity is just the number of lines per unit area.

Now take a close loop of conductor.. which of course in the real world will contain a load such as an LED lamp etc.

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*.

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.

How can you change the flux through the loop? Several options..

1) Change the strength of the magnetic field.
2) Change the size of the loop.
3) Increase the number of loops.
4) Change the orientation of the loop.
5) Move the loop in and out of the magnetic field.

That's it. This is the basis of all conventional electrical generators.

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.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Spooky Sets

Spooky Sets

Math deals with sets. And finite sets have an integer number of members.

But why?

Math is all about extending definitions and concepts. So why does the number of members of a set need to be an integer?

What about a rational number, or an irrational number. Or if you want very spooky sets what about a complex number!

How could this extension be done? I don't know.

And if x is the number of members would we still have the number of subsets = 2^x

Content written and posted by Ken Abbott abbottsystems@gmail.com

Shannon Entropy Explained - Fast

Shannon Entropy Explained - Fast

Suppose you have a system that has n discrete states, where n is an integer.

And suppose the probability of finding the system in a given state is the same for all states.

Then is you solve n=2^x for the variable x you've just found the Shannon Entropy of the system.

Shannon Entropy is also known as Information Entropy.

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.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Stephen Hawking Black Hole Formula

Stephen Hawking Black Hole Formula

Hawking applied Quantum Mechanics to Black Holes in order to derive his famous formula..


S is Entropy and A is the area of the event horizon. All the other things in the formula are constants of nature.

So the formula says the entropy of a Black Hole is directly proportional to the area of its event horizon.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The classification of integers by missing primes

The classification of integers by missing primes

If p1,p2,p3,.. is the sequence of prime numbers then any positive integer can be written:

n=pi^a1*p2^a2*p3^a3...pq^aq

where q is a positive integer and ai are integers greater than or equal to zero.

Of course, if ai=0 then pi^ai=1 and we say that the prime pi is "missing" from n.

I wonder if it would be instructive to classify integers by their missing primes?
As a special case: if n is prime then it has the maximum possible number of missing primes.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Human Retina - Explained in 90 Seconds

The Human Retina - Explained in 90 Seconds

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.

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.

The retina contains about 100 million rod cells.

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!!!!!

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!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Are Extra Dimensions a Dead End?

Are Extra Dimensions a Dead End?

Extra Dimensions are much talked about. They are central to String Theory. The LHC is looking for them (with no success so far).

Now some think Gravitational Waves may be the key to finding them.

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.

So we're looking for something that does not exist - literally.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Optical Computers - When Photons Replace Electrons

Optical Computers - When Photons Replace Electrons

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.

Now a research group says they have done just that.. photon gates

Photons already dominate long haul data transmission (optical fiber). But a photon computer would be truly revolutionary!!!!!!

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.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Relationship Between Mathematics & Physics

The Relationship Between Mathematics & Physics

Understanding the relationship can eliminate much confusion.

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.

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.

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.

Heck, it's even conceivable that something other that mathematics will ultimately be better at modeling reality. Very unlikely, but conceivable.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Building 3D Space

Building 3D Space

Suppose I have a collection of identical balls. Each has 6 connectors on its surface - 2 red, 2 green and 2 blue.

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.

Now I specify a simple constraint: ever ball must have all its connectors filled.

Bingo - the system is a 3D system. Or is it?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Spin Inversion Symmetry (SIS)

Spin Inversion Symmetry (SIS)

SIS is a conjecture I made about elementary particle physics. It says every elementary particle in the Standard Model (except for the Higgs) of spin s has a partner of spin 1/s.

So the bosons (spin 1) in the Standard Model are their own partner, but the fermions (spin 1/2) in the Standard Model have spin 2 partners.

Of course, for this to work we need at least one spin 2 particle. Maybe more. So gravity appears automatically.

I have no theoretical justification for SIS, but I do have a rather amusing story of how I came up with it.

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.

Then I made a strip with 4 half turns and discovered something rather elegant.. it will naturally "flip" into a double thickness Mobius strip.

In other words a strip with 4 half turns (spin 2) naturally flips into a strip with 1 half turn (spin 1/2).

That's a Boson to Fermion transition. Think of gravity (a Boson spin 2 structure) "condensing" into spin 1/2 Fermions.

So there you have it - a desktop version of SIS. All you need is paper, scissors, a touch of glue, and a vivid imagination!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Metrics in Physics - An Unexplored Resource?

Metrics in Physics - An Unexplored Resource?

We could probably do a lot in Physics by simply "tweeking" the metric.

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.

Even General Relativity, THE metric theory, modifies the metric using coefficients but still retains q=2 and never even considers changing that!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Holographic Sphere

The Holographic Sphere

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 an example.

I did. And the result is both simple and elegant.

Consider objects that have their surface area equal to their volume.

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!

Doing the same for a sphere in 3D we get r=3. Which is the dimension of the space!

Is this always true for spheres? Yes!

Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)

Then the surface of a sphere of radius r is the set of points with:
x1^2+x2^2+x3^2+....+xn^2=r^2

and the volume consists of all the points with:
x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2

The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)

So equating them gives r=n.

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.

So we can say in general..

"In n-dimensional Euclidean space a sphere with surface area=volume has radius n"

The radius of the sphere is equal to the dimension of the space!

This may be related to the Holographic Principle because it equates surface area and volume.

Content written and posted by Ken Abbott abbottsystems@gmail.com

CERN - LHC to Get Major Upgrade

CERN - LHC to Get Major Upgrade

CERN has officially begun work on a major upgrade to the Large Hadron Collider (LHC) to boost luminosity.

When the upgrade is complete in 2026 the LHC be able to collect data at almost 10X its current rate.

CERN says..

"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."

Content written and posted by Ken Abbott abbottsystems@gmail.com

A Prime Number Conjecture in Binary

A Prime Number Conjecture in Binary

Take any prime number p and write it in binary format.

From this prime now generate a series of numbers using this simple algorithm:

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.)

If p is m digits long in binary you will generate 2*m+1 new binary numbers. (btw: some may are duplicates.)

CONJECTURE: At least one of these new numbers is prime.

Let me illustrate the process with a simple example.
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)..

1010=10
1011=11
1011=11
1001=9
1101=13
1001=9
1101=13

So in this case we've generated two new prime numbers, 11 and 13.

A note on duplicates:
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.

Content written and posted by Ken Abbott abbottsystems@gmail.com

When General Relativity Fails

When General Relativity Fails

General Relativity sets gravity apart from all other fundamental interactions by claiming it's simply a curvature of spacetime. It's not even a force!

This is why General Relativity can't be reconciled with Quantum Mechanics.

So here's a strange thought. What if spacetime simply ceased to exist (as we currently know it) in certain circumstances?
Then General Relativity would fail.

What might these "certain circumstances" be?
I don't know. But a great place to start is the interior of a spinning black hole.

Content written and posted by Ken Abbott abbottsystems@gmail.com

A Special Sphere in n-Dimensional Space

A Special Sphere in n-Dimensional Space

It's interesting to consider objects that have their surface area equal to their volume.

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!

Doing the same for a sphere in 3D we get r=3. Which is the dimension of the space!

Is this always true?

Yes. Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)

Then the surface of a sphere of radius r is the set of points with:
x1^2+x2^2+x3^2+....+xn^2=r^2

and the volume consists of all the points with:
x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2

The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)

So equating them gives r=n.

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.

So we can say in general..

"In n-dimensional Euclidean space a sphere with surface area=volume has radius n"

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.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Making Prime Numbers - A Conjecture

Making Prime Numbers - A Conjecture

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.)

Conjecture: This process will always generate at least one new prime number.

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.

Examples in base 10
503 is prime. If I insert 2 before the 3 I get 5023 which is prime.
20129 is prime. If I insert 5 at the beginning I get 520129 which is prime.

Example in binary
Take the number 3 in binary 11=3.

If I insert a 1 at the beginning (or in the middle, or at the end) I get 111=7 which is prime.
If I insert a 0 in the middle I get 101=5 which is prime.

If you write a program to test this conjecture I would love to hear about your results.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Sphere in n-Dimensional Space

Sphere in n-Dimensional Space

Here's the elegant result clearly explained.

Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)

Then the surface of a sphere of radius r is the set of points with:
x1^2+x2^2+x3^2+....+xn^2=r^2

and the volume consists of all the points with:
x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2

Then the Theorem is simply this..

"In n-dimensional Euclidean space a sphere with surface area=volume has radius n"

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.

If you're interested in the proof it goes like this..

The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)

So equating them gives r=n.

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.

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Day Einstein Got Lucky

The Day Einstein Got Lucky

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.

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.

It was a huge success. But how could Einstein generalize it?

Easy.. just do the same thing for reference frames that are accelerating. Makes total sense.

So Albert went about the usual business that had been so successful before. But then he suddenly realized something.

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.

A huge bonus!!!!

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.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Was Einstein Toying With Us?

Was Einstein Toying With Us?

First we invent the idea of a coordinate system. That's very helpful.

Then we frame all our physics in terms of it.

But we know that the laws of physics must be covariant - i.e. independent of our choice of coordinate system.

General Relativity showed that, and that alone would have been a very nice achievement.

But no, Einstein wove into it a description of gravity - placing gravity in a totally unique position.

No wonder we can't unify gravity with the rest of physics!!!!!!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Fluid Dynamics of Spacetime

The Fluid Dynamics of Spacetime

The unification of the two great theories of the 20th Century - Quantum Mechanic (QM) and General Relativity (GR) - still seems far away.

Why?

I think both theories will require major modification before they can be united.

The modification will be in how they treat spacetime.

In QM spacetime is a stage on which the action occurs. In GR spacetime is the action!

That's a massive difference.

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.

So, the EFE (Einstein Field Equations of General Relativity) are the "Navier-Stokes Equations" of spacetime - an excellent description at the macro level.

But at the micro level of spacetime we have what? Nobody knows. Our theories are not there yet.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Fermat's Last Theorem - The Search for a Generalization

Fermat's Last Theorem - The Search for a Generalization

One elegant approach is to write the theorem in the language of metric spaces..

Let (x1,x2,..,xn) be a point in a n-dimensional vector space.

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.

This allows a generalized version of Fermat's Last Theorem to be written as follows..

"An integer point (x1,x2,..,xn) is never an integer distance (r) from the origin when q>n"

It's interesting to note that the generalization holds when the metric parameter (q) exceeds the dimensionality of the space (n).

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.

But is this generalized version true?

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.

Content written and posted by Ken Abbott abbottsystems@gmail.com

When Gravity Fails

When Gravity Fails

Einstein's General Theory of Relativity is now over 100 years old and still remains our best description of gravity.

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.

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.

At some point this granularity of spacetime will become apparent - the center of a rotating black hole could be an example.

So, it's not that you can't solve the EFE in this situation - it's that you can't even write them down!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Size of the Universe

The Size of the Universe

It takes light about 100,000 years just to cross our Galaxy. And the Universe contains about 100 billion Galaxies.

So it's huge, right?

Maybe, maybe not.

Suppose the speed of light was changed. So it took a second to cross our Galaxy. Then the Universe would be a much smaller place.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Taxicab Geometry

Taxicab Geometry

Imagine a rectangular lattice and the only way to move around is to go from node to node by horizontal and vertical movements. Diagonal movements are not allowed.

Think of a Taxicab on the Manhattan street grid. It certainly can't drive diagonally through a block!

So the node point (m,n) is at a distance m+n from the origin (m and n are integers of course) and the metric on the space is therefore d(m,n)=m+n

A "circle" centered on the origin is a diamond and pi=4 exactly.

In this grid each node is connected to exactly 4 others and pi=4. Is this just a coincidence?

Is there a grid where each node is connected to exactly 3 others? Yes, a hexagonal grid has this, and pi=3 exactly.

Is there a grid where each node is connected to exactly 6 others? Yes, a triangular grid has this, and pi=6 exactly.

The General Case This geometry is not constrained to grids (rectangular, hexagonal or otherwise). So long as the connectivity is correct the whole thing could be a mesh piled in a giant heap on the floor!

There are just two rules: You can only move between nodes along a connection, and the (minimum) number of connections between 2 nodes is the distance between the nodes.

It's incredibly general. It's topological. It's all about connectivity. Examples: the "rectangular grid" is a mesh with connectivity=4 and the "hexagonal grid" is a mesh with connectivity=3.

A fun way to imagine this geometry: Think of space as a fishing net. The knots in the net are the points of space, the snippets of reality. And the rope between the knots are the connections. It does not matter how you handle the net - throw it on the ground in a heap if you wish - the topology is unchanged. So reality is unchanged.

Could we build physics on such a mesh and if so what would it look like?

Content written and posted by Ken Abbott abbottsystems@gmail.com

Spin Inversion Symmetry

Spin Inversion Symmetry

Spin Inversion Symmetry (SIS) is currently a conjecture.

It says that every elementary particle in the Standard Model (except for the Higgs) of spin s has a dual particle of spin 1/s. Clearly, that demands spin 2 elementary particles, so gravity appears naturally.

For example:

Applying SIS to the 3 neutrinos (electron neutrino, muon neutrino, tau neutrino), and assuming charge is conserved, we get 3 neutral spin-2 bosons.

Applying SIS to the 3 leptons (electron, muon, tau), and assuming charge is conserved, we get 3 charged spin-2 bosons.

Applying SIS to the 6 quarks, and assuming charge is conserved, we get 6 charged spin-2 bosons.

That's a total of 12 spin-2 bosons, 9 charged and 3 neutral.

The photon is interesting. It's spin-1 so the photon is its own dual. The same is true for the W and Z bosons that mediate the weak force, and also for the gluon that mediates the color force.

SIS is still a conjecture. But if true the implications are very deep. It says the universe is awash in spin-2 bosons and the particles we currently know in the Standard Model are only a fraction of the particles out there.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Does Hawking Radiation Exist?

Does Hawking Radiation Exist?

Is it possible that Hawking Radiation does not exist?

Hawing Radiation from a black hole is supposed to happen when the event horizon separates a pair of virtual particles. One falls into the black hole but the other escapes and appears as radiation.

Current theories assume space is a continuum, meaning the position of the event horizon can be specified with infinite accuracy. Suppose this was not the case. Suppose the event horizon was fuzzy. Simply not enough resolution to separate a virtual pair of particles. Then there would be no Hawking radiation.

And let's not forget, Hawking Radiation has never been experimentally verified.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Laws of Information

Laws of Information

Suppose I told you this..

"I weigh 175 lbs."

I just gave you some information. But what exactly is information, and can it be defined in a quantitive way to become a key concept in math and physics? Let's look more closely at what I just gave you..

It removed an uncertainty. Meaning, before you read the statement you were uncertain about my weight. Reading the statement removed your uncertainty. So perhaps information is simply the removal of uncertainty. Great, we've defined information!

Not so fast..

I gave you the information by publishing it in a blog. So you were not the only person to receive it, thousands of others did also. Perhaps some of those people were my family. But they already know this, so for them it did not remove an uncertainly. Which means for them it was not information!

So, our definition is true for some people but not for others. That's a bad definition. We need something better.

OK, let's forget this whole thing. I regret giving you the information, so I'll just delete it.

Not so easy. I can delete it from the blog, but thousands have read it and they remember it. I can't delete that. So perhaps we've discovered the first law of information..

"Once received, information can never be deleted."

Oh boy, information is getting complex. What kind of stuff is this?

It gets more interesting. The above statement implies there's some information that's never received. Can this be true? If it's never received, ever, then how do we know it exists? We don't. So perhaps we should restrict our definition of information to information that's received. Then our first law of information gets even simpler..

"Information can never be deleted."

Does this mean the information content of the universe is constantly increasing, for the simple reason that information cannot be deleted?

Let's look more closely at what happened when the above information was received. The person receiving the information scanned it and committed it to memory. So they can recall it anytime they want. We don't know exactly how memory works, but we do know that something must have changed in the brain's neural structure in order to store this information. And no change comes for free, a small amount of energy was required to process and store the information. So perhaps we have a second law of information..

"Receiving information expends energy."

This statement is interesting for two reasons. First, it shows there's a relationship between information and energy. Second, it indicates a possible quantitative definition of information.. perhaps we can relate the amount of information in a message to the amount of energy needed to receive the message.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Counting is Not Always Easy

Counting is Not Always Easy

Counting is not always easy, even for a small number of objects.

Here are the famous Borromean Rings. How many linkages are there?

Take any two rings and look at the linkage between them - there is none.

But you cannot pull these rings apart, so they must be linked. How many linkages are there?

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Nature of Spacetime

The Nature of Spacetime

We tend to think of space as nothing - simply a void with no properties. But General Relativity says otherwise. It says spacetime can get up to all sort of tricks!

Consider the case of light..

In a vacuum light travels at about 186,000 miles per second. That's incredibly fast, right?

No, it's incredibly slow.

For example, it takes light 100,000 years just to cross our Galaxy. And the Universe contains billions of Galaxies.

So why is light so slow?

Think of spacetime as something "tangible", something that provides "resistance to motion".

So light has difficulty plowing through spacetime.. which is why it's so slow.

And the situation with objects is even worse. Which is why you have to apply a force to move an object.

Moving through spacetime is like wading through molasses!

Content written and posted by Ken Abbott abbottsystems@gmail.com

The Man Who Counted Beyond Infinity

The Man Who Counted Beyond Infinity

Georg Cantor was a mathematician who proved something quite amazing - there are numbers bigger than infinity!

He called these numbers "transfinite numbers" and he even developed an arithmetic for working with them. He denoted them by the Hebrew letter "aleph".

So what did Cantor do?

He formalized counting. He started with the integers {1,2,3,...} and asked what other sets could be placed in 1-to-1 correspondence with the integers. Instead of just saying there are an infinite amount of integers he denoted the number of integers by aleph0 and developed an arithmetic that in many ways treated aleph0 as a regular number. But he went further..

He showed that the rational numbers (fractions) could be placed in 1-to-1 correspondence with the integers. So counterintuitively, there are only as many rational numbers as there are integers. Not more!

But when it comes to irrational numbers, there are many more. He called this number aleph1 and he showed that it was different and bigger than aleph0. He proved that the number of subsets of the set of integers {1,2,3,...} is also aleph1 and he produced this amazing result..

aleph1=2^aleph0

He even asked if there was an aleph number between aleph0 and aleph1.

In his lifetime Cantor was ridiculed, not by the general public, but by his fellow mathematicians!

Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life.

Today Cantor's work is part of any university math curriculum and is regarded as one of the most beautiful pieces of mathematics ever created. It stands apart from most advanced math because you don't need to know much math to understand it. In fact, all you need to know is how to count!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Quantum Mechanical Spin - The most fundamental thing?

Quantum Mechanical Spin - The most fundamental thing?



If you look at the list of elementary particles in the Standard Model you'll see that each has a property called spin. There are only a few values.. 0, 1/2, 1 (and the hypothetical graviton - not in the Standard Model - has spin 2).

Spin divides all elementary particles into two radically different groups. Spin 1/2 particles are Fermions. Spin 0,1,2 particles are Bosons.

And spin is conserved, there is no known process that can change the spin of an elementary particle.

So perhaps spin is the most fundamental thing.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Is String Theory a Dead End?

Is String Theory a Dead End?

Oh sure, it produces some very nice ideas.

But it requires "multiple dimensions". The last I heard was 26.

Hey, with 26 free parameters to adjust I could produce some nice results also!

Suppose multiple dimensions don't exist (even at the Planck level). Suppose all we had was 3. You know, like we currently have.

Is String Theory taking us down the wrong path?

Content written and posted by Ken Abbott abbottsystems@gmail.com