Beauty of Math - Fractals

This is the second part in a series of posts about this subject - beauty of math. This one is about fractals. You are also advised to read the first part - Algebraic surfaces.
Fractals are very common in our world. In fact most of the things we see can be called fractals, although they are not mathematical fractals.

Fractals were first discovered by Maldenbrot. He was a British mathematician. The discovery was triggered by a rather practical question - how can we measure the coast of Britain effectively?
He reasoned that if we will take a ruler with marks of 25 km long, we will get some length but it will not be correct, because the ruler was too large. So lets take a smaller one, 5 km. But still the measurement is not accurate. There are small parts of the coast line that were overlooked when the measurement was taken. The length we get with this ruler is however larger than with the previous one, because a smaller ruler allows us to get closer to the real coast line.
If by making the ruler smaller and smaller we get better approximations of the length of the coast, in order to get the actual length we need to use a ruler of zero length. But if this happens then the length will be infinite. However, the coast line has a finite length - you can take a tour around all of it.

If the coast line was a mathematical line, the measurement would be easy and independent of the ruler length. It is also not a plane. Thus its dimension is neither 1 or two. Which means that it must have a dimension that is not a whole number - a fractal.

From this thought we get the two main descriptions of a fractal - it is shape that has a fractional dimension. Also as it was with the coast, fractals have self similarity at high magnification.

The following videos are example of fractals. They were generated by a program called XaoS.

This one is the so called Koch snowflake








The last one is the Sierpinski carpet.

The quality of this videos is not would I would like it to be. But they give a good example to the beauty of fractals. If you want to see more fractals, you can also check my picasa album: Fractals.

Surprising or not?

This post can be regarded as part two of the previous post, importance of education. I didn't intend to make two post about this subject but a few hours after publishing the previous post I stumbled on an excellent example of our current attitude to education:

You probably wonder why I told that it shows our attitude. The answer is simple - the only way to get such result is to not listen to the teacher.
I don't know if this story is true or not, but it still shows very one of the main problems of the current education system in developed countries. Those who don't want to study will not study, they will simply sit in the classroom. It is so because they don't have real motivation to study, they think that they have all they need now and that they will have all what they need in the future without studying.

In the third world countries the situation is different. The people there see education as a way out of poverty. Also in Japan this problem is solved rather creatively. There you must study until you get a bachelor degree. And unless you have a medical condition that makes impossible or difficult to study, your parents will have to pay fines if your don't do well in school. This system has it own problems, and I am not sure that given the choice I would prefer to study in under such rules. But this is a lesson - there is a way to change people mind about importance of education.

Importance of education

If you have any doubts in it please see the following quotes. I think they show the point pretty well.

"Be sure and put some of those neutrons on it."
-Mike Smith, Baseball pitcher, ordering a salad at a restaurant

"We are not without accomplishment. We have managed to distribute poverty equally."
-Nguyen Co Thatch, Vietnamese foreign minister

"Men, I want you just thinking of one word all
season. One word and one word only: Super Bowl."
-Bill Peterson, football coach

"The word 'genius' isn't applicable in football. A genius is a guy like Norman Einstein."
-Joe Theisman, NFL football quarterback and sports analyst

"I've read about foreign policy and studied-I know the number of continents."
-George Wallace, 1968 presidential campaign

"It isn't pollution that's harming the environment. It's the impurities in our air and water that are doing it."
- Former U.S. Vice-President Dan Quayle

"Please provide the date of your death."
-from an IRS letter

"We're going to turn this team around 360 degrees."
- Jason Kidd, upon his drafting to the Dallas Mavericks

The first and the last quote captures the point especially well. While it is perfectly OK that certain physical or mathematical words are used on a daily basis, it is rather hilarious when people use them without understanding what they are saying.

A month or two ago, a very interesting law was proposed (by Arab Knesset members) in Israel. According to this law everyone who has a political office or is compaining for one, should know how to read and write. This may sound strange but in areas where there is an Arab majority, the mayors often don't have basic education - they even don't know how to read and write. The law didn't pass. The reason - we are too democratic and cannot take the right to vote or to be elected on such basis.

Update: there is a part two to this post: Surprising or not?.

Beauty of math - Surfaces

This is going to be a first post about this subject - beauty of math. This one is about algebraic surfaces.

While many people either don't understand or even hate math, I tend to see it's beauty. To be honest I don't understand those who don't see it. While part of this beauty is indeed hidden, and not everyone will see it, a lot of it is pretty obvious. All you need to do is just change your mindset and be ready to receive something new.

In this post I will talk about one of the most obvious examples of math beauty - surfaces. While it may sounds bizarre at first, to understand what I am talking about take a look at this first:

x³ z + x² + yz³ + z⁴ = 3xyz

yz(x²+y-z) = 0

The equations below the images are the functions that create them. You can find more images like these here.

Now after you saw this what do you think? Does it change your attitude to math? I doubt that. If you are reading my blog you probably don't hate math, but what do you think about it?

For me the beauty in these images is their simplicity. It is surprising how such simple equations can describe such elegant surfaces. Moreover this is a general fact about math - physical laws can be written by simple and elegant equations. There is even a humorous story about this fact. Paul Dirac, widely regarded as one of the greatest physicists of all time, was asked once to lecture about philosophy of physics. He simply stood up and wrote on the board - "physical eqautions should have mathematical beauty". In these pictures we see exactly this - simplicity of equations and the complexity of the result.

The next post in this series is: Beauty of Math - Fractals.

New semester old semester

As I already wrote there was a long strike in all of the Universities in Israel. This was the longest such strike ever - 89 days. It ended this week. Also this week is the last week of the first semester..
I suppose Israel is the only place where after such strike the semester is not canceled and no compensation is payed to the students. Moreover, we still have to do our exams, on the courses that were taught despite the strike, while studying other courses. The semesters will be shortened also, so we will have to learn in 11 weeks what is normally taught 14 weeks. It was also announced today that due to financial problems there will be no TA classes for one of the courses next week.
For me this is good actually. It looks like I will be able to add a course to this semester, and overall I don't have too much to study. So I am not complaining.
There is no timetable for now but I will probably will have less than 20 hours per week.

For the blog this is bad news unfortunately - I will have much less time to work on it especially during the 2-3 next weeks. I hope that I will manage to use all the time I have wisely.

Natural units

One of the most important questions in both physics and mathematics is: How do we measure things? In mathematics this is mostly settled. We have a number system which is good enough for all of our needs. It allows us to assign numerical representation to distance and it is also easy to do algebraic calculation with it. Also, on a graph it doesn't really matter what the numbers stay for - they can refer to meters or to microns (or to bytes). It begins to matter only when we what to actually understand what the graph says us. That is, when we want to use mathematics to describe not only itself but also the world around us.

Describing the physical world is the main goal of physics. So lets look how physics deals with our question.
Currently we have the M.K.S system - meter, kg, second. It is a system that works very well. But is it natural?
Surprisingly this units are not only not natural, but they are not even well defined. The meter for example, was supposed to be 1/40000 of the earth equator. It turned out that the equator length was not calculated correctly, so the length of the meter is not related to it.
For a long time there was a special metal bar 1 meter long, that was kept at a constant temperature. It was called "meter" - and it is all the connection to the physical world that this system has and ever had. The meter is not well defined because it is impossible to build perfect replicas of this metal bar, they will all be slightly different in size and shape. Currently it is defined as the distance traveled by light in 1/299,792,458 of a second.
This is also true for kg. Initially it was defined as the 1/1000 of the weight that one cubic meter of water had at sea level. Well this weight is hardly a natural constant. Also it cannot be natural because it was defined using meters, and they are not natural.
The second is also not natural. Initially the division of the day into 24 hours and then into 60 minutes and 60 seconds was due to the fact that this was the easiest relation to build the watch for. You can read more about measuring time in my post about calendars.

In other ares of physics, we have other units. But most of them are also unnatural. An example on a natural unit used in physics is kelvin. The kelvin temperature scale is defined by pure reason based on motion of mater. Also since Kelvin can be converted to Celsius by simple addition of 273, Celsius is also a natural unit. the only difference between the two is that Celsius has a different point denoted by zero.

The reason why we have so many unnatural units is twofold. Firstly, when those units where invented natural units were unknown. Secondly many natural units are two small. For example the electron charge can be considered a natural unit of charge. But it is only 1.6021765 × 10⁻¹⁹ coulomb. It is simply too small to be used.

Google page rank

Waked up today, went to my comp and then suddenly - my stumble upon blog has a page rank of 3. It was on 1 for a long time.. Good news. Then I went and checked this blog and got "no page rank information available"?!? It was one since the beginning... The some is with my math pages site.
I submitted it to Google a long time ago. I don't have a site map but still, I am not sure I understand why this happened. All my pages are indexed.

Probably it is because there are no links to this blog. At least not links that Google knows about. I guess it will take some time, but I will get a page rank.
Google updates its page rank ones in three moths, so it will take some time...

Also for some reason I have problems with Technorati. I claimed both my blogs there some time ago but then suddenly they stop showing my updates. Even if I pinged them manually it didn't help. It become even weirder. Two days ago I logged in and suddenly it showed that instead of my stumble upon blog I own some weird blog I never heard of - and it is in Chinese(???). I even don't speak Chinese. I deleted it and tried to claim my blog again. The site reported me that they have technical difficulties so claiming is disabled.
Today I tried to claim my blog again - and got this:


It just all had to happen in the some time, didn't it?

Even when I tried to add a photo to my stumble upon blog yesterday, for some reason the automatic photo blog didn't work for it. I thought- well I will I just upload it to picasa, and then hotlink it from there. Didn't work either. The picture refused to show up. Also the picasa web site was very slow yesterday. I wonder if they had network problems or perhaps traffic overload? Eventually I uploaded the photo to photobucket.

Also the post count for my blog on the blogger dashboard is wrong. It shows one less then it should.. It started 2 maybe 3 days ago. I hope it is not the internet trying to say me that it hates me..

Get rational, be real

A long time ago I stumbled upon this little piece of art... It is not something special, but yet it made me laugh then and it still makes me laugh.

On a more serious note it is interesting that most constants both in math and in physics are not rational numbers. An exception is, for example, the speed of light, which while usually referred to be 300000 km/s but, is in fact exactly 299792458 m / s.
The reason for this is simple. Our number system is not natural - it is just the most easy to use system. It is possible I suppose to change it, but what for? It works well enough. The only problem is that as we discover new constants we may end up without symbols for them. After all there is only a finite number of symbols we can remember and use, so what if the number of such constants is greater? For a human being this will not be a problem - we can work only on a little part of the whole science, so we will not need to know all of the symbols. Eventually we will simply have double meanings for the some sign, according to where it is used. But it would be impossible to create a computer program, capable to use and display all of them.

Geometry trick

As I already wrote once, I like strange theories and paradoxes, but I believe that there is always some underlaying logic. For example look at this image:
Do you see the reason for this? There is nothing wrong with this picture. Moreover this is exactly what you will get if you will make your own triangle.

The explanation is very simple. The area of all of the smaller triangles is equal, in both pictures, to the original one. But if you will look carefully you will see that that the slope, of the second figure, is different.
Update: This wasn't clear to some of the readers, so I wrote a more detailed explanation.

Such things are the reason why in math everything must be proven. We cannot just trust our intuition. It was relatively easy to spot the reason in this case, but it is not always easy. Geometry we can at least visualize, and even make pictures. When the question is totally abstract it is much more difficult to understand it - especially if our intuition is wrong about it.

A few days ago I overheard two students talking about their linear algebra exam. There was a question in the exam to find a basis for some vector space. The student found it, but didn't prove it is indeed a basis. He didn't get any points for the question naturally. And he was angry about it...
But without proof there is nothing. Even if the answer is correct, without proof it is worthless. Such approach is the only way in which paradoxes can be solved and mistakes can be avoided

Copyright infringement

First of all a disclaimer: This post contains information that may be used to obtain illegal copies of music, videos, books etc. All of this information is presented here for informational purposes only. By reading it you agree that I am in no way responsible for any damage resulting from using any methods described in this post.

Now lets get to the main idea.
In this post I am going to discuss a method that can be used to obtain any piece of copyright work, that can be uploaded to the internet, for free.
It is a very simple method, although it may require a lot of time. It all starts with numbers, real numbers. As you probably know the numbers can be divided in two groups - rational and irrational numbers. Surprisingly irrational numbers can be used to get staff for free..

The difference between rational and irrational numbers, simply put, is that irrational numbers have infinite decimal expansion - and it doesn't repeat itself as the decimal expansion of 1/3 for example. An example for such a numbers is . It decimal expansion is approximately 3.14159.... Now what will happen if we will write it down in binary? We will get an infinite string of numbers 1 and 0.

After reading this you probably already understand what I mean, but lets see an example. As I said you can use this to get free copies of music. Suppose what you want is to get the latest song of your favorite artist. This song can be stored as a mp3 file. In this form it is also s string of 1's and 0's. Just like . But there is a very important difference - it is finite. It is very long but it is still finite. Therefore it is contained in . From some point the 1's and 0's in string will be exactly the some as the 1's and 0's of the mp3 file. All you need to do is to find this point. After finding this point you can make a new mp3 from this part of the string. Moreover since you didn't copy the first mp3 but created it yourself it belongs to you and not to the artist.

It may seem impossible and a joke. But in fact it is not. While today this is no more then a fun thought, all we need is more powerful computers to achieve this. And then all we will need to do is to break in chunks and see what they turn to be if converted by appropriate program.
Finding a specific song in this way is of course impossible.

Visualizing large numbers

This is a very hard thing to do. While it is easy to think about small numbers it is very difficult to imagine large numbers. It is largely because they seem to be totally unrealistic and not related to reality. For me it is sometimes difficult to visualize even not very large numbers - 100 for example. And visualizing say 10^50 is nearly impossible for everyone.

A bit of trivia: what is the largest named number?

Surprisingly it is a very well known one - googolplex. It is .
Why I said it is well known? Well the Google main office complex is named after it. So it is a very well known number (for such a huge and totally not connected to reality number) .
At this point you are probably wondering why I said that it is totally not connected to reality, that is, there is nothing in the physical world that it describes. After all the universe is so huge. So why wouldn't it contain a googolplex of something? However it turns out that there is only about atoms in the whole universe. This is almost zero compared to the googolplex!!

When I first learned that there is only atoms I actually said: "and this is all? Such a small number... " I said so because I couldn't visualize it. It had no meaning for me, and I new that there were numbers much bigger.

Lets see another example. Suppose we take a sheet of paper 0.1 mm thick and we will cut it in half, stack it and then cut in half again. We will repeat this 100 times.
In the end we will have a stack of paper sheets. Since the stack height is representing the number, it should help us visualize it, right? Well lets see how high will the stack be.

=1,267,650,600,228,229,401,496,703,205,376

This is in 0.1 mm. In light years it is about 13.4 billion light years. This is close to the radius of the whole universe...

The problem with visualization of such numbers should be rather obvious now. Nothing we have in our world can help us visualize them. But what about smaller numbers - 10^6 for example?
Such numbers are possible to visualize using small objects. But it is still difficult.
For example you can visit this site to see how a 10^18 pennies looks like.

For most of us this is not an issue. We don't use such numbers were often, and we are not asked to visualize them. However, there is an interesting underlaying fact in all this. Those numbers can be written, we can use them in equations (I see no reason why but it is possible) but we have no use for them. They don't describe anything in our world. Then why do they exist? And do they exist at all, if they are useless? As I already said, I believe that math is discovered and that it has a "life of its own". While the existence of such numbers is not a proof to this view, it is still something to think about - if most of the numbers that we have are of no use to us then why we invented them, if they were invented by us?

Computer quotes

This is just a short break from a mathematic theme. While there is more then enough serious staff to write about, I don't want my blog to be a boring place. And especially I don't want it to become a blog that requires an advanced degree to read. I will blog about such things but I am not going to focus on them.
So I went over my bookmarks to see what funny staff I recently found on the web. For some reason most of it was about computers.

Please don't laugh too much (it is still a math blog):

I've noticed lately that the paranoid fear of computers becoming intelligent and taking over the world has almost entirely disappeared from the common culture. Near as I can tell, this coincides with the release of MS-DOS. (Larry DeLuca)
(I suppose that after that nothing could convince people that computers can become more intelligent then people.)

AI has been brain-dead since the 1970s. (Marvin Minsky)

When you say: "I wrote a program that crashed Windows", people just stare at you blankly and say: "Hey, I got those with the system -- for free." (Linus Torvalds)
(I don't like this type of jokes very much but I use Ubuntu and it is much more stable and responsive then windows)

It's not the technology, folks, it's the people. When we trace [the errors] back, it's always human error. (Bob Herbold of Microsoft)

What would you rather have to plow a field — two strong oxen or 1,024 chickens? (Seymour Cray)

Java is the most distressing thing to happen to computing since MS-DOS. (Alan Kay)

There's a simple way to find out if an operating system has been well designed. When you get an error message, go to the help system and look up the exact words in that message to see if there was enough of a concept of an architecture that they have a consistent vocabulary to talk about what's broken. (Bill Joy)

"I don't like it, and I'm sorry I ever had anything to do with it."
(Erwin Schrodinger talking about quantum mechanics. Probably he loved the cat)

"Physics is not a religion. If it were, we'd have a much easier time raising money."
(Leon Lederman. So is math..)

"What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school... It is my task to convince you not to turn away because you don't understand it. You see my physics students don't understand it... That is because I don't understand it. Nobody does."
(Richard P. Feynman (1918-1988), QED, The Strange Theory of Light and Matter, Penguin Books, London, 1990, p 9. To those who don't he is Nobel prize laureate)

"Heavier-than-air flying machines are impossible."
(Lord Kelvin, president, Royal Society, 1895.)

"X-rays will prove to be a hoax."
(Lord Kelvin, while president of the Royal Society)

"First you guess. Don't laugh, this is the most important step. Then you compute the consequences. Compare the consequences to experience. If it disagrees with experience, the guess is wrong. In that simple statement is the key to science. It doesn't matter how beautiful your guess is or how smart you are or what your name is. If it disagrees with experience, it's wrong. That's all there is to it."
(Richard Feynman, from a PBS show on Dr. Feynman. He was describing to his class how to look for a new law of physics)

"The great tragedy of science - the slaying of a beautiful hypothesis by an ugly fact."
(T H Huxley (1887-1975))


In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite.
(Paul Dirac1902-1984)

“If at first you don’t succeed; call it version 1.0″

“Be nice to geeks, you’ll probably end up working for one.” (Bill Gates)

“It’s a little-known fact that the Y1K problem caused the Dark Ages.”

“The nice thing about standards is that there are so many to choose from.”

“A computer lets you make more mistakes faster than any invention in human history - with the possible exceptions of handguns and tequila.”

“The glass is neither half-full nor half-empty: it’s twice as big as it needs to be.”

“The box said ‘Requires Windows 95 or better’. So I installed LINUX.”

“My software never has bugs. It just develops random features.”

“A Life? Cool! Where can I download one of those?” (I hope this was never said for real..)

“No keyboard detected. Press F9 to continue.”

“User Error. Please replace user and press any key to continue.” (We need this error dialog. From my experience with it a lot of computer problems come from this type of errors. And it is something that you at least shouldn't enjoy explaining to the user..)

Strange theories

Recently I stumbled on some strange and surprising theories. Personally I like a lot to read about theories or facts that defines normal logic. However, while I do agree that the world is a strange place I think that it follows some logic. Perhaps we cannot see this logic right now with the tools we have but, it is still there. And if it is there it can be found. Eventually.

The first theory is a very straight forward one. It simply asks the question - is there an opposite to absolute zero? Now we all know that absolute zero is -273 Celsius. At this temperature there is no movement at all. But is there are a maximum temperature?
The question may seem a bit strange, but well, the answer is even strange.
While nobody knows exactly what it is, there are 4 contenders. The first one is plank temperature.
This is the temperature believed to be at the beginning of our universe. It is about 10^32 kelvin. the other two contenders are 10^30 and 10^17.
However the last contender is a real surprise - it is 0 kelvin, that is -273 C. And no this is not a typo. There are actually people who believe that absolute zero can also be the maximum possible temperature. While it sounds shocking the underlaying idea as I see it, is very simple: we don't know what happens at absolute zero. The laws of physics we know stop working there. The some think happens when we reach plank temperature. Thus going both ways we get to the same situation.
You can read about it more here.

The second theory is .. well it is barely a theory actually. It is more like a mind game. Before saying what it is about we will need some background:
While we never met with an alien civilization, many speculations about this subject exists. One of such speculations is an attempt to rank them according to their development level. This whole attempt is based mostly on our current understanding of physics so it may be right but it also may be completely wrong.
According to the current rank system there are four ranks of civilizations.
A rank one civilization uses all the energy of its planet. A rank two uses all the energy of its star. And rank three - of the whole galaxy.
As you can see we are not even rank one civilization. We only recently started to use wind and solar energy and such use is hardly widespread. In fact electricity is not widespread enough yet. There are areas on earth that are almost totally untouched by civilization.
Now when we now this we can discuss the theory I mentioned. We as a civilization are in constant threat of a catastrophe capable of destroying all of us. An ice age or an asteroid, and we will disappear. A rank one civilization is mush more stable. A change in weather cannot destroy it. With such energy at its disposal it should be capable of controlling the weather. But still even such civilization can be destroyed by an asteroid or another sufficiently great catastrophe. A rank two civilization can be destroyed only by a supernova. Rank three has colonized all the galaxy so even this is not a threat for it. But what if the universe has an end? This is, what if the universe itself will be destroyed?
The theory I mentioned stays that a type three civilization should have enough energy and knowledge to create a new universe for itself. And thus, such a civilization is truly immortal. Perhaps you don't see anything weird in this line of thought. But to me the idea of creating a new universe seems strange. It is one thing when such claims are made from an religious perspective. But here this is not the case.

The third one is probably the most bizarre. It says that time is slowing down and will stop at some point. It is offered as an alternative to dark energy. At lest the theory claims that our planet will be long gone by then...
You can read about it here.

Calendars and science

This is the post I promised to write about calendars.
We are all familiar with the Gregorian calendar. It is the most used calendar today. However it is not the only one. In Israel for example we have another calendar. And while it is not used as much as the Gregorian (even in Israel), in many places we have dates written according to both Gregorian and Hebrew calendars. The reason why it is still used is mainly religious. The Gregorian calendar is usually regarded as Christian calendar - after all it begins from Jesus birth. Therefore, it is only natural for the orthodox Jews to be against its use.

The Gregorian calendar is an excellent one, in my opinion. However it and all the other calendars suffer from one major problem - they are not universal. I don't mean by this that they cannot by used in other parts of the universe. This is also true but it is only another aspect of the some fundamental problem.

Lets start from the beginning, taking the Gregorian calendar as an example.
First of all, we have the following - a year is 365 or 366 days, divided into twelve months of varying length. The month is usually considered to have 4 weeks and a week is 7 days. You must have noticed that even this description is not perfect - the year can have different number of days and so are months. We have of course rules according to which all these numbers are set in each year. But still the definition is not perfect - while it is not too difficult to make all of the necessary calculations the fact that the devision is not obvious shows that there is something that causes this.

Why we have 365 days in the year? It is the period in which earth orbits the sun. Thus for every planet we have a different duration of the year. Even the duration of one day is different from planet to planet. Therefore, our calendar would be entirely useless on other planets. If lets say we will colonize Mars one day the Gregorian calendar will be of little use there. But if we will make a different calendar for Mars, we will have problems communicating since we will be living according to different calendars. And this is even more important for computers that for humans. So what can be done?

Lets go deeper. What is time? And how can it be measured? It is very difficult to answer this question. And since according to relativity theory, time, gravitation, space and speed are all connected, it is almost impossible to give a really good definition..
In the Gregorian calendar the shortest period of time is one day. This is its basic time unit. But a day is not a good choice for a measure unit because it length changes. Even on earth the length of the day is different at different seasons. In fact all of the problems with the Gregorian calendar and with other calendars come from this simple fact - a day is not a natural measure unit for time.

But then what is? Well one option is the so called Planck time. It is the time needed for light to cross a distance equal to the Planck length. Since the speed of light is invariant the time will also be always equal. The problem with it - it is too small. It is only about 10^-44 sec.
The only other option is to use a system of calendars. This is one calendar for every scale. In this way there can be one calendar for earth and one for mars - and all the communications between the two planets can be done according to the main calendar of the solar system.
While this brings more complexity, this is probably the best choice if we want a calendar that is naturally connected at least to the planet it is used on..