# Tag Archives: mathematics

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## The Princeton Companion to Mathematics.

Princeton University Press just published the Princeton Companion to Mathematics.
I learned about this book while I was reading a blog post on Timothy Gower’s  first blog post.

I have read quite a bit and I think is a wonderful book.  While it does lack a lot of detail (by this I mean there is practically no demonstration in the book). It does excel at giving a general view at what mathematics is all about.

If you ever wonder

What is mathematics?

or what do mathematicians do. This book is a good start. The book is a compilation of essays in different topics in mathematics many written by first class mathematicians including Timothy Gowers and Terence Tao both recipient of the Fields Medal in mathematics the equivalent to the Nobel prize and many others.

I believe the book should be part of any mathematician or aspiring mathematician library.

The book covers mathematical history and also mathematics itself. Some parts of the book could be read by high school students but for the great mayority is necesary to have at least and undergrad degree to be able to understand it. I wonder if a new Ramanujan found this book if he will be able to reinvent the whole of mathematics from this book?

The book is selling for 66 dollars at amazon from the 99 dollars publisher’s price so is a good bargain.

Curiously the shell image in the cover of this book depicts a section of the Nautilus shell long believe to be related to the Fibonacci golden ratio but I have learn from God plays dice that this is not so!

I assume that the believe relation between the Chambered Nautilus shell and Fibonacci golden ratio was the motivation to place the image on the cover as an example of Mathematics appearing in nature.

It is interesting to see also this video hosted at the Clay mathematics site

Timothy Gowers The importance of Mathematics and it is also available at Clay Videos.

for other post click isallaboutmath

PD:

The Chambered Nautilus shape does seems to be related to the Logarithmic Spiral.

In the back flap of the dust jacket is stated that the reason to place the Chambered Nautilus Shell image on the cover is due to its relation with the Fibonacci Sequence something I had speculated in my review above.

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## One easy problem and One not so easy problem.

A few post ago we used Mathematica to draw the altitudes of an arbitrary triangle of given coordinates. Now we are solving these other two problems The first figure the red lines represent the medians and on the second figure the yellow lines represent the angle bisectors. So our problem consist in finding the Mathematica code to produce these figures.

Let us do the easy one first. To draw a triangle and the medians we will use the Lineal Bezier equation we have use before and since we are interested in finding the mid points on each side the resulting Mathematica code is very simple.

a = {0, 0};
b = {3, 0};
c = {1, 2};
BAB[t_, a_, b_] := (1 – t) a + t b;
Graphics[{Background -> Black,
{{Thick, Blue, Line[{a, b, c, a}]},
{Thick, Red, Line[{c, BAB[1/2, a, b]}]},
{Thick, Red, Line[{a, BAB[1/2, c, b]}]},
{Thick, Red, Line[{b, BAB[1/2, c, a]}]}
},
Inset[Text[Style["A", White, Italic, Large]], {-.1, 0}],
Inset[Text[Style["B", White, Italic, Large]], {3.1, 0}],
Inset[Text[Style["C", White, Italic, Large]], {1.1, 2.1}]}]

Now for the angle bisectors it is a bit harder. We are going to need some property the bisectors satisfy that will allow us to get the coordinates of the intersection of each bisector with each of the sides. One property that could help us is this one.

Theorem: For a triangle ABC If CQ is the bisector thru the angle ACB then AC/CB=AQ/QB.

Notice we will need to find some distances between sides that are given by coordinates so the function EuclideanDistance will be of help. Since we can easily compute AC/CB we need to find Q in AB such that AQ/QB is equal to AC/CB but this is not difficult to achieve if we use the function Nearest. We can guess the best value of Q by building a table where Q is going from A to B and then we pick the best value that approaches to the ratio AC/CB. We can accomplished that with the following Mathematica Code.

a = {0, 0};
b = {3, 0};
c = {1, 2};
ac = EuclideanDistance[a, c];
bc = EuclideanDistance[b, c];
ab = EuclideanDistance[a, b];
cc = ac/bc;
BAB[t_, a_, b_] := (1 – t) a + t b;
N[cc];
tabl = Table[
EuclideanDistance[a, BAB[t, a, b]]/
EuclideanDistance[b, BAB[t, a, b]], {t, 0.000001, 1, 0.001}];
nearest = Nearest[tabl, N[cc]];
Flatten[Position[tabl, First[nearest]]]

that will give us the value 443 that we will use in conjunction with the 0.001 doing similarly for the other sides of the triangle we get the very compact solution

a = {0, 0};
b = {3, 0};
c = {1, 2};
BAB[t_, a_, b_] := (1 – t) a + t b;
Graphics[{Background -> Black,
{{Thick, Blue, Line[{a, b, c, a}]},
{Thick, Yellow, Line[{c, BAB[443*0.001, a, b]}]},
{Thick, Yellow, Line[{a, BAB[428*0.001, c, b]}]},
{Thick, Yellow, Line[{b, BAB[486*0.001, c, a]}]}
},
Inset[Text[Style["A", White, Italic, Large]], {-.1, 0}],
Inset[Text[Style["B", White, Italic, Large]], {3.1, 0}],
Inset[Text[Style["C", White, Italic, Large]], {1.1, 2.1}]}]

Again this time Nearest comes to the rescue and helps us get the best value from a list of possible values!

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## Some Phun Links!

These are the links to the Phun scenes on our lecture

Are you having phun Yet?

Binary Computer

Binary Computer with decimal conversion

Pascal’s Triangle

Fibonacci Sequence

Triangular Numbers Sequence

Sieve of Eratosthenes

Livio’s Multiplication Machine

If you like to play or modify the Scene above in the Phun simulator you can download the Phun program from

Phun download

Posting from
www.isallaboutmath.com

Have Phun!

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## Triangular Numbers (Part III)

Going back once more to the original problem of finding a formula to compute the $n$ triangular number.

$T_n=1+2+3+ \cdots +n$

Triangular numbers suggest by their name the idea of geometry and it should be interesting to try apply some geometric reasoning to solve the problem. As we explained in the first lecture triangular numbers are obtained when we arrange a number of stones in an equilateral triangular shape. It just happens that it is very convenient for us to re arrange those stones in another triangular shape. That is right angle triangle. In that way we are able to produce triangular numbers and it become very simple to arrange the stones in a two dimensional array of stones that could be easily counted. Now we will see how the geometric demonstration works for $T_4.$

First, we duplicate the number of stones so we have $2 T_4$ and conveniently rotating the second $T_4$ we can joint the first $T_4$ and we have as a result a rectangular shape.

The number of stones in that rectangular shape is very easy to compute. It will be the product of the number of stones in two sides. In this case $2 T_4=4(4+1)$ and since we have duplicated the original number $T_4$ now we need to divide by 2 and that will us the value $T_4=4(4+1)/2.$ The same procedure could be carry out for 100 stones or any other number. We do not suggest you do this literally for 100 stones, but you should be able to play this same argument in your mind.

In the previous example we were extremely close to form a square number. We actually missed this by just one row. That is why we have the $(4+1)$ factor. A natural question to ask is.

What should we add to any triangular number $T_k$ to get a square number?

We can see it in the diagram. What we need is to add the prior triangular number! We can express this algebraically with the following relation

$T_k+T_{k-1}=k^2$

What this formula is saying is that the sum of two consecutive triangular numbers is an square number. We can verify that this is true since $T_1+T_{2}=1+3=4$ and also $T_2+T_{3}=3+6=9.$ It is easy to see from the geometric configuration why the sum of two consecutive triangular numbers is an square. Since $T_k$ is something we are interested in finding then the relation seems to be also a kind of equation where the unknown to be found is $T_k$. Notice also that $T_{k-1}$ appears on the equation. Naturally if we know how to compute $T_k$ we then also know to compute $T_{k-1}.$ Equations of this type are known as recursive equations. The last row of $T_k$ will be the diagonal since subtracting the diagonal the resulting triangular number is equal to $T_{k-1}.$ We can also write another relation

$T_k=T_{k-1}+k.$

Now these two equations are defining $T_k$ in terms of the prior triangular number $T_{k-1}.$ These type of relations are called recurrence equations and we will discussed them in more detail in some other lecture. For now let us try and see if we can solve the first equation.

$T_k+T_{k-1}=k^2$

(This is a partial transcription of the Video Lecture Triangular Numbers (III) the video continues displaying a Solution)

The complete video lecture can be seen at Triangular Numbers (III)

There is a prior post similar to this at Triangular Numbers (I)

There is a prior post similar to this at Triangular Numbers (II)

This is a blog posting from www.isallaboutmath.com

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## Triangular Numbers (Part II)

In the prior lecture we have shown how Gauss solved the problem of

finding the sum $1+2+3+ \cdots +100.$

In Gauss solution we reduce the problem of finding the sum of different natural numbers to the problem of finding the sum of 50 equal numbers. It is very natural in mathematics to generalized concepts and results. A natural small generalization of the prior result will be to

find the sum $1+2+3+ \cdots +n.$

of the first $n$ natural numbers. It is also a natural impulse to try and solve similar problems with analogous solutions. In this case we will try to solve the problem using the same idea as Gauss with a very small modification. As in the prior lecture let us call $T_n=1+2+3+ \cdots +n$ the $n$ triangular number. We can easily re arrange the order of the elements in the addition in reverse order. Therefore we can write $T_n=n+(n-1)+(n-2)+ \cdots +1.$ If we add this two equalities we get

$2 T_n=(n+1)+(n+1)+(n+1) \cdots +(n+1)$

as in Gauss solution to the problem we have found again that adding a number from the beginning of the sequence to one number from the end of the sequence the sum stays constant. The right hand side of the equality is also very easy to compute. Since we have $n$ of those terms therefore

$T_n=n(n+1)/2$

This solution was easy to find because the solution is base on the same idea as Gauss’s solution. For the special case $T_n=1+2+3+ \cdots +100$ we have already point out why Gauss’s solution works and the same is true here. The solution works by translating the problem of finding the sum of different numbers to the problem of finding the sum of equal numbers and we are able to produce the equal numbers by conveniently re arranging the numbers in the sequence. In both cases we are dealing with sequences of consecutive natural numbers.

Could we generalize this a little bit more?

Yes, we can. What if instead of a sequence that starts on one we get a sequence that starts on $a_1$ and we obtain the next element by adding a constant natural number $d.$ So the elements of this progression will be

$a_1, a_{2}=a_1+d, a_{3}=a_1+2d, ..., a_{n}=a_1+(n-1)d$

this progression is a bit more general than the sequence of natural numbers. First, it does start on an arbitrary number and the difference of two consecutive terms is $d$ instead of 1. Progressions that satisfy this conditions are called arithmetic progressions. Example of arithmetic progressions are

$1, 2, 3, ...,100$

In this case the first element is 1 and the value for $d$ is also 1.

Another example is

$2,4,6, ...,200$

In this other example the first element is 2 and the increment is by $d=2$. So we obtained each term by adding 2.

Can we find a formula for the sum of the first $n$ terms of the arithmetic progression?

That is to find $a_1+a_2+a_3 \cdots +a_n$

where $a_k=a_1+(k-1)d$ for $k$ from 1 to $n$ to find this formula we suspect that we may be able to find some invariant as before …

(This is a partial transcription of the Video Lecture Triangular Numbers (II) the video continues displaying a Solution)

The complete video lecture can be seen at Triangular Numbers (II)

There is a prior post similar to this at Triangular Numbers (I)
This is a blog posting from www.isallaboutmath.com

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## Standing on the shoulders of Giants.

On Project Gutenberg, Math Books and Google Books and DJVU.

As many of you know Project Gutenberg is the brain child of Michael S. Hart.

The web site for Project Gutenberg is at

http://www.gutenberg.org/wiki/Main_Page

It collects in one place public domain works. The number of works of literature they have collected so far at Project Gutenberg is about 22,000. They use volunteers to proof read the converted works. The current process is a laborious one even with the use of very sophisticated technology.

A book is scanned and the imaged pages are feed into a computer OCR program usually ABBY Reader then a text file is produce and the page images and the text file are presented to volunteers in a process called distributed proof reading.

The Project Gutenberg mainly includes works in the English languages, only very few works on other languages are included. Another big disappointment with Project Gutenberg is that very few mathematical works are part of the project. This is not for lack of available material but I will guess for the difficulty in translating mathematical notations into $\LaTeX$ or MathML.

Meanwhile some other options have appear that are trying to filled this vacuum. One other choice is the project initiated by Google. The project that I am referring to here is the Google Books. In this project Google has put a large collection of works this time including a big collection of mathematical books. The books from big university libraries are being scanned. Giving us back this treasures of old.

The works available for free on the net in Adobe PDF Reader format are the works of the greatest mathematicians in history. From the completed works of Gauss to Euler’s to Augustin L Cauchy,Evariste Galois, J Lagrange, Camille Jordan, Serre etc.

While the work done by Google is commendable they are still some issues. Some of the works have not being scanned properly and the pages are crooked and some of the pages the fonts are not readable. On the other hand it is great to be able to do a Google search on all the available classical books! There will be hopefully a day when we can just do research on existing material from the comfort of one’s own home.

Another similar effort is done by

the million book Project

similar in spirit to Google Books and to Project Gutenberg. Mainly all this projects are using the established Adobe PDF file format to publish the works. Unfortunately the file size for adobe PDF files is quite large since most of them are stored as images. For the average book the file size goes from 10 to 25 megabytes. This will be a problem for those with very slow connections to the internet to be able to access this great works!

On the other hand another technology similar to Adobe PDF have emerge. It’s produced by Lizardtech and name of the file format is DJVU (Pronounced “Deja Vu”!)

Their reader can be downloaded freely from

http://www.lizardtech.com

A big collection of mathematical works using the DJVU technology is freely accessible to those able to read Russian.

One is able to find a wonderful collection of math and physics Russian books at

ИНТЕРНЕТ БИБЛИОТЕКА (Internet Library)

there you can find books like

Р.Курант, Г.Роббинс. Что такое математика?

translation to the Russian of Courant and Robbins What is Mathematics? and many more jewels for free.

The majority of the books in the collection are oriented towards elementary mathematics.

You may encounter wonderful books by Yaglom, Perelman, Kolmogorov, Lovachevsky , Euclid even more modern books by Prasolov on Geometry and Topology and all that is needed is the DJVU reader a good internet connection and a bit of patience.

Below you will find a few samples of some classics available at Google Books.

Galois Completed Works at Google

Euler’s Differential Calculus

Agustin Louis Cauchy Cours D’Analyse

Joseph Louis Lagrange Traité de la résolution des équations numériques de tous les degrés.

Camille Jordan Traité des substitutions et des équations algébriques

and many more classics works by N. H. Abel, Jacobi, S. Lie, just to name a few more.

Another important source of mathematical classics is at

Hope you will enjoy the free availability of very hight quality mathematical material!

This is a blog posting from www.isallaboutmath.com

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## Creating beautiful images

visited the Walter’s Museum in the nearby city of Baltimore Maryland. I was attracted to the museum by the Archimedes palimpsest. Unfortunately for me they did not even had a page on display of the famous Archimedes Palimpsest but taking advantage that I have done my trip there I decided to see the museum’s collection of illuminated Manuscripts.

All I can say is

Those monks really knew how to make something beautiful!

After Johannes Gutenberg invention, that is the movable types press, books became more ordinary items that many more people could afford and a big revolution happened when books became more accessible to people.

The down side to this is that books were no longer hand crafted but produce by machines and therefore not unique and not so beautiful! The quality of the images in books degraded very much.

Naturally there is exceptions to this.

Donald Knuth a mathematician and computer Scientist not happy with the way things had turned out for the second printing of his masterful treatise on Computer Algorithm “The Art of Computer Programming” decided to created two wonderful computer systems that are widely use today to produce once again not just beautiful Books but even better, beautiful Math books. Those programs are $\TeX$ and METAFONT. $\TeX$ is use for formatting pages and METAFONT was use by Knuth to create the Fonts to be use by $\TeX$.

Leslie Lamport another mathematician and computer scientist created a set of $\TeX$ macros and the result was called $\LaTeX$. Similar transformation happened also to METAFONT. The original intention for METAFONT was to be use in the creation of beautiful Fonts that could be use by $\TeX$ but sometime some programs are use in ways their original authors did not have in mind. Such is the case of METAFONT and METAPOST.

John D. Hobby transformed MetaFont into MetaPost.

How does METAFONT or for that matter METAPOST work?

METAPOST allows one to create Postscript files of beautiful figures. Using Postscript is important since is a language that most printers understand and using other programs one is able to easily translate postscript into other graphic formats.

Some times in the internet one is able to find small little treasures hidden in some hole.

While I was pondering on how to write this entry I was feeling tempted to start like

“In a hole in the ground there lived a hobbit. Not a nasty, dirty, wet hole, filled with the ends of worms and an oozy smell, nor yet dry, bare, sandy hole with nothing in it to sit down on or to eat: it was a hobbit-hole, and that means comfort.”

This is the first beautiful paragraph of “The Hobbit” from Tolkien.

I wanted to begin that way because it almost seems like out of the hole in the ground I found John D. Hobby’s program in the web.

There is no need for anybody to own the program anymore!

Let the web do it for you!

Well here is the site to the METAPOST previewer on the web

http://www.tlhiv.org/mppreview/

This site was created by yet another Mathematician! :-)

I have to say thank you Troy for placing this beauty on the net for all to use!

When you create a METAPOST images. You are creating a sort of computer assisted drawing. The same kind of drawing you do when you are using a computer program like the freely available Inkscape or Adobe’s Illustrator with mouse clicking and moving points but using METAPOST you communicate your intentions to the computer with words. Yes, you are writing in English or a subset of it, describing with this meta language what you like for MetaPost to do.

Henderson’s interface to METAPOST looks like this

Now let us do a small METAPOST program using Henderson’s Interface to MetaPost.

Draw an arbitrary triangle and the altitudes for this triangle.

By the end of this exercise we should end up with something that looks like this

The Metapost program for such an image will be like this

pair A,B,C,AA,BB,CC,H;
A=(0,0); B=(3cm,0); C=(1cm,2cm);
AA – A = whatever * (B-C) rotated 90;
AA = whatever [B,C];
BB – B = whatever * (A-C) rotated 90;
BB = whatever [A,C];
CC – C = whatever * (A-B) rotated 90;
CC = whatever [A,B];
H = whatever [A,AA];
H = whatever [B,BB];
draw A–AA withcolor red;
draw B–BB withcolor red;
draw C–CC withcolor red;
draw A–B–C–cycle withpen pencircle scaled 1bp withcolor blue;
draw H withpen pencircle scaled 2bp withcolor red;
label.lft(btex \$A\$ etex, A);
label.rt(btex \$B\$ etex, B);
label.urt(btex \$C\$ etex, C);

A program like this is not difficult to create, once you have master the rudiments of the METAPOST language. Let us explain line by line to see how we created the figure above.

The first line tells METAPOST of our intention to use 7 points A,B,C,H , AA,BB,CC.

A=(0,0); B=(3cm,0); C=(1cm,2cm);

In this line we have specified the Cartesian coordinates for point A,B and C. Now the tricky part is the following two lines

Here we are coercing METAPOST to do some Math for us.

AA – A = whatever * (B-C) rotated 90;
AA = whatever [B,C];

We are giving some mathematical intentions to Metapost. If we call AA the point at the feet of the altitudes then AA-A is the segment between AA and A. Notice the “whatever” is use to denote some factor we will let the computer figure out!

B-C is the segment between B and C and is rotated 90 degrees, now to tell Metapost that AA is somewhere in the segment BC we use the second line that is

AA = whatever [B,C];

and so on. The rest of the program is almost self explanatory but if you have wet your appetite and will like to do some magic with METAPOST I direct you to the Manual for more information about how to use the METAPOST program.

I hope you will enjoy the comfort of this Hobby hole as much as I did!

see John D Hobby A User’s Manual for METAPOST

also

see Troy Henderson A beginner’s guide to METAPOST for creating high-quality graphics .

you may also find very enjoyable this book about Metafont from the horse’s mouth. The Metafont Book.

follow this for Donald Knuth site at Stanford

for a collection of examples of Metapost programs and their output

This is a blog posting from www.isallaboutmath.com

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## A marriage made in heaven. Combining the power of Google Reader and Yahoo Pipes to read math blogs.

A while back while surfing the web I discover Google Reader and it was nice to use Google Reader to place links to web sites and blogs I was interested in reading. Google Reader is what they call and aggregator. Once you setup a folder and add a few items to it then by clicking on that folder you get to see a list containing all the items you have added to the folder.

The analog to this is like having an input box for reading and placing papers to read there. Some advantages of this computer box or folder is that it gets updated with new entries from the sources you have selected.

Here is a screen shoot of what Google Reader looks like

well, all is fine with Google Reader until you decided to filter or order the information or in general display the information in any of many multiple different ways that can be displayed.

Then again while searching on the web I discover yahoo pipes

this is another free internet service but this one is by yahoo and this service allows you to graphically create a mash up feed that could contain all the things I was missing from using Google Reader alone!

Here is screen shoot of what a yahoo pipes program looks like

So as you can see you are actually programming when you use Yahoo pipes but you are doing it with a graphical interface and using pipes to make connections.

In this particular case I wanted to create a Yahoo pipes that will contain a few feeds I was interested in reading and I wanted to sort them following the publication date.

Only takes about a minute to do such programming task with yahoo pipes!

and then I grab my pipe url

something that looks like this

and inserted that into one of my Google Reader folders and Voilà!

I am now able to sort and filter and do much more to RSS feeds!

So amazingly we can now combine the technology from two different companies

to produce with very little effort things that could make the web do the searching and retrieving of information for us!

One more thing

Your Yahoo Pipes can be shared with other people so you can send them the Url of your pipes so others could use it! They could even cloned and change it so modifying and customizing it to their own taste!

Obviously this can be apply to any other topic.

hope you enjoy reading this is a blog posting from www.isallaboutmath.com

for more info on Yahoo pipes check out Tim O’Reilly’s blog on Yahoo pipes.

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## Hello world!

This is the isallaboutmath.com blog!

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