zentangle challenge # 118

Here is my response to this week's challenge. It was done in a bit of haste so I am not very happy with .
the result. Hope you like it

Zentangle Challenge # 117

This week's challenge is to use the tangle created byRick......which  reminded me of the Sierpinski's triangle ( a fractal  named after the Polish Mathematician Waclaw Sierpinski  who described it in 1915. ) .In this pattern triangles (arrows with the tail cut off ? arrowheads ?) of varying sizes are placed in opposite directions giving a pretty pattern.( See Fig below for a series of steps for drawing it.)

So here is my response for this week's......It's Shway , arrowheads and Sierpinski's triangles  all the way ..

Zentangle Challenge #114

I am seeing stars again!  The eight pointed star is commonly used in art and architecture and I have used 2nd stellation of the octagon. for creating my response to this week's challenge.

So here is my starry tile

 I love both the mathematical and aesthetic perspective of stars. I had written and posted on my blog about "Stellations in 2D & 3D" some months back  in my column here

The octagon has two stellation , one being the true star polygon{8/3)and the other being a compound of two squares ) also called the Star of Lakshmi . {8/2) a symbol  which figures in Hinduism  where it represents  Ashtalakshmi  the eight forms, or "kinds of wealth", of the goddess Lakshmi. A similar symbol Rub el Hizb  is a Muslim symbol found on a number of emblems and flags. You can see that the two overlapping squares is embedded in the 2nd stellation

Here is a collage of some  octagon stars embellishing Islamic Architectures in India

Don't forget to visit http://iamthedivaczt.blogspot.in/ to see the responses of other's to this week's challenge

Note: Some of the  picture above are  taken from the internet and used  only for education and inspiration . If I have violated  any copyright issues , please send me comments and I will remove the associated picture.

Fun with Mathematics - Euler - Platonic Solids -Graph theory and the "God's equation"

Today is Leonarda Euler's,  306^th  birthday and Google celebrates it with the wonderful doodle shown above... depicting some of the important mathematical discoveries he made. Euler ( pronounced"oiler") is considered as one of the most prolific mathematician of our times .  

He was born on 15th April 1707 and contributed immensively to the fields of geometry , graph theory , trignometry , number theory and other areas of physics . At the age of 28 ( in 1735) he  lost sight in his right eye ,  and in  1766  he lost sight in his left eye. Despite being blind he continued to publish his results aided by his phenomenal memory and by dictating his discoveries to his assistants.

It is said that Euler could create mathematics faster than most people can write it, which put immese pressure on his scribes He published over 800 papers in his lifetime. Even though he lost vision in both his eyes, his productivity continued to increase.

Several countries have paid homage to this great mathematician and physicist by releasing  postage stamps and currencies on his birth anniversaries. Here are a few which I have taken from the internet.
 
 
 









I came across Euler's work  several times when I was investigating links between art and mathematics, and some puzzles and games (we played as children ) 

When I was studying and creating beautiful paper  models ( particularly the Platonic Solids) , I came across one of Euler's discoveries .... V - E + F =2 ...called Euler's formula which expresses the the relationship between the vertices , edges and faces of the polyhedron. I have written about this in my earlier post Fun with Mathematics: The art and science of Platonic solids which shows that this formula applies to Platonic solids ( discovered by the Greek mathematician , Plato in 5th century BC) . However, it was only in 18th century AD  that the relationship  between the edges , faces and the vertices was discovered by Euler.  ( The Google-Doodle shows this formula and 2 of the 5 Platonic solids ). If you wish to make yourself a set of Platonics solids from paper you will find templates at my post on Fun with Mathematics -Platonic solid

The other place where I came across Euler's work  was in graph theory when I was investigating the paper and pencil game of drawing a specific line pattern ( graph) consisting of several dots ( vertices)  joined by lines( edges) with the following rule ...... " Starting from any  dot ( vertex)  , can one draw the complete line pattern by  traversing  each line( edge) only once ?  ".  I remember trying to solve several such line diagrams as a child.

These patterns,( which we played with as children )  are mathematically called Euler's path or Euler's circuits and belongs to a branch of mathematics called graph theory. 


Here are a few such line patterns ( graphs), which I have collected from the internet  for you to try and see which of these line diagram can be traversed without lifting the pencil from the paper. These will surely bring back your memories of your childhood !

Which of the patterns above  can you draw without lifting the pen from the paper and without traversing any line twice ?

Using graph theory you can easily analyse which of these graphs can be drawn without violating the above rule. This has connections to the famous  problem known as the Seven Bridges of Königsberg. which Euler solved and is considered to be the first theorem of graph theory  ( you can see the same depicted in the Google Doodle (...at  bottom left  of the doodle. ).
I will write more about Euler's path / circuit  in my next post on graph theory  including the famous Traveling salesman problem , the puzzle called Instant Insanity ... and the relevance of graph theory in many field.  

The most enigmatic equation which Euler discovered is  called the "Euler's identity"  and is considered as   "most beautiful theorem in mathematics" ... and at time called ..."The God's equation" 

 The equation contains nine basic concepts of mathematics — once and only once — in a single expression. These are: e (the base of natural logarithms); the exponent operation; π; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero.

Benjamin Peirce, Professor at Harvard described  it as " .... it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."    

Someone has remarked : "What could be more mystical than an imaginary number interacting with real numbers to produce nothing?"

Here is a short video on the famous equation

Note: Some of the  picture above are  taken from the internet and used  only for education and inspiration . If I have violated  any copyright issues , please send me comments and I will remove the associated picture.

Zentangle Challenge # 113

It's a " Square within a square, within a square ,within a square" challenge ! This was an easy one. A geometrical figure filled with tangles does not require much pondering on composition and can be done mindlessly! ( Isn't that what meditation is all about ?....no mind ( mindless?) , relaxed but alert !)

So here is my "mindless"  response to this week's challenge

Don't forget to visit http://iamthedivaczt.blogspot.in/ to see the responses of other's to this week's challenge

Zentangle Challenge # 112

The first Monday of every month is the "Use my Tangle ( UMT) challenge ...and this week's challenge is use the  tangle "Tuxedo" created by Ledenzer, ( the creator of the most gorgeous zentangles on the net )  You can find the instructions for this tangle here

This was a little tough for me as I couldn't find the right spacing of the dots / shading / colouring combination . So I tried a few variation , but couldn't get the beautiful effect Ledenzer has achieved in her tile using her own tangle .

Here is my response to this week's challenge

Don't forget to visit http://iamthedivaczt.blogspot.in/ to see the responses of other's to this week's challenge

Zentangle Challenge #111

Mooka is a lovely tangle and I love playing around with it....flows beautifully  and is very soothing!

Here is my response to this week's challenge

Don't forget to visit http://iamthedivaczt.blogspot.in/ to see the responses of other's to this week's challenge