My Mathematical Art

This is one of my first pieces of art given to my wife a few years ago to mark our 34 years together. It contains rectangles whose sides are all numbers from Fibonacci sequence -- the largest being 13 x 13 and the smallest ones are 1 x 1. One is gray, one is green, two are beige, three are yellow, and five are garnet. This artwork captured so many aspects of our lives, including the 13 x 13 box that represented the passage from a reading at our wedding. The art represents many elements of our family -- some include the two of us, the three children, and the family of five.

Mathematics, like art, is representational, symbolic, moving, and it stimulates the sense of wonder. It is why I so strongly believe that when we see mathematics, we see its beauty -- for it’s art and when we see art, we see its beauty -- for  it is mathematics.

And in the end, three things endure: faith, hope, and love, but the greatest of these is love.

--1 Corinthians  13:13

A Timeless Connection

This is during a visit to the SF Museum of Art in summer of 2009. The Mondrian piece was one of his composition works of the 1930s.

The connection of mathematics and art is timeless. Line, shape, proportion (including phi!), design and symmetry, shifts, tessellations, transformations, fractals, and patterns, patterns, and even more patterns.

 Mathematics is the language in which God has written the universe.
 -- Galileo Galilei
      1564-1642


 

1.61803444782168...

Phi (~1.61803...) is an extremely fascinating number. What other number is one less than its square and one more than its reciprocal?

The golden spiral can be created by making adjacent squares of Fibonacci dimensions. For instance, the square to the left has each side measuring 21 units, the square in the bottom right has a side of 13 units, the top right is an 8 unit by 8 unit square. The top middle square;s sides are 5 units each, and so forth. The entire rectangle below measures 21 by 34 units, representing the golden rectangle.

Piet Mondrian

Another artist who saw the beauty of mathematics and art's close connection was Piet Mondrian (1872 – 1944). The Dutch artist made use of simple geometrical shapes and primary colors. His believed that all shapes can be created with the simple geometric shapes, such as the rectangle, while all colors are the result of combinations of red, blue, and yellow. 

While his early works were influenced by his spiritual and philosophical beliefs, his work shifted around 1919 upon his return to Paris when he began his popular grid-like artwork. The golden rectangle is one of the basic shapes that appear in Mondrian's art, such as the Composition with Gray and Light Brown (1918) and Composition with Red, Yellow, and Blue (1926).

Every true artist has been inspired more by the beauty of lines and color and the relationships between them than by the concrete subject of the picture. 

– Piet Mondrian

Lewis Carroll

Charles Lutwidge Dodgson was born on this day in 1832. He was a mathematics professor at Oxford University in London who wrote books about the subject, especially symbolic logic and puzzles. But more than anything, he exemplified "connections" and was better known as Lewis Carroll, the author of the classic Alice's Adventures in Wonderland.

Not only was he a brilliant mathematician and noted author,  he was a man of religion who would include his humor, puns, and logic in his sermons. His was also an artist, becoming a well-known photography in the field's infancy, as well as an inventor.

He died thirteen days before his sixty-sixth birthday in 1898.

"Can you do addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?" "I don't know," said Alice. "I lost count." 

 from Lewis Carroll's Through the Looking Glass

Touch

The post placed yesterday morning regarding the premiere of the television program Touch is gone. I went in to move a number and, the next thing I know, the text and chart are no longer online.

Go figure. I will update this "math minute" soon.

A Thought on Teaching

Quote: Einstein

Architecture and Phi

The connection that exists between mathematics and art is evident in architecture and design as demonstrated in city landscapes.

Over  600 years ago, Luca Pacioli's De Divina Proportione (The Divine Proportion) included the work of drawings made by Leonardo Da Vinci who called the ratio the sectio aurea (Latin for the golden section).

In 1950, the architect Le Corbussier popularized the modulator system  that was based on the golden ratio. He felt that there was a sense of balance, harmony, and order to phi.

Design is where science and art  break even.
--Robin Mathews

The Face & Phi

The golden ratio is also evident in the human face. Just as you would not go up to someone on the street to measure the distance from their belly button to the ground, it is highly recommended that you don't pull out a tape measure to check these ratios out. Besides, many people may not like being told that they may not have the "perfect" face as assessed by artists, scientists, mathematicians, and others.

I was not surprised when I learned that the distance of my two upper front teeth divided by the height of these teeth yields the golden ratio. The golden ratio is also found in the length of face (D) / width of face (d), the distance between the middle of the lips and where the eyebrows meet (D)/ length of nose (d),  the length of mouth (D) / width of nose (d), and the length of face (D) / distance between tip of jaw and where the eyebrows meet (d). 


The beauty doesn't stop there. The golden ratio is one more than its reciprocal, meaning that two of the numbers in series, say 610 and 987 when divided (987/610) gives us 1.61803... and its reciprocal (610/987) is 0.61803... 


If you have an hour and a half or so, please visit the YouTube link to experience Keith Devlin's presentation, Fibonacci & the Golden Ratio Exposed, that was a part of the Math Encounter series hosted by the Museum of Mathematics.

Search: Devlin Math Encounters

 

Vitruvian Man

 Da Vinci's Vitruvian Man demonstrates his passion for the  "ideal" proportional relationship that exists throughout the average human body and that come very close to the golden ratio. For instance, take your height (D) and divide it by the distance from the floor to your naval (d) and the quotient will be equal to or almost 1.6180 -- the golden ratio. Try it and see, but don't stop there.

Measure the distance between your finger tip and your elbow (D) and divide it by the distance between the wrist and the elbow (d). Viola, the golden ratio,  Check out the distance between your navel and knee (D) compared to the distance between the knee and the end of the foot (d). Again, the golden ratio.
 

Don't stop yet. Check out the hand that has been helping you cruise through the internet today. As you'd expect, you'll see another of the many examples of the golden ratio, Notice that your fingers have three sections. The proportion of the first two to the full length of the finger gives the golden ratio 1.6180. Compare your middle finger to your little finger and, once again, you'll find the golden ratio. 

Google the number 1.6180 and see what's out there. You will be amazed and I hope that when you see mathematics, see its beauty -- for it is art and when you see art, see its beauty -- for it is mathematics. It's all about the connections; for though there are many, they're one. 

on 01-21-2012 

visit 1C 12:20

1 Corinthians 12:20

The Golden Ratio

Leonardo DaVinci extensively used mathematics in his artwork. The dimensions of the room and the table of his "The Last Supper" and his portrait of the Mona Lisa were based on the Golden Section, which was known in the Renaissance period as The Divine Proportion. The golden ratio is referred to as "phi" and is approximately equal to 1.6180. This ratio can be found in architecture, including the pyramids, and throughout nature: the spiral in the nautilus, snails, pine cones, pineapples, and sunflowers. Even your fingers have elements of the golden ratio.  

Another Italian mathematician Leonardo Pisano Bigollo, best known as Fibonacci, demonstrated that the golden ratio consists of the sum of the two numbers before it as shown below. 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...

Fibonacci found that by dividing  any number in his sequence by the one before it, for example 21/13 or 987/610, the quotient is always close to 1.6180, the golden ratio. 

Rhombicuboctahedron

Vi Hart's balloon polyhedra reminded me of Leonardo da Vinci's rhombicuboctahedron (right), first  printed in the Divina Proportione.  This Archimedean solid has eight triangular and eighteen square faces, along with 48 edges. There are 24 identical vertices, with one triangle and three squares meeting at each.  

Described as a Italian polymath, da Vinci was a painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist, and writer. He was a genius who, perhaps more than that of any other person, epitomized the Renaissance humanist ideal -- a man of insatiable curiosity with an endless and inventive imagination.

"Whoever despises the high wisdom of mathematics nourishes himself on delusion.” 
            – Leonardo DaVinci

Vi Hart

Vi Hart is a talented musician and recreational mathemusician whose hobby is mathematics, with special interests in symmetry, polyhedra, and surreal complexity. Her work (and fun) usually results in collaborative research examining more closely subjects such as computational geometry, theoretical computer science, or as mathematical art.

Please visit www.vihart.com and check out the wealth of mathematical wonders and muses. Especially noteworthy is her binary dance (found on YouTube), polyhedra balloons, Harry Potter septet, and geometric food. The link will definitely convince visitors of her passion for mathematics and its many connections, as well as leave them in awe with her many talents.

Math Encounters

One of the artists whose work is found on the Bridges site (www.bridgesmathart.org/) is George Hart. Dr. Hart is Chief of Content at the Museum of Mathematics in Manhattan. He has had an extensive multi-disciplinary career that has made use of his many talents including his awesome skills as a sculptor, mathematician, computer scientist, engineer, writer, and educator.

While the Museum of Mathematics is expected to open in late-2012, it offers Math Encounters,  a public presentation series celebrating the spectacular world of mathematics. The next session, The Shape of Space: An Exploration of Multiconnected Universes, is scheduled for Wednesday, February 1, The presentations are designed to help increase the public's understanding mathematics by sharing the many connections the subject has in our lives. For more information, visit their site.

Bridges

If you are interested in the connection of mathematics and art, one site that you will want to regularly visit is http://bridgesmathart.org/. There you will find information about the organization, their conferences, archives of information from previous years (especially the art work!), and many other resources of interest.

Introduction

This blog was created as an outlet for a retired educator who has a little time on his hands and enjoys sharing what's on his mind. While I often have a chance to present to fellow teachers, the blog will offer me an opportunity to do more. During my most recent workshop, the emphasis was on the mathematics and art connection in which the parting message was "when you see mathematics, see its beauty -- for it is art and when you see art, see its beauty -- for it is mathematics."

Come back for some of the links and connections, as well as the chance to share. I hope this blog serves as a means to inspire, to connect, and to nurture the sense of wonder, the spirit of curiosity, and the gift of creativity.

Announcement

Mr. Kotter's blog will begin soon.