Chess and Mathematics

 

 

 Once upon a time many centuries ago in India, according, a sage gifted a game of war, which he had invented, to a king. This game was  Chess . 

 The king was extremely delighted to receive this game. He told the sage to ask for any gift he wished. The sage asked the king to give him as many grains of wheat that would fill the chess board in a manner that the first square contains one grain, the second 2 grains, the third 4 grains, the fourth 8 grains, and doubling the number of grains in each successive square till he reached the last square i.e. the sixty-fourth square.

The king smiled at the sage's request for such a small simple gift, when the sage could have asked for anything expensive or precious. The king soon realised that the entire annual grain production wouldn't satisfy the sage's demand. Actually the  wheat demanded  is more than the entire world's wheat production for more than 2000 years. The king was horrified that he wouldn't be able to honour his word. But the kind sage forgave the king. The king  told that he was even more delighted by the sage's mathematical trick then the game he had gifted him.

How much do you think is the total number of grains demanded by the sage from the king?

Here, the number of grains in each successive squares are doubled. That is, the common ratio amongst the number of grains in successive squares is 2. Such sequences of numbers which bear a common ratio amongst successive terms are called Geometric progression.

2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + … + 2^61 + 2^62 + 2^63 = 2^64 – 1

 It is 18446744073709551615.   

İst.h/Hatice Uysal Çatmakaş/Salihli Şehit Mustafa Serin AİHL

Kirigami: The Ancient Art of Paper Cutting and How Artists Are Keeping It Alive

When you think of Japanese paper crafts, origami is likely the first art form to come to mind. If you're familiar with this age-old practice, you know that a work of origami art is created by manipulating a single sheet of paper with nothing but a series of strategic folds. While this is the most well-known approach to the ancient art form, there are also adaptations. Kirigami, a variation of origami, offers a bit more creative freedom by allowing artists to cut, clip, and snip their paper creations.


Today, kirigami is kept alive by contemporary crafters. Before skimming through some these modern-day models (and even learning how to create your own), it's important to contextualize the craft and learn about its history.

What is Kirigami?

The term kirigami is derived from two Japanese words: kiru (to cut) and kami (paper).

Like traditional origami, kirigami is created from a single sheet of paper and incorporates folds. Sometimes, these folds are incorporated into the finished product, culminating in a three-dimensional piece.

Often, however, these folds are temporary; once the contorted paper has been cut, it is unfolded. Then, new folds are made, and the process is repeated until the work is complete, resulting in a flat work of art.

The History of Kirigami



Like origami, kirigami has roots in China, where paper was invented around 105 CE. In the 6th century, the Chinese began using colored paper to create decorative cut-outs—a practice known as jiǎnzhǐ. Initially, these paper creations were intended to honor gods and ancestors. However, in the 14th century, jiǎnzhǐ evolved into an activity practiced by women and children for leisure.



    





Undoubtedly inspired by jiǎnzhǐ, the Japanese began cutting paper into decorative works of art in the 7th century. Like the Chinese, they used a special kind of paper derived from mulberry plant fibers that had been “soaked in clear river water, thickened, and then filtered through a bamboo screen” by hand.

Origami Applications in the Past and Present

 

Lately, the art of origami has spread widely, and has caught the attention of young people and adults because it is an interesting and useful art that develops the skills. Origami is simply the art of folding paper to make shapes of different sizes and colors. Origami only uses flat white or colored sheets without the need of scissors or glue; it just uses simple tools for folding or flattening paper. The Japanese used origami for the first time in 700 CE following the invention of paper. Some historians believe that the Chinese were the first to use it; however, origami flourished in Japan.

Origami Techniques

The origami uses several folding techniques, which could be basic as the Mountain Fold and the Valley Fold, intermediate as the Rabbit Ear Fold and the Petal Fold, or more complex as the Unsink and the Closed Unsink. Several techniques could be merged for making multiple and unique forms.

The applications of origami evolved over the years for entertainment and educational purposes; yet, in the middle of the last century, there has been a quantum jump in its application that included several scientific fields. Let us get to know some of these applications in the past and the present.

Origami Applications in the Past

In the past, origami was used in decoration. For example, it was used to decorate letters and gifts, and to make animal and bird shapes, and toys using colored sheets and simple folding techniques.

The applications of origami in that period was simple and was only for entertainment. On the other hand, origami in the recent time is applied in several areas that may surprise you!

Origami Applications in the Present Time

1. Architecture and Civil Engineering: Origami is used in the manufacturing of miniature models of bridges and stadiums.
2. Medicine: Paper folding techniques are used to embody the cell membrane and protein, and are used in modeling DNA samples.
3. Mechanical Engineering: Origami techniques were used in the pipes of the Japanese high speed trains. In-pipe supports were used to absorb the excess pressure; thus, reducing the risk of accidents.
4. Mathematics: Some teachers currently use origami as an educational tool that helps in clarifying geometry, explaining fractions, and solving problems.
5. Space technology: There are plans to launch an aircraft made using origami techniques into space. Tests have already been implemented in 2008, but it has not been launched yet.

National Museum of Mathematics

 

The National Museum of Mathematics  is a museum dedicated to mathematics in Manhattan, New York City. It opened on December 15, 2012.  It is the only museum dedicated to mathematics in North America, and features over thirty interactive exhibits. The mission of the museum is to "enhance public understanding and perception of mathematics". The museum is known for a special tricycle with square wheels, which operates smoothly on a catenary surface.

Exhibition programs

1 -  Math Midway is a traveling exhibition of math-based interactive displays. The exhibits include a square-wheeled tricycle that travels smoothly over an undulating cycloidal track; the Ring of Fire, which uses lasers to intersect three-dimensional objects with a two-dimensional plane to uncover interesting shapes; and an "organ function grinder" which allows users to create their own mathematical functions and see the results. After making its debut at the World Science Festival in 2009, Math Midway traveled the country, reaching more than a half million visitors. The Midway's schedule included stops in New York, Pennsylvania, Texas, California, New Jersey, Ohio, Maryland, Florida, Indiana, and Oregon. In 2016, the Math Midway exhibit was sold to the Science Centre Singapore.

2 - Math Midway 2 Go (MM2GO) is a spinoff of Math Midway. MM2GO includes six of the most popular Math Midway Exhibits. MM2GO began traveling to science festivals, schools, community centers, and libraries in the autumn of 2012.

3 - Math Encounters is a monthly speaker series presented by the Museum of Math and the Simons Foundation. The lectures initially took place at Baruch College in Manhattan on the first Wednesday of each month, but moved to MoMath's visitor center at 11 East 26th Street in March, 2013. Every month a different mathematician is invited to deliver a lecture. Lecturers have included Google's Director of Research Peter Norvig, journalist Paul Hoffman, and computer scientist Craig Kaplan. Examples of topics are "The Geometry of Origami", and "Mathematical Morsels from The Simpsons and Futurama". The lectures are meant to be accessible and engaging for high school students and adults. The first lecture occurred on March 3, 2011. Twenty unique lectures had been delivered as of December 2012.

4 - Family Fridays began in April 2014 and occur once a month. MoMath and Time Warner Cable launched the initiative to provide free mathematical opportunities to low-income families in the form of an event series with new activities and presentations each month. In 2017, the sponsorship was taken over by Two Sigma.