Wednesday, October 24, 2012

Math Adventures with Penrose, the Mathematical Cat

As I have said earlier, math can be fun. Puzzles, games and stories are some ways in which we can indulge in mathematical adventures. Stories, especially work very well with children. And more so if they are stories of a mathematical cat!

A mathematical cat? What's that? Penrose is a mathematically gifted cat, who belongs to Theoni Pappas, the author of the stories.
pic courtesy flipkart

The Adventures of Penrose
the mathematical cat        

The Further Adventures of Penrose
the mathematical cat   

Fractals, Googols and other
mathematical tales
Written by Theoni Pappas
Published by World Wide Publishing/Tetra
Ages: 8+

At the beginning of 'The Adventures...', she tells us the story of how she realised that the cat was interested in math. She then wrote down all of Penrose's stories for children- these stories tackle one concept at a time, and can be read as stand-alone stories, too.

The stories point out the association of math with the things we encounter in the real world: in nature and in man-made things. Pappas is that rare mathematician, who breaks down the overwhelmingly complex into easily digestible bites, makes them palatable to all- children and adults alike. The books take into consideration the youngest possible reader, and the harried parent, who would much appreciate these handy stories to delve into the breathtakingly beautiful, yet sometimes alarming world of math.

pic courtesy

One day, Penrose finds his mistress playing with soap bubbles. He runs to play with them, and is amazed with the perfect sphere shapes, that magically change shapes as they coalesce. He discovers, (as did my daughter A, while playing with soap one day instead of washing her hands quickly and coming for dinner) that soap bubbles are always perfect spheres when discrete, as the sphere is the most stable form occurring in nature with the maximum volume for the smallest surface area - no matter what the shape of the bubble blower. Or that when they coalesce to form a cluster, they tessellate to form regular shapes once they stabilise. That if you draw a plane through this cluster, they will look like a cut-section of a honeycomb.

What is tessellation  Figures that fit together perfectly without any space in between. They could be the same figure or different figures. Nature creates many beautiful designs using this concept. Designs that are often copied by man in the arts.Quilts, tangrams, optical illusions like stereograms, tile designs, jigsaw puzzles, rangoli designs, kaleidoscope patterns, knitting patterns, even the hexagons on a football are examples of tessellation  And in these days of computer generated images, the possibilities are myriad.

Another day, Penrose comes across a snail in its shell. He is mesmerised by the shell- it is such a beautiful pattern. In conversation with the snail, Penrose discovers the secret of the golden rectangle- a very common concept that occurs in nature, creating breathtakingly beautiful designs. The ratio of length to breadth always tends to 1.618.

pic courtesy
Start with a 1 by 1 square. Then, add another square of the same size. Subsequently, add squares whose sides are equal to the longest side of the existing rectangle. This can go on infinitely in both directions- increasing as well as decreasing. The rectangle that is formed every time is the golden rectangle. The numbers also form an interesting pattern-

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

These can be written as each consecutive number being the sum of the two numbers preceding it.

1, 0+1, 1+1, 1+2, 2+3, 3+5, 5+8, 8+13, 13+21, 21+34,...

This sequence is also called the Fibonacci sequence. How does this tie up with a snail's shell? If you draw a curve inside each square in the figure, and join all the curves, they form what is called the golden spiral, that can extend infinitely inwards as well as outwards.

pic courtesy

pic courtesy google images
More examples occurring in nature? Whorls of a flower, arrangement of seeds in spirals in flowers like the sunflower, branching of a tree, may check this out at howstuffworks.

Art copying nature? Check this out. A very modern use of the golden rectangle- the credit card..

More of what occurs in nature? Fractals. What are those, and where do they come in daily life? Penrose was amazed when he found out how ubiquitous they were, as we are, too when we dig a little deeper. The definition says- duplication of a form inwards or outwards. So this, by definition, will incorporate and go beyond the golden spiral or Fibonacci numbers.

fractals from basic shapes
pic courtesy
Ever seen a snowflake grow? The original nidus decides what design it will form- so that no two snowflakes are the same. Or just any crystals grow? Think clouds, galaxies, fern fronds, diffusion and movements of fluids, music, population explosion, measuring length of coastlines...the list could go on.

pic courtesy google images
Does Penrose encounter only these slightly unusual math concepts? Not at all. He comes across the very basics that our children might encounter on a daily basis too- numbers, the magic of prime numbers, number line, conventions in math, number processes, pi, irrational and rational numbers, graphs, estimates, probability, two-and-three-dimensional figures, x - the unknown in algebra, the concept of zero...and much more.

Each short story, illustrated with pictures of Penrose and simple figures for the math concept, ends with a puzzle which we are left to solve along with the cat. Pappas, of course, gives a very helpful solutions and explanations section at the end of each book that completely de-mystifies the concept at hand. There is also a list of internet links for those interested in further exploration of each topic.


Should we end this post with a fun activity for kids?

Check out the stereograms at this site. For those new to the concept, you need to stare at the image, and slowly relax, letting your eyes focus on a point behind the picture. It's easy once you master the technique, and can be loads of fun.


Choxbox said...

Trust you to dig out gems like these!

Arundhati said...

WOW! Have to get hold of these books!

sathish said...

I wonder about the name Penrose. I guess they have named based on the famous mathematician Roger Penrose.

Choxbox said...

@Sathish: And totally coincidentally I have The Emperor's New Mind in my hand as I am reading your comment and replying!

Sheela said...

Thank you, Sandhya, for bringing such treasures to us! I am much-intrigued and can't wait to get my hands on these books.

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