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Posted by admin on July 12th, 2008 in Uncategorized

A very important pattern makes a brief appearance after running the R-pentomino for 774 steps:

It doesn’t last long in these stormy surroundings, but you can enter it into the applet by itself and watch what it does. First it moves to the right, but after a few steps, it produces the still life called a beehive and turns back to the left. Then it produces another beehive to its left and turns right again. Conway called this pattern the queen bee shuttle for this reason. Unfortunately, the next time around, it crashes into the first beehive it created. But we can get rid of unwanted beehives using a block:

If you run this in the applet, you will see that the beehive goes away but the block remains. By putting a block on either side of the queen bee, we can cause it to shuttle back and forth indefinitely. This is a period-30 oscillator, since the shuttle moves for 15 steps in each direction. Later, we will see how this pattern was used to answer an important question about Life.

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Posted by admin on July 12th, 2008 in Uncategorized

Some of the most common objects in Life remain the same from step to step. No live cells die and no new cells are born. Conway, who is fond of making puns, called this kind of object a “still life.” You can observe several of these objects by running the R-pentomino in the applet. For an object to be a still life, every live cell most have 2 or 3 live neighbors, and every dead cell may have any number of neighbors except 3.

The most common still life is called the block. It is simply a 2×2 square of live cells:

You will see it appear many times as you run the R-pentomino. Every live cell has exactly three neighbors, but no dead cell has more than two neighbors.

Some other still lifes you will see are:
beehive
boat
ship
loaf

Look for these in the applet, and try to understand why these objects remain the same. You can design still lifes by hand, and it makes an interesting puzzle. Some people with a lot of experience are good at designing still lifes, but usually computer search is used to find new ones. All still lifes of up to 20 live cells have been enumerated this way

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Posted by admin on July 12th, 2008 in Uncategorized

Life is one of the simplest examples of what is sometimes called “emergent complexity” or “self-organizing systems.” This subject area has captured the attention of scientists and mathematicians in diverse fields. It is the study of how elaborate patterns and behaviors can emerge from very simple rules. It helps us understand, for example, how the petals on a rose or the stripes on a zebra can arise from a tissue of living cells growing together. It can even help us understand the diversity of life that has evolved on earth.

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Posted by admin on July 12th, 2008 in Uncategorized

Life was invented by the mathematician John Conway in 1970. He choose the rules carefully after trying many other possibilities, some of which caused the cells to die too fast and others which caused too many cells to be born. Life balances these tendencies, making it hard to tell whether a pattern will die out completely, form a stable population, or grow forever.

Life is just one example of a cellular automaton, which is any system in which rules are applied to cells and their neighbors in a regular grid.

There has been much recent interest in cellular automata, a field of mathematical research. Life is one of the simplest cellular automata to have been studied, but many others have been invented, often to simulate systems in the real world.

In addition to the original rules, Life can be played on other kinds of grids with more complex patterns. There are rules for playing on hexagons arranged in a honeycomb pattern, and games where cells can have more than two states (imagine live cells with different colors).

Life is probably the most often programmed computer game in existence. There are many different variations and information on the web. (See the Paul Callahan’s home page for more information.)

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Posted by admin on July 12th, 2008 in Uncategorized

Life is played on a grid of square cells–like a chess board but extending infinitely in every direction. A cell can be live or dead. A live cell is shown by putting a marker on its square. A dead cell is shown by leaving the square empty. Each cell in the grid has a neighborhood consisting of the eight cells in every direction including diagonals.

To apply one step of the rules, we count the number of live neighbors for each cell. What happens next depends on this number.

• A dead cell with exactly three live neighbors becomes a live cell (birth).
  
• A live cell with two or three live neighbors stays alive (survival).
  
• In all other cases, a cell dies or remains dead (overcrowding or loneliness).
  

Note: The number of live neighbors is always based on the cells before the rule was applied. In other words, we must first find all of the cells that change before changing any of them. Sounds like a job for a computer!

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Posted by admin on July 12th, 2008 in Uncategorized

The Game of Life (or simply Life) is not a game in the conventional sense. There are no players, and no winning or losing. Once the “pieces” are placed in the starting position, the rules determine everything that happens later. Nevertheless, Life is full of surprises! In most cases, it is impossible to look at a starting position (or pattern) and see what will happen in the future. The only way to find out is to follow the rules of the game.

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Posted by admin on July 12th, 2008 in Uncategorized

Spirograph

You will need a Java capable browser to see these graphics.

What is a Spirograph?
A Spirograph is formed by rolling a circle inside or outside of another circle. The pen is placed at any point on the rolling circle. If the radius of fixed circle is R, the radius of moving circle is r, and the offset of the pen point in the moving circle is O, then the equation of the resulting curve is defined by:

x = (R+r)*cos(t) - (r+O)*cos(((R+r)/r)*t)

y = (R+r)*sin(t) - (r+O)*sin(((R+r)/r)*t)

Here is how you can use the controls in this Spirograph applet:

* The first three scroll bars in the control panel let you change R, r and O respectively. You can change these values between 1 and 100.
* You can use the next three scroll bars to change the color of the drawing. These scroll bars change the red, green and blue values of the color (in the range 0-255) respectively.
* The last scroll bar lets you choose the number of iterations for the Spirograph.
* You can use the Random button to select random values for the radii and color. The number of iterations is not changed by the Random button.

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Posted by admin on July 12th, 2008 in Uncategorized

The word FRACTAL was invented by Benoit Mandelbrot. Fractals are interesting because as you zoom in closer, the pattern is just as beautiful and complex as when you start.

Learn about fractals and create your own beautiful fractal images by following the links below.

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Posted by admin on July 12th, 2008 in Uncategorized

This revised edition of Helping Your Child Learn Math was made possible with the contributions of many people. We would like to acknowledge and pay special thanks to Peggy Quinn Caliguro, Wilma Greene, and Paulette Lee of the Office of Educational Research and Improvement for their guidance in getting this book to print. A special thanks to Ed Esty for his valuable insights, critique, and constant support throughout the development of this publication, Kim Schied Silverman of the U.S. Department of Education for the cover design and layout, and the illustrator, Roberta Toth.

Particular acknowledgments go to Joy Belin, Adriana DeKanter, Cynthia Hearn Dorfman, Lance Ferderer, Ricardo Hernandez, Carole Lacampagne, Robert LeGrand, Diane Magarity, Steve Perkins, Linda Rosen, Patricia Ross, Barbara Vespucci, Linda Roberts, and Judy Wurtzel of the U.S. Department of Education. Our thanks, also, to all of those inside and outside the U.S. Department of Education who contributed their time, effort, and expertise to help produce this book.

We are indebted to the staffs of: the Office of Educational Research and Improvement’s Media and Information Services; Office of the Assistant Secretary, OERI; Planning and Evaluation Service; the National Library of Education; Office of Public Affairs; Office of the General Counsel; Office of Vocational and Adult Education; Office of Special Education and Rehabilitative Services; Office of Reform Assistance and Dissemination; and the Office of Intergovernmental and Interagency Affairs for the important roles they played in helping to bring this book to print.

We are especially grateful to the following organizations for their invaluable review of this publication: Andy Clark, Portland Public Schools; Eileen Erickson, National Council of Teachers of Mathematics; Sue Ferguson, Partnership for Family Involvement in Education; Alice Gill, American Federation of Teachers; Steven Jordan, University of Illinois at Chicago; Kay Luzier, National PTA; Shirley McBay, Quality Education for Minorities Network; Clarence Miller, Johns Hopkins University; Freida Nash, Algebra Project, Howard Elementary School; Cuca Robledo-Montecel, International Development Research Association; Virginia Thompson, Family Math, Lawrence Hall of Science, University of California; Linda Wilson, University of Delaware; Roger Sharp, National Education Association.

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Posted by admin on July 12th, 2008 in Uncategorized

Games Magazine, P.O. Box 10147, Des Moines, Iowa 50347. The adult version of Games Junior (see below). Older children may prefer this to Games Junior.

Dynamath. Scholastic. Available from the school division. Filled with many different activities that involve all strands of math. Children in grade five particularly like this. Nine publications are sent each school year. $5.00 for the subscription.

Games Junior, P.O. Box 10147, Des Moines, Iowa 50347. A challenging and fun magazine filled with all different kinds of games that give children hours of “brain workouts.” Appropriate for ages 7 and up.

Math Power. Scholastic. Available from the school division. Exciting and inviting, this magazine is filled with many activities that involve all types of math. Good for grades 3 and 4. Nine publications are sent each school year for $5.00.

Puzzlemania. Highlights, P.O. Box 18201, Columbus, Ohio 43218-0201. Includes puzzles involving words, logical thinking, hidden pictures, and spatial reasoning. The cost is about $7.50 per month.

Zillions. Consumer Reports, P.O. Box 54861, Boulder, Colorado 80322. Children’s version of Consumer Reports. Shows math in the real world and offers children the opportunity to see how gathering data and information can lead to good decisionmaking. The cost is approximately $2.75 per issue.

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