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New Online Math Game – Continuum

January 31st, 2011 by Math Tricks | No Comments | Filed in Fractals, Games, Golden Rectangle, Math Games, Math Geometry, Math Software, online math games

Continuum – Action Math Game!

math games - continuum

The new online math game is out, and I hope you will enjoy it!

For this game, I included several math aspects.  The most obvious one is the use of fractals for the backgrounds.  I think they add a level of depth to the play fields – and are really cool to look at!

Next, I made all the bricks in the game golden rectangles.  Well, as close as I could anyway, because I had to use integer units in the design.  The dimensions of the bricks are 41 x 25 pixels, giving a ratio of 1.64 width to height, which is just a tad more than the actual value of approximately 1.61803399.

Lastly, I made as targets in the game polygons, which contain interfering balls after the polygons are destroyed.

This game is a little more mainstream from the others I have made for Math Tricks, but I think it is the most enjoyable.  Here is a game-play screen shot:

continuum - online math game

Anyway, it is online, and you can play it here.  Please feel free to give me any comments and suggestions!

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Snow Days, The Koch Snowflake, and God

January 31st, 2010 by Math Tricks | 5 Comments | Filed in Fractals, Koch Snowflake

As a child, there was nothing like the first snowfall of the year.  It was a sign that the joyous upcoming holiday season was upon us.  Looking around at the growing accumulation, one would begin to anticipate snowball fights and sledding down white, glistening hills.  If you caught a snowflake on your tongue, the cold, crisp beauty of the event would precipitate within your soul.

Now consider the snowflakes themselves.  Their beautiful geometric patterns have fascinated us.  Taken individually, they seem to us lovely enough, but considering the innumerable variety of complex patterns, one is almost overwhelmed with this marvel of nature.

Snowflake patterns do form as a consequence of the physical properties of water, but this does not take away from our appreciation of the astonishing symmetry within them.  Indeed, one can come away with a feeling of the tight bond between mathematics and nature.  To many, a picture of a snowflake represents order within nature.   Simply by looking at a snowflake, you may be convinced that there is an underlying order in our universe.

Natural snowflakes can be imitated mathematically with fractal patterns.  Helge von Koch was a pioneer in fractal mathematics, and in 1904, he came up with a fractal pattern (the Koch snowflake) that resembled a snowflake:

The Koch Snowflake1

Constructing this fractal is an easy process.  You begin with an equilateral triangle.  For each side of the triangle, divide the side into thirds.  For the middle third, draw an equilateral triangle.  When complete, repeat this process for each triangle of your new construct.  This process is illustrated in the figure below:

Construction of the Koch Snowflake2

When properly considered, there is more to the intrinsic beauty within a natural snowflake or one constructed using fractal mathematics.  If you consider the Koch snowflake carefully, you will realize that there is much more to it than what you can see.  Indeed, if you analyze the Koch snowflake mathematically, you will even be able to glimpse of the handiwork of God!

Let me show you two mathematical properties of the Koch snowflake that, taken together, are really very remarkable philosophically.  First, what is the area of the Koch snowflake?  I will not derive the formula for you – I will just give it to you here for iteration k to n:

where s is the length of each side of the original equilateral triangle.

Now, as n approaches infinity, the area of the Koch snowflake becomes a finite value, and is equal to:

So what about the length of the perimeter of the Koch snowflake?  The length of the perimeter at iteration n is given by:

Now, as n approaches infinity, can you see what happens to the length?  Amazingly, the length of the perimeter of the Koch snowflake becomes infinite!  So the Koch snowflake is a mathematical object with finite area bounded by an infinite boundary – how awesome is that?!

There are many things I like to think about when I philosophize about the nature of the universe, and the mathematical properties of the Koch snowflake is one of them.  I like to think of the Koch snowflake as a metaphor for existence outside of our realm of perception.  We are all bounded by the limits of our lives, but is there something beyond our existence that we sometimes fail to see?  Look at the Koch snowflake, and you may see, evidence of intelligence, and what may be.

1Illustration by Wrtlprnft: commons.wikimedia.org/wiki/User:Wrtlprnft

2Illustration by Wxs: commons.wikimedia.org/wiki/User:Wxs

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