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Spirolaterals

February 27th, 2012 by Math Tricks | No Comments | Filed in Math Art, Spirolaterals




Spirolaterals – A Perfect Match for Doodlers!

Have you ever had to wait in an office for a long time for an appointment without a book?  Sure, they may have magazines at the dentist’s office or what have you, but chances are that they are months old.  What do you do to entertain yourself while you wait?  One solution is to draw spirolaterals – a great way to draw interesting math art with a few simple rules – plus a notepad, pen, and patience!

Spirolaterals are spiral structures generated by setting up simple rules for drawing lines, and performing reiterated iterations of these rules.  Using complex algorithms, the number of spirolaterals that can be generated may indeed be infinite.  These structures were first investigated in 1968 by Hal Abelson and his colleagues.  Spirolaterals are open or closed, and can range from simple squares and rectangles to complex curves.

A very simple spiral is a square.  The rules for drawing a square are:

Draw a line of length one, rotate your paper by 90 degrees and repeat.  Repeat procedure until a closed design is generated.

I have outlined these steps using the excellent spirolateral program at http://math.fau.edu/MLogan/Pattern_Exploration/Spirolaterals/SL.html:

Iteration 1

spirolateral square 1

 

 

 

 

 

 

 

Iteration 2

spirolateral square 2

 

 

 

 

 

 

 

Iteration 3

spirolateral square 3

 

 

 

 

 

 

 

Iteration 4

spirolateral square 4

 

 

 

 

 

 

 

So now what do you get if you use the above algorithm, but change the angle to 60 degrees?  You get a geometric figure which has
an angle of 60 degrees between all of its line segments:

hexagon spirolateral

 

 

 

 

 

 

That’s right – a hexagon!

If you vary the line length as you progress in your iterations, much more interesting shapes can be made.  Of course, you can do all of this roughly with pen and ink, but here I will generate the designs with the program mentioned above.

Using variable lengths 1 and 2 (here it is set to letter codes MZ, which codes for 13 and 26, which is the same as 1 and 2, but the program draws the design bigger this way – LOL!), and an angle of 120 degrees, we get this design:

1 2 120 spirolateral

 

 

 

 

 

 

 

This one was drawn using 1 5 9 lengths, and 35 degrees.  It reminds me of the designs i used to make with the Spirograph toy when I was a kid:

1 5 9 35 spirolateral

 

 

 

 

 

 

Length 1 (actually, it is set to Z, or 26) and 30 degrees:

1 30 spirolateral

 

 

 

 

 

 

1 2 3 4 5 6 7 and 35 degrees:

1 2 3 4 5 6 7 35 spirolateral

 

 

 

 

 

 

Give it a try, either with pen and paper or by using the program.  It really fun and interesting, and you get a sense of how patterns should develop before you actually draw them.

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An Introduction to Vedic Mathematics

February 2nd, 2012 by Math Tricks | No Comments | Filed in Math Tricks, Vedic math




Quickly Calculating the Product of Two 2-Digit Numbers

A resource about math tricks would not be complete without mention of the techniques of Vedic mathematics.  Vedic math was introduced by Bharati Krishna Tirthaji Maharaja in the first half of the 1900s, and are a collection of sutras which allow the user to quickly solve mathematical problems very efficiently.  Tirthaji claimed that these sutras were found while studying ancient Hindu writings, but confirmation of his explanation has never been made.

Vedic math can be used such that calculations can be performed mentally or very quickly using single-line notation.  This article will demonstrate the Vedic math technique of quickly calculating the product of two 2-digit numbers.

In this example, 32 will be multiplied by 43:

32
x43

First take the product of the right-most digits and multiply them.  Then, write the result under and to the right of the multiplication set:

32
  |
43
6

Next, take the product of each diagonal digits and add them together.  Write the sum to the left of the first result.  I will show this step in two parts:

32
 X
43
(3×3)+(4×2)   6

Which is the same as:
32
 X
43
17  6

Lastly, take the product of the left-most digits, and write the result to the left of the first two results:

32
 |
43
12  17  6

Almost done – now if you have any remainder in the tens columns, be sure to add it to the ones columns to the result to the left.  In our example, the “17” has a remainder of “1” in the tens columns, so add 1 to 12:

32
43
13  7  6

Now just “squeeze” the results together, and you will have your answer:

32
43
1376

And, indeed, if you use a calculator, you will find that 32 x 43 = 1376.

 

And now for a few more examples:

12×16:

12
16
1   2+6   12

12
16
1  8  12

192 answer

 

16×36:

16
36
3   24   36

16
36
3   24   36

16
36
3  27  6

16
36
5  7  6

576 answer

 

25×97:

25
97
18  45+14  35

25
97
18  59  35

25
97
18  62  5

25
97
24  2  5

2425 answer
Practice some more of your own on paper to get the technique down, and then try to do some in your head.  With a little practice, you will be able to do two-digit calculations quickly in your head!

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