Spiral (Grades 5-8)


click on meIntroduction

This spiral was studied by Archimedes in about 225 BC in a work On Spirals. Although it had already been considered by his friend Conon, it is often called the Spiral of Archimedes. Archimedes was able to work out the lengths of various tangents to the spiral.


Activity
Supplies: index cards, rulers, pencils, scissors, tool(see instructions below)
Calculations: simple scientific calculator (for example TI-34)

Making the Tool
The tool used in this exercise is made from a 4 by 6 sq. in. index card with one inch marked on its short side. This tool is used to draw right triangles that have one leg which is one inch long.

Task 1.
Draw a spiral built from right triangles that have one leg 1 inch long. (See picture.) Start with a right triangle that has both legs 1 inch long. Using the tool, draw the next triangle, starting with the hypotenuse of the previous one. Stop before the triangles overlap.

Remark:
Show students how to use the tool efficiently. Precision is important; students must use sharp pencils and draw lightly. At the very end they may color and decorate their drawings.

Task 2.
Measure the rays radiating from the center of the spiral. Form the table as follows. (The numbers below were taken from a specific drawing.)

number of radius, n:

length in inches:

decimal equivalent:

square root of number n:

1

1

1

1

2

1 7/16

1.4375

1.4142

3

1 13/16

1.813

1.732

4

2

2

2

5

2 5/16

2.313

2.236

6

2 9/16

2.563

2.449

7

2 3/4

2.750

2.645

8

2 15/16

2.938

2.828

9

3 1/16

3.063

3.000

...

...

...

...

16

4

4

4



Compute the decimal equivalents in the table as follows:
1 13/16 is computed by

[13] [/] [16] [+] [1] [=].

 


Explanation
The n-th triangle has legs with length √n and 1 inches, and hypotenuse √(n+1) inches, by the Pythagorean Theorem.




Remark:
If students do not know the Pythagorean Theorem, this may be its first introduction.

Question:

How fast does the spiral curl around?

Solution:
Adding the n-th triangle increases the angle by arctan (1/√n).  Thus, for each n up to some chosen number n’, compute arctan(1) + arctan(1/√2) + ... + arctan(1/√n’).  Make a table of values, and compare them with the angles measured on your drawing.

 

The program (using a TI-34II calculator)

 

[2nd][FIX][0]

OP1=A+1A

OP2=B+tan-1(√(A)-1B

 

Now initialize variables A and B:

0A

0B

 

Now run the program:

Repeat

OP1

OP2 

Each OP2 on the display gives the new angle.  Record the values in a table.

 

Example Table:

n:

angle in degrees:

1

45

2

80

3

110

4

137

5

161

6

183

7

204

8

223

...

...

 


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Webpage Modified by Aous Manshad
Last Modified: May 31, 2006