Problem: The Nautilus Spiral – Sequence of Triangles

We are shown the Nautilus Spiral formed by a sequence of ever increasing isosceles right triangles…

Triangle 1 has each leg = 1, its hypotenuse is Ö2, Triangle 2 has each leg = Ö2 cm and its hypotenuse = . We are asked to make a table of values for the legs, the hypotenuse, and the Area of this sequence of isosceles right triangles and indicate the explicit and recursive functions, diagrams and graphs.

Below is the sketch of the sequence of isosceles right triangles.

The Nautilus Spiral (sequence of isosceles right triangles)

The hypotenuse of the isosceles right Triangle #1 becomes the leg of the isosceles right Triangle #2, then the hypotenuse of Triangle #2 becomes the leg of Triangle #3 and so on.

Process: We will construct a five-column table of values for the Stage Number, the Base, the height, the hypotenuse, and the Areas of these triangles. Then by inspection and some calculation, we’ll generate both the recursive and explicit functions. It’s Process Continued: obvious that as the stage gets greater, the base, the height, the hypotenuse, and the Area calculation will increase.

The 5 column Table starts with Stage 0 (which we’ll fill in after seeing the pattern), by filling in the data for the five fields – Stage Number, base, height, hypotenuse, and the Area of each of the isosceles right triangles. Then with the help of our calculators, we’ll use the explicit equations to help with the graphing of the Hypotenuse and the Area as a function of the Stage Number.

Stage Number	Base	Height	Hypotenuse	Area of the isosceles right triangle
0			= 1
1
2				(·)/2 =
3	2=	2=
4
5
6
….
….. k Explicit			=
Recursive	=	=	Where 1	Where

Solution (continued):

Using the calculator, the graph of Stage Number vs. Hypotenuse has these values.

In column #1, we have Stage Number, column #2 is Hypotenuse, and column #3 is the Area.

Stage Number	Hypotenuse	Area
	=
0	1	1/4
1	==1.4	1/2
2	= 2	1
3	=2.83	2
4	=4	4
5	=5.66	8
6	=8	16
7	=11.31	32
8	= 16	64
9	=22.63	128
10	=32	256
15	=181.02	8192
20	=1024	262,144
25	5792.6	8,390,000
50	=3.36*	2.8*
100	=1.1*	3.2*

Solution (continued):

Conclusion:

The explicit functions for the base, height, hypotenuse, and area are as follows:

, , = , .

Explicit functions are useful for (1.) graphing, and (2.) to determine the values for the base, height, hypotenuse, or the Area for “any” Stage Number. As an example, to see how the “base” changes with the stage number, we would enter the equation into our calculators where y is the “base” and x is the “stage number.”

Our calculator would then display: (0, .70711), (1, 1), (2, 1.4142), (3,2),

(4, 2.824), (5, 4), (6, 5.6569), etc., and we could then graph the base vs. stage if we wished to. Consider Stage 15 using an algebraic approach,

(a.) The base, = ;

(b.) And the height = 128 since the right triangle is isosceles;

(c.) The hypotenuse, = = = ·= ·. In other words, one sees or concludes that the hypotenuse can always be calculated for any Stage triangle by multiplying the base (or the height) by the sqrt (2) = ;

(d.) The area, = = 8192.

In words,

(a.) To calculate the triangle’s “base” or the “height,” you subtract “1” from the stage, divide result by 2, finally raise “two” to that power;

(b.) To compute the hypotenuse for any stage number, you raise “two” to the stage number divided by 2;

(c.) To figure the area for any stage number, you raise “2” to the stage number decreased by 2

Recursive functions are used to determine the k^th term of a sequence

when the k-1^st term is already known. Also, recursive functions are useful in determining algebraically the connection (relationship) between consecutive values of the

function being investigated. The recursive functions for each of the four variables are:

Conclusion (continued):

(a.) For the base, the recursive function is: . In words, you multiply the preceding “base value” by to determine the current base for the stage number under investigation;

(b.) Similarly, the height’s recursive function is: . To find the height for any stage number, you multiply the preceding height’s value by ;

(c.) The recursive function for the hypotenuse is: . The hypotenuse for any stage number can be calculated by multiplying the prior hypotenuse by ;

(d.) Finally, the recursive function for the Area is:

Summarizing, the recursive function: we see that if any of the values (for the base, height, or hypotenuse) are known for the current stage number, we can simply multiply each of them—the base, height, or hypotenuse – by to calculate the base, height, or hypotenuse for the next stage. This can be confirmed by looking at the table on page 2.

For the area, the recursive function tells us that successive areas can by calculated by repetitive multiplication by 2.

This is a very nice “function” problem to introduce into Algebra II classes. It demonstrates the interesting connections between the base, height, and hypotenuse of ever increasing isosceles right triangles.

To use “The Nautilus Spiral” problem in the classroom, I believe that some time would need to be spent with the students in developing expertise with integer and fractional exponents, both negative and positive, an understanding of operations with radical values, all combined with knowledge of the graphing calculator.