Square Root Spiral Art

[Frida Kosofsky]

Althoughall of Theodorus' work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus.[2]

Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.[3]

In 1958, Kaleb Williams proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into a line, they will never pass through any of the other vertices of the total figure.[4][5]

Theodorus stopped his spiral at the triangle with a hypotenuse of 17 \displaystyle \sqrt 17 . If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.

It is convenient to place the spiral in the complex plane. The -th triagle has sides , and . We denote the vertices as and and the angle at the origin by . A typical component triangle is shown in the figure below. From the figure it is clear that

The spiral of Theodorus is such that each loop is approximately the same distance from the preceding one. We recall that for the Archimedean spiral , the distance between consecutive windings is always . We can approximate the square-root spiral by . In the figure below, the left panel shows the first 530 vertices . In the right panel, a spiral of Archimedes is superimposed on these. We see that as increases, there is ever-closer agreement between the two spirals.

Classroom Activity (Constructing the 'square root spiral'): Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2 perpendicular to OP1 of unit length. Now draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in this manner, you can get the line segment Pn-1Pn by square root spiral drawing a line segment of unit length perpendicular to OPn-1. In this manner, you will have created the points P2,P3,.....,Pn,....., and joined them to create a beautiful spiral depicting 2,3,4,.....

a square root spiral is a spiral formed by many right angledtriangles, where, the hypotenuse of each triangle is the squareroot of a number.the first triangle is always an isoseles triangle,with its hypotenuse equal to square root of 2.

Last week, Ms. Nihan, the 8th math teacher posted drawings created by students who were studying a geometrical pattern based on the work of a 5th Century Greek mathematician, Theodorous of Cyrene. He developed a pattern called the Spiral of Theodorous, a square root spiral composed of contiguous right triangles.

As I continue to think about my new job in September and plan for what I want to hang in my classroom, I am exploring the Pythagorean Theorem. The Pythagorean Theorem is probably one of the most well-known or well-remembered theorem in math. It is often taught in both algebra and geometry. In algebra it lends to working with exponents and roots and in geometry with triangles. I have seen memes saying how people did not use the Pythagorean theorem today, but I have also been told by many people that they have used it in their lives from building a new deck and woodwork to programming and more. Although math has real life applications and was mostly discovered to explain the world, much of math is taught to help develop the brain of our children. In high school the brain is just beginning to truly develop its logic skills and math is huge in helping with this. The Pythagorean theorem also is mentioned (incorrectly) in the Wizard of Oz. Yes, it is this famous!

Although it is named for the Greek mathematician, Pythagoras, it was known throughout the world before his time. It is referenced in Ancient Egypt and Babylon (around 1900 BC). Apparently, it did not become as well known until Pythagoras stated it. There are many proofs of this theorem and some of them like the one below is a visual proof.

Now Pythagoras was a teacher and philosopher in the 6th century BC. He had a group of followers known as the Pythagoreans. The Pythagorean were a secretive group and many of their discoveries were stated under the name of Pythagoras and not an individual. In Mathematical Scandals, Theoni Pappas shares the story of the death or at least expulsion from the Pythagoreans of Hippasus of Metapontum. It is believed by some that he was pushed overboard or put to death at sea for his discovery (and proof) of the square root of 2 as the length of the diagonal of a square with side length of 1 and that the square root of 2 is not a rational number. The Pythagoreans believed all numbers were whole or could be written as a ratio of whole numbers. It was scandalous that there was an irrational number. For more about the Pythagoreans check out here.

Mathematical Scandals is a fun book to add some scandalous history to your math classes! There are many fun stories that relate to different areas of math. Now I have been focusing on mathematical art and things I can make for my classroom as well as projects I can have my students make. In my search for mathematical art, I discovered the Spiral (or Wheel) of Theodorus. It is also called the Square Root Spiral and Pythagorean Spiral. It gives a visual of the square roots in numerical order.

It is actually pretty easy to make on your own and involves some great Pythagorean theorem use!! I made this project sheet for kids to calculate the square roots using the Pythagorean theorem and then to create their own spiral by starting with a different sized triangle rather than the legs being 1 unit.

I however decided to make mine as true spirals of Theodorus. To create it you start with an isosceles triangle with legs of one unit. The hypotenuse of the triangle will be the square root of 2. Using the hypotenuse as one leg of the next triangle you draw a second leg perpendicular to the first that is one unit long. Then continue this as you add triangles to your spiral.

In The Right Triangle Quilts chapter, there are different activities that include working with a proof of the Pythagorean theorem, Pythagorean triples, relationship with the Fibonacci Sequence and more. Then there are three right triangle "quilt" projects. I decided to try the Pythagorean Triples Quilt. Pythagorean triples are three whole numbers that create the sides of a right triangle. The first being 3, 4, 5. This is probably the most well-known one.

I found this project on-line that helps ask more questions for the kids and combines both "quilts". I found the table for the Spiraling Pythagorean Triples very useful! I love that it has kids look for patterns in the Pythagorean triples used in the quilt. Now my Pythagorean Triple "Quilt" I used 3/4", 1", 1 1/4" for my first triangle or my scale is 1/4" : 1". I literally divided all the lengths of the triples by 4 inches. With my scale, my largest triangle has a leg that is 28 inches! I used poster board for the black, blue and purple since the triangles were too long for the paper I had. (The Dollar Tree sells colored poster board for 89 cents!)

1) Let point A be the center of the circle of radius which we seek to dissect into rings.

2) Construct a unit of length of .

3) Draw . Then draw a segment of the same unit length perpendicular to at point B. Join A to the free end B1 of this new segment.

4) Draw a further segment of length perpendicular to and join its free end to A.

5) Repeat this procedure times; in figure 1. Draw circles with A as center with each of the segments beginning with A as their respective radii. This completes the required dissection.

One can see from the figure that the construction results in a sequence of side-sharing right triangles whose hypotenuses increase successively as the square roots of the natural numbers. Their legs of unit length trace out a spiral path. Hence, this figure might be termed the square root spiral (SRS). One may also see that as the number of dissections we are left with thinner and thinner rings that are essentially the circumference of the circle: .

Before we get to that we shall first create a further generalization of the discrete SRS: After having constructed the initial discrete SRS as described above, reflect (invert) the point B which initiates the spiral on the hypotenuse to get a new point. Then reflect the outer end of the hypotenuse on the hypotenuse; then reflect the outer end of the hypotenuse on the 2 hypotenuse, so on. This yields the second branch of the discrete SRS (in red in figure 2). So the Davis problem in its more general form requires one to interpolate a smooth curve through both the branches of of the discrete SRS.

The approach taken by Davis to solve it along with the solution has some striking parallels the story one of the great problems in the history of modern mathematics (described by Davis himself). Hence, we shall take detour to look a bit at that famous problem. The factorial function was originally discovered by Hindu mathematicians. For instance, it is clearly provided by the Kashmirian polymath finance minister of the Seuna Yādava rulers Śārṅgadeva in his work on the theory of Hindu music the Saṃgīta-Ratnākara in 1225 CE. This original form of the factorial function is organically described as the serial product of natural numbers: . In the first half of the 1700s it was noticed that these discrete points of the factorial function seemed to define a curve. But the question was how does one find the intermediate points of the curve like say 2.5!. The interest in this type of interpolation problem was likely initiated by Newton in England and passed on to his junior associate, the Frenchman de Moivre. In course of his study of probability de Moivre discovered the first continuous function that was an approximate fit to the discrete factorials:

; where is a constant.

His junior associate Stirling after some experimentation refined the value of the constant as leading to many people wrongly attributing the formula to him instead of de Moivre.

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