Right Triangle Spiral

There are a number of right triangle spirals; this one is based on the sequential series of square roots. The construction itself is an interesting and beautiful graphic design.

The Pythagorean theorem, on which this construction is based, is a very interesting subject. Many books have been written on this subject alone, and an internet search for Pythagorus or Pythagorean theorem will produce thousands of interesting websites. The construction provides many opportunities for creative use, as "study sheets" like other projects in these pages, and as topics for student reading and research.

The spiral is a nice graphic just as it is, but is more visually interesting if folded to produce a 3-dimensional "geometric plaything"! Students always enjoy projects that allow them to be creative, to draw, cut, fold and create.

But this project is far more than a plaything; research has shown that the very act of writing the theorems helps students to remember them.

How to create a spiral of sequential square roots:

1) Construct an isosceles right triangle ABC with side 1 unit. The hypotenuse AC will then by equal to the square root of 2.

2) Construct line at point C perpendicular to segment AC. Construct segment CD on this new line, equal in length to segment AB. Construct segment CD on this new line, equal in length to segment AB. Connect points A and C. Construct a line at C perpendicular to AC. Construct segment CD equal in length to segment AB or BC. Then construct segment AD. AD will then be equal to the square root of 3 (use the Pythagorean Theorem to calculate this length).You now have 2 right triangles, with one leg of the second triangle formed by the hypotenuse of the first triangle. You now have two right triangles, as shown below:

3) Construct a line at C perpendicular to AC. Construct segment CD equal in length to segment AB or BC. Then construct segment AD. AD will then be equal to the square root of 3 (use the Pythagorean Theorem to calculate this length). Construct a line at D perpendicular to AD. Construct segment CD equal in length to segment AB or BC. Then construct segment AD. AD will then be equal to the square root of 3 (use the Pythagorean Theorem to calculate this length).You now have two right triangles, as shown below:

Continue this process until you have 11 right triangles as shown below. If you have used Sketchpad, hide all the points.If you have used a compass and straightedge, trace the segments onto a clean sheet of paper, without the construction marks.

This spiral can be used as a "reviewee": if you cut it out and fold on the each line in alternating directions (using origami terminology, this means alternately folding up and then down) you will have a small "package" that you can carry in your pocket to review in your spare time.

Pleated:

Folded into a small "package":

And, if you like, you can color the other side of the spiral to create a beautiful graphic as shown below:

"Mathematics possesses a beauty cold and austere, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." Bertrand Russell