The Shipley School

Objectives:

Practice with compass and straight edge

Explore a geometric representation of square roots, deepening understanding

Introduce students to spirals, curves that are seldom studied in traditional textbooks

Develop an awareness of the historical context for the study of irrational numbers and spirals

Recognize spirals in nature and appreciate the mathematics inherent in them

Historical Perspectives:

The root spiral generated in the exercise below has been attributed to Theodorus of Cyrene (~465-399 BC). Theodorus was Plato’s tutor. In Plato’s Theaetus, Socrates makes reference to Theodorus proving that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, and 17 were irrational. Plato raises the question "Why did Theodorus stop at Ö 17?". One possible answer is that the Ö 17 hypotenuse belongs to the last triangle that does not overlap the figure.

In 1958 E. Teuffel proved several more interesting facts about the root spiral. If the procedure for generating the spiral is continued indefinitely so that the figure overlaps, no two hypotenuses will coincide. In other words, they will never lie directly on top of each other. Also, if the sides of unit "one" length are extended forever, they will not pass through any of the other vertices of the total figure.

The Spiral of Theodorus approximates the Logarithmic Spiral. By the early 1600’s the logarithmic spiral was being studied in depth. See Historical Perspectives II in Appendix A for additional information.

Prerequisites:

Students should be able to construct a perpendicular to a line through a point on the line with a compass and straight edge. However, the lesson would still be effective if modified for use with a protractor and ruler. Students should have been introduced to the Pythagorean Theorem and square roots.

Audience:

This lesson is intended for students in a Standard or Advanced Geometry course with a forty minute period. In most texts, the Pythagorean Theorem is discussed in later chapters. Hence, this lesson would come later in the year. With an emphasis on Part B of the procedures, this lesson could also be effective with Algebra I students that have begun their study of square roots.

Materials Needed:

Compass, straightedge, several overheads of figures provided, an overhead picture of a nautilus and of a spiral galaxy (you can print these directly from several sources on the web), an overhead projector, precut uncooked spaghetti or coffee stir straws.

Additional Recommended Materials: a timeline of mathematicians, pictures of mathematicians discussed in the historical perspectives, a slide rule.

Procedure:

A. Constructing a Root Spiral (15 minutes)

1. Post on the board: Construct an isosceles right triangle. Instruct students to start near the middle of their page and to make the legs about one inch long.
2. Discuss with students the length of the hypotenuse. (Ö 2 times the length of the leg) Explain that for the rest of the exercise each leg will represent a "one" unit.
3. Post on board: Using your same Ö 2 segment as a leg, construct a right triangle with legs Ö 2 and one.
4. Encourage students to chat with themselves about how to do this before demonstrating or posting Figure One on the overhead.
5. Discuss the length of the new hypotenuse. (Ö 3) Justify with the Pythagorean Theorem. Ask "If we repeat this procedure, what will our next hypotenuse be?" (Ö 4, or 2) "How can we check?" (copy our unit segment onto the new hypotenuse twice, and/or use Pythagorean Theorem.)
6. Post on board: Using your same Ö 3 segment as a leg, construct a right triangle with legs Ö 3 and one. (i.e. repeat the procedure)
7. Post Figure Two on the overhead. You may need to demonstrate the construction again. It helps to have students turn their paper so that the Ö 3 leg is horizontal.
8. Discuss what will happen if the procedure is continued. (segments of length Ö 5, Ö 6, and Ö 7 etc will be generated)
9. Have the students continue repeating the procedure until a segment of length Ö 8 is generated (if time permits). You may post Figure Three as students work.
10. As some students finish before others, and are not engaged in helping their peers, place a picture of a chambered nautilus on the overhead. Ask those students that are finished to generate a list of spirals found in nature. (galaxy, sheep horn, uncoiling fern, whirlpool etc)

B. Discussion of root lengths (5 minutes)

1. Before class: Using uncooked spaghetti (or a similar, thin, stick manipulative) cut several pieces that match your overhead segments of one, Ö 2 ,and Ö 3 units
2. Lay the pieces on the diagram, and then side by side. Ask students to compare the lengths. (students should notice that the Ö 2 is about one and a half times longer than the one. They should also notice that the Ö 3 is longer still.)
3. Ask "Does Ö 2 plus Ö 3 equal Ö 5?" Have students chat with themselves and then each other.
4. Lay the Ö 2 and Ö 3 pieces along the Ö 5 length of the overhead diagram. (notice that Ö 2 plus Ö 3 is longer than Ö 5)
5. Using a simple calculator, give a decimal approximation of Ö 2, Ö 3, and Ö 5. Show arithmetically that 1.414+1.732 > 2.236
6. Repeat steps 3-5 with the Ö 6 length on your diagram.
7. Ask "what does Ö 2 plus Ö 2 equal?" Have students chat with themselves and each other.
8. Lay a pair of Ö 2 lengths along the Ö 8 length of the overhead. Have students explain why they are equal. (Ö 8 simplifies to 2Ö 2) Demonstrate that they are NOT equal to Ö 4.

C. Relating to Occupations and Historical Perspectives (15 minutes)

1. Place a picture of a chambered nautilus shell on the overhead. Ask students if they know what it is a picture of. ( A nautilus is a sea creature that creates its shell. It adds a larger chamber as it grows too big for the previous space. By the time is has completed a full growth rotation the final chamber is three times as large as the original chamber) Ask students to describe any similarities and differences from the spiral they generated. (The root spiral is bent instead of smooth. Both generally preserve their shape as enlarged.)
2. Point out to students that biologists often look for mathematical patterns in the nature they study. Ask students where else a biologist might notice spirals. (Students who finished their constructions first will hopefully have already generated a list.)
3. Explain that the "root spiral" they generated is an approximation of the Equiangular, or Logarithmic, spiral of the nautilus. Explain why it is called equiangular (lines drawn from the center intersect the sides at the same angle…see figure four).
4. With figure three on the overhead, ask students how far they think they could repeat their procedure before the diagram begins to overlap. (answer…up to Ö 17)
5. Share information with students about Theodorus of Cyrene (see first paragraph of Historical Perspectives).
6. Explain what it means to be irrational, or "incommensurable". Explain that square roots were problematic for the Pythagoreans.
7. With Figure Three still on the overhead, share some of the Teuffel theorems with students. (See Historical Perspectives.) Point out how recently these theorems were proved, and that they (students) could also still discover new math related to old figures.
8. Ask students how they might transform their root spiral into a "smooth" curve like the nautilus. (one possible solution is to use their compass to draw approximate arcs between hypotenuses. Another solution might be to make their "one" unit very small.)
9. Discuss with students the fact that these strategies would be approximations. You might point compare the method of reducing the original "one" unit to the method of exhaustion used to approximate the circumference of a circle (by inscribing polygons of increasingly shorter sides).

Optional: Including this in your lesson would be especially effective if your students are simultaneously studying the Scientific Revolution and Enlightenment in their History classes. However, much of this lesson would be more appropriate for Algebra II students that have started their study of logarithms.

10. Inform students that the Equiangular Spiral, as a function, was first studied in depth in the early 1600’s. The location of each point was defined by its distance to the origin along a line perpendicular to the tangent at that point.
11. Share the information contained in the Historical Perspectives II in the Appendix. A collection of pictures to use as accompanying overheads is also provided in the Appendix, although printing directly from a web page source would produce higher quality overheads. See the list of links for sources for these pictures.
12. If you have a slide rule to show students, I recommend it. (You might point out to students that slide rules were used like calculators up through the 1960’s. Some students may have seen them used in the Apollo 13 movie.)

D. Closure

1. Place a picture of a spiral galaxy on the overhead.
2. Read the following excerpt from Plato’s Timaeus: "…when all the stars which were necessary for the creation of time had attained a motion suitable to them…the motion of the same made them all turn in a spiral".
3. Ask students to chat with themselves about how the pure shapes we study in Geometry relate to the real shapes of the universe around us.

In the late 16^th century and early 17^th century, aristocratic amateurs (as opposed to the professional class we have today) conducted science in Europe. An educated gentleman would dabble in the studies of optics, time measurement, navigation, mechanics, and surveying. Often he was a craftsman, or employed craftsmen, making his own scientific apparatus. Exploration of the newly discovered territories provided the impetus and cash for developing improved navigational instruments and strategies. The belief that the natural world could be defined in terms of mathematical laws also inspired these philosopher scientists. The astronomer Johannes Kepler (1571-1630) wrote in 1605, "My Aim is to show that the celestial machine is to be likened not to a divine organism but rather to a clockwork." Mathematicians of the time frequently found themselves at odds with established church doctrine, despite their professed faith.

In 1614, an Englishman named John Napier (1550-1617) developed logarithms in order to manipulate trigonometric functions. Trigonometry was used frequently for navigation or surveying. Logarithms helped simplify the calculation of true motion and direction of a ship moving in a current. Napier also applied these principles to the military arts, such as the placement of artillery quadrants. Both of these applications were of vital importance as European navies set out to conquer new territories. Many scientific gentlemen readily adopted Napier’s tool for quickly calculating exponents of small quantities. Within four years, the slide rule was developed using logarithmic scales, making calculation easier for hundreds of years. Logarithms were of such universal significance that by 1650 Sie Fong-tsu, a student of Jesuit missionaries, published the first Chinese treatise.

Rene Descartes (1596 -1650) applied Napier’s logarithms in his quest to describe the universe in mechanical terms. Descartes believed that all natural phenomena followed mathematical laws. (Descartes was the first philosopher to regularly use the expression "laws of nature", as opposed to "principles".) He described the motion of stellar masses as a vortex, with eddies producing lesser vortices such as planets and their moons. Within this construct, all motion is relative and Descartes could claim that the earth was "relatively" motionless. While Descartes believed that this description of earth’s relation to the universe did not contradict church doctrine, his books were banned (although popular!). A court order arrived during his funeral to prevent an oration. He was buried in the section of the graveyard reserved for the unbaptized.

Descartes applied the logarithmic spiral to his study of falling bodies and vortices. In "On the Nature of Curved Lines", Descartes described categorical differences between two classes of curves: geometric and mechanical. Straight lines, circles, ellipses, parabolas, and hyperbolas were considered geometric because they could be determined from one motion. Thus, he suggested, these curves should be included in the study of geometry. The spiral and the quadratrix (used to trisect an angle or square a circle) belonged in the study of mechanics because the motion of two objects defined them.

Several other famous mathematicians contributed to the study of Logarithmic Spirals. Jacob Bernoulli (1654-1705), a professor at the university in Basle, Switzerland dubbed the spiral "Spira Mirabilis", or "the wonderful spiral" because it shows up so many places in nature. He requested that the spiral be engraved on his tomb with the phrase "Eadem mutata resurgo", which translates as "I will arise the same, though changed". Ironically, the spiral on his tombstone appears to be an Archimedean spiral as opposed to a logarithmic spiral. Kepler made the connection between the Fibonacci sequence and the logarithmic spiral in 1611. In 1645 Evangelista Torricelli developed a method for constructing a straight line segment with length equal to that of a logarithmic spiral arc. He also showed that arc lengths on the logarithmic spiral are finite, even if they wrap around the center point an infinite number of times.

Four mice start from the corners of a box and move continually towards one another. The curves their paths trace are logarithmic. A nice visual of this (for other n-gons) is provided on-line through Eric’s Treasure Troves of Science. (See web address above.)

Elementary students enjoy this activity. Take any regular polygon. Divide each side into equally spaced intervals. Using a straightedge, connect the first mark of one side to the last mark of the adjacent side. Connect the second mark to the second to last mark of the adjacent side, and so on. A complete description of this activity can be found in Dale Seymore’s book Line Designs from Creative publications, 1968. Sample pictures are also available on-line through Eric’s Treasure Troves of Science. (See above.)

A good source book on the methods of constructing celtic spiral is Celtic Art by George Bain, published by Dover Publications. The Solstice Project produces a video titled The Sun Dagger that documents a spiral celestial calendar carved into a butte in New Mexico by the Ancient Puebloans (Anasazi Indians).

Divide a circle into n equally spaced concentric rings. Further cut the circle into n equal angles (pie-shaped pieces). Connect successive intersections of ring and radius to produce a spiral. This construction provides a nice connection with Descartes classification of Spirals as mechanical; The location of each point is determined by two factors. Archimedes defined his spiral by rotating a ray uniformly while simultaneously moving a point out the ray.

Graphing spirals on the TI-83

The polar equation for an Archimedean Spiral is r = kq , where k is a constant. The equation for an Equiangular Spiral is r = ae^bq , where a and b are constants.

Using an overhead of an Archimedean spiral and some bird seed, demonstrate that the area of one circuit is one third of the area of the circle on the larger radius.

This is a concise encyclopedia of mathematics with an alphabetical listing and a search feature.

The Geometry Center at the University of Minnesota has innumerable exciting materials for a mathematics teacher.

This provides accounts of the lives of 17^th & 18^th century mathematicians, adapted from a book by W.W.Rouse Ball.

On this page there is a nice photo of a chambered nautilus with the Equiangular Spiral superimposed. On their home page (MacTutor History of Mathematics Archive) you can access a variety of other mathematical history pages, timelines, and pictures of mathematicians.

A rather mystical web site, but this page includes several nice pictures of spirals in clouds, galaxies, and conch shells. It also includes a well illustrated demonstration of how to construct a Golden Mean Spiral, with links to an algebraic proof of the construction.

This site provides an interestng index of famous curves with their related involutes, evolutes, pedals, etc.

Boorstin, Daniel, The Discoverers, Vintage Books, New York, 1983

Burton, David, History of Mathematics, William Brown Publishers, Iowa, 1991

Davis, Philip, Spirals from Theodorus to Chaos, A.K.Peters, Massachusets, 1993

James,Robert et al., Mathematics Dictionary 4^th ed., Van Nostrand Reinhold, NewYork, 1976

Smith, D.E. History of Mathematics, Dover, New York, 1951

Turner, Anthony, Early Scientific Instruments, Europe1400-1800, Sotheby’s Publications, NewYork, 1987