Wooden Models of the Conic Sections

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Francisco Treceño
Valladolid-Spain

Models of the Conic Sections in Wood

Creations by a Wood Lathe Craftsman

The 2010 Spanish movie spectacular "Agora" centered on the female mathematician Hypatia (~400 AD) and the emergence of Christianity in Alexandria inspired Francisco Treceño. He created wooden models of the conic sections in wood. These models have found an appreciative audience, especially from those in mathematics and science.

Following a National Curve Bank tradition of supporting craftsmen producing models of mathematically related forms, we invited Treceño to join Steve Mathias ( Sphericons ) and Greg Frederickson ( Penrose Tilings ) in uploading images of their work. In addition, Treceño offers a video.

Watch as the cone is decomposed to reveal the planar circle, ellipse, one sheet of a hyperbola and parabola.

Video

Demonstrations using mathematical models have a strong tradition in American mathematics education. In visiting older university mathematics departments across the US students may see showcases of models created in earlier years by professors ardently seeking to demonstrate properties of the conics. Many talks at MAA meetings included demonstrations with models.

Similarly, models are often displayed in European departments. In particular we remember smiling at seeing the Platonic solids on display in Basel, Switzerland, home of Euler and the "Euler Formula."

Treceno's video adds to this tradition in 2012.

Recall the study of cones in Western Civilization dates to Apollonius (262-190 BC) and other early Greeks.

The circle, ellipse, parabola and hyperbola are "sections" of a cone.

Apollonius of Pergamon named the Conic Sections - the parabola, ellipse, and hyperbola. Moreover, he chose the ancient Greek names of "ellipse" and "hyperbola" for their meaning in relationship to the "parabola." Interestingly, the corresponding literary terms are parable, ellipsis, and hyperbole.

Archimedes (287-212 B.C.) and Apollonius (262 - 190 B. C.) investigated spirals and the conics centuries before Fermat and Descartes, but the "Ancients" did not have the advantage of symbolic algebra or analytic geometry.

References
For a MATHEMATICA® reference, see < http://mathworld.wolfram.com/topics/ConicSections.htm >.

For a Maple® source, open < http://www.maplesoft.com/support/help/Maple/view.aspx?path=Definition%2Fconicsection> .

Jennings, George A., Modern Geometry with Applications, Springer-Verlag, 1994, p. 83 - 113.

Stewart, James, Calculus, 7th ed., Brooks/Cole: CENGAGE Learning, 2012, pp. 670 - 687.

For a review of the movie on the life of Hypatia, the first woman mathematician we know by name,
type "Agora" in the search box of < http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=3496 >.

"It is difficult for us today to comprehend how Apollonius could discover and prove hundreds of beautiful and difficult theorems without modern algebraic symbolism. Nevertheless, he did so, and there is no record of any later Greek mathematical work that approaches the complexity and intricacy of the Conics."

Victor J. Katz, A History of Mathematics, 2nd ed, Addison, Welsey, 1998, p. 118.

Be sure to visit the Pergamon Museum if you are in Berlin to see a spectacular reproduction of the ancient city.