Model of Cones with a Common Vertex by Richard P. Baker, Baker #508 (a Ruled Surface)

Wheeler designed of “cones with a common vertex.” Four of these are in the Smithsonian collections. They are Wheeler’s numbers 504 (MA.211257.098), 505 (MA.211257.099), 507 (MA.211257.100), and 508 MA.211257.101).

Reference:

Richard P. Baker, “Mathematical Models,” Iowa City, Iowa, January, 1931, p. 10.

This string model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.

The typed part of a paper label on the bottom of the wooden base of this model reads: No. 508 (/) OSCULATING CONTACT; ROOTS 3. Model 508 appears on page 10 of Baker’s 1931 catalog of models as “Osculating contact” under the heading Cones with common vertex. It is explained on page 9 that the "3" listed for the model refers to “the multiplicity of the roots.”

Baker’s string models always represent a special type of geometric surface called a ruled surface. A ruled surface, sometimes called a scroll, is one that is swept out by a moving line. This model shows two ruled surfaces, a circular cone whose lines are represented by red thread and an elliptical cone whose lines are represented by blue thread. These two cones share a vertex and each vertical side of the model shows a red circle and a blue ellipse. The cones meet and cross each other along two lines. The meeting is visible on each of the sides where there are two holes that share a red and a blue thread. The crossing is clearly visible at one of the holes, while the other is at a point of osculating contact.

The term osculating derives from the Latin for kissing. In plane geometry, mathematicians use the term to refer to a point at which two curves (1) share a tangent line, (2) are on the same side of that common tangent line, and (3) have the same curvature. Since curvature measures how much a curve curves at a point, a straight line has zero curvature at every point. The curvature of a circle is also the same at every point and its numerical value is 1 divided by the radius of the circle so a circle with a small radius has a large curvature and a circle with large radius has a small curvature. An ellipse has a constantly changing curvature that is smallest where the curve looks straightest. On either vertical side of this model the point of osculating contact has multiplicity three as a point of intersection of the circle and the ellipse. Although these curves cross at that point of osculating contact, neither curve crosses the shared tangent line at that point.

Currently not on view