Geometrical Concepts from Constructions, Models, and Investigations Summary
Thursday, July 18, 2002
Our group was very excited by the geometry in Romulo's salt-pouring demonstration this morning. We talked initially about the triangle and the quadrilateral.
Jim challenged the group with a related ruler-only problem: Suppose a triangle is drawn on a sheet of paper. You have a pencil and an ordinary ruler for drawing lines. Construct with these tools the angle bisectors of the triangle and the incenter of the circle.
The group jumped right on it and found a solution method. Connections were pointed out with the salt; namely, that for the triangle and quadrilateral, the ruler draws level curves of the salt. Jim did a related demo with a traveling set of lines parallel to the sides of the triangle. He (modestly) pointed out that in his Geometry Through the Circle he has a chapter devoted to "Strips" that provide a Sketchpad simulation of the ruler (without mentioning salt).
We also discussed the salt-pouring with the circular hole and connected this with the construction of conics by paper-folding or centers of sliding tangent circles. The height of the curve.was conjectured to be proportional to the distance from the boundary circles or lines. Troy then suggested a related 3D model where a tangent sphere of variable size rolls between two cylinders. The height of the center should be the height of of the top ridge of salt.
Next, resisting our tangential tendencies, we tore ourselves away from this mathematical topic and discussed the abstracts for our projects and the shorter version needed for our presentation this afternoon. We then adjourned to the lab to complete preparing the presentation.
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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.