## Investigating Geometry Summary

### Monday - Friday, July 6 - 10, 2009

In Week 2, we continued examining the geometry of a sconce that Jim had photographed in his bathroom! The overall shape was square. In addition to an inscribed circle, the square contained two pairs of tangent semicircles whose diameters were sides of the square. We determined arc lengths and the angles at which the arcs intersected. This led to an investigation of generalized triangles whose sides could be line segments or arcs. Although the interior angle sum of these generalized triangles is not constant, we discovered that the sum of the exterior angle measures plus the signed arc angles is 360 degrees. We connected this discovery to the constant sum of the exterior angles of a polygon.

Joe Malkevitch of CUNY gave a guest lecture on taxonomies of quadrilaterals based upon the work of Branko Grünbaum
Classification of Convex Quadrilaterals [PDF format]
His more general approach used angle and side congruences, particularly how many of each and the general arrangement thereof. This taxonomy generates many more kinds of quadrilaterals than we have names for. For example, quadrilaterals with three congruent sides and two congruent adjacent angles currently have no name. We also looked at proofs eliminating some possible combinations.

We also considered a figure made of a circle, with two radii and two intersecting tangents forming a kite with its diagonals. The problem was posed to construct the same figure given different sets of points as objects as initial data.

We discussed parameters of the workgroup projects, split into project groups, and identified general areas of inquiry. One group is focusing on justifying constructions and another is exploring topics related to polygon angles. We spent the rest of our time developing our project work.

During our investigations, Jim pointed out that some historically significant geometry books by Mabel Sykes, significant at least to geometry education in the United States, are available for free on http://books.google.com.

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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.