The Geometry Forum: Constructing Geometry on the Internet - sum95

Geometry Forum Summer Institute - July 9-15, 1995

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Daily Summary
Tuesday, July 11, 1995

It's been a busy day here at Swarthmore College. The early morning was spent checking e-mail and the like, followed by some more instruction about html. The primary lesson involved the construction of Forms on the Web. Many of the participants added forms to their home pages.

A form is a highly interactive aspect of the Net in which data, information, messages, etc. can be entered by the visitor and then automatically sent on to the form's creator. One example of a form can be found at the site where a 6th grade math class asks for help in conducting an experiment (The Great Penny Toss, A visitor to their page is asked to enter results from coin flips into their form. These data are presumably added to an ongoing record, and eventually conclusions are passed on to all those who participated.

Steve and Annie warned us that there is often some work that goes into making forms easy, friendly and interesting. As we did with home pages, we cut and pasted pre-existing models of forms. Also, we already have here an AppleScript that transforms information entered in forms into normal, readable e-mail text.

We were honored this afternoon to hear a talk by Don Shimamoto, the Chair of the Swarthmore Math Department. Don teaches a unique geometry class here for majors and non-majors alike (also required for those obtaining their teaching certification). This class does a lot of work with The Geometer's Sketchpad, and thus was of great interest to our workshop. The purpose of today's talk was to provide the teachers here with an idea of possible project that they might bring to their own classrooms. In the process, the participants learned a lot of geometry themselves, as well as greater proficiency with Sketchpad.

Don began by demonstrating some problems that he has designed for areas other than geometry. As an example, there's a calculus max/min problem in which one has to find the longest possible ladder that will fit around a corner. In Sketchpad, you can manipulate the size of the ladder and actually attempt to drag it around the corner! Don showed us how the problem actually becomes a minimization problem if looked at the right way.

As an demonstration of an Analysis problem, we used Sketchpad to iterate a pattern within a square to observe that in the limit, a continuous curve fills the entire square (space-filling curves).

Most of Don's talk was devoted to Hyperbolic Geometry. He introduced us to this non-euclidean geometry, an interesting property of which is that there are infinitely many lines through a given point that are parallel to a given line. The basic definitions in Hyperbolic Geometry are as follows:

  • parallel lines are non-intersecting lines
  • the plane is the interior of a disk
  • lines are diameters or arcs that are perpendicular to the boundary of the disk
  • angles are measured as the euclidean measure of the angle between the tangents to the arcs at the point of intersection

The immediate result that we observed was that the sum of the interior angles of a triangle was not 180 degrees, and was not even constant.

The next logical question, one that Don poses to his students, is how to construct a hyperbolic version of Sketchpad. He has essentially done this by providing a series of hyperbolic script tools that can be used in the interior of any disk. We spent some time experimenting with drawing lines and segments, perpendiculars (the same Euclidean concept exists) and reflections.

Our final activity was to look at some students' projects. The first was a demonstration of Morley's Theorem, and the second was an exploration of polygons in Hyperbolic Geometry. It is impossible to construct a rectangle because of the fact that triangles' angles sum to less than 180 degrees. However, it is possible to have polygons of 5 or more sides with interior angles all equal to 90 degrees.

I hope all this is reasonably clear. Please feel free to add to or correct any portion of my reports.

Eric Sasson
Geometry Forum

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July 1995