(TCPS 9) Novel Introductions to Non-Euclidean Geometry

Thursday, August 4, 2016, 1 PM – 2:55 PM, Hyatt Regency Columbus, Union A


This session invites presenters to share interesting ways in which to introduce undergraduate students to non-Euclidean geometry. These “tastes” of geometry may be demonstrations, in-class activities, projects, proofs, or ways in which to guide undergraduates to explore and to learn about non-Euclidean geometries. but not those related to differential geometry or (low-level) graduate courses. Those discussing demonstrations or in-class activities are encouraged to share key portions. Presenters should discuss the facets of their approaches which highlight the differences between the geometry being explored and the Euclidean geometry with which undergraduates are familiar. Information regarding prerequisite topics and related areas with which students have difficulty should be discussed as should follow-up topics and problems, if any, experienced when using this approach. Presenters are invited to discuss how they have modified their approaches over time and to share information about successes, failures, and student reaction. Abstracts should include the type of geometry being examined, a brief description of the aspects of this geometry which are introduced, the theorem, if appropriate, the software or application, if any, which may be used, and what makes this approach a unique introduction to non-Euclidean geometry. Those whose presentations are dependent upon software or tablet explorations must provide their own laptop or tablet.


Organizer:  Sarah L. Mabrouk, Framingham State University



1 PM – 1:15 PM

Bending Students' Intuition

Thomas Q. Sibley, St. John's Universtiy, College of St. Benedict          Presentation File


Physical models enable students to expand their geometrical intuition from Euclidean geometry to spherical and hyperbolic geometries. In particular, in less than a class period drawing and measuring on saddle shaped surfaces, my students consistently make several conjectures. As we develop hyperbolic geometry, they are able to prove these conjectures. I find this approach a more successful initiation than analytical models, such as the Poincare disk model. (Later I do introduce analytic models.)

·        Protractor



1:20 PM – 1:35 PM

Concrete Conics and Pencils in Projective Geometry

Michael Hvidsten, Gustavus Adolphus College           Presentation File


Students often have difficulty in conceptualizing properties of Projective Geometry. Many of the concepts, such as pencils of lines and points, seem to have no intuitive connection to student's perception. This is unfortunate, as Projective Geometry is one of the most beautiful and elegant ideas in mathematics. This talk will demonstrate a project that is used in the presenter's geometry class where students investigate how pencils of lines naturally lead to conic sections in Euclidean Geometry. By extending this investigation to the circle model of Euclidean Geometry, students see how all conic sections are essentially the same in Projective Geometry. This investigation makes apparent how the idea of pencils of points and conics arise naturally in Projective Geometry.

·        Michael Hvidsten’s Home Page:  http://homepages.gac.edu/~hvidsten/

·        Geometry Explorer

·        Geometry with Geometry Explorer



1:40 PM – 1:55 PM

Explorations using Cinderella

Ruth I. Berger, Luther College


Cinderella is an easy to use dynamic software program which allows for constructions in Euclidean, Hyperbolic, and Elliptic geometries. Hyperbolic geometry uses the Poincare disk model. In Elliptic geometry the screen shows a transparent sphere where, because a “point” consists of a pair of antipodal points, the user sees the other part of their constructions on the back side of the sphere. The menu selection in Cinderella is similar to Geometer's Sketchpad: plot points, lines, segments, midpoints, drop perpendicular from P to l, measure segments and angles, and of course dynamically move your construction around. Cinderella also has a built in calculator. My course focusses on proofs, but almost every week I have an exploratory Cinderella lab, so students can get a feeling for these other geometries and make conjectures. Well known Euclidean results are verified while students get used to the menu items needed in the construction, then they explore the same constructions in the other geometries. Lab activities include: Measure the angles of a triangle and compute the angle sum. Measure the sides of a right triangle and check if c2=a2+b2. Do the perpendicular bisectors of a triangle always intersect? If so, what is the relevance of that point? What about the angle bisectors? Construct an equilateral triangle for a given adjustable side length, measure the angles. Construct a rectangle (quadrilateral with 4 right angles). Measure the angles and the sides of a Saccheri quadrilateral. Do parallel (non-intersecting) lines exist? If so, do they have equidistant points, or a common perpendicular? The non-pro version of Cinderella is sufficient for my classroom use, it can be downloaded for free at Cinderella.de.

·        The Interactive Geometry Software Cinderella



2 PM – 2:15 PM

Introducing Spacetime Geometry: Relativity on Rotated Graph Paper

Roberto Salgado, University of Wisconsin, La Crosse          Presentation File


The Minkowski Spacetime Diagram is an essential tool for understanding Special Relativity. However, its non-euclidean geometry makes it difficult to interpret. Since the circle is replaced by a hyperbola asymptotic to the light cone, displacements of equal size in Minkowski geometry do not look equal and directions that are perpendicular in Minkowski geometry do not look perpendicular. To help students see these features, we present an approach that allows us to draw and calibrate a Minkowski spacetime diagram using ordinary graph paper rotated by 45 degrees. The boxes in the grid (called “clock diamonds”) represent units of measurement corresponding to the ticks of an inertial observer’s light clock. We show that many quantitative results from special relativity can be read off a spacetime diagram simply by counting boxes, with very little algebra. We demonstrate this using visualizations created with GeoGebra.

·        GeoGebra

·        Roberto Salgado’s GeoGebra page (robphy)

·        Physics Forums Insights Blog Post:  Relativity on Rotated Graph Paper

·        DOI Reference for the paper “Relativity on Rotated Graph Paper”



2:20 PM – 2:55 PM




Ruth I. Berger, Luther College

Michael Hvidsten, Gustavus Adolphus College

Roberto Salgado, University of Wisconsin, La Crosse

Thomas Q. Sibley, St. John's Universtiy, College of St. Benedict



Sarah Mabrouk, Framingham State University




This page was created and is maintained by S. L. Mabrouk, Framingham State University.

This page was last modified on Thursday, August 11, 2016.