### SLO 9 Summary

Just as visiting another country can offer one a richer perspective on their own culture, the study of non-Euclidean geometries can help students to develop a deeper understanding of Euclidean geometry. The term “non-Euclidean geometry” is interpreted broadly here, referring to any geometry different from Euclidean, including spherical, hyperbolic, incidence, and taxicab geometries, among others. The choice of which non-Euclidean geometries to consider might depend on the demands of a particular GeT course, but all non-Euclidean geometries offer rich opportunities to explore and visualize novel and engaging worlds. Learning the properties of non-Euclidean geometries puts preservice teachers in the position of their students who may be learning Euclidean geometry for the first time. Moreover, non-Euclidean geometries challenge student assumptions about what is “true” or “obvious” in Euclidean geometry (e.g. the shape of a parabola or the angle sum in a triangle). In this way, exploring non-Euclidean geometries naturally encourages questions of “Why?” and “How?” and supports the development of mathematical thinking, especially the need for justification.

### SLO 9 Narrative

Just as visiting another country can offer us a richer perspective on our own culture, so too can the study of non-Euclidean geometries help students to develop a deeper understanding of Euclidean geometry. While it is natural for students to be uncomfortable working in a geometry that varies from their intuition, non-Euclidean geometries afford an opportunity to explore and visualize novel worlds that can engage their imagination. In addition, learning the rules of these geometries puts students that plan to teach in the position of their students who may be learning Euclidean geometry for the first time. The choice of which different non-Euclidean geometries to consider might depend on the demands of the GeT course, but all offer new perspectives on familiar geometric objects and relationships.

In our everyday experience, we regularly encounter multiple geometries. Buildings tend to be Euclidean. We expect floors and ceilings to be planar. Outside, the horizon reminds us that we live on a sphere. Our visual field routinely processes distant objects as smaller than comparably sized things that are nearby, just as they could be represented in projective geometry. In the car, we measure distance with a taxicab metric. Fans of science fiction may even encounter images and ideas of hyperbolic geometry. Notably, non-Euclidean geometries can be viewed through two different lenses: geometrically, as spaces that are physically different from Euclidean space, or axiomatically, as spaces in which different axioms are true [see SLO 4].

The amount of time devoted to non-Euclidean geometries can vary widely depending on factors including audience, instructor preference, and institutional expectations. For a class consisting primarily of preservice teachers, a substantial amount of Euclidean content is necessary, though at least some non-Euclidean geometry is recommended. In a comparative geometries course, it would be natural to consider several different non-Euclidean geometries, while a class that focuses primarily on Euclidean geometry might include a brief survey of some non-Euclidean examples or focus on one flavor for a longer period of time. In any case, what follows are some of the learning opportunities offered by each.

Incidence Geometries are useful for getting a sense of how theorems follow from a set of axioms. These involve a reduced set of axioms and perhaps make it easier to introduce some principles of proof-writing [see SLO 1] in that context. Taxicab geometry is an easily described alternate geometry that can lead to rich mathematical exploration. Here we note that when we speak of “non-Euclidean geometry,” we mean this broadly, referring to geometries that are different from our usual notion of Euclidean two-or three-dimensional space. In taxicab geometry, we change our usual definition of distance in the plane. Rather than using a Pythagorean measurement, we measure the distance between two points as the sum of the absolute differences of their Cartesian coordinates. This radically changes the form of objects that are defined in terms of distance. For example, a circle (the set of points at a given distance from a given point) no longer appears round. Ellipses, hyperbolas, and parabolas provide an even greater challenge!

Spherical geometry offers the advantage of being (fairly) easy to visualize (or hold in your hand). As a more accurate representation of the surface of the planet than a flat Euclidean world, it has relevance. An introduction to spherical geometry immediately challenges our understanding of the undefined term “line” [see SLO 5 on definitions] and our belief that between any two points there can be drawn a unique line. Other explorations might have students consider parallel lines on the sphere or the angle sum of a triangle.

Taxicab and spherical geometry serve well as examples of non-Euclidean geometries that can be explored at any point in a GeT course. Hyperbolic geometry can be as well, though its close relationship to Euclidean geometry is, perhaps, best appreciated when students are more experienced with axioms and axiomatic systems [see SLO 4]. Hyperbolic geometry differs from Euclidean geometry only in a parallel postulate. In Euclidean geometry, we assume there is exactly one parallel through a given point not on a given line. In hyperbolic geometry, we adopt a different parallel postulate, so that there are multiple lines through a given point parallel to a given line. Changing this axiom is, in fact, how hyperbolic geometry was first developed historically1 [see SLO 7]. Moreover, we can simply remove that axiom altogether to end up with a third geometry: Neutral geometry. Comparing the mathematical properties of these three geometries and their interplay leads to rich discourse. It is also worth noting that although hyperbolic geometry may arise most naturally from this axiomatic change, it can also be viewed geometrically as a space of constant negative curvature. In this sense, it provides an instructive example of a non-Euclidean geometry having properties different from Euclidean geometry.

The relationship between Euclidean, hyperbolic, and Neutral geometry can be made explicit by proving the equivalence of parallel postulates in Neutral geometry. For example, transitivity of parallelism (“Two distinct lines each parallel to a third line are parallel to each other.”) is logically equivalent to Euclid’s fifth postulate in Neutral geometry. Proving that equivalence, or one similar, can be a valuable experience by strengthening student understanding of proof [See SLO 1]. These proofs are demanding but generally relatively brief.

Rectangles (quadrilaterals with four right angles) are among our most familiar geometric objects. However, while the existence of rectangles can be easily established in Euclidean geometry, it cannot be proven that they exist in hyperbolic geometry or in spherical geometry. In both of those cases, a pair of lines can share, at most, one common perpendicular line—making a rectangle impossible. In neutral geometry, we can neither prove nor disprove their existence. Likewise, we can show that similar, non-congruent triangles do not exist in hyperbolic geometry. Examples such as these differentiate between the geometries and demonstrate the necessity of Euclid’s fifth postulate. In this way, they can help strengthen student understanding of axiom systems [see SLO 4].

Hyperbolic geometry offers opportunities for students to strengthen their facility with geometric straightedge and compass constructions [see SLO 8] through an exploration of hyperbolic geometry models. Since both the Poincaré and Klein models of hyperbolic geometry are situated within Euclidean geometry, constructions of “lines” and “perpendiculars” in these models translate to Euclidean constructions. These constructions can cover a range of complexity, from the trivial (e.g., constructing a “line” in the Klein disk) to the highly involved (e.g., “dropping a perpendicular” in the Poincaré disk). Dynamic geometry software [See SLO 6] is an excellent resource here, as there is a wealth of online tools available that automate some of the most difficult constructions. This has the added benefit of encouraging students to engage in higher-order thinking on constructions in ways that were not possible only a few years ago.

References

Greenberg, M. J. (2008) Euclidean and Non-Euclidean Geometries: Development and History. W.H. Freeman.

Henderson, D. W., & Taimiņa, D. (2020) Experiencing Geometry: Euclidean and Non-Euclidean with History. 4th Ed. Project Euclid, DOI: 10.3792/euclid/9781429799850.

Hofstadter, D. (1999) Gödel, Escher, Bach: an Eternal Golden Braid, 20th Anniversary Edition. Random House.

Krause, E. F. (1987) Taxicab Geometry: An Adventure in Non-Euclidean Geometry. Dover.

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1See (Hofstadter, 1999, Ch. 4, pp. 88–93), for an accessible discussion of this history written for a general audience.