Partialy Paradoxist Smarandache Geometries

A paradoxist Smarandache geometry combines Euclidean, hyperbolic, and elliptic geometry into one space along with other non-Euclidean behaviors oflines that would seem to require a discrete space. A class of continuous spaces is presented here together with specific examples that emibit almost all of these phenomena and suggest the prospect of a continuous paradoxist geometry

Partially Paradoxist Smarandache Geometries

A paradoxist Smarandache geometry combines Euclidean, hyperbolic, and elliptic geometry into one space along with other non-Euclidean behaviors of lines that would seem to require a discrete space. A class of continuous spaces is presented here together with specific exmples that exhibit almost all of these phenomena and suggest the prospect of a continuous paradoxist geometry.Comment: To appear in Smarandache Notions Journa

PAPER MODELS OF SURFACES WITH CURVATURE CREATIVE VISUALIZATION LABS BALTIMORE JOINT MATHEMATICS MEETINGS

Abstract. A model of a cone can be constructed from a piece of paper by removing a wedge and taping the edges together. The paper models discussed here expand on this idea (one or more wedges are added and/or removed). These models are flat everywhere, except at the “cone points,” so the geodesics are locally straight lines in a natural sense. Non-Euclidean “effects ” are easily quantifiable using basic geometry, the Gauss-Bonnet theorem is a naturally intuitive concept, and the connection between hyperbolic and elliptic geometry and curvature is clearly seen. 1. Objectives and Notes The notion that a geometric space can be manipulated is an idea that I would like to instill in students. A number of behaviors of lines/geodesics can be found by constructing a variety of surfaces. I believe that this can be of value, as it is in topology where metric spaces with marginally intuitive properties are readily available. All of the models described in the labs are essentially 2-manifolds, so the notion that there are many accessible manifolds will hopefully be carried by the student into a study of differential geometry or topology. The local geometry of these paper models corresponds directly to the geometry of smoothly curved surfaces, so they can be used as an introduction to a study of Riemannian geometry. Geodesic

Partially Paradoxist Smarandache Geometries Howard Iseri

A paradoxist Smarandache geometry combines Euclidean, hyperbolic, and elliptic geometry into one space along with other non-Euclidean behaviors of lines that would seem to require a discrete space. A class of continuous spaces is presented here together with specific examples that exhibit almost all of these phenomena and suggest the prospect of a continuous paradoxist geometry