Now we haveseentwo majordifferencesbetweentaxicabgeometryand Euclideangeometry. Basedon a
new definitionof the “distancebetweentwo points”in taxicabgeometry, the look of a circlehas changed,
and one of the fundamentalpostulatesaboutthe midpointhas changed.
Theseshortexamplesillustratehow one smallchangein the rulesresultsin differentrulesfor manyparts
of the taxicabgeometrysystem.
Othernon-Euclideangeometriesapplyto othersituations,suchas navigatingon the globe,or findingthe
shapeof surfaceson bubbles.All of thesedifferentsystemsof geometryfollowpostulatesand definitions,
but by changinga few key rules(eitherthe postulatesor definitions)the entiresystemchanges.For example,
in Taxicabgeometrywe see that by changingthe definitionof "the distancebetweentwo points"we also
changedthe meaningof midpoint.
LessonSummary
In this lesson,we exploredone exampleof non-Euclideangeometry. Specifically, we havelearned:
- Wherenon-Euclideangeometryconceptscomefrom.
- How to find taxicabdistances.
- How to identifyand understandtaxicabcircles.
- How to identifyand understandtaxicabmidpoints.
Thesewill help you solvemanydifferenttypesof problems.Alwaysbe on the lookoutfor new and interesting
waysto applyconceptsof non-Euclideangeometryto mathematicalsituations.
PointsTo Consider
Now that you understandlines and angles,you are goingto learnabouttrianglesand their specialrelation-
ships.
LessonExercises
Solveeachproblem.
- Whatis the taxicabdistancebetweenpoints and in the diagrambelow?