- Identifyand understandtaxicabmidpoints.
Introduction
Whatif we changedthe rulesof a populargame?For example,whatif battersin baseballgot five strikes
insteadof three?How wouldthe gamebe different?How wouldit be the same?Up to this point,you have
beenstudyingwhatis calledEuclideangeometry. Basedon the workof the GreekmathematicianEuclid,
this type of geometryis basedon the assumptionthat givena line and a pointnot on the line, thereis only
one line throughthat pointparallelto the givenline (this was one of our postulates).Whatif we changed
that rule?Or whatif we changedanotherrule (suchas the ruler postulate)?Whatwouldhappen?Non-Eu-
clideangeometryis the term usedfor all othertypesof geometricstudythat are basedon differentrules
than the rulesEuclidused.It is a largebodyof work,involvingmanydifferenttypesof theoriesand ideas.
One of the mostcommonintroductionsto non-Euclideangeometryis calledtaxicabgeometry.Thatwill
be the principalfocusof this lesson.Thereare manyothertypesof non-Euclideangeometry, suchas
sphericaland hyperbolicgeometrythat are usefulin differentcontexts.
BasicConcepts
In previouslessons,you havelearnedto find distanceson a plane,and that the shortestdistancebetween
two pointsis alwaysalonga straightline connectingthe two points.This is true whendealingwith theoretical
situations,but not necessarilywhenapproachingreal-lifescenarios.Examinethe map below.
Imaginethat you wantedto find the distanceyou wouldcoverif you walkedfrom the cornerof 1
st
and
to the cornerof 3rdand. Usingthe kind of geometryyou havestudieduntil now, you woulddrawa straight
line and calculateits distance.