Geometry: An Interactive Journey to Mastery

Bending the Axioms—New Geometries

Lesson 36

Topics
x Spherical geometry.
x Taxicab geometry.
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x taxicab geometry: If distances between two points A ͼ௘a 1 , a 2 ௘ͽDQGB ͼ௘b 1 , b 2 ௘ͽDUHPHDVXUHGE\WKH
sum of the horizontal and vertical displacements, dAB(, ) | b a 11 | |b a2 2|, rather than via the
Pythagorean theorem, and all other structures of planar geometry are left the same, then the geometry
that results is called taxicab geometry.
Formula
x The area of a spherical triangle with interior angles of measures x°, y°, and z° is
xyz 720 180 utotal surface area of the sphere.

Summary
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the like, while still obeying all but one of the postulates of geometry. That way, you can develop a feel for the
degree to which a particular postulate in geometry is important and see how things drastically change if that
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brand new geometry called taxicab geometry.
Example 1
In taxicab geometry, what are the distances between the following pairs of points?
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E௘ͽ P ͼ௘í௘ͽDQGQ ͼ௘í௘ͽ
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