Transcript Taxicab Geometry

TAXICAB GEOMETRY
An exploration of Area and Perimeter
Within city blocks.
EUCLIDEAN GEOMETRY
Type of geometry generally taught in High School
 Named after mathematician Euclid, circa 300 BC
 Created under several assumptions

2 dimensional plane containing points, lines, circles,
angles, measures and congruence
 5 Postulates Defining Euclidean Geometry

5 POSTULATES
We can draw a unique line segment between any
two points.
 Any line segment can be continued indefinitely.
 A circle of any radius and any center can be
drawn.
 Any two right angles are congruent.
 If a straight line crossing two straight lines
makes the interior angles on the same side less
than two right angles, the two straight lines, if
extended infinitely, meet on that side on which
are the angles less than the two right angles.
(The Parallel Postulate).

THERE ARE OTHER TYPES OF GEOMETRY!!!
Here are some examples:

SPHERICAL GEOMETRY
Based on a sphere, instead of a coordinate plane
 Think of Geometry based on the Globe

How is it possible for a person to walk 10 miles south, then
10 miles west, and then 10 miles north, and return to their
home?
 Consider if their home is on the North Pole!


TAXICAB GEOMETRY
WHAT IS TAXICAB GEOMETRY?
Introduced by Eugene F. Krause
 Redefines Euclidean Distance

Euclidean Distance – A line segment represents the
shortest distance from one point to the other.
 “As the Crow Flies”
 Formula comes from Pythagorean Theorem

D  ( x1  x2 ) 2  ( y1  y2 ) 2
DEFINING TAXICAB DISTANCE
Consider the
coordinate plane as
city blocks.
 Taxicab may only
travel along the city
streets.
 Distance is found by
counting how many
units one must travel
to get from one point
to another, moving
only horizontally and
vertically.

WHY IS TAXICAB GEOMETRY BENEFICIAL?
Students can relate.
 Taxicab Treasure Hunt
 Address alternatives to Euclidean Geometry.

Often people do not even know they exist.
 Focuses on need for set rules, postulates and
definitions.


Explore common ideas in a new parameter.
Conic Shapes, Perpendicular Bisectors, etc.
 Circles:

ACTIVITY 1
Sketch Triangle with
Vertices: A (-3, 2),
B (1, 4), and C (2, 2)
 Find Euclidean
Distance of all 3 sides
 Find Taxicab Distance
of all 3 sides


Sketch the path you
took from each vertex
to find the distance
ACTIVITY 2 – PERIMETER AND AREA

Find Euclidean Perimeter of the triangle.

Find Taxicab Perimeter of the Triangle.

Find the Euclidean Area of the Triangle.

Can we find Taxicab Area? Explain what this
means and how you would find it.
Ticket out
the Door
1.
Describe a situation when it would make most
sense to use Taxicab Geometry. Explain.
2.
Describe a situation when it would make most
sense to use Euclidean Geometry. Explain.