Adventures in
Non-Euclidean Distance
Ricky Bobinchuck
Tashauna Thompson
Tosha Pelfrey
How would you find the shortest
distance between the two points on
the two dimensional plane below?
Pythagorean Theorem?
a b c
2
3 4 c
2
9  16  c
2
2
2
2
2
25  c
2
25  5
The distance between the two points is 5 units.
The Distance Formula
The shortest distance between two points is a
straight line.
The distance formula finds the shortest
distance between two points when working in a
two dimensional plane.
In Euclidean Geometry, can you find
another path that is equal to the
shortest distance?
Nope, in Euclidean Geometry
only one path is the length of
the shortest distance.
Get Real!
When you are traveling in a car the
shortest distance between two locations
is still a straight line, but this path may
not be an option to travel.
Distance Depending on the Space
• In the 19th century,
Hermann Minkowski,
proposed a family of
metrics where the notion
of distance is different
depending on the space in
question.
• Minkowski’s ideas helped
Albert Einstein to develop
his Theory of Relativity.
"Taxicab Geometry" First Coined
• Karl Menger was the first to
use the term “taxicab” to
describe Minkowski’s metric
in the booklet You Will Like
Geometry.
• In 1952, Menger had an
exhibit of taxicab geometry
in the Museum of Science
and Industry in Chicago.
Taxicab Geometry
•Non-Euclidean Geometry
•Route a cab would take
•Can’t cut though buildings
•Limited to streets – gridlines and one-way
traffic flow
Taxicab Distance
Shortest path between 2 points
• least number of blocks a taxi must
travel along streets
• points at intersections (x, y)
• coordinates are integers
Example
What’s the shortest taxicab
distance from (0, 0) to
(1, 2)?
4
Taxicab Distance Activity
GeoGebra File
1.
2. Taxicab Distance Formula
d T  | x 2  x1 |  | y 2  y 1 |
Taxicab Geometry
in Middle School
Gangster Grid Game
The objective of this game is to see who can catch the gangster first.
Rules
1. Each player chooses a point where their gangster is hiding on their game board
(keep it a secret).
2. Player one should choose the ordered pair that he or she thinks the gangster
is located on the opponent’s board.
3. Player two will tell player one the taxicab distance that he or she is away from
the gangster.
4. Now player two will choose the ordered pair that he or she thinks the gangster
is located on the opponent’s board.
5. Player one will tell the taxicab distance that he or she is away from the
gangster.
6. Repeat steps until the gangster is located.
7. The first to locate the gangster is the winner.
8. Replay the game until time is called.
Hint: Keep track of how many blocks you are away from the gangster.
Recap…
Distance
▫ The shortest path between any two points.
Euclidean Geometry
▫ There is one path, a straight line, that describes the shortest
distance between two points.
 Found using
Taxi Cab Geometry
▫ The shortest path between two points is restricted by gridlines
and streets.
 Found using d T  | x 2  x1 |  | y 2  y 1 |
▫ Is there only “one” shortest path between two points????
How many Paths can you find?
How many Paths are there?
Is the number of Paths related to the distance?
How many paths
can you find
between points
A(0,0) and B(2,2)?
Think Combinations “!”
• Is there a way that we can find the number of paths?
• Using combinations, we can find the number of
paths with the formula for C ( m , k ) ; where m is the
distance between two points:
m
m!
  
 k  ( m  k )! k !
m=distance
We defined the distance between two points earlier as the
horizontal change (x) plus the vertical change (y).
yk
x
n
Algebra!
• Now we can define distance in terms of n and k,
where n represents our x coordinate and k our y.
• If
m
m!
  
 k  ( m  k )! k !
• Substituting n+k for the distance m we have:
n  k 
( n  k )!
( n  k )!

 

n! k !
 k  ( n  k  k )! k !
Let’s check it out…
n  k 
( n  k )!
( n  k )!

 

n! k !
 k  ( n  k  k )! k !
 2  2  ( 2  2 )!
4!

 

2!2!
2!2!
 2 

4  3  2!
2!2!

43
2

12
6
2
How many paths did you say we can find for
points A and B?
Answers
Will that always work?
( n  k )!
n! k !

(1  3 )!
1!3!

4!
1!3!

24
6
4
Do you notice anything?
(  n   k )!
 n!  k !

( 2  1)!
2!1!
k
n

3!
2!1!

6
2
3
Find the number of paths!
(|  n |  |  k |)!
|  n |!|  k |!

(| 2 |  |  1 |)!
| 2 |!|  1 |!

3!
2!1!

6
3
2
Excel File
Number of Shortest Paths
Number of Shortest Paths at Each Intersection
Notice anything familiar?
Euclidean Geometry VS. Taxicab Geometry
Taxicab
Euclidean
d 
( x 2  x1 )  ( y 2  y 1 )
2
The shortest path
is a straight line.
2
Geometry
Distance Formula
Shortest Path(s)
Coordinate Plane
Only one shortest path
Points
Lines
d T  | x 2  x1 |  | y 2  y 1 |
The shortest path(s) are the
path(s) where the least
number of blocks are travel
to get from one point to the
next.
Usually has multiple
shortest paths
Number of paths
found with the
formula  n  k  ( n  k )!


k
 

n! k !
Foldable
Amazing Race
1.
2.
3.
4.
5.
Label the cover of your booklet with your name and group color.
Each group should choose a runner.
The runner must present the correct answer and show work to get the next question.
At the end of the race, the runner will turn in everyone's work.
All booklets must be complete in order to be the winner and receive your prize.