Adventures in Non-Euclidean

Distance

Ricky BobinchuckTashauna ThompsonTosha Pelfrey

How would you find the shortest distance between the two points on the two dimensional plane below?

Pythagorean Theorem?

525

25

169

43

2

2

222

222

c

c

c

cba

The distance between the two points is 5 units.

5 un

its

The Distance Formula

The shortest distance between two points is a

straight line.

The distance formula finds the shortestdistance between two points when working

in atwo dimensional plane.

Nope, in Euclidean Geometry only one path is the length of the shortest distance.

In Euclidean Geometry, can you findanother path that is equal to theshortest distance?

Get Real!When you are traveling in a car theshortest distance between two

locationsis still a straight line, but this path

maynot 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 DistanceShortest path between 2 points• least number of blocks a taxi must travel along streets•points at intersections (x, y)•coordinates are integers

ExampleWhat’s the shortest taxicabdistance from (0, 0) to(1, 2)?

4

Taxicab Distance Activity

|||| 1212 yyxxdT GeoGebra File 1.

2. Taxicab Distance Formula

Taxicab Geometry in Middle School

Gangster Grid GameThe objective of this game is to see who can catch the gangster first.

Rules1.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

▫ Is there only “one” shortest path between two points????

|||| 1212 yyxxdT

How many Paths can you find?

How many Paths are there?Is the number of Paths related to the distance?

How many pathscan you findbetween pointsA(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 ; where m is the distance between two points:

),( kmC

!)!(

!

kkm

m

k

m

m=distanceWe defined the distance between two points earlier as the

horizontal change (x) plus the vertical change (y).

x

y

n

k

Algebra!•Now we can define distance in terms of n

and k, where n represents our x coordinate and k our y.

•If

•Substituting n+k for the distance m we have:

!!

)!(

!)!(

)!(

kn

kn

kkkn

kn

k

kn

!)!(

!

kkm

m

k

m

Let’s check it out…

How many paths did you say we can find for points A and B?

!!

)!(

!)!(

)!(

kn

kn

kkkn

kn

k

kn

62

12

2

34

!2!2

!234

!2!2

!4

!2!2

)!22(

2

22

Answers

Will that always work?

!!

)!(

kn

kn4

6

24

!3!1

!4

!3!1

)!31(

Do you notice anything?

k

n

!!

)!(

kn

kn3

2

6

!1!2

!3

!1!2

)!12(

Find the number of paths!

|!|!||

|)!||(|

kn

kn3

2

6

!1!2

!3

|!1|!|2|

|)!1||2(|

Excel File

Number of Shortest Paths

Number of Shortest Paths at Each Intersection

Notice anything familiar?

Euclidean Geometry VS. Taxicab Geometry

Distance Formula|||| 1212 yyxxdT

Geometry

Shortest Path(s)The shortest path is a straight line.

The shortest path(s) are the path(s) where the least number of blocks are traveled to get from one point to the next.

Only one shortest path

Usually has multipleshortest paths

Number of paths found with the formula

Coordinate Plane

Points

Lines

Euclidean Taxicab

212

212 )()( yyxxd

!!

)!(

kn

kn

k

kn

Foldable

Amazing Race

1. Label the cover of your booklet with your name and group color.2. Each group should choose a runner.3. The runner must present the correct answer and show work to get the next question.4. At the end of the race, the runner will turn in everyone's work. 5. All booklets must be complete in order to be the winner and receive your prize.