Taxicab Geometry (A Story of Alex and Blake)Taxicab Geometry, An Adventure in Non-Euclidean Geometry...

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TaxicabGeometry(A Story ofAlex andBlake)

DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

NewApartment(on a budget)

NewApartment(almost fair)

The By-Pass

The Break-Up

The End

Taxicab Geometry(A Story of Alex and Blake)

Don Hickethier

Flathead Valley Community College

MCTM Conference, MissoulaOctober 19, 2017

dhicketh@fvcc.edu1 / 18

DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Distance Between Alex and Blake

Alex and Blake met at a local coffee shop, The DailyDose.After some small talk they realize they live in the same area.

Find the distance they each walk from their apartments to TheDailyDose.

Find the distance between their apartments.

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Distance

The Taxicab distance, dT , between two points is the number ofvertical and horizontal units between the two points.

dT (A,D) =

dT (B,D) =

dT (A,B) =

Are A, B and D collinear?

Does the route matter?

How many paths are there from Blake’s apartment to the DailyDose? (Another talk on Pascal’s Triangle.)

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Distance

dT (A,D) =

dT (B,D) =

dT (A,B) =

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Distance

dT (A,D) =

dT (B,D) =

dT (A,B) =

3 / 18

DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Distance

dT (A,D) =

dT (B,D) =

dT (A,B) =

3 / 18

DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Distance

dT (A,D) =

dT (B,D) =

dT (A,B) =

How many paths are there from Blake’s apartment to the DailyDose?

(Another talk on Pascal’s Triangle.)

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Distance

dT (A,D) =

dT (B,D) =

dT (A,B) =

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Midpoint Between Alex and Blake

After many conversations over coffees Alex and Blake realizethey are quite compatible and both very broke. They decide toget an apartment together. To be totally fair to each otherthey decide they need to find an apartment that is equaldistance from each apartment.

Find all of the possible locations to look for an apartment.

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Midline

The midline is the set of points equidistant from two givenpoints.

d(M,A) = d(M,B)

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Equal Distance from the The Comfy Complex

After much searching they settle on a place with a much bettername than facilities, The Comfy Complex. Again in an effort tobe totally fair to one another they decide to look for new jobsthat are equal distance from the Complex.

Find all locations that are 4, 8, 12 and 16 units from C.

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Circle

A circle is a set of points that are equidistance from a fixedpoint.

d(P,C) = r

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Total Distance to Work

Ironically Alex and Blake found jobs at Almost Antiques andthe Busy Bakery, just below their old apartments. Now thatthey both have jobs they decide they can look for morefavorable housing than the Comfy Complex. Still on a bit of abudget they decide that the total combined distance they travelto work must be no more than 30 units.

Find the boundary of the search region for a new apartment.

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Ellipse

An ellipse is a set of points such that the sum of the distancesfrom two fixed points is a constant.

d(P,A) + d(P,B) = s

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Difference Traveled

Not having much luck in the house search Alex and Blakedecided that as long as one person did not have to travel 6more units to work than the other, that would be okay.

Find the new boundary for their search.

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Hyperbola

A hyperbola is a set of points such that the difference of thedistances from two fixed points is a constant.

|d(P,A)− d(P,B)| = k

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Between Busy Bakery and the By-Pass

Still not having much luck in the house search they realize thatthe new by-pass has just been completed making it much easierto get to Almost Antiques. This time Alex and Blake decidethat as long as the distance from the Busy Bakery and theby-pass is the same that would be pretty much fair.

How far is it from the Busy Bakery to the by-pass?

Find the new possible search region.

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Between Busy Bakery and the By-Pass

Still not having much luck in the house search they realize thatthe new by-pass has just been completed making it much easierto get to Almost Antiques. This time Alex and Blake decidethat as long as the distance from the Busy Bakery and theby-pass is the same that would be pretty much fair.

How far is it from the Busy Bakery to the by-pass?

Find the new possible search region.

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab Parabola

A parabola is a set of points equal distances from a fixed pointand a line.

d(P,A) = d(P,L)

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Divided SpaceThree Ways

After weeks of fruitless searches disagreements on the “fairest”way to locate a house begin to surface. Alex and Blake realizethey really had nothing in common other than coffee. Theydecide to split up. Not wanting to accidentally run into eachother at the Daily Dose they decide they will each look forhousing no more than midway between their work, the DailyDose and the others work.

Find the new areas they may isolate themselves for their finalhousing searches.

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Thanks for Playing

Taxicab Geometry, An Adventure in Non-Euclidean GeometryEugene F. Krause, Dover

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Taxicab is Non-EuclideanSAS fails

Given a 1-1 correspondence between the vertex set of twotriangles. If two sides and the included angle of the firsttriangle are congruent to the corresponding parts of the secondtriangle, then the correspondence is a congruence.

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Applications and Further Study

City and Emergency Planning

Amazon robots

3 Dimensional Taxicab

Trigonometry in Taxicab

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DonHickethier

NewAcquaintances

Meeting HalfWay

Finding Jobs

The By-Pass

The Break-Up

The End

Euclidean vs Taxicab

dE(A,D) =√

(a1 − b1)2 + (a2 − b2)2

dT (A,D) = |a1 − b1|+ |a2 − b2|

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Taxicab Geometry (A Story of Alex and Blake)Taxicab Geometry, An Adventure in Non-Euclidean Geometry...

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