Taxicab Geometry is an easily described example of a non-Euclidean metric space. Many theorems in Euclidean Geometry have analogues in Taxicab geometry. This presentation begins by looking briefly at the some theorems in Euclidean combinatorial geometry. Among these will be some results on Erdös type distance problems, Borsuk’s Theorem, and a few others. The Taxicab versions of these results are then examined. The presentation is primarily expository.

1.
Excursions
in
Combinatorial
in
Taxicab
Geometry
Math
Fest
2015
John
Best
Summit
University
of
Pennsylvania
jbest@summitu.edu

2.
>
Points
–
The
coordinate
plane
ℝ!
.
>
Distance
Function
–
𝑑! 𝑥!, 𝑦! , 𝑥!, 𝑦! = 𝑥! − 𝑥! + 𝑦! − 𝑦!
>
This
is
the
Taxicab
plane
–
denoted
by
ℝ!
, 𝑑! .
>
Euclidean
plane
is
denoted
by
ℝ!
, 𝑑!
𝑑! 𝑥!, 𝑦! , 𝑥!, 𝑦! = (𝑥! − 𝑥!)! + (𝑦! − 𝑦!)!
What
is
Taxicab
Geometry?

3.
Fig.
1.
Unit
Circle
in
the
Taxicab
plane
What
is
Taxicab
Geometry?

4.
Look
for
Taxicab
versions
of
theorems
from
combinatorial
Euclidean
Geometry.
Motivation

5.
Theorem
1
(Erdös-­‐Anning)
If
an
infinite
set
of
points
in
the
Euclidean
plane
determines
only
integer
distances,
then
all
the
points
lie
on
a
straight
line.
Proof
Reference:
Ross
Honsberger,
Mathematical
Diamonds
,
MAA,
2003

6.
Fig.
2.
Infinitely
many
(almost
linear)
points
in
Taxicab
plane
with
integer
distances.

7.
Fig.
3.
Infinitely
many
(very
non-­‐linear)
points
in
Taxicab
plane
with
integer
distances.
There
seems
to
be
no
Taxicab
version
of
Erdös-­‐Anning.

8. Theorem
2
a)
There
are
no
four
points
in
the
Euclidean
plane
such
that
the
distance
between
each
pair
is
an
odd
integer.
b)
The
maximum
number
of
odd
integral
distances
between
𝑛
points
in
ℝ!
, 𝑑!
is
𝑛!
3
+
𝑟(𝑟 − 3)
6
where
𝑟 = 1, 2, 𝑜𝑟 3
and
𝑛 ≡ 𝑟 𝑚𝑜𝑑 3

9. Proof
Reference:
a)
Jiri
Matousek,
Thirty-­‐three
Miniatures-­‐Mathematical
and
Algorithmic
Applications
of
Linear
Algebra,
American
Mathematical
Society,
2012
b)
L.
Piepmeyer,
The
Maximum
Number
of
Odd
Integral
Distances
Between
Points
in
the
Plane,
Discrete
and
Computational
Geometry
16
(1996),
pp.
113-­‐115

10.
Fig.4.
Four
points
in
Taxicab
plane
with
pairwise
odd
distances.
(0,0),
(5,0),
(.5,
2.5),
(.5,
-­‐8.5)
6
distinct
distance
Theorem
2a
fails
in
Taxicab
plane

11. Theorem
2T
a)
There
are
no
five
points
in
the
Taxicab
plane
such
that
the
distance
between
any
two
is
an
odd
integer.
b)
The
maximum
number
of
odd
integral
distances
between
𝑛
points
in
ℝ!
, 𝑑!
is
3𝑛!
8
+
𝑟(𝑟 − 4)
8
where
𝑟 = 1, 2, 3, 𝑜𝑟 4
and
𝑛 ≡ 𝑟 𝑚𝑜𝑑 4.

12. Proof:
John
Best,
Odd
Distances
in
the
Taxicab
Plane,
In
Preparation.

13.
Def.
1:
A
graph
is
an
ordered
pair
G=(V,
E)
consisting
of
a
nonempty
set
V
of
vertices
together
with
a
set
E
of
unordered
pairs
of
distinct
vertices
called
edges.
A
complete
graph
on
|V
|=m
vertices
is
a
graph
in
which
every
pair
of
distinct
vertices
is
connected
by
a
unique
edge.
A
complete
graph
is
denoted
by
K
m.
A
Little
Graph
Theory

14.
Def.
2:
A
complete
bipartite
graph
is
a
graph
whose
vertices
can
be
partitioned
into
two
subsets
V
and
W
such
that
no
edge
has
both
endpoints
in
the
same
subset,
and
every
possible
edge
that
could
connect
vertices
in
different
subsets
is
part
of
the
graph.
If
|V|=m
and
|W|=n,
we
denote
the
graph
by
K
m,n.
Def.
3:
A
unit
distance
graph
is
a
set
of
points
V
in
a
metric
space
with
an
edge
connecting
two
vertices
iff
the
distance
between
the
points
equals
1.
A
Little
Graph
Theory

15. Theorem
3
The
complete
graph
K4
and
the
complete
bipartite
graph
K2,3
are
not
unit
distance
graphs
in
ℝ!
, 𝑑! .
Proof:
That
K2,3
is
not
a
unit
distance
graph
in
ℝ!
, 𝑑!
is
a
consequence
of
that
fact
that
two
distinct
unit
circles
can
intersect
in
at
most
two
points.
To
see
that
K4
is
not,
consider
an
equilateral
triangle
of
side
length
1,
and
show
that
there
cannot
be
a
4th
point
at
distance
1
from
the
three
vertices.
∎

16. Theorem
3T
The
complete
graph
K4
and
the
complete
bipartite
graph
K2,3
are
unit
distance
graphs
in
ℝ!
, 𝑑! .
The
complete
graph
K5
is
not
a
unit
distance
graph
in
ℝ!
, 𝑑! .
Proof:
The
set
of
points
0,0 , 1,0 ,
!
!
,
!
!
,
!
!
,
!!
!
show
that
K4
is
a
unit
distance
graph
in
the
Taxicab
plane.

17. To
see
that
K2,3
is
a
unit
distance
graph
in
ℝ!
, 𝑑!
consider
the
sets
𝑉 = 0,0 , 1,1
and
𝑊 =
!
!
,
!
!
,
!
!
,
!
!
,
!
!
,
!
!
Calculations
show
that
the
Taxicab
distance
from
any
point
in
V
to
any
point
in
W
equals
1,
and
the
Taxicab
distance
between
points
in
V
and
W
is
not
1.
The
last
assertion
follows
from
Theorem
2T.
∎

18.
Fig.
5.
Unit-­‐Distance
K2,3
in
Taxicab
plane
(black
edges,
green
&
red
vertices).

19. Def.
4:
Let
S
be
a
bounded
set
of
points
in
either
ℝ!
, 𝑑!
or
ℝ!
, 𝑑! .
The
diameter
of
S
is
the
number
𝛿 = 𝑠𝑢𝑝 𝑑!(𝑎, 𝑏) 𝑎, 𝑏 ∈ 𝑆

20. Theorem
4
(Jung’s
Theorem)
Every
finite
set
of
points
in
ℝ!
, 𝑑!
with
diameter
𝛿
can
be
enclosed
in
a
circle
with
radius
𝑟 ≤
!
!
.
Proof
Reference:
Hans
Rademacher
and
Otto
Toeplitz,
The
Enjoyment
of
Mathematics,
Dover
Publications,
1990

21. Theorem
4T
Every
finite
set
of
points
in
ℝ!
, 𝑑!
with
diameter
𝛿
can
be
enclosed
in
a
Taxicab
circle
with
radius
𝑟 ≤
!
!
.
Proof
Reference:
V.
Boltyanski
and
H.
Martini,
Jung’s
Theorem
for
a
pair
of
Minkowski
Spaces,
Adv.
Geom.
6
(2006),
pp.
645-­‐650

22.
Let
F
be
a
plane
figure
with
diameter
𝛿
.
The
Borsuk
Conjecture
in
the
plane
asks
for
the
fewest
number
of
pieces
F
can
be
cut
into
so
that
each
piece
has
diameter
less
than
𝛿
.
We
denote
this
number
by
a(F)
Borsuk
Conjecture
in
the
Plane

23. Theorem
5
(Borsuk)
For
any
figure
F
in
ℝ!
, 𝑑! ,
with
diameter
𝛿,
𝑎(𝐹) ≤ 3
Proof
Reference:
V.
Boltyanski
and
I.
Gohberg,
The
Decomposition
of
Figures
into
Smaller
Parts,
The
University
of
Chicago
Press,
1980.

24. Theorem
5T
For
any
figure
F
in
ℝ!
, 𝑑! ,
with
diameter
𝛿,
𝑎(𝐹) ≤ 4
Equality
is
obtained
iff
the
convex
hull
of
F
is
a
dilation
of
the
Taxi
unit
circle
by
a
factor
of
!
!
(and
possibly
a
translation.)
Proof
Reference:
Same
as
Theorem
5.

25.
Fig.6.
Borsuk
decomposition
of
Taxi
Unit
circle.
Four
parts
of
diameter
1.

26.
1.
Investigate
these,
and
other
Euclidean
theorems,
in
higher
Taxi
dimensions
ℝ!
, 𝑑! , 𝑛 ≥ 3.
2.
Investigate
these,
and
other
Euclidean
theorems,
in
other
metrics,
such
as
Chinese
Checker
Metric,
Generalized
Absolute
Value
Metric,
etc.
Future
Research