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Excursions in Combinatorial in Taxicab Geometry
Math Fest 2015
John Best
Summit University of Pennsylvania [email protected]
> 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?
Fig. 1. Unit Circle in the Taxicab plane
What is Taxicab Geometry?
Look for Taxicab versions of theorems from combinatorial Euclidean Geometry.
Motivation
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
Fig. 2. Infinitely many (almost linear) points in Taxicab plane with integer distances.
Fig. 3. Infinitely many (very non-‐linear) points in Taxicab plane with integer distances. There seems to be no Taxicab version of Erdös-‐Anning.
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
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
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
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.
Proof:
John Best, Odd Distances in the Taxicab Plane, In Preparation.
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
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
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. ∎
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.
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. ∎
Fig. 5. Unit-‐Distance K2,3 in Taxicab plane (black edges, green & red vertices).
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
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
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
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
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.
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.
Fig.6. Borsuk decomposition of Taxi Unit circle. Four parts of diameter 1.
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
