What are the uses of taxi geometry

Taxi geometry T-points T-segment T-straight line Condition for permissible distances

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1 Taxi geometry 2 We have already pointed out in Sect. 1.1 that Hermann Minkowski () had the idea for a taxi geometry in which he introduced a model for the primitive terms point and straight line that deviated from Euclidean geometry and another Distance term used. In this chapter we follow Minkowski's footsteps and embark on an interesting journey of discovery. The starting point of our considerations is a city with a uniform street grid in the form of a square grid in which a taxi can move from one intersection to another without detours. We understand the intersections of this T-city (taxi city) as T-points and the route of the taxi between the T-points A T and B T as a T-segment, which lies on a T-straight line g T. Fig. 2.1 shows three of the possible routes the taxi can travel between T-points A T and B T, i.e. H. three T-segments between these T-points. If we understand Fig. 2.1 and all the following figures in this chapter as part of the city map of the T-City, then we can formulate the condition for permissible routes of the taxi in such a way that it is only allowed to move horizontally and vertically in one direction. When driving from A T to B T, the taxi may only move in the east and north, when driving from B T to A T only in the west and south. If, as agreed, we regard every permissible path between the T-points A T and B T as a T-segment, then we discover the first strange peculiarities: Two different T-points can be connected to one another by several T-segments, i.e. That is, several T-lines exist between two different T-points. This property already deviates from Springer-Verlag GmbH Germany 2017 J. Wagner, Insights into Euclidean and non-Euclidean geometry, DOI / _2 97

2 98 2 Taxi geometry from an axiom of E geometry (Euclidean geometry), therefore T geometry is a non-Euclidean geometry. T-segments can look different than E-segments (segments in Euclidean geometry). The T-points are not close because they align with the grid. All T-segments between two different T-points have the same length. If we assign the length 1 to each square side of the grid, then all T-sections between the T-points AT and BT in Fig.2.1 have the length 7, since the taxi has to drive through four units in the horizontal direction and three units in the vertical direction . The following question is obvious: How many T-segments are there between two T-points that are separated from each other by h horizontal and v vertical units? A suitable counting strategy is already required in Fig. 2.1 to determine this number, because there are considerably more T-segments than we have drawn. In Fig. 2.2 we have entered the number of paths under all T-points that lead from A T to the respective T-point when the taxi moves in the direction of B T. Fig. 2.1 Some T-sections between T-points A T and B T B T A T Fig. 2.2 Number of T-sections between T-points A T and B T BT A T

3 2 Taxi geometry 99 We determine the following regularities from Fig. 2.2 (the second property results from systematic counting initially as a conjecture that can be proven by complete induction): For all T-points that are on the horizontal or the vertical T. -Laying straight with respect to the starting point, exactly one path leads in each case. The number of paths from the starting point to a northeastern inner T-point of the grid is equal to the sum of the paths from the starting point to the western and southern neighboring points of the T-point under consideration. For example, the number of paths from the starting point A T to the destination point B T is the sum of 20 and 15, since exactly 20 and 15 paths lead to the western or southern neighboring point of B T. The regularities discovered in Fig.2.2 also come across in other contexts: when binomials are raised to the power of the coefficients of the summands, the relationships are analogous to those when determining the number of paths in T-geometry. (a + b) 1 = 1 a + 1 b (a + b) 2 = 1 aab + 1 b 2 (a + b) 3 = 1 aa 2 b + 3 abb 3 (a + b) 4 = 1 aa 3 b + 6 a 2 babb 4 All outer coefficients are equal to 1. Each inner coefficient is the sum of two coefficients from the term of the preceding power. For example, the second coefficient results in the fourth power as the sum of the first and second coefficient of the third power: 4 = This coefficient indicates the number of possibilities to form the term a 3 b: a a a b, a a b a, a b a a, b a a a. In fact, in Fig. 2.2 we discover Pascal's triangle along the secondary diagonal of the lattice, which also contains the coefficients when binomials are raised to the power. With a Bernoulli chain of length four there are four possibilities to achieve the result 3 hits and 1 rivet, since the sequence of the test outcomes does not matter: TTTN, TTNT, TNTT, NTTT. The number of these possibilities is equal to the number of possibilities to get the term a 3 b when building the fourth power of a binomial. Both the number of possibilities of obtaining the term a k b n k when raising the binomial n-fold (a + b) and the number of k hits and (n k) rivets in a Bernoulli chain of length n is

4 100 2 Taxi geometry () n = k () n = n k n !. The number of T-segments (n k) is analogous to this! k! between two T-points that are separated by h horizontal and v vertical units: h + v h + v () () =. The already determined value () () = = 7 results for the number of distances between the T points A T and B T! 4 3 3! 4! = = 35. After these exhausting number determinations we turn back to geometric questions of T-geometry. Based on the representation of two T-lines g T and h T in Fig. 2.3, we find the following interesting peculiarities: The T-lines g T and h T have six common T-points. With the T-delta A T B T we discover a geometric object that has no equivalent in Euclidean geometry (in spherical geometry we will again encounter a delta, which, however, looks different from the T-delta). We are certainly curious whether the axiom of parallels in E geometry also applies to T geometry, Fig. 2.4 helps us to clarify this fact. The axiom of parallels in Euclidean geometry does not apply in T geometry, since we find several parallels to a T line g T through a T point P T outside of this T line. These parallels have a different appearance than in E geometry, but they all run through the T point P T and they each have no point in common with the T line g T. Fig. 2.3 T-straight lines B T ht AT g T

5 2 Taxi Geometry 101 Fig. 2.4 T-parallels to the T-straight line g T through T-point P T P T g T Fig. 2.5 Example 1 for a T-center perpendicular A T B T Another interesting field of investigation are geometric locations as sets of points with a characteristic property. We then consider two different geometric locations. First Geometric Location Set of all T-points that are equidistant from two given T-points. In e-geometry, the geometrical location sought is referred to as a perpendicular, perpendicular or line symmetry. Fig. 2.5 shows a T-center vertical. Our result in Fig. 2.5 looks like a perpendicular to the E-geometry (in contrast to E-points, however, the T-points are not located

6 102 2 taxi geometry close). It seems that this geometrical place does not have any special features. But be careful: So far we have only considered one special case in which the two T-points A T and B T lie on a special T-straight line! Even a small variation holds the first surprise in store: If the two T-points A T and B T are still on a horizontal T-line, but have an odd-numbered distance from one another, then there is no T-center perpendicular at all! So forewarned, we continue to vary the mutual position of the T-points A T and B T. When looking at Figs. 2.6 and 2.7 we notice that the first geometric location we have chosen has very interesting properties (in Fig. 2.7 we have drawn points equidistant from A T and B T in the same color). Second Geometric Location Set of all T-points that are equidistant from a given T-point. In planar E geometry, the geometrical location sought is a circle. Fig. 2.8 shows a T-circle around the T-point MT with T-radius r T = 3. In the meantime, our astonishment is certainly limited that a circle looks different in T-geometry than in E-geometry . The circumference of a T-circle is formed by four T-segments, for the representation of which we have several options. In Fig. 2.9 we have decided on a variant that is somewhat similar to an E-circle. Each of the four T-segments, each representing a quarter of the circumference of a T-circle with T-radius r T, has the T-length 2 r T, since each of these T-segments connects two T-points, which are located in horizontal as well as vertical direction Fig. 2.6 Example 2 for a T-center vertical BTAT

7 2 Taxi Geometry 103 Fig. 2.7 Example 3 for a T-center perpendicular B T A T Fig. 2.8 Differentiating T-circle M T by r T units. With this we can calculate the value of the circle number π T in T-geometry: π T = u T 2 r T = 4 (2 r T) 2 r T = 4. Also in T-geometry this results for every T-circle the same circle number, but its value is simply 4, i. that is, the hard-to-understand

8 104 2 Taxi-Geometry Fig. 2.9 Circumference of a T-circle M T There is no transcendence of the circle number in the E-geometry in the T-geometry. At the beginning of this chapter, we stated that there is a different concept of distance in T geometry than in E geometry. We have already taken this into account in our investigations by taking into account that the taxi can only drive along streets. In e-geometry we are used to the fact that there are also diagonal connections between two points. If we introduce a Cartesian coordinate system in each of the two geometries, then we can calculate the distance between two points using the point coordinates: In E geometry, according to the Pythagorean theorem, we get d (a, B) = (x A x B) 2 + (y A y B) 2. In T-geometry we get d (a T, BT) = x TA x TB + y TA y TB. Both distance functions have the properties of a metric, since they are positively definite because of d (x, y) 0 and d (x, y) = 0 x = y, and because d (x, y) = d (y, x) they are symmetrical , because of d (x, y) d (x, z) + d (z, y) satisfy the triangle inequality. Therefore, the T-geometry is sometimes characterized in the literature as a geometry in which the taxi metric applies.

9 2 Taxi geometry 105 Comment Sometimes the taxi metric is also referred to as the city block metric, Mannheimer metric or Manhattan metric, since the city of Mannheim and the Manhattan district in New York have a uniform street grid in the form of a square grid. The T-geometry invites further considerations, e.g. B. through the investigation of further geometric locations such as ellipses, parabolas or hyperbolas, the investigation of the properties of n-corners, the use of a triangular grid instead of the square grid, the transition from the plane into space. We refrain from working on these questions, since in this short chapter we primarily dealt with the thematization of a non-Euclidean geometry, which was possible with astonishingly little effort. With that we are ready to get to know other non-Euclidean geometries.

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