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Kevin's Corner

Taxicab Geometry > Euclid's Axioms

Euclidean geometry is an axiomatic system, meaning all features of the geometry can be derived from a small set of assumptions. When exploring non-Euclidean geometries, it is useful to determine which of the Euclidean axioms, or properties derived from these axioms, is no longer true. Since the time of Euclid, other aximoatic systems have been proposed that are equivalent to Euclid's but are more explicit and make fewer unstated assumptions. The axioms of David Hilbert (1862-1943) are an example.

Taxicab geometry only fails one of the axioms or postulates of Euclidean geometry, but it does so in grand style. In Euclidean geometry, if two sides and the included angle of two triangles are congruent, then the triangles are congruent. This is called the SAS (side-angle-side) property for triangles. (Other congruence properties are ASA and AAS.) This assumption does not hold in taxicab geometry, and one example can prove it fails dramatically. In Figure 1, two sides and all three angles of the two triangles are congruent (see the Angles page for more about taxicab angles). But, the third sides are not congruent. So, taxicab geometry not only fails the SAS property, but even an ASASA property does not guarantee congruence of triangles. (Note that this is the case for both traditional and pure taxicab geometry.)

FIGURE 1: Two triangles that satisfy the property ASASA but are not congruent
(distances are taxicab and angle measures are shown in t-radians).

In taxicab geometry, there is precisely one congruent triangle property. It relies on the fact that even in taxicab geometry, the sum of the angles of a triangle is always the same: 4 t-radians. The only congruent triangle property is SASAS: only if all three sides and two angles of two triangles are congruent will the triangles be congruent.

References
[1] Dawson, Robert J. MacG., Crackpot Angle Bisectors!, Mathematics Magazine, Vol. 80, No. 1 (Feb 2007), pp. 59-64.
[2] Krause, Eugene F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry, 1975.
[3] Thompson, Kevin and Tevian Dray. Taxicab Angles and Trigonometry, The Pi Mu Epsilon Journal, Worcester, MA. Vol. 11, No. 2 (Spring 2000), pp. 87-96.

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