axiomatic geometry

Axiomatic geometry can be traced back to the time of Euclid. In his bookElements, written back in the 300’s B.C., Euclid gave five rules, or postulates, describing howpoints, lines, line segments, etc behave as they are ordinarilyperceived. Based on these postulates, he set out to prove hundreds ofproperties. Today, these properties are under the field of study known asplane Euclidean geometry, more popularly known as high school geometry.The systematic and axiomatic approach to proving geometric facts is what makes his Elements one of the most important contributions to mathematics.

One key feature of Euclid’s axioms is the abundance of what are known today asthe undefined terms. Geometric notions such as points, lines, andcircles are mentioned in his axioms but never clearly defined. For example, Euclidcalled a “point” as “that which has no part”. But what the meaning of “part” wasnever clarified. It is because of this abundance of undefined terms, Euclid’s postulates bytoday’s mathematical standards lack rigor. While some undefined terms give no seriousproblems, others create holes in proofs which are unacceptable. In the late 19th century,David Hilbert published his classic Foundations of Geometry, putting Euclid’s postulateson a more solid ground. In the book, he broke down Euclid’s postulates into fivegroups of axioms:

1.
incidence axioms
2.
order axioms
3.
congruence axioms
4.
continuity axiom
5.
parallel axiom (http://planetmath.org/ParallelPostulate)

These axioms have been shown to be independent of each other, in the sense that no one axiom can beproved from the rest, and consistent, in the sense that no contradictions can be derived from them. Theseaxioms today serve as the foundation of plane Euclidean geometry.

Since Hilbert’s work, the natural next step is to look at geometries that are not Euclidean. In other words,geometric models that lack one or several of the “Euclidean axioms” above. The result has been the manyexotic geometries that, surprisingly, have found applications in other places.

单词	AxiomaticGeometry
释义	axiomatic geometry Axiomatic geometry can be traced back to the time of Euclid. In his bookElements, written back in the 300’s B.C., Euclid gave five rules, or postulates, describing howpoints, lines, line segments, etc behave as they are ordinarilyperceived. Based on these postulates, he set out to prove hundreds ofproperties. Today, these properties are under the field of study known asplane Euclidean geometry, more popularly known as high school geometry.The systematic and axiomatic approach to proving geometric facts is what makes his Elements one of the most important contributions to mathematics. One key feature of Euclid’s axioms is the abundance of what are known today asthe undefined terms. Geometric notions such as points, lines, andcircles are mentioned in his axioms but never clearly defined. For example, Euclidcalled a “point” as “that which has no part”. But what the meaning of “part” wasnever clarified. It is because of this abundance of undefined terms, Euclid’s postulates bytoday’s mathematical standards lack rigor. While some undefined terms give no seriousproblems, others create holes in proofs which are unacceptable. In the late 19th century,David Hilbert published his classic Foundations of Geometry, putting Euclid’s postulateson a more solid ground. In the book, he broke down Euclid’s postulates into fivegroups of axioms: 1. incidence axioms 2. order axioms 3. congruence axioms 4. continuity axiom 5. parallel axiom (http://planetmath.org/ParallelPostulate) These axioms have been shown to be independent of each other, in the sense that no one axiom can beproved from the rest, and consistent, in the sense that no contradictions can be derived from them. Theseaxioms today serve as the foundation of plane Euclidean geometry. Since Hilbert’s work, the natural next step is to look at geometries that are not Euclidean. In other words,geometric models that lack one or several of the “Euclidean axioms” above. The result has been the manyexotic geometries that, surprisingly, have found applications in other places.
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