Rare Earth hypothesis: Difference between revisions
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In 1932, [[George David Birkhoff|G. D. Birkhoff]] created a set of four [[postulate]]s of [[Euclidean geometry]] sometimes referred to as '''Birkhoff's axioms'''. These postulates are all based on basic [[geometry]] that can be confirmed experimentally with a [[Vernier scale|scale]] and [[protractor]]. Since the postulates build upon the [[real number]]s, the approach is similar to a [[model theory|model]]-based introduction to Euclidean geometry. Other often-used axiomizations of plane geometry are [[Hilbert's axioms]] and [[Tarski's axioms]]. | |||
Birkhoff's axiom system was utilized in the secondary-school text ''Basic Geometry'' (first edition, 1940; see References). | |||
Birkhoff's axioms were also modified by the [[School Mathematics Study Group]] to provide a new standard for teaching high school geometry, known as [http://userpages.umbc.edu/~rcampbel/Math306/Axioms/SMSG.html SMSG axioms]. | |||
==Postulates== | |||
'''Postulate I: Postulate of Line Measure'''. | |||
A set of points {''A, B'', ...} on any line can be put into a 1:1 correspondence with the [[real number]]s {''a, b'', ...} so that |''b'' − ''a''| = ''d''(''A, B'') for all points ''A'' and ''B''. | |||
'''Postulate II: Point-Line Postulate'''. | |||
There is one and only one line, ''ℓ'', that contains any two given distinct points ''P'' and ''Q''. | |||
'''Postulate III: Postulate of Angle Measure'''. | |||
A set of rays {''ℓ, m, n'', ...} through any point ''O'' can be put into 1:1 correspondence with the real numbers ''a'' (mod 2''π'') so that if ''A'' and ''B'' are points (not equal to ''O'') of ''ℓ'' and ''m'', respectively, the difference ''a''<sub>''m''</sub> − ''a''<sub>''ℓ''</sub> (mod 2π) of the numbers associated with the lines ''ℓ'' and ''m'' is <math>\angle</math>''AOB''. Furthermore, if the point ''B'' on ''m'' varies continuously in a line ''r'' not containing the vertex ''O'', the number ''a''<sub>''m''</sub> varies continuously also. | |||
'''Postulate IV: Postulate of Similarity'''. | |||
Given two triangles ''ABC'' and ''A'B'C' '' and some constant ''k'' > 0, ''d''(''A', B' '') = ''kd''(''A, B''), ''d''(''A', C' '') = ''kd''(''A, C'') and <math>\angle</math>''B'A'C' '' = ±<math>\angle</math>''BAC'', then ''d''(''B', C' '') = ''kd''(''B, C''), <math>\angle</math>''C'B'A' '' = ±<math>\angle</math>''CBA'', and <math>\angle</math>''A'C'B' '' = ±<math>\angle</math>''ACB''. | |||
==See also== | |||
* [[Foundations of geometry]] | |||
==References== | |||
*Birkhoff, George David. 1932. "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)," ''[[Annals of Mathematics]] 33''. | |||
*Birkhoff, George David and Ralph Beatley. 1959. ''Basic Geometry'' 3rd ed. Chelsea Publishing Co. [Reprint: American Mathematical Society, 2000. ISBN 978-0-8218-2101-5] | |||
[[Category:Axiomatics of Euclidean geometry]] | |||
[[Category:Elementary geometry]] | |||
Revision as of 04:31, 29 January 2014
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry. Other often-used axiomizations of plane geometry are Hilbert's axioms and Tarski's axioms.
Birkhoff's axiom system was utilized in the secondary-school text Basic Geometry (first edition, 1940; see References). Birkhoff's axioms were also modified by the School Mathematics Study Group to provide a new standard for teaching high school geometry, known as SMSG axioms.
Postulates
Postulate I: Postulate of Line Measure. A set of points {A, B, ...} on any line can be put into a 1:1 correspondence with the real numbers {a, b, ...} so that |b − a| = d(A, B) for all points A and B.
Postulate II: Point-Line Postulate. There is one and only one line, ℓ, that contains any two given distinct points P and Q.
Postulate III: Postulate of Angle Measure. A set of rays {ℓ, m, n, ...} through any point O can be put into 1:1 correspondence with the real numbers a (mod 2π) so that if A and B are points (not equal to O) of ℓ and m, respectively, the difference am − aℓ (mod 2π) of the numbers associated with the lines ℓ and m is AOB. Furthermore, if the point B on m varies continuously in a line r not containing the vertex O, the number am varies continuously also.
Postulate IV: Postulate of Similarity. Given two triangles ABC and A'B'C' and some constant k > 0, d(A', B' ) = kd(A, B), d(A', C' ) = kd(A, C) and B'A'C' = ±BAC, then d(B', C' ) = kd(B, C), C'B'A' = ±CBA, and A'C'B' = ±ACB.
See also
References
- Birkhoff, George David. 1932. "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)," Annals of Mathematics 33.
- Birkhoff, George David and Ralph Beatley. 1959. Basic Geometry 3rd ed. Chelsea Publishing Co. [Reprint: American Mathematical Society, 2000. ISBN 978-0-8218-2101-5]