Absolute geometry: Difference between revisions
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absolute geometry assumes that lines do not loop, but they do in elliptical geometry |
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'''Absolute geometry''' is a [[geometry]] that does not assume the [[parallel postulate]] or any of its alternatives. Its theorems are therefore true in [[non-Euclidean geometry|non-Euclidean geometries]], such as [[hyperbolic |
'''Absolute geometry''' is a [[geometry]] that does not assume the [[parallel postulate]] or any of its alternatives. Its theorems are therefore true in [[non-Euclidean geometry|non-Euclidean geometries]], such as [[hyperbolic geometry]], as well as in [[Euclidean geometry]]. In [[Euclid's Elements]], the first 28 Propositions avoid using the parallel postulate, and therefore can be included in absolute geometry. It is sometimes referred to as '''neutral geometry''', as it is neutral with respect to the parallel postulate. |
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Absolute geometry is an example of an incomplete postulational system. Consider the statement "The sum of the angles in every triangle is equal to two right angles". This is not provable in absolute geometry, because if it was, it would be true in hyperbolic geometry, and the sum of the angles in a hyperbolic triangle is less than two right angles. However, the negation of the statement, that there exists a triangle whose angles don[[']]t add up to two right angles, is not provable either, because if it was, it would be provable in Euclidean geometry, and the sum of the angles in Euclidean geometry [[is]] always [[two]] right angles. Therefore this proposition is undecidable in absolute geometry. |
Absolute geometry is an example of an incomplete postulational system. Consider the statement "The sum of the angles in every triangle is equal to two right angles". This is not provable in absolute geometry, because if it was, it would be true in hyperbolic geometry, and the sum of the angles in a hyperbolic triangle is less than two right angles. However, the negation of the statement, that there exists a triangle whose angles don[[']]t add up to two right angles, is not provable either, because if it was, it would be provable in Euclidean geometry, and the sum of the angles in Euclidean geometry [[is]] always [[two]] right angles. Therefore this proposition is undecidable in absolute geometry. |
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Revision as of 03:29, 19 August 2006
Absolute geometry is a geometry that does not assume the parallel postulate or any of its alternatives. Its theorems are therefore true in non-Euclidean geometries, such as hyperbolic geometry, as well as in Euclidean geometry. In Euclid's Elements, the first 28 Propositions avoid using the parallel postulate, and therefore can be included in absolute geometry. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate.
Absolute geometry is an example of an incomplete postulational system. Consider the statement "The sum of the angles in every triangle is equal to two right angles". This is not provable in absolute geometry, because if it was, it would be true in hyperbolic geometry, and the sum of the angles in a hyperbolic triangle is less than two right angles. However, the negation of the statement, that there exists a triangle whose angles don't add up to two right angles, is not provable either, because if it was, it would be provable in Euclidean geometry, and the sum of the angles in Euclidean geometry is always two right angles. Therefore this proposition is undecidable in absolute geometry.
See also
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