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FIELD EXTENSIONS AND THE CLASSICAL COMPASS AND STRAIGHT-EDGE CONSTRUCTIONS
"... Abstract. This paper will introduce the reader to field extensions at a rudi-mentary level and then pursue the subject further by looking to its applications in a discussion of some constructibility issues in the classical straight-edge and compass problems. Field extensions, especially their degree ..."
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Abstract. This paper will introduce the reader to field extensions at a rudi-mentary level and then pursue the subject further by looking to its applications in a discussion of some constructibility issues in the classical straight-edge and compass problems. Field extensions, especially
Logic of ruler and compass constructions
- Computability in Europe 2012
, 2012
"... Abstract. We describe a theory ECG of “Euclidean constructive geometry”. Things that ECG proves to exist can be constructed with ruler and compass. ECG permits us to make constructive distinctions between different forms of the parallel postulate. We show that Euclid’s version, which says that under ..."
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Cited by 1 (1 self)
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Abstract. We describe a theory ECG of “Euclidean constructive geometry”. Things that ECG proves to exist can be constructed with ruler and compass. ECG permits us to make constructive distinctions between different forms of the parallel postulate. We show that Euclid’s version, which says
Constructive Geometry
, 2009
"... Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics ” leads to the development of a first-order t ..."
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Cited by 4 (1 self)
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Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics ” leads to the development of a first
FOUNDATIONS OF EUCLIDEAN CONSTRUCTIVE GEOMETRY
"... Abstract. Euclidean geometry, as presented by Euclid, consists of straightedge-andcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics” leads to the development of a first ..."
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first-order theory ECG of “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rather close in spirit to the Elements of Euclid. Using Gentzen’s cut-elimination theorem, we show that when ECG proves an existential theorem, then the things proved to exist can be constructed
A New Look At Euclid's Second Proposition
- The Mathematical Intelligencer
, 1993
"... There has been considerable interest during the past 2300 years in comparing different models of geometric computation in terms of their computing power. One of the most well known results is Mohr's proof in 1672 that all constructions that can be executed with straight-edge and compass can be ..."
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Cited by 1 (1 self)
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There has been considerable interest during the past 2300 years in comparing different models of geometric computation in terms of their computing power. One of the most well known results is Mohr's proof in 1672 that all constructions that can be executed with straight-edge and compass can
Construction
"... This document is a compilation of abstracts of 20 research papers presented at the 51st Annual Meeting of the National Council of Teachers of Mathematics. Six reports concern methods of instruction, eight investigate patterns of learning, three deal with evaluation of attitudes, and three reports co ..."
The Parallel Postulate in Constructive Geometry
, 2009
"... Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics ” leads to the development of a first-order t ..."
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algebraic theories of geometric constructions, it does not have a test-for-equality construction. In previous work [3], we have shown that ECG corresponds well to Euclid’s reasoning, and that when it proves an existential theorem, then the things proved to exist can be constructed by Euclidean ruler-and-compass
Synthesizing geometry constructions
- In PLDI
, 2011
"... In this paper, we study the problem of automatically solving ruler/compass based geometry construction problems. We first introduce a logic and a programming language for describing such constructions and then phrase the automation problem as a program synthesis problem. We then describe a new progr ..."
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Cited by 29 (16 self)
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In this paper, we study the problem of automatically solving ruler/compass based geometry construction problems. We first introduce a logic and a programming language for describing such constructions and then phrase the automation problem as a program synthesis problem. We then describe a new
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