Questions tagged [axiomatic-geometry]
Questions about axiomatic systems for geometry. Use this tag if you're looking for a proof starting directly from some set of axioms (e.g., Hilbert's axioms for Euclidean geometry), or if you have a question about the axioms themselves.
161 questions
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On Line-Plane Perpendicularity in Euclidean Geometry
Due to a course of euclidean geometry that I enrolled to complete my graduation degree, in which we study plane euclidean geometry from the axiomatic point of view, I've decided — for fun! — to try on ...
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Is a mapping that maps parallelograms to parallelograms a collineation (on an affine plane)? [closed]
It is easy to show that in a $(\mathcal{P},\mathcal{L})$ affine plane any collineation maps any parallelogram to a parallelogram. But is it true that if a $\mathcal{P} \to \mathcal{P}$ bijection maps ...
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Parallel-perpendicular proof in purely axiomatic geometry
Günter Ewald's book "Geometry: an introduction" approaches geometry in a purely formal axiomatic way. He defines parallel and perpendicular lines purely axiomatically, without reference to ...
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Why does an arbitrary plane contain three non-collinear points by the Hilbert's axoims?
I'm working through Hilbert's The Foundations of Geometry and I'm stuck on what seems to be a fundamental proof regarding the axioms of incidence.
From Axiom I.3
"There exist at least two points ...
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How to define orientation from the axioms of Euclidean geometry
My question originates from this answer by Kulisty, where it is stated that the notion of orientation can be defined using only the axioms of neutral geometry. I am particularly interested in proving ...
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List of Results Provable in Absolute/Neutral Geometry?
I'm looking for a list of results that can be proven using the axioms of Euclid's Elements, but without using the parallel postulate, either explicitly or implicitly. From my search so far, I've ...
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Which system of axioms is used most often in modern geometry?
We have many different systems of axioms in geometry and from observations we most often use Euclidean ones. Euclid's postulates are insufficient, but the Hilbert system seems overloaded and redundant....
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Questionable ontology in Hilbert's Foundations of Geometry
The theory presented in Hilbert's Foundations of Geometry is clearly a many-sorted theory, since the ontology includes at least two sorts of objects: points (obviously), but also point-sets, i.e. ...
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Is the Euclidean Distance Conceptually Equivalent to the Pythagorean Theorem?
We often hear statements like, "I never use the Pythagorean Theorem after my studies." In response, I sometimes argue—half-jokingly—that if one is familiar with the concept of Euclidean ...
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Artin's Affine Space axioms in higher dimensions
I'm working through a portion of Artin's Geometric Algebra to understand an axiomatic definition of affine spaces which I need for another text I am going to read. The axioms he gives only describe an ...
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Can we prove the converse of Euclid's fifth postulate in neutral (i.e. non-euclidean) geometry?
In this video from Mount St Mary's University (California), the author affirms (and it is also said in the description of the video) that he proves the converse of Euclid's fifth postulate in neutral ...
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Incidence axioms in Euclidean space and their visual representation
My prof. listed 9 axioms of incidence but I will list them group by group since they can be merged:
Every line consists of at least 2 points.
There exists at least one line such that it contains 2 ...
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How axioms of inner product ensure that an instantiation/realization capture notion of angle correctly?
Axiomatic definition of inner product can lead to various instantiations like euclidean inner product or complex inner product or weighted inner product etc.
Whatever the special case, we can be sure, ...
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reflexive property of congruence
While exploring basic geometry with my artist friend, she raised a question that neither of us could answer satisfactorily. Until today I thought that Hilbert's axiomatization of Euclidean geometry is ...
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How does "Pasch axiom" in Tarski geometry relate to the usual Pasch axiom?
In Tarski's axiomatization of planar geometry (see here and there), there is an axiom called "Pasch axiom" and formulated as:
$(Bxuz \land Byvz) \rightarrow \exists a\, (Buay \land Bvax)$
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