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.
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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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Euclid's fourth postulate [closed]
I am currently reading through Roger Penrose's The Road To Reality. A question about Penrose's explanation for the necessity of Euclid's fourth postulate (Chapter 2, pg. 28-29 in pdf). Penrose writes:
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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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Am I incorrectly interpreting axiom I, 3 of Hilbert's Foundations of Geometry?
Context
Hilbert's third axiom of incidence is:
There exist at least two points on a line. There exist at least three points that do not lie on the same line.
(https://en.wikipedia.org/wiki/Hilbert%...
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Absolute Scale and Distance Function in Hyperbolic Geometry
I have a question about intuition on key differences between axiomatic hyperbolic and Euklidean geometries. More precisely I'm pondering about last sentence of following explanations from wikipedia:
[...
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Prove that the union of all lines containing a point A is the plane.
I need to prove this with the knowledge of incidence and order axioms.
Let $X$ be a set with all the points forming all the lines containing the point $A$, and let $Y$ the set of all points in the ...
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Is Pieri's ternary equidistance relation sufficient to axiomatize arbitrary Riemannian geometries?
Robinson [1] proved that given the primitive notion of points, Pieri’s primitive ternary equidistance relation is sufficient to axiomatize Euclidean, elliptic, and hyperbolic geometries. Is Pieri’s ...
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straightedge and compass construction in Hartshorne's Geometry: Euclid and Beyond
Do you think that the following straightedge and compass construction as a single step is valid?
Given line segment AB and a point C on line L, for a given side of C, find point D such that $AB \cong ...
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Is it possible to take S.S.S. Congruence criterion as a postulate and prove S.A.S. and A.S.A. through it?
In all of the treatments of elementary Euclidean geometry which I've seen so far, the section about triangle congruences introduces S.A.S. criterion as the basic postulate from which A.S.A. and S.S.S. ...
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A problem with Hilbert's "Foundations of Geometry" I.2 axiom.
In the Hilbert's Foundation there is a $I.2$ axiom:
Any two distinct points of a straight line completely determine that line; that is, if $AB = a$
and $AC = a$, where $B \not= C$, then is also $BC = ...
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Axiomatizing higher-dimensional geometries
While there are various axiom systems for two and three-dimensional geometries (Hilbert, etc.), it seems not at all clear that this axiomatic approach generalizes well to more than three dimensions. ...
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Axiomatic system that differentiates between straight lines and curved lines
I was thinking about a formal system that involves a set of primitives along with a property that, when applied to the model of planar one-dimensional objects, can distinguish between straight lines ...
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Correspondence between real numbers and points of a line
Consider this fact that we all know from school mathematics:
There is a one to one correspondence between real numbers and points of a line.
But the problem is I have never seen a rigorous proof of ...
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Prove that the radius of the incircle of one triangle has two times the radius of the incircle of another triangle with Euclidean geometry.
"Let ABCD be a square. Let CDP be an equilateral triangle inside the square. Let Q be the intersection of the line drawn through points A and P and the side BC. Let R be a point such that the ...
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