All Questions
Tagged with euclidean-geometry incidence-geometry
7 questions
11
votes
3
answers
557
views
Was the small Desargues Theorem known to ancient Greeks?
My question concerns the classical Desargues Theorem and its simplest version
The small Desargues Theorem: Let $A$, $B$, $C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...
- 42k
3
votes
1
answer
212
views
Another implication of the Affine Desargues Axiom
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
- 42k
23
votes
3
answers
2k
views
Why do all incidence theorems follow from Pappus' theorem?
In Hilbert and Cohn-Vossen's ``Geometry and the Imagination,"
they state in the last paragraph of Chapter 20 that "Any
theorems concerned solely with incidence relations in the
[Euclidean projective]...
- 418
6
votes
1
answer
295
views
Does any real projective plane incidence theorem follow from axioms?
Is it known whether any projective geometry statement that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the standard projective axiomatics?
...
- 308
2
votes
0
answers
169
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theories where angles exist without a metric
The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted ...
- 2,041
5
votes
0
answers
119
views
What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?
Here are two facts that are superficially similar.
Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons.
Feit-Higman Theorem: The only finite ...
- 1,389
2
votes
0
answers
281
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Axiomatization of the incidence geometry of the Euclidean plane
There are several well-known axiomatizations of Euclidean plane geometry, the language of which is usually considered to include at least the relations of
incidence (point-line, point-segment, or ...
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