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Results tagged with euclidean-geometry
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user 61536
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.
13
votes
Is every connected subgroup of a Euclidean space closed?
A small amendment of the result of Hidehiko mentioned in the answer of Moishe Kohan: Theorem 2.2 in this paper implies that every continuum-connected subgroup of $\mathbb R^n$ is a (closed) linear sub …
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12
votes
1
answer
443
views
Is each cover of the plane by lines minimizable?
A cover $\mathcal C$ of a set $X$ by subsets of $X$ is called
$\bullet$ minimal if for every $C\in\mathcal C$ the family $\mathcal C\setminus\{C\}$ is not a cover of $X$;
$\bullet$ minimizable if $\ …
- 41.8k
5
votes
1
answer
252
views
Is the action of $SO(n)$ on the sphere $S^{n-1}$ ballanced?
A subset $B$ of a group $G$ is called balanced if $gBg^{-1}=B$ for all $g\in G$.
An action of a group $G$ on a metric space $X$ is called ballanced if for each non-empty balanced subset $B\subset G$ …
- 41.8k
0
votes
Accepted
Another implication of the Affine Desargues Axiom
Just to close this question as answered, I will give a sketch of the proof of the parallelity of the lines $\overline{zc}$ and $\overline{yb}$.
As was noticed by @AlexRavsky, it suffices to prove that …
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3
votes
1
answer
210
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\ …
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8
votes
1
answer
275
views
Almost convex combinations in $\mathbb R^n$
Working on some problems in the $C_p$-theory I discovered the following simple but amazing
Fact. For any subset $A\subset \mathbb R^n$, non-zero vector $a\in \bar A\subset\mathbb R^n$ and $\varepsil …
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11
votes
1
answer
221
views
The set of boundary vectors of compact convex body has empty interior
Let $K$ be a compact convex body in the Euclidean space $\mathbb R^n$ and $\partial K$ be its topological boundary in $\mathbb R^n$.
Definition. A vector $\mathbf v\in\mathbb R^n$ is called $K$-bounda …
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1
vote
The set of boundary vectors of compact convex body has empty interior
Here is a proof of Conjecture in dimension $n=3$.
Let $K$ be a compact convex body in $\mathbb R^3$. For $k\le 3$, let $Gr(k,3)$ be the space of $k$-dimensional linear subspaces of $\mathbb R^3$. So, …
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6
votes
1
answer
366
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Desargues ten point configuration $D_{10}$ in LaTeX
I want to draw the Desargues configuration $10_3$ in LaTeX using the standard picture environment, which allows only lines with the slopes $n:m$ where $\max\{|n|,|m|\}\le 6$. Is it possible? If not, t …
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10
votes
1
answer
514
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A projective plane in the Euclidean plane
Problem. Is there a subset $X$ in the Euclidean plane such that $X$ is not contained in a line and for any points $a,b,c,d\in X$ with $a\ne b$ and $c\ne d$, the intersection $X\cap\overline{ab}$ is …
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11
votes
3
answers
553
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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 $a,a'\in A$, $b,b'\in B$, $c,c'\in C$ …
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2
votes
Was the small Desargues Theorem known to ancient Greeks?
In fact, I asked this question hoping to find some appropriate name that can be legally used for naming affine spaces satisfying the small Desargues Theorem. Now I have learned that in Projective Geom …
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16
votes
1
answer
587
views
A textbook on foundations of geometry in spirit of Tarski
I am interested in a textbook for studying (and teaching) foundations of geometry in the spirit of Tarski. I know that there is a rather old German book [W. Schwabhäuser, W. Szmielew, A. Tarski, Metam …
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