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Questions tagged [euclidean-geometry]

For questions on geometry assuming Euclid's parallel postulate.

The geometry of Euclid is based on five axioms (Euclid called them postulates). Any geometry based on the first four of these is called an absolute geometry. The fifth one states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

It was observed by Proclus that, in the presence of the the other four postulates, Euclid's fifth postulate can be replaced by Playfair's axiom:

Given a line and a point not on it, then one and only one line parallel to the given line can be drawn through the point.

The independence of the parallel postulate and its equivalent formulations from the first four axioms was shown by Beltrami in 1868.

Another alternative definition is that two lines are parallel if every perpendicular extended from one meets the other as a perpendicular.

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What is the difference between Euclidean and Cartesian spaces?

For me those are references to the same thing. On Wikipedia there are references to both but I still don't see the difference. http://en.wikipedia.org/wiki/Cartesian_coordinate_system#Cartesian_space Is there any difference? If yes, is there a…
Aalex Gabi
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Why is Euclidean geometry scale-invariant?

In Euclidean geometry, I frequently use concepts related to invariance under scaling. For example, I know that if two squares have different side lengths, the ratio of their side lengths is the square root of the ratio of their areas. I became…
Mark Eichenlaub
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A circle in the plane contains at most four lattice points?

Let $\cal C$ be a circle in ${\mathbb R}^2$ : $\cal C=\lbrace (x,y)\in{\mathbb R}^2 | (x-x_0)^2+(y-y_0)^2=r^2\rbrace$ for some constants $x_0,y_0,r$. What is the maximal number of points that can be contained in ${\cal C}\cap {\mathbb Z}^2$ ? I…
Ewan Delanoy
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In classical geometry why is a line considered to be parallel to itself?

A definition in classical geometry (for example, Birkhoff's formulation, but I suppose it could be all of them) is that a line is always considered to be parallel to itself. I understand this is probably for convenience, but in my mind since two…
Ryan
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Right triangle inscribed in a square. Find the square area?

I hope it's valid to ask for "a more neat solution" of a problem on this network, despite the fact that I don't have a strict definition of the word "neat". Here is the square and the right triangle inscribed in it. I did the following: $$AC = Ah +…
MMa
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How to make a perpendicular construction in 3 moves?

I've been playing Euclid: The Game for some time now. I'm quite addicited to it, trying to get all the records now. Suprisingly, I'm not able to get a record for some really early level. In Level 4 (http://euclidthegame.com/Level4/), I'm trying to…
Pjotr
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Can Euclid prove that a straight line is the shortest distance between two points?

From Euclid's definitions, postulates, and common notions, can you prove that a straight line is the shortest distance between two points, or is that basically an assumption of the way lines are measured? Here is an online copy of much of the text…
David Gudeman
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How to plot $N$ points on the surface of a $D$-dimensional sphere roughly equidistant apart?

Let's say I have a $D$-dimensional sphere with a radius $R$. I want to plot $N$ number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where those points are exactly, just that they…
diracleo
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What's the average width of a convex polygon?

If one computes the average width of a triangle, then one gets $(s_1+s_2+s_3)/\pi$, where $s_1$, $s_2$, $s_3$ are the side lengths. I did this by brute force, using an integral which went through an interval of length $\pi$. For each angle…
David Richter
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What regular polygons can be constructed on the points of a regular orthogonal grid?

Besides a square, what regular polygons can be constructed so that the points of that polygon lie on the points of a regular, planar, orthogonal grid? Besides a triangle and hexagon, what regular polygons can be constructed so that the points of…
oosterwal
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Is the ratio of the side and at least one diagonal of a rhombus always irrational?

The ratio between the side of a square $ c = 1 $ and its diagonal is $ \frac 1 { \sqrt 2 } $; a square is a type of rhombus. The ratio between the side $ c = 1 $ of a rhombus, with angle $ a = \frac \pi 3 $ and its longest diagonal $ AC $ is $ \frac…
adrienlucca.net
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How many 1-balls are needed to cover the 2-ball in an $n$-dimensional Euclidean Space?

Consider $\mathbb{R}^n$ and its usual Euclidean norm given by the distance $d(x,y) = \sqrt{\sum_{i=1}^n (x_i-y_i)^2}$. Let $B(y,1) = \{ x\in \mathbb{R}^n : d(x,y) \leq 1 \}$ be the closed 1-ball around $y$ and $B(0,2) = \{x\in \mathbb{R}^n: d(x,0)…
Santiago C.
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Fit a equilateral triangle on three arbitrary parallel lines with an edge and compass

How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?
qed
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Maximum volume of tetrahedron with given face areas

Everyone knows Heron formula for the area of a triangle as a function of its sides. Moving to three dimensions, what is the maximum volume of a tetrahedron with given four face areas $a$, $b$, $c$, and $d$? Obviously, this is an optimization…
Dmitry Baksheev
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Regular polygon inside another?

Inspired by this question, I was wondering if one can generalize to the case of an $n$-gon. For example, when $n=5$ we have this picture: where $ABCDE$ is a regular pentagon, $AA_1=BB_1=\cdots=EE_1$. Also $A,A_1,B_1$ are colinear in that order,…
Quang Hoang
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