Questions tagged [euclidean-geometry]
For questions on geometry assuming Euclid's parallel postulate.
1,780 questions with no upvoted or accepted answers
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Where are the other Pythagorean Theorems?
One way to prove the Pythagorean Theorem is by noticing that the altitude of a right triangle divides it into two pieces similar to itself. The theorem immediately follows from the fact that areas ...
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New shape? Any literature on a cuboid with proportions of $1: \sqrt{2}:2-\frac {1} {\sqrt {2}} $?
I found a 3d shape that shares properties similar to the golden rectangle or root 2 rectangle. It is a rectangular prism that has a repeating self symmetry when subdivided, somewhat similar to the ...
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A non-self-intersecting unit side length polygon in a unit square has odd number of sides unless it is the square itself
This is the same question as here in MO.
I have a conjecture, it is like this:
Suppose there is a non-self-intersecting polygon lies inside a closed square of length $1$. The polygon has every side ...
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Number of circles in configuration
Consider the $n^2$ lattice points $(i, j)$, where $1 \leq i, j \leq n$.
Let the number of circles that pass through at least 3 points of this set be $C(n)$. What is a good way to count this? Is there ...
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Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?
In the web game Ancient Greek Geometry, there are challenges to construct regular polygons and circle packings using ruler and compass constructions. The game measures the number of line and circles ...
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Is "Escherian metamorphosis" always possible?
This question is motivated by Escher's series of Metamorphosis woodcuts (see e.g. here), where one tesselating tile is gradually transformed into another. Basically, this is a precise way of asking ...
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Does the Sequence formed by Intersecting Angle Bisector in a Pentagon converges?
Now asked on MO here.
Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $A_{n+1}B_{n+...
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What makes a geometric construction more or less stable?
As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
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Optimal Compass and Straightedge Constructions
I was recently looking over some Islamic geometry patterns, and was struck by the complexity of the constructions needed to create seemingly simple patterns. This got me wondering regarding optimal ...
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Can one embed $\frac{n^2(n^2-1)}{12}$ planes in $\Bbb R^n$ in a symmetric way to determine the Riemann curvature tensor?
To explain my question I need to discuss two notions from differential geometry (the Riemann curvature tensor and sectional curvature), but this is really a linear algebra / Euclidean geometry ...
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A triangle's sides are integers, and its circumradius is a prime number. Prove that the triangle is right-angled.
The lengths of the sides of a triangle are integers, whereas the radius of its circumscribed circle is a prime number. Prove that the triangle is right-angled.
Solution: We'll use three well-known ...
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Characterization of Convex Polygons
John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons.
Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not ...
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Smooth tiling of the plane
For my master thesis, I solved a PDE under the assumption of the domain being smooth and small. I wanted to patch these domains and solutions somehow together, hoping that I can get a global result.
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Which objects can be Minkowski halved?
The Minkowski Sum of two subsets $A,B \subset \mathbb{R}^n$ is
$$A \oplus B = \{a + b | a \in A, b \in B\}$$
For a given $A$, is there some condition that tells me when I can find a $B$ such that $A = ...
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Rational point inside a rational polygon
I have the following conjectures.
Conjecture 1:
Hypotheses:
Let $P = (v_1, v_2, …. v_n)$ be a (convex or concave) polygon drawn on a plane.
The lengths of the edges $(v_1, v_2)$, $(v_2, v_3)$ ... $(...
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