Questions tagged [plane-geometry]
Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper
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How can one construct a four-coloring of a tiling of the plane with Smith, Myers, Kaplan, and Goodman-Strauss's aperiodic monotile?
This is motivated by the new paper of Smith, Myers, Kaplan, and Goodman-Strauss, wherein they define the existence of an aperiodic monotile. Clearly their tiling is not three-colorable, so we have ...
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Sets of evenly distributed points in the Euclidean plane
Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection
with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite?
If the answer is yes, can $P$...
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The space of triangles that fit inside a given triangle, parametrized by edge lengths
Given a triangle T with sides a, b, and c, describe its "fitting set," the set of all points (x,y,z) in 3-dimensions for which a triangle with sides x, y, z exists that fits in T.
Such a set lies in ...
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Six points on an ellipse
Can you prove the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with centroid $G$. Let $D,E,F$ be the points on the sides $AC$,$AB$ and $BC$ respectively , such ...
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spiral of Theodorus
A long time ago when I was in college I read about making a spiral out of right triangles with sides 1 and $\sqrt{N}$. (A google search seems to indicate that this is called the Spiral of Theodorus.)
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Optimal wireframe sphere
Suppose you have a length $L$ of metal pipe at your disposal,
and you would like to build a wireframe unit-radius sphere,
by bending, cutting, and welding the pipe into a connected structure $F$.
Your ...
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Interpolating between disks in the plane
Below, a "disk" means a compact subspace $D \subset \mathbb R^2$ whose boundary is a smooth simple closed curve.
Task: Find a procedure which takes as input a pairs
of disks
$
D_0 \subseteq ...
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Dao's theorem on six circumcenters associated with a cyclic hexagon
This questions from Ngo Quang Duong's paper
In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...
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Koebe–Andreev–Thurston theorem - where can I find a proof?
Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two ...
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Connectedness in the plane
There are several open problems in topology which concern connectedness and subsets of the plane. The biggest of these is undoubtedly:
Question. Does every non-separating plane continuum have the ...
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'Eigenvectors' of evolute operation
The evolute of a curve is the locus of its centers of curvature.
The evolute of some plane curves is a scaled, or scaled and
reflected/rotated, version of that curve.
For example, the evolute of a ...
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Characterization of discs
Let $D$ be a bounded simply connected region (open subset homeomorphic to the disc)
in the plane, containing the origin.
Suppose that for every line $L$ through the origin the intersection $L\cap\...
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Including a Jordan arc into a Jordan loop (Can the Magi go home by another way?)
The title refers, of course, to Matthew (2:12) ''And being warned in a dream not to return to Herod, they departed to their own country by another way''. To be honest, it is not that specific ...
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Complexity of planar scissor congruence
Two (not necessarily convex) poygons of equal area are scissor-congruent, i.e. both can be cut along a finite number of straight lines or segments into isometric pieces.
What can be said about the ...
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The status of the journal “Forum Geometricorum”
The online journal Forum Geometricorum is a sort of central organ of elementary geometry (mainly triangle geometry and related topics). It has been published regularly since 2000 but seems to have ...
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Subset of the plane that intersects every line exactly twice
In a comment to this question, Tim Gowers remarked that using the axiom of choice, one can show that there exists a subset of the plane that intersects every line exactly twice (although it has yet to ...
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A claim on partitioning a convex planar region into congruent pieces
Let us define a perfect congruent partition of a planar region $R$ as a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent (ie any piece can ...
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Generalizing square wheels rolling on inverted catenaries
It is not uncommon to see in a science museum a bicycle with
square wheels that rides smoothly over a washboard-like
surface made from inverted catenary curves (e.g., at the Münich museum).
The square ...
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Connected components $0-1$ matrices
Let $M$ be a $0-1$ matrix.
Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...
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"Transcendental tilings": Do they exist?
Let $T$ be a tiling of the plane.
Fix an origin and shoot a ray $r$ from the origin.
Mark off points $p_i$ along $r$ separated by unit distance.
Compute from $r$ a binary number $0 < b(r) < 1$ ...
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Probability that random cubic polynomials meet in a square
Let $p_1(x)$ and $p_2(x)$ be cubic polynomials with
random coefficients in $[-1,1]$.
I wanted to compute the probability that $p_1$ and $p_2$
share at least one point within
the square $[-1,1]^2$.
Of ...
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Is the area of a polygonal linkage maximized by having all vertices on a circle?
Consider a (non-stellated) polygon in the plane. Imagine that the edges are rigid, but that the vertices consist of flexible joints. That is, one is allowed to move the polygon around in such a way ...
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Decomposing the plane into intervals
I posted this on Stack Exchange and got a lot of interest, but no answer.
A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so ...
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Dodecahedral rolling distance
Let a dodecahedron sit on the plane,
with one face's vertices on an origin-centered unit circle.
Fix the orientation so that the edge whose indices are $(1,2)$ is horizontal.
For any $p \in \mathbb{R}...
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What is the area of the biggest open convex set inside the unit square not containing k points?
Given $k\in \mathbb N$, and $k$ points inside the unit square there should be an arrangement that minimizes the area of the biggest open convex set inside the unit square not containing these points.
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Connecting a compact subset by a simple curve
Let $K$ be a compact subset of $\mathbb R^n$ with $n\ge 2$ (say if you like $n=2$, which is possibly sufficiently representative).
Q: Does there exist a closed simple curve $u:\mathbb S^1\to\mathbb R^...
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Triangle with largest perimeter in a convex region
What is the largest value of $r$ such that the following statement is always true?
"Let $C$ be a convex region with area $1$. There must exist a triangle contained in $C$ whose perimeter is at least ...
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Metrically homogeneous subsets of the plane
A metric space $M$ is metrically homogeneous if for every pair of points $x, y \in M$ there is an isometry $f$ of $M$ onto $M$ such that $f(x)=y$. What is known about metrically homogeneous spaces? ...
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A generalization of the law of tangents
The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
Let $a$, $b$, and $c$ be the lengths of the three ...
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Yau's problem: Construct a triangle given a side, an angle, and an angle bisector
In Shing-Tung Yau's autobiography The Shape of a Life, he mentions a problem that he came up with as a teenager.
Suppose you know the length of one side of a triangle, one angle, and the length of ...
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An open triangle problem in plane geometry
Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following:
Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is ...
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Polygons uniquely inducing arrangements
A beautiful, relatively recent result is that,
Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$.
In a simple arrangement, every pair of lines intersect ...
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Schoenberg's rational polygon problem
"A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by a ...
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Tiling with incommensurate triangles
Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...
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Packing rectangles: Does rotation ever help?
Dominic van der Zypen posed an interesting Box stacking problem.
This is a spin-off question.
Let a collection of rectangles $r_1,\ldots,r_n$ be given by their side lengths in $\mathbb{R}$.
Let $R$ ...
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Convex curves with many inscribed triangles maximizing perimeter
A classical nice result of Euclidean geometry states that the triangles maximizing the perimeter among all inscribed triangles of a given ellipse constitue a one-parameter family. Precisely, for each ...
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How can we find n points on a plane so that as many pairs of points as possible have the same distance?
There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
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Monochromatic point sets in two-colored plane
Which are the configrations $P\subset \mathbb{R}^2$ of points, such that the following property holds:
Property M (for Monochromatic): Every two-coloring of $\mathbb{R}^2$ contains a monochromatic ...
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Geometric construction of the fourth intersection points of two conics
In general, two conics in the plane intersect at most 4 points. Suppose three of those points are given as $A,B,C$. Then let $c_1$ be the conic passing through those three points and $D_1,E_1$. Let $...
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Perimeter-halving center of a convex shape
Let $P$ be a convex polygon (or any convex body in $\mathbb{R}^2$)
with perimeter of length $1$. Call a chord $c$ of $P$ perimeter-halving
if half the perimeter lies to one side of $c$
(and so half to ...
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2-layer tilings with a center-of-gravity constraint
I've encountered a tiling problem with a physical constraint that
might place it outside the literature on tiling.
"Tiling" is a bit of a misnomer; it is a special type of cover.
All the tiles are ...
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Reorienting a ladder among $\mathbb{Z}^2$ poles
Imagine an object, which I'll call a ladder $\cal{L}$, a "racetrack" shape
composed of a rectangle of length $L$ capped at either end by
semicircles of radius $r$; so it is $L+2r$ tip-to-tip.
View ...
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Optimization of points on a plane
Suppose we have $n$ points on a plane. Let $D$ be the sum of the squares of all the pairwise distances between the points. Let $A$ be the area of the convex hull. What is the minimum possible value of ...
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A forked plane continuum
I came up with this question while trying to solve the following MO one:
Does every connected set that is not a line segment cross some dyadic square?
Suppose $C$ is a plane continuum (i.e. a ...
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Formula for "cointersection" of three circles?
I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?
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Chord arrangement that avoids confining small or large disks
These two questions are two-dimensional variations on this recent MO question,
"Threading pinholes in the wall of cylinder to pass through an internal coordinate."
Noam Elkies suggested that even a 2D ...
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Axiom of choice and a set in the plane that intersects every line in two points
In this question Subset of the plane that intersects every line exactly twice someone ask for a reference of a paper where they proof the result : ''There exist a subset of the plane that intersects ...
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Fractal plane continuum with order $\aleph_0$?
Continuum means compact and connected.
Define the order of a point $x$ in a continuum $X$ to be the least cardinal $\alpha$ such that $X$ has a neighborhood base of open sets at $x$ with no more ...
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Do cut-length-minimizing equidissections exist?
Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...
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Which finite sets could be packed into a square?
This question is inspired by an interesting visualization of the finite levels of von Neumann's hierarchy on Adam P. Goucher's blog, Complex Projective 4-Space.
The problem starts with a two-...
