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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Does this property characterize straight lines in the plane?
Take a plane curve $\gamma$ and a disk of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\...
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Does this geometry theorem have a name?
Start with a circle and draw two tangent circles inside. The (black) inner tangent lines to the smaller circles intersect the large circle. The (red) lines through these intersection points are ...
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Can an arbitrary collection of circles of total area 1/2 fit into a circle of area 1?
Assume the circles are actually open disks, otherwise two circles each of area $\frac{1}{4}$ wouldn't fit into the circle of area 1.
This seems like it should be true, thinking about packing density,...
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Polynomial roots and convexity
A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but no one had ever ...
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Extending a line-arrangement so that the bounded components of its complement are triangles
Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that $\{L_1,\dots,L_m\}...
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Parabolic envelope of fireworks
The envelope of parabolic trajectories from a common launch point is itself a parabola.
In the U.S. soon many will have a chance to observe this fact directly, as the 4th of July is traditionally ...
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Which theorems have Pythagoras' Theorem as a special case?
Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...
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Tiling the plane with incongruent isosceles triangles
It is not difficult to tile the plane with incongruent triangles.
One could tile with equilateral triangles, and then partition
each equilateral into three triangles, displacing their common
...
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About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals
Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$.
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Can Morley's theorem be generalized?
Morley's theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.
In a talk some years ago, David Rusin made the provocative ...
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Term for "uncheckable constructions"
Is there a term for "uncheckable geometric constructions"?
Say, Angle Trisection and Doubling the Cube are checkable;
i.e., if the answer is given one can do finite Compass-and-straightedge ...
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Tiling of the plane with manholes
Some shapes, such as the disk or the Releaux triangle can be used as manholes,
that is, it is a curve of constant width.
(The width between two parallel tangents to the curve are independent of the ...
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Is there a subset of the plane that meets every line in two open intervals?
Using the Axiom of Choice, it is possible to construct a subset of the plane that meets every line in two points (these are called "$2$-point sets"). What if, instead of points, we ask for two open ...
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How fast are a ruler and compass?
This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO.
Consider the standard assumptions ...
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Is the ratio Perimeter/Area for a finite union of unit squares at most 4?
Update: As I have just learned, this is called Keleti's perimeter area conjecture.
Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
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Blocking light with mirrored convex objects
There is a long-unsolved problem posed by Janos Pach,
sometimes known as the enchanted forest problem,
which asks if it is possible to block a point light source
in the plane
from reaching
infinity by ...
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A question about subsets of plane
Is there a subset $X$ of plane with two points $x, y$ such that each one of $X \setminus \{x\}$, $X \setminus \{y\}$ is isometric to $X$? I tried hard to construct a counterexample but failed.
Sorry ...
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Why is it so hard to prove Toeplitz' conjecture?
I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) ...
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Pinball on the infinite plane
Imagine pinball on the infinite plane, with every lattice
point $\mathbb{Z}^2$ a point pin.
The ball has radius $r < \frac{1}{2}$.
It starts just touching the origin pin, and shoots off at angle $\...
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Are there infinitely many "generalized triangle vertices"?
Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
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Tetrahedron insphere iteration
I know that iterating the following incircle construction approaches an equilateral triangle in the limit:
Starting with any triangle $T$, one forms $T'$ by connecting ...
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What upper bounds are known for the diameter of the minimum spanning tree of $n$ uniformly random points in $[0,1]^2$?
Let $P$ be a pointset consisting of $n$ uniformly random elements of $[0,1]^2$. It is known that the diameter (greatest number of edges in any shortest path between two points) of the Delaunay ...
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Trapped rays bouncing between two convex bodies
At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...
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Does greedy circle packing exhaust the measure of every bounded open set in the plane?
The greedy circle packing of a bounded region in the plane is the result of placing at each stage the largest possible disk into the region that remains uncovered.
The greedy circle packing of a ...
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Aperiodic monotile without reflections?
The recently discovered amazing aperiodic monotile (or "einstein") of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss tiles the plane only if reflections of the ...
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Happy ants never leave compact domain?
I am curious if the following seemingly simple question has an easy answer?
Consider an ant population of $N$ ants that lives in $\mathbb R^2$. Each ant can be labeled by some coordinate $x\in \mathbb ...
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Straight-line drawing of regular polyhedra
Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane).
For example, ...
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Erdős, Harary, Tutte's "dimension of graph": Progress in last 48 yrs?
I just ran across this delightful paper by an amazing triumvirate:
Paul Erdős, Frank Harary, and William Tutte. "On the dimension of a graph." Mathematika 12.118-122 (1965): 20.
(Cambridge link)
...
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Minimal pizza cutting
Given a circle, we want to divide it into $n$ connected equally sized pieces. In such a way that the total length of the cutting is minimal. What can we say about the solution for each $n$. Are they ...
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Hidden points in polygons
Let $h(n)$ be the largest number of mutually invisible points that can be located in a
polygon $P$ of $n$ vertices. Two points $x$ and $y$ are mutually invisible if the segment
$xy$ contains a point ...
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Lightray trapped between two mirror disks: Computation formulation?
I would like to calculate the angle of a ray $r$ from a given
point $p$ such that it gets "stuck" reflecting between
two congruent mirror-disks.
For why there is such a ray, see the (amazing!) answer
...
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All saddles in the unit ball have area $<2\pi$?
Let $M$ be the saddle surface in $\mathbb R^3$ defined by $x^2-y^2+z=0$. For any $r\geq 0$ and $(x_0,y_0,z_0)\in\mathbb R^3$, let $rM+(x_0,y_0,z_0)$ denotes the surface obtained by scaling $M$ by $r$ ...
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Definition of area
I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...
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An ellipse through 12 points related to Golden ratio
I am looking for a proof of the problem as follows:
Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$,...
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Arrangements of points in the plane
Let $p_1,\ldots,p_n$ be a collection of distinct points in $\mathbb{R}^2$, no three of which lie on a line. For each $p_i$, let $\omega_i(p_1,\ldots,p_n)$ be the following ordered list (well-defined ...
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Planar sets where any line through the center of mass divides the set into two regions of equal area.
This question is influenced by the following riddle:
You are given a rectangular set in the plane with a rectangular hole cut out (in any orientation). How do you cut the region into two sets of ...
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Can two-point sets be Borel?
Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914.
I wonder if the following question of ...
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What is the smallest positive integer for which the congruent number problem is unsolved?
The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...
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Is the perimeter of an ellipse with integer axes irrational?
Let $Q$ be an ellipse with integer-length axes $a$ and $b$:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$
The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$:
$4 ...
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The sparsest planar net that captures every unit segment
Let $\cal C = \lbrace C_i \rbrace$ be a collection
of rectifiable curves in the plane with the property that
every unit-length segment meets at least one curve
in at least one point.
Call such a ...
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Isoperimetric-like inequality for non-connected sets
The classical isoperimetric inequality can be stated as follows: if $A$ and $B$ are sets in the plane with the same area, and if $B$ is a disk, then the perimeter of $A$ is larger than the perimeter ...
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Kakeya crossed-needles problem
The Kakeya needle problem asks for the
minimum area planar region in which one can completely turn around a line segment through
a series of translations and rotations. There is no minimum: There are &...
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Applications of visual calculus
Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool.
The basis is
Mamikon's theorem. The area of a tangent sweep is equal to the area ...
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Are Penrose tilings universal? Do aperiodic universal tilings exist?
Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile ...
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Can a shape rolling inside itself reproduce that shape?
Q. Is the circle the only shape that, when rolling inside itself,
has a point that draws out a scaled copy of itself?
Let $C$ be a simple, closed, smooth curve in the plane.
(Likely "smooth" can be ...
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Is "Escherian metamorphosis" always possible?
$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
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Do two new special points in any triangle exist?
There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.
Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $...
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Three squares in a rectangle
One of my colleagues gave me the following problem about 15 years ago:
Given three squares inside a 1 by 2 rectangle, with no two squares overlapping, prove that the sum of side lengths is at most 2. (...
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Lines passing through many points of the form $(c^n,c^m)$
For $c>1$ consider the subset $X\subset \mathbb R^2$ consisting of all points $(c^n,c^m)$ where $n,m\in \mathbb Z$.
Question. Suppose $L\subset \mathbb R^2$ is a line that is not horizontal, not ...
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Ratio of circumscribed/inscribed $(n{-}1)$-gons
As a discrete analog of the MO question,
"Löwner-John Ellipsoid: incribed and circumscribed,"
I've been wondering what might be the maximum ratio
of this quantity?
Let $P$ be a convex ...
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