For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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How to quantify the “uniformity” of a distribution of holes in a surface

I want to try to quantify if the distribution of holes over a surface is uniform or not. The holes can have any given shape and can be arranged in any way over the surface. Three examples are ...
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75 views

The Light beam Problem.

I am puzzled with a problem like this. Suppose you have $N$ lighthouses, each of which has $360/N$ angle of light. Suppose they are arbitrarily allocated on totally flat endless earth (i.e. $R^2$). ...
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135 views

Singularities on a weighted projective curve

Let $C$ be the curve of degree $3$ defined over $\mathbb{C}$ given by $$x(y+z)=y^3-z^3$$ which lives in the weighed projective space $\mathbb{P}(x,y,z)=\mathbb{P}(2,1,1)$. Is the curve singular ?
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42 views

Related Rates - possible textbook error

Recently I've been asked to do some exercises from a textbook but I cannot understand how the author derived the correct answer. Here is the exercise in question: At $8$:$00$ boat $A$ is located ...
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91 views

Twisted nail puzzle framed in terms of algebraic topology?

See here for a description of the puzzle: Twisted Nail Puzzle. My question is, can someone provide a description of the puzzle and its solution in context of the language of algebraic topology?
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66 views

Visualising 3rd degree equations

I know that general second degree curve, i.e. $ax^2 + by^2 + 2hxy + 2gx + 2fy + c=0$ gives us the equation of different cross sections of a cone. Similarly, what does a third degree* curve actually ...
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157 views

resizing rectangle within triangle

Imagine I have a parking lot that changes in width and length and in number of levels, and all of the levels need to be visible to a cctv camera at a fixed position, and I would want the camera to see ...
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94 views

Maybe Stuck Pedal Triangle with geometry problem

Suppose $P$ is any point within an acute-angled triangle,Let $X,Y,Z$ be the feet of the perpendiculars from $P$ onto the sides $BC,CA,AB$ respectively. and $U,V,W$ be where $AP,BP,CP$ meet the sides ...
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63 views

Can ${n \choose 2}$ points be covered by lines determined by $n$ points?

Let $S=\{(a_1,b_1),\ldots,(a_{n \choose 2},b_{n \choose 2})\}$ be ${n \choose 2}$ points on the plane. Does there exist $n$ points, such that the lines determined by the $n$ points cover all the ...
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63 views

Three-colored plane

Let $\Pi$ is cartesian plane with the usual topology. $A, B, C$ are pairwise disjoint subsets $\Pi$, $A\cup B\cup C = \Pi$. Each of these sets is dense in $\Pi$: $\overline{A}=\Pi$, and ...
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84 views

Polygons with coincident area and perimeter centroids

Let $P$ be a simple, planar polygon. Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the closed shape $P$. Define $c_p$ as the perimeter centroid of $P$, the center of gravity ...
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174 views

From the viewpoint of modern geometry, is there a “best” definition of the term “triangle”?

I can think of at least six different possible definitions of the term "triangle" in Euclidean geometry. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three or fewer points. ...
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139 views

From Icosahedron to Pentagonal hexecontahedron (Floret Tessellation)

Inspired by this post: Floret Tessellation of a Sphere I tried to transform myself an icosahedron into its simplest Floret tessellation. But I am having trouble when applying the 'method' given in the ...
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633 views

Any weird 'modular like mathematical space' that behaves like as if it is infinite until a threshold is reached?

(Using the advice from Mathoverflow, I have rephrased and splitted up the question) (Might be a bit layman because I don't have rigorous math term to describe the concept) Generalize it to ...
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58 views

What does the rotation group of $\mathbb{\bar{Q}}^n$ look like?

There's a structural difference between the rotation groups of $\mathbb{Q}^n$ and $\mathbb{R}^n$; in some abstract sense the former is 'small' (discrete?) while the latter is 'large'. I suspect that ...
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64 views

Name for a body that can be completely described using its silhouettes

I'm shooting blind over here because I have no background in this field of mathematics. I assume that if you have a body (in $\mathbb{R}^3$), you can call it convex if any segment from one point ...
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230 views

How to balance learning and researching as a new PhD student?

As a new PhD student, how to balance learning and researching? I am in Australia and here we don't have any course in PhD period. I know I need to learn something about my programme, but sometimes ...
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188 views

Set geometry and inclusion

I would like to prove that the set of the symmetric positive semi-definite matrices which is defined as $$\Delta_2= \{S\in\mathbb{S}_{m,m} \quad \text{s.t.}\quad ...
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126 views

A “Paradoxist Geometry”

This question is about "how badly can we 'break' the laws of geometry and still have something which is deserving of the name geometry?". It is named after something else I saw of the same name which ...
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102 views

A light beam enters a closed room. What is the maximal number of reflections?

I have the following problem: a light beam enters a mirror room with integer coordinates in the plane (consider it as a polygon). One of the walls of the room is removed and the light beam enters the ...
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101 views

Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...
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119 views

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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68 views

The space of paths

Let $ (M,g) $ be a compact Riemannian manifold, and let the path space $ \Omega $ of $ M $ be the set of equivalence class of smooth maps $ \gamma : [0,1] \rightarrow M $ (equivalent under ...
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260 views

Hexagonal circle packings in the plane

I have been reading the paper Spiral hexagonal circle packings in the plane (Alan F. Beardon, Tomasz Dubejko and Kenneth Stephenson, Geometriae Dedicata Volume 49, Issue 1, pp 39-70), which proves ...
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256 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ ...
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148 views

Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...
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335 views

Invariant of matrix under elementary transformations

$\DeclareMathOperator{\rank}{rank}$ Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix $$ B = \begin{bmatrix} A & b \\ b^T & c ...
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474 views

Second derivative of a metric in terms of the Riemann curvature tensor.

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial ...
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184 views

Nontrivial trivial integrals

Consider two propositions in geometry: Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals the ...
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2k views

Lowest surface-to-volume ratio for an uncovered vessel

It is well known that a sphere has the lowest surface to volume ratio. However, a related question is: What is the shape that gives the lowest surface to volume ratio if you do not include the top in ...
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244 views

Points in the plane at integer distances

Does there exist a set of $n$ points $p_1,p_2,...,p_n$ in the plane, all at mutual integer distances to each other, and an $e>0$, such that the following statement holds: For all $a,b$ with ...
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112 views

Can Kollros' Theorem w/ extra turns be seen via inversive geometry?

Kollros' Theorem, with extended turns allowed, says For every ring containing $p$ spheres, there exists a ring of $q$ spheres, each touching each of the $p$ spheres [...] if more than one turn is ...
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311 views

Can “anti-groups” and “anti-manifolds” be constructed? (and other “anti-objects”)

Is it possible to create an "antigroup"? What I mean by this is, given some group G, and some "antigroup" H, then the "free product" of G and H, G*H will equal the "group" (vacuously a group) of no ...
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Functions determine geometry … Riemannian / metric geometry?

Given a compact Riemannian manifold $(M,g)$, is there a subring of $C^{\infty}(M)$ that determines the isomorphism class of $(M,g)$? (In the same way that $C^{\infty}(M)$ determines the diffeomorphism ...
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40 views

Area of convex hull of 6 points greater than twice the sum of the areas of 2 triangles formed by the 6 points

I noticed something interesting but I couldn't prove it to the end. I want to prove that if I have $6$ coplanar points and the area of convex hull of these points is equal to $P$, then I can mark the ...
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71 views

The reverse pizza problem .

The pizza problem is a fairly well-known problem which sounds like this : You have a circular pizza and you need to cut it such that you and your friend would both receive half of the pizza . ...
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33 views

Decomposing geodesic tessellations over a sphere into parallelograms

I'm working with some icosahedron-based tessellations of triangles over the surface of a sphere. Class I and Class II tessellations have a nice property where, cutting along the edges of the ...
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64 views

Number of members of a set in geometry

Suppose we have a $2n \times 2n$ grid of unit squares divided into $2 \times 2$ contiguous squares. Suppose we wish to have a set of these unit squares such that At least one unit square of each 2 x ...
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73 views

Affine-regular hexagon in convex body

An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb ...
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24 views

Can “Taxicab geometry” be given a Hilbert-style axiomatization?

Hilbert's axioms provide a synthetic system for Euclidean geometry. Is it possible to do the same thing for the Taxicab plane? It would seem that one would only need to alter the axioms for ...
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123 views

How to classify this surface

I know that it should be either a sphere, torus, Klein bottle, real projective plane, or a connect sum of any combination of these, but I don't know the steps in identifying what kind of surface ...
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38 views

When are convex polygon tilings Voronoi?

A square is divided into convex polygons. Is this always a Voronoi diagram? If not, what are some simple examples of non-Voronoi tilings? Which of the pentagon tilings are Voronoi? I took a look ...
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24 views

Universal geometry

Is there something analog as to what universal algebra is for algebra for geometry? Some formalism incorporating both analytical geometry, differential geometry and algebraic topology. Or what comes ...
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54 views

Rational distances from the corners of a unit square

Is there a point all of whose distances from the corners of the unit square are rational?
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30 views

Iterated circumcenters - proving collinearity and establishing distance ratios

Let $P_0, P_1, P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle ...
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151 views

Differences and similarities between Euclidean and Minkowski geometry

I am trying to get my head around the differences and similarities between Euclidean and Minkowski plane geometry. AS far as I understand it they are both affine geometries meaning the parallel ...
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57 views

Isomorphism of divisors

Consider the cartier divisor group $CDiv_{T_{N}}(X_{\Sigma})$ defined on the fan $X_{\Sigma}$. I am having trouble proving the following assertion that there is a natural isomorphism ...
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Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
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85 views

Velociraptor escape

This is a mock test problem from xkcd. I think that he should run at the angle bisecting two of the dinosaurs, but then again, one is wounded.
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Constructing Pythagorean Polygons

I ran into this idea of "Pythagorean Polygons" on a problem from Project Euler, and I thought of an interesting question. A "Pythagorean Polygon" is defined as a polygon that is cyclic and has its ...