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

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Intersection of a quadric and cubic in $\mathbb{P}^3$

My question is drawn from Miles Reid's textbook Undergraduate Algebraic Geometry, p. 116. Let $S \subset \mathbb{P}^3$ be a smooth, irreducible cubic. Let $l_1, l_2, l_3, l_4 \subset S$ be ...
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+100

Arithmetic implications of different ways to geometrically construct an Hilbert's curve

I have a question on the relation between the geometric and the arithmetic representation of the Hilbert's space-filling curve. Geometric representation: consider the Hilbert's curve $f_h:[0,1]\...
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25 views

Global Chart implies no cut locus?

If a manifold $M$ admits a global chart, does this imply that there exists a point $p\in M$ such that $Cut_p=\emptyset$? Recall: Definition of $Cut_p$: Let $\mathfrak{C}_p$ be defined as the set ...
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Interesting cube subdivisions: what is going on here, and what are these polytopes?

I was messing around recently with a unit cube. If you draw vertices on the midpoint of each edge of the cube, then connect those points by new edges, you will form the wireframe of what I figured ...
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68 views

Optimal escape route out of a half-space in $\mathbb{R}^3$

In $\mathbb{R}^3$, what is the minimum length of a curve starting at the origin whose convex hull contains the unit sphere centered at the origin? I'm looking for an exact answer or bounds. The ...
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Prove that $\angle{EPF}$ is right angle.

$D$ is any point on side $BC$ of $\triangle{ABC}$. Points $E, I, F$ are respectively the incentres of $\triangle{ADB},\triangle{ABC},\triangle{ADC}$. $P$ is the foot of perpendicular from $I$ on $BC$. ...
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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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83 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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32 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 congruence,...
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147 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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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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68 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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170 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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98 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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64 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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64 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 $\overline{B}=\...
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107 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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182 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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171 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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680 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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60 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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66 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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117 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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191 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 \|S-\big(Y^TY\big)^{1/2}\|_F\leq\...
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128 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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104 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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119 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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121 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 re-...
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289 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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291 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 $$ g:\...
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370 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 \end{...
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522 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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193 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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107 views

Points at Integer Distances in 3-space

With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration of points on the plane such that all pairs of points are at integer distances. [1] For 3-...
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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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247 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 $a^2+b^...
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114 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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313 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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How did Archimedes calculate rational bounds of pi from a 96-gon?

Archimedes famously determined that $223/71 < \pi < 22/7$ using the 96-gons circumscribed by and circumscribing a circle of unit diameter. But I haven't found a reference that explains the final ...
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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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What is so special about natural isomorphism?

We know that all finite dimensional vector spaces of the same dimension are isomorphic. In particular, if $V$ is of finite dimension, then $(\star)$$V^{**} \cong V \cong V^{*}$. However, we know that ...
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minimize the area of convex hull of sum of 3 balls

How should we place 3 balls $B_1,B_2,B_3$ on the plane, if we want to minimize the area of convex hull of $B_1\cup B_2\cup B_3$ ? Balls can have boundary common points only -- the intersection of any ...
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Is there a way to determine the cheapest way to cut a line if each cut costs the current length of a line?

I was reading through an example question for the UNSW Computing ProgComp and found a question they claimed to be impossible to solve without going through all possible solutions. From https://cgi....
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Archimedes Classic Proof for Area of Circle: Love it but can't grasp one aspect…

The proof assumes that:... The perimeter of any CIRCUMSCRIBED regular polygon is GREATER than the circumference of the circle. ie: !http://www.themathpage.com/atrig/Trig_IMG/eval1.gif Is this an ...
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Can a portion of a hypocycloid be a regular polygon?

Hypocycloids are curves that generally don't include straight lines. A significant exception is a hypocycloid with 2 cusps, generated by rolling one circle inside another having twice the radius of ...
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Representing a point in the convex hull

Consider a set of $n$ points $x_i , i= 1 ... n$ belonging to some space $\mathbb{R}^m$. Given a point $p$ in the convex hull of the $x_i$ it is well known that we can represent $p$ as $ p = \sum _i ^ ...
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Prove that the circle contains the polygon.

Given a convex polygon. The circle is constructed for every triple of consecutive vertices of the polygon.We get the n circles. Select the circle with the largest radius. Prove that the circle ...
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88 views

Uniqueness of ruling for a ruled surface

Suppose one has two isomorphic (geometrically) ruled surface $C_1\times \mathbb{P}^1$ and $C_2\times \mathbb{P}^1$, how can we show $C_1$ is isomorphic to $C_2$? Thanks!
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“Algebraic” closure of hyperfields

It is very interesting to see that the notion of field (single-valued addition and multiplication) can be extended to the notion of hyperfield (multi-valued addition and single-valued multiplication). ...