geometry assuming the parallel postulate of Euclid: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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238 views

Hidden geometrical gems in Euclid's Elements?

I am teaching a course in Euclidean geometry at the University of South Carolina, and it seemed highly appropriate and interesting to read Euclid himself. (See here for a wonderful, and completely ...
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156 views

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 ...
6
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83 views

What property does this equation calculate?

It's pretty difficult to Google for the meaning of a formula. This is the equation, it has to do with ellipses and GIS coordinates. $$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$ $a$ ...
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258 views

Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial: The shortest curve between two points is a (straight) line. An elementary school teacher construed it as follows ...
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34 views

Regular polygons constructed inside regular polygons

Let $P$ be a regular $n$-gon, and erect on each edge toward the inside a regular $k$-gon, with edge lengths matching. See the example below for $n=12$ and $k=3,\ldots,11$.       Two ...
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284 views

Counting simple quadrilaterals in a rectangular lattice.

I've been trying to make an algorithm to find the number of all possible simple quadrilaterals in a N*M lattice. I already have a brute force solution but since this is a Project Euler problem I ...
4
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53 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 ...
4
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304 views

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 ...
4
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117 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 ...
4
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174 views

Three properties of the Lebesgue measure on $\mathbb{R}^n$

I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$. It is a non-negative ...
4
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92 views

Unlit region in a Room lined with Mirror

Mathematician Ernst Straus wondered if a room lined with mirror can always be lit with a single match. He (or someone else) discovered that in the following room, light shone from A can't reach B: ...
4
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169 views

Which chapters of Euclid's elements would be helpful for drawing a grid?

I am drawing a $19 \times 19$ grid on my desk. For aesthetic purposes, I don't want to use a ruler. Rather, I want to use Euclidean theorems to 'prove' to myself that such and such line meets at a ...
3
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51 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 ...
3
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44 views

Euclidean geometry and irrational numbers.

I was wondering, given a square that is $1 \times 1$, how can we know that the diagonal is an irrational length geometrically??? We could use the Pythagorean Theorem to see that the diagonal of a ...
3
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76 views

Heron's Formula; an Intuitive or Visual Proof

I've found several proofs for Heron's formula for the area of a triangle in term of its sides, but none of them is simple and intuitive enough to show WHY the formula works. Do you know an intuitive ...
3
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51 views

What do the Purser's theorem says?

Mathworld's entry on Purser's Theorem says the following: Let $t, u$, and $v$ be the lengths of the tangents to a circle $C$ from the vertices of a triangle with sides of lengths $a, b$, and $c$. ...
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67 views

How to determine the length of $x$ in this sketch?

Consider the following construction: Assume that you are given the lengths of the sides $a,b,c,d,e$ as well as the size of the angle $\phi$. The task is to determine the length of $x$ in terms of the ...
3
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189 views

Menelaus's Theorem clarification

If three points, one on each side of a triangle (extended if necessary) are collinear, then the product of the ratios of division of the sides by the points is -1 if internal ratios are considered ...
3
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189 views

Ellipse and circles

Playing with an ellipse I discovered the following properties and am now looking for a nice proof or references. Let's consider an ellipse with foci $A$ and $B$ and let $C$ be a point on it. Let $l$ ...
3
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89 views

Computing the proportion of vectors with the same sign

Let $s \propto (1, \ldots, 1)$ be a d-dimensional unit vector, so that $s_i = 1 / \sqrt{d}$. Let ${\cal V} = \{ u \in {\mathbb S}^{d-1}:u \cdot s = \cos \theta \}$ be a collection of unit vectors that ...
3
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243 views

Euclidean geometry and the Euclidean group

At Wikipedia's Erlangen program I read that "quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups". Some examples are given. But what are examples ...
2
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31 views

Expected area of an inscribed triangle in a sphere

On the surface of a unit sphere, three points $A$, $B$ and $C$ are chosen in the following way: Points $A$ and $B$ are chosen randomly and independently on the whole surface After $A$ ...
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46 views

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and its smallest containing square has side-length 2. What is the smallest possible angle of the polygon? What is its smallest possible area? ...
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22 views

Theorem about two quadrilaterals with parallel edges

I'm looking for a name for the following theorem: If $abAB$ lie on one line and $cdCD$ lie on another line, and furthermore $ac\Vert AC,ad\Vert AD,bc\Vert BC$, then $bd\Vert BD$. One can ...
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37 views

How do we understand and visualize the meridians of a 3-sphere?

The equator of a n-sphere is (n-1)-sphere. Then the equator of a 3-sphere is a 2-sphere. Does this mean the meridian of a 3-sphere is a hemisphere (half 2-sphere)? In contrast, Wikipedia describes the ...
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45 views

Geometric conditions equivalent to a set being the unit circle for some norm

Here's the question, as in the textbook (Real Mathematical Analysis, Pugh). The unit ball with respect to a norm $||\, \cdot \,||$ on $\mathbb{R}^2$ is $$ \{ v \in \mathbb{R}^2 : ||\, v \,|| ...
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31 views

Are a half-disk and a quarter-disk scissors-congruent?

Call two measurable subsets of the plane of equal area scissors-congruent if one can be decomposed into a finite number of measurable pieces which may be reassembled to yield the second. Any polygon ...
2
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77 views

Shining a laser into a mirror maze

I tried to formulate the following problem in a more mechanical way involving soccer balls, but the physics got too unrealistic. I know that what follows could be made more precise, but I hope the ...
2
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118 views

Dynamic Geometry Software for Straight-edge and Compass Constructions

Geogebra is a very good dynamic geometry software. It has so many default tools, e.g. parallel line, angle bisector, tangent to the circle, inscribed and circumscribed circles, etc. But I want the ...
2
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89 views

Megiddo's algorithm for lines of least weighted sum distance from a set of points

I came across the following problem: Given a set of n points (coordinate in 2d plane) within a rectangular space, find out a line ($ax+by=c$), from which the sum of the perpendicular distances of all ...
2
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156 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
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73 views

Are there impossible boolean constructions?

I was reading about logic and I remember, for example: That with the binary $\mathtt{NAND}$ connector can be used to assemble all the other binary connectors - I already know that there are primitive ...
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132 views

Infimum over area of certain convex polygons

Let us identify the Euclidean plane with the complex plane $\Bbb C$. For every $n\ge 1$, consider the following collection of finite sets in $\Bbb C^{n+1}$: $$S_n:=\{(z_0,z_1,\dots,z_n)\in\Bbb ...
2
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84 views

Inequalities involving areas of triangles in the plane, and generalizations

It is known that if $O$ is the origin, and $A, B, C, D$ are points in the first quadrant (in $\Re^2$) ordered such that the sequence of slopes of the lines $OA, OB, OC, OD$ is increasing, then $\Delta ...
2
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62 views

About the relation between a tetrahedra and spheres moving in a tetrahedra

I found the following question in a book: There exists a regular triangle $OAB$ which has edge-length $2$. Let $H, I, J$ be a foot of the perpendicular line drawn from a point $P$ in $OAB$ to the ...
2
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191 views

euclidean distance vs squared euclidean distances in 1 dimension, which one is the best?

I would like to compare the "disparities" between two groups. I define the mean disparity of a group as the mean of all the euclidean distances within the group with both groups presenting their ...
2
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68 views

quadrature formula for a lune

I have been reading about quadrature formulas in the complex plane. On the set $\mathbb{D} = \{|z| < 1\}$ we have $\int_{\mathbb{D}} f(z) dA = \pi f(0)$. Standard result in Harmonic functions. ...
2
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39 views

light geometry and art

I'm a professional artist. I have a question regarding light logic. I have simplified the problem a bit. Imagine a segment in 3D space, AB of known length and a point light illuminating the ...
2
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308 views

Derivative/Jacobian of the matrix logarithm

I need help finding the Jacobian of the matrix logarithm function, i.e. $\log{M} = R$ defined by $e^R = M = V\begin{bmatrix}e^{\lambda_1} & & \\ & \ddots & \\ & & e^{\lambda_n} ...
2
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93 views

Five squares in a box.

Let $A,B,C,D$ and $E$ be five non-overlapping squares inside a unit square, of side lengths $a,b,c,d,e$, respectively. Prove that $$a+b+c+d+e \le 2.$$ (This problem is a continuation of my previous ...
2
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205 views

What is requirement to inscibe sphere in pyramid with quadrilateral base?

In certain math problem the only information about pyramid with quadrilateral base is that you can inscribe sphere inside. What kind of constraints was put on my pyramid ABCDS? Is it something ...
2
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36 views

Euclidian embedding of lines

I'm looking for a way to convert a set of lines in R^3 into points in R^n so that distance between any pair of points points is a good approximation of the distance between corresponding pair of ...
2
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0answers
452 views

Equations of branches of a mind map

Sorry for the long question, but it's not so simple to explain. Consider a mind map like this: I want to draw branches in a cartesian coordinate system. I'd like to find two equations which ...
2
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352 views

Euler's Line of a medial triangle

I have the following problem with a comment below on the steps that I took so far. Here is the example: Let triangle ABC be any triangle. The midpoints of the sides in Triangle ABC are labeled $A', ...
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65 views

Plane tessellation $6^2*3^2$

An article I am reading mentioned "the plane tessellation $6^2*3^2$", I tried looking it up and I found all sort of plane tessellations - but not $6^2*3^2$. However, I did find information about ...
2
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223 views

Geometric interpretation of element by element division of one vector by another

This is my first post here, and I'm not a mathematician, so please go easy on me :) In statistics there is a geometric interpretation of correlation that uses basic vector geometry. This is fairly ...
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188 views

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, then is there a similar thing do happen in Spherical and hyperbolic spaces? In ...
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744 views

History of mathematical symbols, especially the symbol for right angle

Yesterday a child asked me, why (historically) a right angle is denoted by an arc and a dot like in this picture: I dont't know it, but I am interested in it too, so I post this question to this ...
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568 views

Is there a formula for the solid angle at each vertex of tetrahedron?

A tetrahedron has four vertices as much as a triangle has three vertices. A tetrahedron therefore can have four solid angles as much as a triangle can have three angles. I am wondering: Is there a ...
2
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203 views

Generalized Jordan curve theorem (and a related MAIN QUESTION)

Preliminaries A Jordan map is a continuous map $f: [0,1] \rightarrow \mathbb{R}^2$ such that $f(0) = f(1)$ the restriction of $f\ $ to $[0,1)$ is injective A Jordan curve is a subset $\gamma$ of ...