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.
8
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123 views
Who first discovered that the torus supports a flat structure?
Who first recognized that there exists a homogenous metric on the closed genus 1 orientable surface?
8
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162 views
Can we alter Hilbert's axioms to have $\mathbb{Q}^3$ as a unique model?
Can we alter Hilbert's axioms to have $\mathbb{Q}^3$ as a unique model?
The critical axioms seem to be the congruence axioms IV.1 and IV.4, and presumably the line completeness axiom V.2.
But how ...
7
votes
0answers
98 views
Q: Given the graph of $y = \frac{1}{x}$, construct the $(x,y)$ coordinate axes using straightedge and compass
The solution to the problem above is known (see comments for a hint).
What other analytic functions can one substitute for $y = \frac{1}{x}$, and still be able to do so?
6
votes
0answers
114 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 ...
5
votes
0answers
101 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 ...
5
votes
0answers
63 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.
$$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$
$a$ ...
5
votes
0answers
143 views
Minimum cardinality of a difference set in $\mathbb R^n$
Given a finite set $S$ of $m$ points in $\mathbb R^n$ that do not all lie in the same $(n-1)$-dimensional hyperplane, consider the set of difference vectors:
$\{x-y \, | \, x,y \in S\}$
What is the ...
4
votes
0answers
46 views
A numerical coincidence with continued fractions
My brother built a garage that measures 45 feet by 30 feet. To make sure the right angles were accurate, he measured the two diagonals of the rectangle to see that they were equal. In inches,
$$
...
4
votes
0answers
102 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
votes
0answers
76 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
votes
0answers
137 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
votes
0answers
106 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
votes
0answers
140 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
votes
0answers
86 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 ...
2
votes
0answers
116 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
votes
0answers
56 views
Construct a pentagon form the midpoints of its sides
Let $p_{1},p_{2},p_{3},p_{4},p_{5}$ be five points in the euclidean plane such that no set of three of those points lie on the same line. It is easy to prove that there exists a unique pentagon such ...
2
votes
0answers
65 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
votes
0answers
33 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
votes
0answers
225 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
votes
0answers
261 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', ...
2
votes
0answers
58 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
votes
0answers
168 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 ...
2
votes
0answers
144 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 ...
2
votes
0answers
327 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
votes
0answers
147 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 ...
2
votes
0answers
70 views
Solving some geometry problems using involutions
Some geometry problems ( like this and this ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularly interested in ...
2
votes
0answers
211 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 ...
1
vote
0answers
38 views
What is known about $n$ independent random variables that yield a “converse” to uniform sample of a coordinate from a surface of an $n$-sphere?
It's well-known that to sample a coordinate $(Y_1,\ldots,Y_n)$ from a surface of an $n$-dimensional unit-radius sphere, it suffices to generate $n$ independent random variables $X_1,\ldots,X_n$ from ...
1
vote
0answers
45 views
What's a non-standard model of Tarskian Euclidean geometry?
Tarski's axioms (see here: http://en.wikipedia.org/wiki/Tarski%27s_axioms) are a first-order axiomatization of Euclidean Geometry. Now, I believe the standard model for the axioms is the real number ...
1
vote
0answers
31 views
find example from some regularities. euclids elements
I heared that there are 6 basic math abilities.
and one of them is to find an example from regularities.
I want to develop this ability so I tried to find an example from real world.
but I am not ...
1
vote
0answers
41 views
Proof by vectors segments, Euclidean.
Hi I have a midterm this Friday, and our prof gave us this practice midterm, I got $6$ of the $10$ questions but couldn't do the last $4$, could you please help me out in this question, I have no idea ...
1
vote
0answers
56 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.$$
(This problem is a continuation of my previous ...
1
vote
0answers
16 views
3D orientations from distance constraints
I want to determine the relative orientations within a set of rigid 3D objects given some pairwise distances between certain points on pairs of objects. There are sufficient constrains to fully ...
1
vote
0answers
56 views
Partition of open sets in $\mathbb{R}^d$.
Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets.
In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
1
vote
0answers
122 views
Apollonius' circle
A theorem states, that the three Apollonian circles, associated with the given triangle $ABC$ with sidelengths $a \neq b \neq c$ intersect in two points. The proof proceeds by showing that if the two ...
1
vote
0answers
101 views
Evaluating the average distance from a point in the unit disk to the disk
I am interested in finding the average Euclidean distance from a point $(x,y)\in\mathbb{D}_2$, the unit disk $\{(u,v):u^2+v^2\leq 1\}\subseteq\mathbb{R}^2$, to the disk $\mathbb{D}_2$. This amounts to ...
1
vote
0answers
220 views
Prove a very original version of Descartes's circle theorem
Prove:
I define the radius of three mutually externally tangent to be $d,e,f$ respectively. The circle with radius $x$ is internally tangent to all three circles. Then
$$ddeeff+ddeexx+ddffxx+eeffxx ...
1
vote
0answers
71 views
The orientation of a closed discrete curve embedded in a triangle.
The two triangles $xyz$ and $x^{\prime}y^{\prime}z^{\prime}$, shown below, have opposite orientations. A closed curve $abcd$ is embedded in the first triangle ($abcd$). The vertices of the ...
1
vote
0answers
73 views
Polynomial function from $S^3$ to $S^3$ and quaternions
I am searching the polynomial functions from $S^3$ to $S^3$.
We say $g$ is a polynomial function from $S^3$ to $S^3$, if there exists $f_1,f_2,f_3,f_4 \in \mathbb{R}[X_1,X_2,X_3,X_4]$ and
...
1
vote
0answers
431 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 ...
1
vote
0answers
371 views
rotate vector around another vector
If I have vectors $a = (1,0,0)$, $b = (0,1,0)$, and $c=(0,0,1)$ and I want to rotate them counterclockwise at rate $r$ rad/sec around vector (1,1,1). What are the formulas for $a, b, c$?
1
vote
0answers
121 views
Projection of the area of a bounded plane over other bounded plane
I have two bounded planes $\pi$ and $\rho$ in three dimensional space. Each plane is bounded by a coplanar rectangle. How can I find the orthogonal projection area of $\pi$ over $\rho$?
Thanks in ...
1
vote
0answers
126 views
Distance between two unbounded sets
How to calculate the distance between two (possibly unbounded) ranges of positive real numbers? For example, if three guys specify their prices they would pay for a product:
...
1
vote
0answers
169 views
Calculating the Epsilon Neighborhood of line segments in 3d
I am working on a trajectory clustering algorithm (in C++) and one of the steps required in this algorithm is to take a set of 3d line segments (D), and for each line segment (L) in D, to calculate an ...
1
vote
0answers
94 views
Complexity of Counting the number of inducing $n$-gons
Definition: A $n$-gon is simple if it has no self intersection and is in general position if no pair of its edges are parallel.
It is clear that by extending the edges of each simple $n$-gon in ...
1
vote
0answers
140 views
Characterizations of Euclidean space
There are presumably three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:
axiomatically (with axioms concerning dimension)
by the abstract Euclidean ...
0
votes
0answers
41 views
How to prove the property of scalar distribution over vector addition when the vectors are collinear?
$\overrightarrow{a},\overrightarrow{b} \in V^3 , \alpha \in \mathbb{R} $
Prove: $\alpha(\overrightarrow{a} +\overrightarrow{b}) = \alpha\overrightarrow{a} + \alpha\overrightarrow{b}$
When $\alpha = ...
0
votes
0answers
31 views
Distance to a convex polyhedron: about different approaches
I know there are a lot of litterature out there about convex polyhedra and distance computation, but I don't quite catch which one has the best computational complexity in practice and in theory. I ...
0
votes
0answers
40 views
Given 2 outer points of a perfect circle, find the centerpoint
Alright, I hope this makes some sense.
I am using a software that can create arcs.
This arc is defined by:
Begin point
End point
Center of "circle"
The center is supposed to be the center of the ...
0
votes
0answers
26 views
Magnetometer electronics engineering " having values X,Y,Z How to calculate angle from north in 3d
Having values X,Y,Z Mathematics of How to calculate angle from north in 3d?
Thank's in advance
