Tagged Questions

1
vote
0answers
39 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 ...
3
votes
3answers
119 views

Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space?

I have question. Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space ? Please can you give me an advice some book names? Thank you!
0
votes
1answer
52 views

Euclidean geometry applied to Ptolemy geocentric model

I'm looking for a good reference on how Ptolemy used the Euclidean geometry to calculate the planets positions.
16
votes
1answer
293 views

Does there exist a copy of Euclid's Elements with modern notation and no figures?

I am working through Euclid's Elements for fun, but I find the propositions difficult to understand without referencing the provided figures. Unfortunately, the figures usually give away the proofs, ...
1
vote
0answers
58 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 ...
3
votes
1answer
110 views

Is there a geometric proof to answers about the 3 classical problems?

I know that there is a solution to this topic using algebra (for example, this post). But I would like to know if there is a geometric proof to show this impossibility. Thanks.
0
votes
1answer
151 views

Proof of Archimedes Lemma about the Center of Mass

I am looking for a proof of the following statement, known as Archimedes' Lemma: If an object is divided into two smaller objects, the center of mass of the compound object lies on the line segment ...
4
votes
2answers
120 views

Higher-dimensional Extension of Triangle Geometry?

I am currently exploring generalizations of triangle geometry to higher dimensions. I know that "important questions" of Euclidean geometry have been already addressed and is considered obsolete; most ...
7
votes
1answer
109 views

A scalar product in the space of oriented volumes?

Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
9
votes
5answers
898 views

Geometry Book Recommendation?

Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...
0
votes
3answers
277 views

Looking for a book dealing with Euclidean Geometry

Good Day I am looking for a Book that contains detailed proofs of theorems in the field of Euclidean Geometry that are used to solve exams and school work of high school students. theorems like: ...
7
votes
2answers
243 views

Construction of a regular pentagon

In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only. It is the pentagon $ABCDE$ pictured below. I am having trouble seeing why the central angles ...
2
votes
1answer
365 views

Similar Triangle Theorem in the Incommensurable Case

The following is a geometry theorem whose proof is examinable in the Irish 'High School' Exam. Let $\Delta ABC$ be a triangle. If a line $L$ is parallel to $BC$ and cuts $[AB]$ in the ratio $s:t$, ...
8
votes
5answers
418 views

What is the modern axiomatization of (Euclidean) plane geometry?

I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
1
vote
2answers
237 views

High-dimensionality and intuition

Over the years I've come across (usually as a tangential remark in a lecture) examples of how our intuitions (derived as they are from the experience of living in 3-dimensional space) will lead us ...
5
votes
1answer
136 views

Triangle from lengths of angle bisectors

According to http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml it is impossible to construct a triangle from the lengths of its angle bisectors. Is there a more comprehensive account of ...
1
vote
1answer
72 views

Arrangements of congruent rectangles

I have stumbled on an interesting problem. How many congruent rectangles on a plane can be arranged in such a way that they all touch each other but never overlap? I pondered this problem for a few ...
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 ...
9
votes
6answers
3k views

Book recommendation on plane Euclidean geometry

I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more ...
8
votes
1answer
227 views

Chebyshev center = center of mass?

I would like to know for which convex polyhedra $P$ in $\mathbb{R}^3$, is the center of the largest sphere enclosed in $P$ (a.k.a. the Chebyshev center, or the incenter) the same as the center of ...
-2
votes
1answer
323 views

Greek Geometry Algebra [closed]

Tell me everything you know about Euklid, Pythagoras etcetera. I'm about to begin reading about greek algebra used in calculating geometric figures. After reading I'm going to write a 15-20 pages ...
1
vote
1answer
458 views

Distinct Hamiltonian cycles of the icosahedron and dodecahedron

I am seeking a listing of the distinct Hamiltonian cycles following the edges of the icosahedron and the dodecahedron. By distinct I mean they are not congruent by some symmetry of the icosahedron or ...