56
votes
Does this geometry theorem have a name?
Even more is true for this theorem. Check out this drawing from Arseniy Akopyan wonderful book of Geometry in Figures (Second, extended edition, 2017). On page 65 we find Figure 4.7.29)
In the ...
- 10.3k
42
votes
Accepted
Understanding sphere packing in higher dimensions
There are two things you need to understand. The first is how to prove sphere packing bounds via harmonic analysis ("linear programming bounds"). My lecture notes from PCMI 2014 give an exposition ...
- 16.2k
41
votes
Accepted
Which polygons can be turned inside out by a smooth deformation?
This question was explored here:
Lenhart, William J., and Sue H. Whitesides. "Reconfiguring closed polygonal chains in Euclidean $d$-space." Discrete & Computational Geometry 13, no. 1 (1995): ...
- 147k
39
votes
Accepted
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
These are the so-called Regge symmetries, described by T. Regge in a 1970-ish paper. For a bit on it, with references, see the paper
Philip P. Boalch, MR 2342290 Regge and Okamoto symmetries, Comm. ...
- 94.5k
39
votes
A curious relation between angles and lengths of edges of a tetrahedron
Euclidean case
Using the formula for the tan of the half solid angle that Robin Houston quotes, and expressing everything in terms of edge lengths by using the cosine law to convert the dot products, ...
- 2,842
39
votes
Accepted
Automatically solving olympiad geometry problems
Arguably, the so-called "area method" of Chou, Gao and Zhang represents the state of the art in the field of machine proofs of Olympiad-style geometry problems. Their book Machine Proofs in ...
- 70.8k
34
votes
Is orientability a miracle?
To me, it is not that hard to imagine an alternate universe where the fact that $|\pi_0(GL_n(\mathbb{R}))| = |\pi_0(O_n(\mathbb{R}))| = 2$ is an unstable fact that holds for small $n$, but not in very ...
30
votes
A curious relation between angles and lengths of edges of a tetrahedron
This is not an answer to the question, but an experimental observation that suggests a sharper conjecture: it’s only written as an answer because I’d like to flesh it out a bit more than there’s room ...
- 2,726
25
votes
Tetrahedra passing through a hole
Did you ever find any answer to this?
I find it intriguing that figuring out which shapes of holes a given solid object can pass through is widely considered to be a suitable puzzle for 2 year olds, ...
- 351
21
votes
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
To address the scissors congruence question at the end of the post: the Regge symmetries produce tetrahedra which are scissors congruent. This is proved in Section 6 (Theorem 9 and Corollary 10) of
...
- 16.8k
20
votes
Accepted
Emergence of the orthogonal group
Your quote about Cartan thinking of $B_n$ and $D_n$ as 'projective groups..." is actually Cartan describing the lowest dimensional homogeneous space of these groups (except, of course, for a few ...
- 103k
20
votes
Accepted
Is it a new discovery on conic section?
It suffices to consider the case when $\Omega$ is a circumcircle, so let it be.
At first, the points $A_b, A_c, B_c, B_a, C_a, C_b$ lie on a conic if and only if
$$
\frac{AB_a\cdot AB_c}{AC_a\cdot ...
- 93.6k
19
votes
Open problems in Euclidean geometry?
W. Wernick has tabulated 139 triangle construction problems using a
list of sixteen points associated with the triangle. In each case three points are given and the goal is to construct a triangle ...
19
votes
Open problems from antiquity solved with analytic geometry
A classic problem in this category is Alhazen's billiard problem. I reproduce a quote from 100 Great Problems of Elementary Mathematics. The problem could not be solved using compass and ruler ...
18
votes
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
Let me give a geometric interpretation for the case of tetrahedra of volume zero. The statement becomes as follows:
Given four positive numbers $a,b,c,d$ that satisfy the quadrangle inequalities, ...
- 6,137
18
votes
A curious relation between angles and lengths of edges of a tetrahedron
Here goes the elementary proof of the claim by Robert Houston that the quadraples $(P_1^{-1},P_2^{-1},P_3^{-1},P_4^{-1})$ and $(\cot \frac{\Omega_1}2,\cot \frac{\Omega_2}2,\cot \frac{\Omega_3}2,\cot \...
- 93.6k
17
votes
Open problems in Euclidean geometry?
Chromatic Number of the Plane or Hadwiger–Nelson problem asks for the minimum number of colors required to color the plane such that no two points at distance $1$ from each other have the same color.
...
17
votes
Accepted
On the determinant of a class symmetric matrices
Your guess is correct. If the elements outside the diagonal have absolute values less than $1/(n-1)$, the matrix has 'diagonal dominance', thus it is nonsingular.
To make the answer self-contained, ...
- 93.6k
17
votes
Accepted
Axioms for constructive Euclidean geometry
Have a look at Hartshorne's Geometry: Euclid and Beyond. He uses Hilbert's axioms for geometry and discusses (section 11) the following "circle-circle intersection axiom (E)":
Given two circles $\...
- 62.4k
17
votes
Accepted
The 4th vertex of a triangle?
I propose an alternative "fourth vertex".
To paraphrase Sherman's result:
The "fourth side" ($w$) of a triangle is a chord of the circumcircle, is a tangent to the incircle ($\...
- 1,148
17
votes
Is orientability a miracle?
One interesting phenomenon is that there’s a good notion of symmetric tensor category $\mathrm{Rep}(O_t)$ when $t$ is not an integer, but no good notion of $\mathrm{Rep}(SO_t)$ for generic $t$. (...
- 27.2k
16
votes
Accepted
A problem in elementary geometry
Let O be the point of intersection of the angle bisectors of the initial triangle ABC (the incenter). Then the length of OB' is equal to the length of OC, |OC'|=|OA| and |OA'|=|OB|. Now the statement ...
16
votes
A curious relation between angles and lengths of edges of a tetrahedron
Here one can find an algebra-geometric proof of the trigonometric relations. The main point is that given a (say, spherical) tetrahedron $T$ one can construct a rational elliptic surface $X_T$ with ...
- 4,176
16
votes
Accepted
Three circles intersecting at one point
Such things are quick in complex numbers. Let $O=0$ be the origin, $ABC$ be the unit circle. The centroid of $ABC$ is $G=(A+B+C)/3$, the Euler circle is the image of the circle $ABC$ under homothety $...
- 93.6k
15
votes
Is Euclid dead?
It might be interesting to point out that the support for removing axiomatically taught Euclidean geometry (not all geometry) from the school education predates Bourbaki. Oliver Heaviside (1850-1925),...
15
votes
Accepted
What is needed to prove the consistency of Tarski's Euclidean geometry?
In 1999, Harvey Friedman showed how to prove the consistency of Tarski's axioms for geometry in EFA. This is Elementary Function Arithmetic, otherwise known as $I\Delta_0(exp)$, a subtheory of PRA ...
- 18.7k
15
votes
Accepted
Is there a triangle which makes dense set of angles by drawing medians?
The answer to the second question is yes, for any non-flat triangle $T$, the set of angles $A$ is dense in $(0, \pi)$. This follows from a stronger result of Barany et al. [Theorem 1, 1]:
...
- 52.6k
15
votes
Open problems from antiquity solved with analytic geometry
This paper contains a very readable account of Descartes invention of analytic geometry and describes some questions that it can solve reasonably easily compared to methods familiar to the ancient ...
15
votes
Accepted
The abc-conjecture as an inequality for inner-products?
The matrix $L_n$ is positive definite.
Proof. The matrix $G_n$ with entries ${\rm gcd}(a,b)$ is positive definite because of $G=D^T\Phi D$ where $\Phi={\rm diag}(\phi(1),\ldots,\phi(n))$ ($\phi$ the ...
- 49.2k
14
votes
Open problems in Euclidean geometry?
I am reposting my own answer from Not especially famous, long-open problems which anyone can understand for it answers this question as well.
Are there eight points on the plane, no three on a line,...
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