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 ...

45
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 ...

43
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 ...

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): ...

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. ...

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, ...

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 ...

Community wiki

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 ...

26
votes

Accepted

### Is symmetric power of a manifold a manifold?

$\newcommand{\Cone}{\operatorname{Cone}}$Let $d$ be the dimension of the manifold $M$. For $n \geq 2$, I will prove that the symmetric power $SP^n(M)$ is a manifold with boundary for $d=1$, a ...

26
votes

Accepted

### Does a function from $\mathbb R^2$ to $\mathbb R$ which sums to 0 on the corners of any unit square have to vanish everywhere?

The answer is yes, I learned this from the paper that appeared on arXiv literally yesterday https://arxiv.org/abs/2403.01279, they quote
R. Katz, M. Krebs and A. Shaheen, Zero sums on unit square ...

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, ...

22
votes

### Psychological test for Euclidean geometry

The FCI is a "concept inventory" test for classical mechanics. There exist several such tests for mathematics. Some may be implemented as "peer instruction" (as Eric Mazur ...

22
votes

Accepted

### Group generated by two irrational plane rotations

The commutator of any two elements of your group is a translation, so they all commute. So for example if $a$ and $b$ are two elements of your group then $[a,b]$ commutes with $[a^b,b]$. This is a ...

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
...

21
votes

Accepted

### Is there a pyramid with all four faces being right triangles?

This picture of a quadrirectangular tetrahedron is from István Lénárt, `The Right Triangle as the Simplex in 2D Euclidean Space, Generalized to $n$ Dimensions'

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 ...

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 ...

20
votes

Accepted

### Aperiodic monotile without reflections?

The same authors have just released a preprint claiming a positive answer to this question.
EDIT: Here is a picture of the reflection-free aperiodic monotile:
More visualizations and other data are ...

19
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, ...

19
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 ...

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 ...

Community wiki

18
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 $\...

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 \...

18
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 ($\...

18
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 $t$. (...

18
votes

### What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

The area, call it $K$, is indeed maximized when the quadrilateral is cyclic,
in which case $K$ is given by Brahmagupta's
remarkable generalization of Heron's formula:
$K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$ ...

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, ...

17
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 ...

17
votes

### Is there an absolute geometry that underlies spherical, Euclidean and hyperbolic geometry?

I don't know if this is the kind of answer you expect, but:
In the hyperbolic space of dimension $n+1$ one naturally gets all $n$-dimensional constant curvature geometries.
spheres (points at ...

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