We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a tetrahedron T, divided by 2*pi, gives the probability that an orthogonal projection of T onto a random 2-plane is a triangle. More generally, it is shown that the sum of the (solid) inner vertex angles of an n-simplex S, normalized by the area of the unit (n-1)-hemisphere, gives the probability that an orthogonal projection of S onto a random hyperplane is an (n-1)-simplex. Applications to more general polytopes are treated briefly, as is the related Perles-Shephard proof of the classical Gram-Euler relations.
A point in the interior of a planar triangle determines a subdivision into six subtriangles. A triangle with angles commensurable with $π$ is called $π$-commensurable. For such a triangle a subdivision where each of the subtriangles are $π$-commensurable too is called $π$-commensurable. We prove that there are infinitely many $π$-commensurable triangles that do not admit any $π$-commensurable subdivision except the one given by angle bisectors. We count the number of $π$-commensurable subdivisions of triangles. We perform a similar count for Z-degree sub-divisions of Z-degree triangles too. Fi…
Let $M$ be the space of triangles, defined up to shifts, rotations and dilations. We define two maps $f:M\to M$ and $g:M\to M$. The map $f$ corresponds to a triangle of perimeter $π$ the triangle with angles numerically equal to edges of the initial triangle. The map $g$ corresponds to a triangle of perimeter $2π$ the triangle with \emph{exterior} angles numerically equal to edges of the initial triangle. For $p\in M$ the sequence $\{p,f(p),f(f(p)),\ldots\}$ converges to the equilateral triangle and the sequence $\{p,g(p),g(g(p)),\ldots\}$ converges to the "degenerate triangle" with angles $(0…
Given a triangle $Δ$, we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of $Δ$ with respect to area and perimeter. This problem was initially posed by Nandakumar and was first studied by Kiss, Pach, and Somlai, who showed that if $Δ'$ is the smallest area isosceles triangle containing $Δ$, then $Δ'$ and $Δ$ share a side and an angle. In the present paper, we prove that for any triangle $Δ$, every maximum area isosceles triangle embedded in $Δ$ and every maximum perimeter isosceles triangle embedded in $Δ$ shares a side and an angle with $Δ$. So…
After having investigated the geodesic and translation triangles and their angle sums in $\SOL$ and $\SLR$ geometries we consider the analogous problem in $\NIL$ space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in $\NIL$ geometry and we provide a new approach to prove that it can be larger than or equal to $π$. Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of $\NIL$ isoptic surfaces of translation-like segments and as a special ca…
Aperiodic tiling is a well-know area of research. First developed by mathematicians for the mathematical challenge they represent and the beauty of their resulting patterns, they became a growing field of interest when their practical use started to emerge. This was mainly in the eighties when a link was established with quasi-periodic materials. Several aperiodic tilings made of two tiles were discovered, the first one being by Penrose in the seventies. Since then, scientists discovered other aperiodic tilings including the square-triangle one, a tiling that has been particularly useful for t…
Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into 4 triangles by joining the midpoints of its edges. We show the existence of a uniform $δ>0$ such that, at any step of the subdivision, all the triangle angles lie in the interval $(δ, π-δ)$. Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses.
Rubtsovsk Industrial Institute of Altai State Technical University after I.I. Polzunov（阿尔泰国立技术大学波卢佐诺夫鲁布佐夫斯克工业学院）；Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences（俄罗斯科学院弗拉季高加索科学中心南方数学研究所）
How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to students and the general public using different physical models, including chalksphere, crocheted hyperbolic surfaces, curved folding, and polygon tilings. Spherical geometry offers a simple yet surprising introduction to the topic, whereas hyperbolic geometry is an entirely new and exciting concept to most. Non-Euclidean geometry demonstrates how crafts and ar…
A strong triangle inequality in hyperbolic geometry
对于双曲平面中的三角形，令$α,β,γ$分别表示与边$a,b,c$相对的角。另外，设$h$为侧面高度$c$的高度。在假定$α,β, γ$在$(0,π)$区间内一致选取，并给出$α+β+γ<π$的条件下，证明了强三角不等式$a + b > c + h$在79%的情况下成立.为了实现这一点，我们证明了一些理论结果，以确保概率可以计算到任意精度，并且误差可以有界。
We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least $2π$ have minimal area among all nonpositively curved disks of the same perimeter and the same total curvature; (2) geodesic triangles in a Euclidean (resp. hyperbolic) cone of angle at least $2π$ have minimal area among all nonpositively curved geodesic triangles (resp. all geodesic triangles of curvature at most $-1$) with the same side lengths and angles.
A comprehensive analysis of the Snellius-Pothenot problem
Rubtsovsk Industrial Institute of Altai State Technical University after I.I. Polzunov（阿尔泰国立技术大学鲁布佐夫斯克工业学院）；Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences（俄罗斯科学院弗拉季高加索科学中心南方数学研究所）；Drake University（德雷克大学）
