Related papers: The Golden Angle is not Constructible
The difference between the (squared) sides of the Grace-Danielsson inequality for tetrahedra will be represented as a sum of two nonnegative terms. This gives another proof of the inequality. Examining the denominator allows us to…
A proof that there is no $3 \times 3$ magic square constructed with nine distinct square numbers is given.
This paper has been withdrawn, because the argument for generation of Goldstone modes is incorrect.
Let $ABC$ be an equilateral triangle. For certain triangles $T$ (the "tile") and certain $N$, it is possible to cut $ABC$ into $N$ copies of $T$. It is known that only certain shapes of $T$ are possible, but until now very little was known…
Starting with a flat sheet of paper, points can be constructed as the intersection of two folds. The set of constructible points clearly depends on which folds are admissible. In this paper, we study the situation where a fold is admissible…
The construction of an unbounded polyhedron from a "jagged'' convex cap is described, and several of its properties discussed, including its relation to Alexandrov's "limit angle."
The golden ratio and Fibonacci numbers are found to occur in various aspects of nature. We discuss the occurrence of this ratio in an interesting physical problem concerning center of masses in two dimensions. The result is shown to be…
The quadratrix received its name from the circle quadrature, squaring the circle, but it only solves it if completed by taking a limit, as pointed out already in antiquity. We ask if it can square the circle without limits and restrict its…
This article proves that there are no self-intersections in the dragon curve when the unfolding angle is greater than 98.195{\deg}. This is shown by constructing a hull for the dragon curve that is mapped onto itself by the generating…
Closed form solutions for the computation of the solid angle from polygonal cross-sections are well known, however similar formulae for computation of projected solid angle are not generally available. Formulae for computing the projected…
The rectangle condition for a genus $g$ Heegaard splitting of a 3-manifold, defined by Casson and Gordon, provides a sufficient criterion for the Heegaard splitting to be strongly irreducible. However it is unknown whether there exists a…
In this short note, we construct some nontrivial examples of topological biquandle. The key ingredient of the construction is the notion of skew brace.
Richard Guy asked the following question: can we find a triangle with rational sides, medians, and area? Such a triangle is called a \emph{perfect triangle} and no example has been found to date. It is widely believed that such a triangle…
A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of…
It is well known that the set of origami constructible numbers is larger than the classical straight-edge and compass constructible numbers. However, the Huzita-Justin-Hatori origami constructible numbers remain algebraic so that the…
This paper shows that the method of Galperin which had been widely accepted in the literature since 1983 for constructing a non-dense and non-periodic trajectory in a triangle can never work. This is done by showing that all possible…
We construct non-constructible simplicial $d$-spheres with $d+10$ vertices and non-constructible, non-realizable simplicial $d$-balls with $d+9$ vertices for $d\geq 3$.
Consider an orthogonal polyhedron, i.e., a polyhedron where (at least after a suitable rotation) all faces are perpendicular to a coordinate axis, and hence all edges are parallel to a coordinate axis. Clearly, any facial angle and any…
A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…
We provide a complete description of the edge-to-edge tilings with a regular triangle and a shield-shaped hexagon with no right angle. The case of a hexagon with a right angle is also briefly discussed.