Related papers: The Golden Angle is not Constructible

A construction to arbitrarily section a taxicab angle into an equal number of angles in (pure) taxicab geometry is presented.

The status of angles within The International System of Units (SI) has long been a source of controversy and confusion. We address one specific but crucial issue, putting the case that the idea of angles necessarily being length ratios, and…

It is proved that the non-rationality of a generic cubic fourfold follows from a conjecture on the non-decomposability in the direct sum of non-trivial polarized Hodge structures of the polarized Hodge structure on transcendental cycles on…

We show the non existence of manifold structure on certain topological constructions. The main results of this paper are: (1) The wedge of manifold with itself is not a manifold. (2) The cone of a manifold is not a manifold. The main tools…

We construct a faithfully flat algebra over the infinite polynomial ring on an algebraically closed field that is not descendable.

This paper presents formulae for calculation the solid angle of intersecting spherical caps, conical surfaces and polyhedral cones.

Let $\vartheta := \frac{-1+\sqrt{5}}{2}$ be the golden ratio. A golden lattice is an even unimodular $\Z[\vartheta ]$-lattice of which the Hilbert theta series is an extremal Hilbert modular form. We construct golden lattices from extremal…

This note is purely expository. The statement of the Gauss theorem on the constructibility of regular polygons by means of compass and ruler is simple and well-known. However, its proofs given in most textbooks rely upon much unmotivated…

We prove some constructive results that on first and maybe even on second glance seem impossible.

In this paper we prove the transcendence of $\pi$ using Hilbert's method. We also prove that all points constructible with compass and straightedge have algebraic coordinates. Thus we give a self-contained proof that squaring the circle is…

We show that conformal geodesics on a Riemannian manifold cannot spiral: there does not exist a conformal geodesic which becomes trapped in every neighbourhood of a point.

A construction similar to Hagge's construction for circles through the orthocentre is shown to apply for any point.

An example is constructed of a local ring and a module of finite type and finite projective dimension over that ring such that the module is not rigid. This shows that the rigidity conjecture is false.

We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families…

A tetrahedron is called a path tetrahedron, if it has three mutually orthogonal edges that do not intersect at a single point. A tetrahedron is called a 4-ball tetrahedron, if there exists a sphere tangent to all its edges. We derive…

We construct a Jordan curve $\Gamma \subset \mathbb{C}$ so that for any rectifiable arc $\sigma$ with endpoints in distinct complementary components of $\Gamma$, $H^1(\sigma \cap \Gamma) > 0$.

We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of…

We prove the theorem in the title, and prove the theorem for 11 as well as 7. By previous work of others, the problem reduces to a number of cases. The cases not solved already are solved here.

We present a multi-edge-length aperiodic tiling which exhibits 6--fold rotational symmetry. The edge lengths of the tiling are proportional to 1:$\tau$, where $\tau$ is the golden mean $\frac{1+\sqrt{5}}{2}$. We show how the tiling can be…

In this note we show that the crucial orientation condition for commutative geometries fails for the natural spectral triple of an orbifold M/G.