arXiv subjects

George M. Georgiou

Publications and source records attributed to George M. Georgiou.

8 recordsLinked to original sources

arXiv2026

The smallest square tileable by pairwise incomparable integer rectangles

Croft, Falconer and Guy ({Unsolved Problems in Geometry}, Problem~C5) exhibit a tiling of the $27\times27$ square by eight pairwise incomparable integer rectangles and remark that it is not known whether $27$ is the smallest side length of a square that can be tiled by pairwise incomparable integer rectangles, no restriction being placed on the number of tiles. We show that it is: for every integer $n\le26$ and every $k\ge2$, the $n\times n$ square admits no tiling by $k$ pairwise incomparable integer rectangles. The proof combines two structural reductions with an exhaustive search over the $167\,538$ surviving candidate tile sets, carried out by two independently written programs. The complete software, build instructions and output logs are included as ancillary files.

math.CO↗

arXiv2026

Joining Two Pairs of Planar Points by Disjoint Arcs: The Sharp $\sqrt{2}$ Length Bound

Problem F16 of Croft, Falconer, and Guy asks for the least worst-case length needed to join prescribed pairs of points by pairwise disjoint planar arcs, when the distance within each pair is at most one. The book suggests that the answer for two pairs is $\sqrt2$. We prove this exactly. The lower bound is a crossing argument in a square. The upper bound follows from a sharp ellipse lemma. As a consequence, the value proposed in the book for three pairs, $(\sqrt3+1)/2$, cannot be correct under the literal formulation, because the constants are nondecreasing in the number of pairs.

math.MG↗

arXiv2026

Congruent Triangular Faces, Reflections Allowed: Universal Realization and the Minimum Face Count in Problem B22

Problem B22 in Unsolved Problems in Geometry asks which triangles occur as the common face of a convex polyhedron, how many copies are needed, and how they may be arranged. We settle the existence and minimum-face-count questions for triangles in the version that allows reflected copies; we do not classify all attainable face counts, nor the possible arrangements. Every nondegenerate Euclidean triangle occurs: we exhibit an explicit convex polyhedron, combinatorially an octahedron, all eight of whose faces are congruent to a prescribed triangle. We then determine the minimum number of faces for \emph{every} triangle. It is four for an acute triangle; six for a right or obtuse isosceles triangle with side lengths $(λ,λ,β)$ satisfying $λ\sqrt2\leqβ<λ\sqrt3$; and eight in all remaining cases.

math.CO↗

arXiv2026

Five-Point Hyperbolas on Power Curves and Near Lamé-Oval Vertices

Problem A33 of Croft--Falconer--Guy records Reznick's question about infinite plane sets for which every five-point subset determines an ellipse or, respectively, a hyperbola, and suggests that $|x|^{2.001}+|y|^{2.001}=1$ might yield only ellipses. We give two results. First, if $p>2$ and $a>0$, every five distinct points of the power curve $u=ay^p$, $y>0$, determine a nondegenerate hyperbola; bounded subarcs therefore give nonconic rectifiable examples for the hyperbolic side of Reznick's question. Second, every fixed one-sided five-point profile, contracted toward an axial vertex of the Lamé oval $|x|^p+|y|^p=1$, eventually determines a nondegenerate hyperbola. Consequently, the suggested Lamé oval with $p=2.001$ has five-point subsets that determine nondegenerate hyperbolas, so it does not yield only ellipses. Symmetric profiles straddling the same vertex, however, determine ellipses. We also separate these results from the classical osculating-conic criterion supplied by equi-affine curvature.

math.MG↗

arXiv2026

Reciprocal-Polar Linearization and Two Conic Constructions for the Cone Projection

Let $$ f_R(x)=\frac{x}{1+\norm{x}/R},\qquad R>0, $$ be the radial cone projection of the plane onto the open disk $\DR=\{x:\norm{x}<R\}$. Previous work established the Self-Directrix and Confocal-Codirectrix Theorems, according to which $f_R$ maps focal conic arcs to focal conic arcs while preserving the distinguished focus and directrix. This note combines those results with three classical facts recorded by W.~H.~Besant. First, reciprocal polarity with respect to $\partial\DR$ represents every focal conic under consideration as the reciprocal polar of a unique circle, and in this circle representation the nonlinear action of $f_R$ becomes the elementary radius translation $q\mapsto q+R$. Second, the normals at the intersections of a fixed ray with the self-directrix family envelope an explicit parabola. Third, the tangent at $f_R(z)$ is constructed directly from the original point $z$ and the original line, without differentiating $f_R$ or solving for the conic.

math.MG↗

arXiv2026

Straightedge-and-Compass Constructibility of the Reciprocal-$k$-th-Power Law

Given positive lengths $a$ and $b$ and a positive integer $k$, let $c$ be determined by $$ \frac{1}{c^k}=\frac{1}{a^k}+\frac{1}{b^k}. $$ We prove that a single finite unmarked-straightedge-and-compass construction producing $c$ for every pair $a,b$ exists if and only if $k$ is a power of two. Necessity follows by specializing to $a=b=1$ and applying the algebraic-degree obstruction to $2^{1/k}$; sufficiency is established by a recursive construction using third, fourth, and mean proportionals. We give an explicit construction for $k=4$ and its iteration for $k=8,16,\ldots$. We also describe a plane construction, intersecting a line with a Lamé curve, that realizes the same relation geometrically for every real $k>1$, although generally not by classical straightedge-and-compass operations.

math.MG↗

arXiv2026

The Cone Projection $f(z)=\dfrac{z}{1+|z|/R}$: Geometric structure and the Self-Directrix Theorem

The cone projection $f_R(z)=z/(1+|z|/R)$ is a radial homeomorphism from $\mathbb{C}$ onto the open disk $D_R$ of radius $R$, obtained by an elementary cone-and-perpendicular construction (independent of the cone's height) and governed by the reciprocal lens identity $1/|f_R(z)|=1/|z|+1/R$. Its main Euclidean feature is the \emph{Self-Directrix Theorem}: every line $\ell$ not through the origin maps to the focus-side arc of the conic with focus $O$, directrix $\ell$ \emph{itself}, eccentricity $R/d$, and semi-latus rectum $R$, so the single distance $d=\operatorname{dist}(O,\ell)$ fixes the ellipse/parabola/hyperbola trichotomy. The \emph{Confocal--Codirectrix Theorem} extends this from lines to every focal polar locus of a fixed focus--directrix pencil, keeping the focus and directrix while lowering the eccentricity by $1/e\mapsto1/e+δ/R$; the image of a circle, by contrast, is generally a circular quartic rather than a conic. The same lens identity organizes the remaining structure: a curvature-additive composition law and its flow, a raywise cross-ratio structure, and higher-dimensional, metric, and axiomatic results.

math.DG↗

arXiv2019

Encoding and Visualization in the Collatz Conjecture

The Collatz conjecture is one of the easiest mathematical problems to state and yet it remains unsolved. For each $n\ge 2$ the Collatz iteration is mapped to a binary sequence and a corresponding unique integer which can recreate the iteration. The binary sequence is used to produce the Collatz curve, a 2-D visualization of the iteration on a grid, which, besides the aesthetics, provides a qualitative way for comparing iterations. Two variants of the curves are explored, the r-curves and on-change-turn-right curves. There is a scarcity of acyclic r-curves and only three r-curves were found having a cycle of minimum length greater than 4.

math.HO↗