Function Theory on Inflationary Tessellations
This paper studies function theory on periodic and aperiodic inflationary tessellations.
math.CV↗
arXiv subjects
Howard L. Resnikoff
Publications and source records attributed to Howard L. Resnikoff.
An Algebraic Realization of the Taylor-Socolar Aperiodic Monotilings
The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptions equivalent to each of these presentations. It also shows how the monotilings and Taylor triangles pattern that characterizes the aperiodicity can be obtained from just one algebraic equation.
math.DG↗
On Curves and Surfaces of Constant Width
This paper focuses on curves and surfaces of constant width, with some additional results about general ovals. We emphasize the use of Fourier series to derive properties, some of which are known. Amongst other results, we show that the perimeter of an oval is $π$ times its average width, and provide a bound for the radius of curvature of an oval that depends on the structure of the harmonics in its Fourier series. We prove that the density of a certain packing of Reuleaux curved triangles in the plane is $$\frac{2 (π- \sqrt{3})}{\sqrt{15} + \sqrt{7} - 2 \sqrt{3}} \simeq 0.92288$$ which exceeds the maximum density for circles ($\simeq 0.9060$), and conjecture this is the maximum for any curve of constant width. For surfaces of constant width we show that $ ρ_0(P)+ ρ_1(Q)=w$ where the $ρ_i$ are the principal curvatures, $P$ and $Q$ are opposite points, and $w>0$ is the width. Moreover, an ovoid is a surface of constant width $w>0$ if and only if $ρ_{\rm mean}(P)+ρ_{\rm mean}(Q) = w$ where $ρ_{\rm mean}(P)$ is the average radius of curvature at point $P$. Finally, we provide a Fourier series-based construction that produces arbitrarily many new surfaces of constant width.
math.DG↗