English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

A compact circle-packing $P$ of the Euclidean plane is a set of circles whichbound mutually disjoint open discs with the property that, for every circle$S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteqP$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent toboth circles $S$ and $A_{i+1\mod n}.$  We show that there exist at most $13617$ pairs $(r,s)$ with $0<s<r<1$ forwhich there exist a compact circle-packing of the plane consisting of circleswith radii $s$, $r$ and $1$.  We discuss computing the exact values of such $0<s<r<1$ as roots ofpolynomials and exhibit a selection of compact circle-packings consisting ofcircles of three radii. We also discuss the apparent infeasibility of computing\emph{all} these values on contemporary consumer hardware with the methodsemployed in this paper.

On compact packings of the plane with circles of three radii