arXiv subjects
Publications and source records attributed to Jinjin Liang.
By computing all cyclotomic points on some algebraic varieties, we get an independent and efficient way to find all rational $a^3b$-monotiles for the sphere, thereby completing the classification of edge-to-edge monohedral quadrilateral tilings. Both of the previous classifications \cite{lw2} and \cite{cl} depended on many old works of different authors while quite a few typos and gaps were found.
We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with rational angles in degree: they are a one-parameter family of symmetric $a^4b$-pentagonal subdivisions of the tetrahedron with $12$ tiles; a sequence of unique symmetric $a^4b$-pentagons admitting a symmetric $3$-layer earth map tiling by $4m$ tiles for any $m\ge4$, among which each odd $m$ case admits two standard flip modifications; and a unique non-symmetric and degenerate $a^4b$-pentagon admitting a non-symmetric $3$-layer earth map tiling and its standard flip modification with $20$ tiles. The full classification from this series and all induced non-edge-to-edge quadrilateral tilings from degenerate pentagons are summarized with their 3D pictures.
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 of triangles each admitting many 2-layer earth map tilings with $2n$($n\geq3$) tiles, together with rotational modifications for even $n$; a 1-parameter family of triangles each admitting a unique tiling with $8$ tiles; and a sporadic triangle admitting a unique tiling with $16$ tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples.
Pentagonal subdivision gives three families of edge-to-edge tilings of the sphere by congruent pentagons. Each family forms a two dimensional moduli. We describe the moduli in detail.
In this article we investigate a family of nonlinear evolutions of polygons in the plane called the $β$-polygon flow and obtain some results analogous to results for the smooth curve shortening flow: (1) any planar polygon shrinks to a point and (2) a regular polygon with five or more vertices is asymptotically stable in the sense that nearby polygons shrink to points that rescale to a regular polygon. In dimension four we show that the shape of a square is locally stable under perturbations along a hypersurface of all possible perturbations. Furthermore, we are able to show that under a lower bound on angles there exists a rescaled sequence extracted from the evolution that converges to a limiting polygon that is a self-similar solution of the flow. The last result uses a monotonicity formula analogous to Huisken's for the curve shortening flow.