Searcharxiv⌕ Search

arXiv subjects

Erxiao Wang

Publications and source records attributed to Erxiao Wang.

At least 19 recordsLinked to original sources
arXiv2026

Reduction of integer tiles via CRT and base-p digits

Coven and Meyerowitz gave two cyclotomic conditions, (T1) and (T2), which characterize integer tiles whose cardinalities have at most two distinct prime factors. We prove that the same characterization holds without this restriction. The proof uses a reduction in Chinese remainder coordinates: slicing by the lowest base-p digit produces sets with a common tiling complement in a group of order smaller by a factor of p. This reduction preserves the cyclotomic data needed for an induction on the exponents in (T2).

math.NT↗
arXiv2026

Graded strength of comparative illusions is explained by Bayesian inference

Like visual processing, language processing is susceptible to illusions in which people systematically misperceive stimuli. In one such case--the comparative illusion (CI), e.g., More students have been to Russia than I have--comprehenders tend to judge the sentence as acceptable despite its underlying nonsensical comparison. Prior research has argued that this phenomenon can be explained as Bayesian inference over a noisy channel: the posterior probability of an interpretation of a sentence is proportional to both the prior probability of that interpretation and the likelihood of corruption into the observed (CI) sentence. Initial behavioral work has supported this claim by evaluating a narrow set of alternative interpretations of CI sentences and showing that comprehenders favor interpretations that are more likely to have been corrupted into the illusory sentence. In this study, we replicate and go substantially beyond this earlier work by directly predicting the strength of illusion with a quantitative model of the posterior probability of plausible interpretations, which we derive through a novel synthesis of statistical language models with human behavioral data. Our model explains not only the fine gradations in the strength of CI effects, but also a previously unexplained effect caused by pronominal vs. full noun phrase than-clause subjects. These findings support a noisy-channel theory of sentence comprehension by demonstrating that the theory makes novel predictions about the comparative illusion that bear out empirically. This outcome joins related evidence of noisy channel processing in both illusory and non-illusory contexts to support noisy channel inference as a unified computational-level theory of diverse language processing phenomena.

cs.CL↗
arXiv2026

Hexagonal Tiling of the Plane

Since the thesis of K. Reinhardt in 1918, it is well known that there are exactly three types of convex hexagons that can tile the plane. However, the proof of the fact is far from being complete. We prove this fact, under an assumption weaker than the convexity.

math.CO↗
arXiv2026

Tiling of Hyperbolic Surface by Multiple Tiles

Tilings of a surface of negative Euler characteristic by n-gons with n\ge 7 is a finite problem. We develop the algorithm for finding all the tilings for fixed number of tiles and present the calculation for tilings of surfaces of small genus by two tiles. We also discuss the number of distinct edge lengths in multiple tile tilings.

math.CO↗
arXiv2026

Tiling of Hyperbolic Surface by a Single Tile

Tilings of a surface of negative Euler characteristic by n-gons with n\ge 7 is a finite problem. One extreme of the finite problem is single tile tilings. We develop the algorithm for finding all the single tile tilings and present the results for surfaces of small genus.

math.CO↗
arXiv2026

Side-to-side Tiling of the Sphere by Congruent Curvilinear Triangles

The edge-to-edge tilings of the sphere by congruent polygons, where all edges are straight, have been completely classified. We classify the curvilinear version of the similar triangular tilings, where the edges may not be straight, and find that these are the modifications of the straight triangular tilings.

math.CO↗
arXiv2025

Cyclotomic points on varieties and all rational $a^3b$-monotiles

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.

math.CO↗
arXiv2025

Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles

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.

math.CO↗
arXiv2025

Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle

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.

math.CO↗
arXiv2024

Tilings of the sphere by congruent pentagons IV: Edge combination $a^4b$ with general angles

We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with any irrational angle in degree: they are three $1$-parameter families of pentagonal subdivisions of the Platonic solids, with $12, 24$ and $60$ tiles; and a sequence of $1$-parameter families of pentagons admitting non-symmetric $3$-layer earth map tilings together with their various rearrangements under extra conditions. Their parameter moduli and geometric data are all computed in both exact and numerical form. The total numbers of different tilings for any fixed such pentagon are counted explicitly. As a byproduct, the degenerate pentagons produce naturally many new non-edge-to-edge quadrilateral tilings. A sequel of this paper will handle $a^4b$-pentagons with all angles being rational in degree by solving some trigonometric Diophantine equations, to complete our full classification of edge-to-edge tilings of the sphere by congruent pentagons.

math.CO↗
arXiv2024

Tilings of Flat Tori by Congruent Hexagons

Convex hexagons that can tile the plane have been classified into three types. For the generic cases (not necessarily convex) of the three types and two other special cases, we classify tilings of the plane under the assumption that all vertices have degree $3$. Then we use the classification to describe the corresponding hexagonal tilings of flat tori and their moduli spaces.

math.CO↗
arXiv2023

Tilings of the sphere by congruent regular triangles and congruent rhombi

All edge-to-edge tilings of the sphere by congruent regular triangles and congruent rhombi are classified as: (1) a $1$-parameter family of protosets each admitting a unique $(2a^3,3a^4)$-tiling like a triangular prism; (2) a $1$-parameter family of protosets each admitting 2 different $(8a^3,6a^4)$-tilings like a cuboctahedron and a triangular orthobicupola respectively; (3) a sequence of protosets each admitting a unique $(2a^3,(6n-3)a^4)$-tiling like a generalized anti-triangular prism for each $n\ge3$; (4) 26 sporadic protosets, among which nineteen admit a unique tiling, one admits 3 different tilings, one admits 5 different tilings, three admit 2 different tilings, two admit too many tilings to count. The moduli of parameterized tilings and all geometric data are provided.

math.CO↗
arXiv2023

Tilings of the sphere by congruent quadrilaterals II: edge combination $a^3 b$ with rational angles

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of $a^3b$-quadrilaterals with all angles being rational degrees. There are $12$ sporadic and $3$ infinite sequences of quadrilaterals admitting the $2$-layer earth map tilings together with their modifications, and $3$ sporadic quadrilaterals admitting $4$ exceptional tilings. Among them only $3$ quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.

math.CO↗
arXiv2023

Tilings of the sphere by congruent quadrilaterals III: edge combination $a^3b$ with general angles

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of $a^3b$-quadrilaterals with some irrational angle: there are a sequence of $1$-parameter families of quadrilaterals admitting $2$-layer earth map tilings together with their basic flip modifications under extra condition, and $5$ sporadic quadrilaterals each admitting a special tiling. A summary of the full classification is presented in the end.

math.CO↗
arXiv2022

Tilings of the sphere by congruent quadrilaterals I: edge combination $a^2bc$

The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of quadrilateral subdivisions of the octahedron with $24$ tiles together with a flip modification for a special parameter, and a sequence of $3$-layer earth map tilings with $8n$ $(n\ge2)$ tiles together with two flip modifications for odd $n$. We also describe the moduli and calculate the geometric data.

math.CO↗
arXiv2021

Tilings of the Sphere by Congruent Pentagons I: Edge Combinations $a^2b^2c$ and $a^3bc$

We develop the basic tools for classifying edge-to-edge tilings of the sphere by congruent pentagons. Then we prove that, for the edge combination $a^2b^2c$, such tilings are three two-parameter families of pentagonal subdivisions of the Platonic solids, with $12$, $24$ and $60$ tiles. We also prove that, for the edge combination $a^3bc$, such tilings are two unique double pentagonal subdivisions of the Platonic solids, with $48$ and $120$ tiles.

math.MG↗