In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper continues our study of alternating semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices and with two kinds of regular facets occurring in an alternating fashion.

Our main concern here is the universal polytope ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, an alternating semiregular $(n+1)$-polytope defined for any pair of regular $n$-polytopes ${\mathcal{P}},{\mathcal{Q}}$ with isomorphic facets. After a careful look at the local structure of these objects, we develop the combinatorial machinery needed to explain how ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ can be constructed by “freely assembling” unlimited copies of ${\mathcal{P}}$, ${\mathcal{Q}}$ along their facets in alternating fashion. We then examine the connection group of ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, and from that prove that ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ covers any $(n+1)$-polytope ${\mathcal{B}}$ whose facets alternate in any way between various quotients of ${\mathcal{P}}$ or ${\mathcal{Q}}$.

Higher dimensional analogues of the modular group $\mathit{PSL}(2,\mathbb{Z})$ are closely related to hyperbolic reflection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic $k$-rectified regular polyhedra, which is of independent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value $\unicode[STIX]{x1D701}(3)$.

Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz’s square peg problem. We prove Hadwiger’s 1971 conjecture that any simple loop in $3$-space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop. If the loop is rectifiable, there is a rectangle that cuts the loop into four pieces which can be rearranged to form two loops of equal length. (The previous two results are independently due to Schwartz.) A rectifiable loop in $d$-space can be cut into $(r-1)(d+1)+1$ pieces that can be rearranged by translations to form $r$ loops of equal length. We relate our results to fair divisions of necklaces in the sense of Alon and to Tverberg-type results. This provides a new approach and a common framework to obtain inscribability results for the class of all continuous curves.

Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^{d}$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint.

We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geqslant 2$, $d\geqslant 2$, $c\geqslant d$ with $m\geqslant n(c-d)+d$ are integers, and $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^{d}$, we prove that there exists a partition of $\mathbb{R}^{d}$ into $n$ convex pieces which equiparts the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{d-1}$, and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$.

We give an equality condition for a symmetrization inequality for condensers proved by F.W. Gehring regarding elliptic areas. We then use this to obtain a monotonicity result involving the elliptic area of the image of a holomorphic function f.

Let $\mathbf{H}_{\mathbb{H}}^{n}$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group $\text{Sp}(n,1)$ acts on $\mathbf{H}_{\mathbb{H}}^{n}$ by isometries. A subgroup $G$ of $\text{Sp}(n,1)$ is called Zariski dense if it neither fixes a point on $\mathbf{H}_{\mathbb{H}}^{n}\cup \unicode[STIX]{x2202}\mathbf{H}_{\mathbb{H}}^{n}$ nor preserves a totally geodesic subspace of $\mathbf{H}_{\mathbb{H}}^{n}$. We prove that a Zariski dense subgroup $G$ of $\text{Sp}(n,1)$ is discrete if for every loxodromic element $g\in G$ the two-generator subgroup $\langle f,gfg^{-1}\rangle$ is discrete, where the generator $f\in \text{Sp}(n,1)$ is a certain fixed element not necessarily from $G$.

We construct a family of self-affine tiles in $\mathbb{R}^{d}$ ($d\geqslant 2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in $\mathbb{R}^{2}$, and its extension to $\mathbb{R}^{3}$ by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.

We describe the parameter spaces of some families of quadrilaterals, such as parallelograms, rectangles, rhombuses, cyclic quadrilaterals and trapezoids. For this purpose, we prove that the closed $n$-disc $\mathbb{D}^{n}$ is the unique topological $n$-manifold (with boundary) whose boundary and interior are homeomorphic to $\mathbb{S}^{n-1}$ and $\mathbb{R}^{n}$, respectively. Roughly speaking, our main result states that the natural compactifications of the parameter spaces of cyclic quadrilaterals and of trapezoids, modulo similarity, are both homeomorphic to $\mathbb{D}^{3}$.

In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

Let ℍ be the division ring of real quaternions. Let SL(2, ℍ) be the group of 2 × 2 quaternionic matrices $A={\scriptsize{(\begin{array}{l@{\quad}l} a & b \\ c & d \end{array})}}$ with quaternionic determinant det A = |ad − aca−1b| = 1. This group acts by the orientation-preserving isometries of the five-dimensional real hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, ℍ).

We consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. Finally, we establish the quasi-invariance properties of both metrics under quasiconformal maps.

We say that a planar set $A$ has the Kakeya property if there exist two different positions of $A$ such that $A$ can be continuously moved from the first position to the second within a set of arbitrarily small area. We prove that if $A$ is closed and has the Kakeya property, then the union of the non-trivial connected components of $A$ can be covered by a null set which is either the union of parallel lines or the union of concentric circles. In particular, if $A$ is closed, connected and has the Kakeya property, then $A$ can be covered by a line or a circle.

In the full rectangular version of Gilbert's planar tessellation (see Gilbert (1967), Mackisack and Miles (1996), and Burridge et al. (2013)), lines extend either horizontally (with east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a stationary Poisson point process, each ray stopping when it meets another ray that has blocked its path. In the half-Gilbert rectangular version, east- and south-growing rays, whilst having the potential to block each other, do not interact with west and north rays, and vice versa. East- and south-growing rays have a reciprocality of blocking, as do west and north. In this paper we significantly expand and simplify the half-Gilbert analytic results that we gave in Burridge et al. (2013). We also show how the idea of reciprocality of blocking between rays can be used in a much wider context, with rays not necessarily orthogonal and with seeds that produce more than two rays.

The optimistic limit is a mathematical formulation of the classical limit, which is a physical method to estimate the actual limit by using the saddle-point method of a certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. This modified version is easier to handle and more combinatorial than the original one. On the other hand, it is known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at a certain root of unity. This optimistic limit of the colored Jones polynomial was also formulated by the author and others, but it is very complicated and needs many unnatural assumptions. In this article, we suggest a modified optimistic limit of the colored Jones polynomial, following the idea of the modified optimistic limit of the Kashaev invariant, and show that it determines the complex volume of a hyperbolic link. Furthermore, we show that this optimistic limit coincides with the optimistic limit of the Kashaev invariant modulo $4{\it\pi}^{2}$. This new version is easier to handle and more combinatorial than the old version, and has many advantages over the modified optimistic limit of the Kashaev invariant. Because of these advantages, several applications have already appeared and more are in preparation.

We provide a combinatorial characterization of LG(3,6)(${\mathbb{K}}$) using an axiom set which is the natural continuation of the Mazzocca–Melone set we used to characterize Severi varieties over arbitrary fields (Schillewaert and Van Maldeghem, Severi varieties over arbitrary fields, Preprint). This fits within a large project aiming at constructing and characterizing the varieties related to the Freudenthal–Tits magic square.

Extremal problems for quadrangles circuminscribed in a circular annulus with the Poncelet porism property are considered. Quadrangles with the maximal and the minimal perimeters are determined. Two conjectures end the paper.

For any subfield $K\subseteq \mathbb{C}$ , not contained in an imaginary quadratic extension of $\mathbb{Q}$ , we construct conjugate varieties whose algebras of $K$ -rational ( $p,p$ )-classes are not isomorphic. This compares to the Hodge conjecture which predicts isomorphisms when $K$ is contained in an imaginary quadratic extension of $\mathbb{Q}$ ; additionally, it shows that the complex Hodge structure on the complex cohomology algebra is not invariant under the Aut( $\mathbb{C}$ )-action on varieties. In our proofs, we find simply connected conjugate varieties whose multilinear intersection forms on $H^{2}(-,\mathbb{R})$ are not (weakly) isomorphic. Using these, we detect nonhomeomorphic conjugate varieties for any fundamental group and in any birational equivalence class of dimension $\geq $ 10.

It is well known to be impossible to trisect an arbitrary angle and duplicate an arbitrary cube by a ruler and a compass. On the other hand, it is known from ancient times that these constructions can be performed when the use of several conic curves is allowed. In this paper, we prove that any point constructible from conics can be constructed using a ruler and a compass, together with a single fixed nondegenerate conic different from a circle.

We investigate the ray-length distributions for two different rectangular versions of Gilbert's tessellation (see Gilbert (1967)). In the full rectangular version, lines extend either horizontally (east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a Poisson point process, each ray stopping when another ray is met. In the half rectangular version, east- and south-growing rays do not interact with west and north rays. For the half rectangular tessellation, we compute analytically, via recursion, a series expansion for the ray-length distribution, whilst, for the full rectangular version, we develop an accurate simulation technique, based in part on the stopping-set theory for Poisson processes (see Zuyev (1999)), to accomplish the same. We demonstrate the remarkable fact that plots of the two distributions appear to be identical when the intensity of seeds in the half model is twice that in the full model. In this paper we explore this coincidence, mindful of the fact that, for one model, our results are from a simulation (with inherent sampling error). We go on to develop further analytic theory for the half-Gilbert model using stopping-set ideas once again, with some novel features. Using our theory, we obtain exact expressions for the first and second moments of the ray length in the half-Gilbert model. For all practical purposes, these results can be applied to the full-Gilbert model—as much better approximations than those provided by Mackisack and Miles (1996).

Perfect colouring of isonemal fabrics by thick striping of warp and weft and the closely related topic of isonemal prefabrics that fall apart are reconsidered and their relation further explored. The catalogue of isonemal prefabrics of genus V that fall apart is extended to order 20 with designs that can be used to weave cubes with colour symmetry as well as weaving symmetry.