Related papers: A system for constructing relatively small polyhed…

Origami design principles are scale invariant and enable direct miniaturization of origami structures provided the sheets used for folding have equal thickness to length ratios. Recently, seminal steps have been taken to fabricate…

We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple…

There are many structures, both classical and modern, involving convex polygonal geometries whose deeper understanding would be facilitated through interactive visualizations. The Ipe extensible drawing editor, developed by Otfried Cheong,…

We investigate the folding problem that asks if a polygon P can be folded to a polyhedron Q for given P and Q. Recently, an efficient algorithm for this problem has been developed when Q is a box. We extend this idea to regular polyhedra,…

The regular polyhedra have the highest order of 3D symmetries and are exceptionally at- tractive templates for (self)-assembly using minimal types of building blocks, from nano-cages and virus capsids to large scale constructions like glass…

An extension of the Tamari lattice to the multiplihedra is discussed, along with projections to the composihedra and the Boolean lattice. The multiplihedra and composihedra are sequences of polytopes that arose in algebraic topology and…

Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module…

We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to…

Any solid object can be decomposed into a collection of convex polytopes (in short, convexes). When a small number of convexes are used, such a decomposition can be thought of as a piece-wise approximation of the geometry. This…

We propose an algorithmic framework of a pluripotent structure evolving from a simple compact structure into diverse complex 3-D structures for designing the shape transformable, reconfigurable, and deployable structures and robots. Our…

The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a square rhomboid that is an example of non-series-parallel graphs. Our intention is to simplify the expressions of square rhomboids…

We describe how to construct a dodecahedron, tetrahedron, cube, and octahedron out of pvc pipes using standard fittings.

We discuss well known geometric constructions via paper-folding. The note is written primary for school students.

Self-assembly of nanoscale synthetic subunits is a promising bottom-up strategy for fabrication of functional materials. Here, we introduce a design principle for DNA origami nanoparticles of 50-nm size, exploiting modularity, to make a…

Recently, the notions of self-orthogonal subspace codes and LCD subspace codes were introduced, and LCD subspace codes obtained from mutually unbiased weighing matrices were studied. In this paper, we provide a method of constructing…

We analyze the problem of folding one polyhedron, viewed as a metric graph of its edges, into the shape of another, similar to 1D origami. We find such foldings between all pairs of Platonic solids and prove corresponding lower bounds,…

Special orthogonal matrices with rational elements form the group SO(n,Q), where Q is the field of rational numbers. A theorem describing the structure of an arbitrary matrix from this group is proved. This theorem yields an algorithm for…

A quad-mesh rigid origami is a continuously deformable panel-hinge structure where planar, rigid, zero-thickness quadrilateral panels are connected by rotational hinges in the combinatorics of a grid. This article provides a comprehensive…

This note presents a procedure of constructing a higher dimensional sphere map from a lower dimensional one and gives an explicit formula for smooth sphere map with a given degree. As an application a new proof of a generalized…

This article is concerned with an example of complex planar geometry arising from flat origami challenges. The complexity of solution algorithms is illustrated, depending on the depth of the initial analysis of the problem, starting from…