
Journal for Geometry and Graphics 23 (2019), No. 2, 245258 Copyright Heldermann Verlag 2019 Foldable and SelfIntersecting Polyhedral Cylinders Based on Triangles Jens Wittenburg Institute for Technical Mechanics, Karlsruhe Institute of Technology, Kaiserstr. 12, 76128 Karlsruhe, Germany jens.wittenburg@kit.edu [Abstractpdf] \def\rV{{\rm V}} An infinitely long strip of paper is divided by a zigzagging line into congruent triangles with side lengths $1$, $a$ and $b$. On both rims of the strip the vertices $\rV_k$ of the triangles are labeled from $\infty$ to $+\infty$ with a shift $n$ such that $(\rV_0 \rV_1 \rV_n)$ is a representative triangle. Along the sides of the triangles folds with alternating fold angles are made. Under certain conditions on $a,b$ and $n$ and with appropriately chosen fold angles it is possible to bring every vertex $\rV_k$ on the upper rim in coincidence with the vertex $\rV_k$ of equal name on the lower rim. The resulting body is a polyhedral cylinder (PC). The vertices are distributed at equal intervals along a helix on the surface of a circular cylinder. For given lengths $a$ and $b$ up to $(n2)$ PCs can be formed. There are foldable PCs and selfintersecting PCs. In the case $n=4$ selfintersecting PCs consist of a core body with congruent nonconvex pentagonal faces and of an infinite number of congruent tetrahedra, each tetrahedron in edgetoedge contact with the core body along three edges. Keywords: Polyhedral cylinder, core body, foldability, flexible polyhedra, periodic framework. MSC: 52C25; 53A17, 51M20 [ Fulltextpdf (912 KB)] 