Computer Geometry: Rep-Tiles with a Hole
- Computer ScienceThe Mathematical Intelligencer
A cube is an 8-rep-tile: it is the union of eight smaller copies of itself. Is there a set with a hole which has this property? The computer found an interesting and complicated solution, which then…
A 3D rep-tile is a compact 3-manifold X in R that can be decomposed into finitely many pieces, each of which are similar to X, and all of which are congruent to each other. In this paper we classify…
3D cache-oblivious multi-scale traversals of meshes using 8-reptile polyhedra
- Computer Science
A stack-assignment scheme that allows the Haverkort element traversal to use stacks for storing the input data, the output data, and the temporary data of vertices and elements and a constant-number-of-stacks solution is developed that can compete with and outperform numerical simulations using cache-optimization techniques based on loop blocking when running on CPUs or GPUs.
SHOWING 1-10 OF 21 REFERENCES
Reptilings and space-filling curves for acute triangles
- MathematicsDiscret. Comput. Geom.
No face-continuous space-filling curve can be constructed on the basis of reptilings and gentilings of acute triangles, leading to the following conclusion.
On the shape of tetrahedra from bisection
We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes j-0, 91, g2, ..., which has the following properties: (1) Each mesh Sfn is…
On the Nonexistence of k-reptile Tetrahedra
- BiologyDiscret. Comput. Geom.
It is proved that for d=3, k-reptile simplices (tetrahedra) exist only for k=m3, which partially confirms a conjecture of Hertel, asserting that the only k- reptile tetrahedr are the Hill tetrahedral simplices.
VI.—Division of Space by Congruent Triangles and Tetrahedra
It is proposed to investigate the various ways in which it is possible to divide the plane into congruent triangles, and space of three dimensions into congruent tetrahedra. The method of inquiry…
Which Tetrahedra Fill Space
Filling space by fitting congruent polyhedra together without any gaps is one of the oldest and most difficult of geometric problems, and has a fascinating history. It arose first in ancient times in…
Three Infinite Families of Tetrahedral Space-Fillers
- PhysicsJ. Comb. Theory, Ser. A
THIS volume gives a systematic treatment of the subject of spherical trigonometry, based on the sound foundation of Todhunter rearranged and amplified. While the merit of the original work is…