On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4

A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d = 2, triangular k-reptiles exist for all k of the form a^2, 3a^2 or a^2+b^2 and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d ≥ 3, have k = m^d, where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d = 3, k-reptile tetrahedra can exist only for k = m^3. We then prove a weaker analogue of this result for d = 4 by showing that four-dimensional k-reptile simplices can exist only for k = m^2

No acute tetrahedron is an 8-reptile

An $r$-gentiling is a dissection of a shape into $r \geq 2$ parts which are all similar to the original shape. An $r$-reptiling is an $r$-gentiling of which all parts are mutually congruent. This article shows that no acute tetrahedron is an $r$-gentile or $r$-reptile for any $r < 9$, by showing that no acute spherical diangle can be dissected into less than nine acute spherical triangles.Comment: updated text, as in press with Discrete Mathematics, Discrete Mathematics Available online 10 November 201

Red refinements of simplices into congruent subsimplices

We show that in dimensions higher than two, the popular "red refinement" technique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original one

On simplicial red refinement in three and higher dimensions

summary:We show that in dimensions higher than two, the popular "red refinement" technique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original one

Perfect tilings of simplices

In the present work we study the problem of k-reptile d-dimensional simplices. A simplex is called a k-reptile if it can be tiled in k simplices with disjoint interiors that are all congruent and similar to S. The only k-reptile simplices that are known for d 3 have k = md, where m 2. We also show an idea of Matoušek's proof of nonexistence of 2-reptile simplices of dimensions d 3. We correct a mistake in the proof. Then we give several geometric observations for k = 2. At the end we prove that there is no 3-reptile simplex for d = 3

Perfect tilings of simplices

V předložené práci se zabýváme problémem k-samodlážditelnosti d-dimenzionálních simplexů. Simplex S je k-samodlážditelný, pokud se dá rozdělit na k navzájem shodných simplexů (s disjunktními vnitřky), jež jsou navíc podobné původnímu simplexu S. Jediné dosud známé k-samodlážditelné simplexy v dimenzi d 3 jsou pro hodnotu k = md, kde m 2. V práci nastiňujeme Matouškův důkaz neexistence 2-samodlážditelných simplexů pro d 3, který poté opravíme. Uvádíme několik vlastních geometrických postřehů pro k = 2. Na závěr dokazujeme, že v prostoru dimenze 3 neexistuje 3-samodlážditelný simplex.In the present work we study the problem of k-reptile d-dimensional simplices. A simplex is called a k-reptile if it can be tiled in k simplices with disjoint interiors that are all congruent and similar to S. The only k-reptile simplices that are known for d 3 have k = md, where m 2. We also show an idea of Matoušek's proof of nonexistence of 2-reptile simplices of dimensions d 3. We correct a mistake in the proof. Then we give several geometric observations for k = 2. At the end we prove that there is no 3-reptile simplex for d = 3.Katedra aplikované matematikyDepartment of Applied MathematicsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult