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

Computer geometry: Rep-tiles with a hole

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 could be simplified. We discuss some problems of computer-assisted research in geometry

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

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

Phytochemistry and pharmacology of the genus Drypetes: A review

Aims: Traditional medicinal use of species of the genus Drypetes is widespread in the tropical regions. The aim of this review is to systematically appraise the literature available to date on phytochemistry, ethnopharmacology, toxicology and bioactivity (in vitro and in vivo) of crude extracts and purified compounds. Ethnopharmacological relevance: Plants of the genus Drypetes (Putranjivaceae) are used in the Subsaharan African and Asian traditional medicines to treat a multitude of disorders, like dysentery, gonorrhoea, malaria, rheumatism, sinusitis, tumours, as well as for the treatment of wounds, headache, urethral problems, fever in young children, typhoid and several other ailments. Some Drypetes species are used to protect food against pests, as an aphrodisiac, a stimulant/depressant, a rodenticide and a fish poison, against insect bites, to induce conception and for general healing. This review deals with updated information on the ethnobotany, phytochemistry, and biological activities of ethnomedicinally important Drypetes species, in order to provide an input for the future research opportunities. Methods: An extensive review of the literature available in various recognized databases e.g., Google Scholar, PubMed, Science Direct, SciFinder, Web of Science, www.theplantlist.org and www.gbif.org, as well as the Herbier National du Cameroun (Yaoundé) and Botanic Gardens of Limbe databases on the uses and bioactivity of various species of the Drypetes was undertaken. Results: The literature provided information on ethnopharmacological uses of the Subsaharan African and Asian species of the genus Drypetes, e.g., Drypetes aubrévillii, D. capillipes, D. chevalieri, D. gerrardii, D. gossweileri, D. ivorensis, D. klainei, D. natalensis, D. pellegrini (all endemic to Africa) and D. roxburghii (Asian species), for the treatment of multiple disorders. From a total of 19 species, more than 140 compounds including diterpenes, sesquiterpenes, triterpenes (friedelane, oleanane, lupane and hopane-type), flavonoids, lignans, phenylpropanoids and steroids, as well as some thiocyanates, were isolated. Several crude extracts of these plants, and isolated compounds displayed significant analgesic, anthelmintic, antidiabetic, anti-emetic anti-inflammatory, antioxidant, antiparasitic, central nervous system depressant, cytotoxic, and insecticidal activities both in vitro and in vivo. Some toxicities associated with the stem, bark, seed and leaf extracts of D. roxburghii, and the flavonoid, amentoflavone, isolated from the stem extract of D. littoralis as well as D. gerrardii, were confirmed in the animal models and in the rat skeletal myoblast cells assays. As a consequence, traditional medicine from this genus should in future be applied with care. Conclusions: Plants of this genus have offered bioactive samples, both from crude extracts and pure compounds, partly validating their effectivity in traditional medicine. However, most of the available scientific litteratures lacks information on relevant doses, duration of the treatment, storage conditions and positive controls for examining bioefficacy of extract and its active compounds. Additional toxicological studies on the species used in local pharmacopeia are urgently needed to guarantee safe application due to higth toxicity of some crude extracts. Interestingly, this review also reports 10 pimarane dinorditerpenoids structures with the aromatic ring C, isolated from the species collected in Asia Drypetes littoralis (Taiwan), D. perreticulata (China), and in Africa D. gerrardii (Kenya), D. gossweileri (Cameroon). These compounds might turn out to be good candidates for chemotaxonomic markers of the genus

Summer 2011 Research Symposium Abstract Book

Summer 2011 volume of abstracts for science research projects conducted by Trinity College students

Viewing paleontology through a geochemical lens: 2 case studies

Geochemistry has become increasingly important to the field of paleontology. This thesis comprises two geochemical case studies examining questions at very different scales. The first examines possible controls on the formation of deposits where exceptional preservation of non-biomineralized tissues occurs throughout the Phanerozoic by the creation and analysis of a database of Phanerozoic Konservat-Lagerstatten. Potential controls examined include the influence of marine geochemistry, the availability of fossiliferous strata and evolutionary events. The second case study involves the application of Raman microspectroscopy in pathology and paleopathology. Raman microspectroscopy is used to demonstrate biotic origin for a new type of trace fossil, and to study the mineralogical changes that occur in bone presenting with osteomyelitis and hyperplastic bone marrow disease