Hinged Dissections Exist

We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously without self-intersection to form any polygon in the collection. This result settles the open problem about the existence of hinged dissections between pairs of polygons that goes back implicitly to 1864 and has been studied extensively in the past ten years. Our result generalizes and indeed builds upon the result from 1814 that polygons have common dissections (without hinges). Our proofs are constructive, giving explicit algorithms in all cases. For two planar polygons whose vertices lie on a rational grid, both the number of pieces and the running time required by our construction are pseudopolynomial. This bound is the best possible, even for unhinged dissections. Hinged dissections have possible applications to reconfigurable robotics, programmable matter, and nanomanufacturing.Massachusetts Institute of Technology/Akamai Presidential FellowshipNational Science Foundation (U.S.) (Graduate Research Fellowship

Estudo antropométrico e dietético de nadadores competitivos de áreas metropolitanas da região sudeste do Brasil

Com o objetivo de caracterizar os ábitos alimentares de nadadores competitivos de dois clubes das cidades do Rio de Janeiro e de São Paulo, foram estudados atletas, 30 homens e 37 mulheres pertencentes às equipes juvenil (15-17 anos) e seniors (18-25 anos). Concomitantemente à avaliação antropométrica, foi feito o inquérito alimentar mediante os métodos de registro alimentar, recordatório de 24 horas e freqüência de consumo alimentar. Os resultados mostraram semelhança antropométrica entre as duas faixas etárias para cada um dos sexos, semelhança dos alimentos prevalentemente ingeridos com aqueles referidos pela população da área metropolitana das duas cidades e semelhança entre os suplementos alimentares ingeridos pelos atletas dos dois clubes. A ingestão calórica observada foi quase o dobro da referida para a população daquelas localidades, mas equivalente à descrita para nadadores de outros países. Os lanches, entre refeições, contribuíram com 25-28% da ingestão calórica global e a maior contribuição energética alimentar foi dada pelo grupo de cereais. Apesar de contribuir com quase 50% do aporte calórico diário, a ingestão glicídica foi menor que a recomendada para atletas competitivos (55-60%). A ingestão protéica de 2,5-3,0 g/kg/dia superou em 100% o valor recomendado para atletas e ultrapassou os 15% na participação calórica diária. A ingestão lipídica foi considerada elevada, particularmente a de gordura saturada. O aporte de micronutrientes foi acima do referido para as populações locais em função da ingestão calórica elevada, mas mostrou-se relativamente inadequado em magnésio, ferro e vitaminas A e D. Conclui-se que o padrão alimentar dos nadadores estudados não difere, muito, do da população local, distinguindo-se pelo elevado e freqüente consumo de alimentos energéticos, maior no sexo masculino que no feminino.The daily food intake and alimentary frequency of competitive swimmers of two clubs representative of the two largest cities in Brazil (S.Paulo and Rio de Janeiro) were studied. The 30 males and 37 females studied belonged to the swimming categories junior (15-17 yrs old) and senior (18-25 yrs old). Food intake obtained from self-register and 24 hour-recall showed similar results and therefore the 24 hour-recall was used for comparisons with the literature. Ages within both sexes were anthropometrically similar. The most frequently eaten foodstaffs were similar to those quoted by the metropolitan population of the respective areas. Moroever, the athletes from both cities reported similar food-supplements. The energy intake was found to average almost double the estimated value for the respective metropolitan populations, but were similar to results obtained for swimmers all over the world. Snacks between meals supplied 25-28% of the overall energy intake. The main caloric source were cereals. However, despite their contribution of almost 50% of the total energy intake, carbohydrates attained a level of only 55-60% of that recommended for competitive athletes. on the other hand, the protein intake found (2.5-3.0 g/kg/day) exceeded the recommended values by 100%. The fat intake particularly of satured fat, was also considered high. The swimmers' intake of micronutrients was well abone that quoted by the regular population of these metropolitan areas, when related to the caloric intake, but the values of Mg, Fe and vitamins A and D were somecohot lower than those recommended. This is due to the higers energy intake of the swimmers. From these overall results it may be concluded that the alimentary pattern of the swimmers studied does not differ greatly from that of the local population, being distinguished by its higher and more frequent intake of caloric foods rather than by its qualitative aspects