Tilings of an Isosceles Triangle

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N trianglescongruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we study the case of isosceles (but not equilateral) ABC. We study three possible forms of the tile: right-angled, or with one angle double another, or with a 120 degree angle. In the case of a right-angled tile, we give a complete characterization of the tilings, for N even, but leave open whether N can be odd. In the latter two cases we prove the ratios of the sides of the tile are rational, and give a necessary condition for the existence of an N-tiling. For the case when the tile has one angle double another, we prove N cannot be prime or twice a prime.Comment: 34 pages, 18 figures. This version supplies corrections and simplification

Review of: Only Connect: Discovery Pathways, Library Explorations and the Information Adventure

A novel concept for the world of publishing in this field, Only Connect has been produced primarily as an electronic book to incorporate multimedia resources. Initial viewing of the printed version proved somewhat disconcerting due to the lack of an index and the ‘chapter chooser’ which encourages a non-linear engagement with the book. However, after reading the introduction, this becomes completely logical as there is no sense of a clear progression between chapters. This book has been designed around a range of interpretations and experiences of information literacy (IL) from a number of perspectives. The chapter chooser recommends starting to read at any point and navigating in any order. In fact, the only theme that links these chapters is that of a form of information literacy

Triangle Tiling V: Tilings by a tile with integer sides

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. We wish to understand possible tilings by completely characterizing the triples (ABC, T, N) such that ABC can be N-tiled by T. In particular, this understanding should enable us to specify for which N there exists a tile T and a triangle ABC that is N-tiled by T; or given N, to determine which tiles and triangles can be used for N-tilings; or given ABC, to determine which tiles and N can be used to N-tile ABC. This is the fifth paper in a series of papers on this subject. The previous papers have reduced the problem the case when T has a 120 degree angle and integer side lengths. That is the problem we take up in this paper. We are still not able to completely solve the problem, but we prove that if there are any N-tilings by such tiles, then N is at least 96. Combining this results with our earlier work, we can remove the exception for a tile with a 120 degree angle, obtaining definitive non-existence results. For example, there is no 7-tiling, no 11-tiling, no 14-tiling, no 19-tiling, no 31-tiling, no 41-tiling, etc. Regarding the number N=96: There are several possible shapes of ABC, and for each shape, we exhibit the smallest N for which it is presently unknown whether there is an N-tiling. For example, for equilateral ABC, the simplest unsolved case as of May, 2012 is N=135. For each of these minimal-N examples, the tile would have to have sides (3,5,7)

Amenities and the returns to human capital

A demonstration that regional differences in the returns to human capital do not necessarily imply structural differences in regional labor markets, but could be reflecting compensation for regional differences in amenities.Labor market ; Wages

Interurban comparisons of the quality of life

A methodology is developed for constructing quality-of-life comparisons for metropolitan areas in which the full bundle of an area's attributes is valued, rather than the typical method of focusing on individual characteristics.Cost and standard of living ; Regional economics

Productivity growth and the decline of manufacturing in large metropolitan areas: 1959-78

An examination of the role of productivity differences in explaining the decline of manufacturing activity in large metropolitan areas relative to the rest of the country, with special attention given to the decline of the large metropolitan areas of the Manufacturing Belt.Manufactures ; Industrial location