Euclidean Model Research Articles

Euclidean and Topological Models: Two Modes of Structuring the Art Curriculum

Euclidean and Topological Models: Two Modes of Structuring the Art Curriculum

The Counts of Radio Sources

Following the advice of Professor Zel'dovich, I will begin with two platitudes – first, when we consider the implications of the counts of radio sources, we must look at all the counts of radio sources at all frequencies; second, when we attempt to evaluate the implications of the source counts, we should use proper cosmological models and not rely on the local Euclidean model.

The relation between hierarchical and euclidean models for psychological distances

In one well-known model for psychological distances, objects such as stimuli are placed in a hierarchy of clusters like a phylogenetic tree; in another common model, objects are represented as points in a multidimensional Euclidean space. These models are shown theoretically to be mutually exclusive and exhaustive in the following sense. The distances among a set ofn objects will be strictly monotonically related either to the distances in a hierarchical clustering system, or else to the distances in a Euclidean space of less thann — 1 dimensions, but not to both. Consequently, a lower bound on the number of Euclidean dimensions necessary to represent a set of objects is one less than the size of the largest subset of objects whose distances satisfy the ultrametric inequality, which characterizes the hierarchical model.

An analysis of individual differences in similarity judgments about complex random forms

A variation of the method of graded dichotomies was used to obtain judgments of interstimulus similarity for all pairings of 21 12-turn random shapes. Using the Tucker-Messick procedure, a six-dimensional spatial model, representing individual variation in performance on the task, was derived. Six “idealized individuals” (IIs) were isolated. A five-dimensional Euclidean spatial model was obtained for each II using the Shepard-Kruskal scaling technique. The resulting psychological dimensions of perceived form were related to two sets of physical measures on the forms: the factor space suggested by Brown and Owen and a set of more basic physical measures. These basic measures allowed interpretation of most of the psychological dimensions and revealed differences between IIs in patterns of cue utilization.

The Forum: What Should Become of the High School Geometry Course?: Transformations in High School Geometry

The treatment of geometry in the high schools today is remarkably similar to the Euclidean model set down more than twenty-three centuries ago. In the light of the tremendous development of mathematics in modern times, it would certainly seem that some revitalizing of secondary level geometry is long overdue. This is true despite the fact that the reforms recommended by the SMSG about ten years ago have been almost universally adopted.

Euclidean Models of Projective Spaces

Euclidean Models of Projective Spaces

Distance-based cluster analysis and measurement scales

The clear recognition of the correct geometrical model is particularly important for problems involving measurements and distances, both on general methodological grounds and as a means of verifying the empirical significance of the mathematical conclusions. The addition of axioms or hypotheses which are not pertinent to the essence of the problem can only lead to results that are, at best, misleading. In many cases, the measurement scales which the social scientist has at his disposal are too weak to support a non-trivial metric structure. However, it is also true that metric notions apply to much more general situations than those for which the Euclidean model is relevant. This generality makes it possible to choose distances that not only satisfy the axioms, but also closely correspond to the notion of proximity which the scientist considers relevant to his problem. Whenever a distance function can be introduced, cluster analysis and other classification techniques will derive at least two advantages from a consistent use of metric notions. One advantage is the possibility of using concepts and results from the rich literature of metric spaces. Secondly, a mathematically more precise formulation of the problem of clustering will facilitate the discovery of suitable algorithms with well-defined properties. In this respect, graph theory and related disciplines such as mathematical programming have much to contribute to the development of numerical taxonomy.

THE COUNTS OF RADIO SOURCES

Following the advice of Professor Zel’dovich, I will begin with two platitudes — first, when we consider the implications of the counts of radio sources, we must look at all the counts of radio sources at all frequencies; second, when we attempt to evaluate the implications of the source counts, we should use proper cosmological models and not rely on the local Euclidean model.

Judgments of similarity and spatial models

Ss judged the similarity between all pairs of stimulus objects under 3 conditions: when the objects were (a) Munsell 5R color patches varying in value and chroma; (b) parallelorams varying in size and tilt; and (c) circles-with-radius varying in diameter and angle of radius. For each set of judgments, the pattern of deviations from the Euclidean model was used to diagnose the most appropriate spatial model. The results confirm previous findings that the Euclidean space is appropriate for judgments of color patches, but that the city block space is appropriate for judgments of geometrie forms which vary on perceptually distinct dimensions.

Classification Space: A Multivariate Procedure For Automatic? Document Indexing And Retrieval.

A conceptual approach to linguistic data processing problems is sketched and empirical illustrations are presented of the major software components- indexing, storage, and retrieval-of a document processing system which offers, in principle, the advantages of complete automation, unlimited cross- indexing, effective sequential retrieval, sub-documentary indexing reflecting heterogeneity of subject matter within a document, and a procedure for automatically identifying retrieval requests which would be inadequately handled by the system. The indexing schema, designated as a "Classification Space" consists of a Euclidean model for mapping subject matter similarity within a given subject matter domain. A schema of this kind is empirically derived for certain fields of Engineering and Chemistry. A set of five related empirical studies provide convincing evidence that when appropriate experimental procedures are followed a very stable C-Space for a given content domain can be constructed on a surprisingly small data base. Other empirical studies demonstrate specific computational procedures for effective automatic indexing of documents in a C-Space, using a relatively small system vocabulary. One study demonstrates that a C-Space maps subject matter relevance as well as subject matter similarity, and thereby pro- motes effective sequential retrieval ; this result is also shown under conditions of automatic indexing. Negative results are found in an attempt to use the structural linguistic distinction of subject and object as a means of improving techniques for automatic indexing.

A Generalized Sorting Strategy for Computer Classifications

AGGLOMERATIVE hierarchical methods of computer classification all begin by calculating distance-measures between elements. The hierarchy is then generated by subjecting these measures to a sorting-strategy, which depends essentially on the definition of a distance-measure between groups of elements. In nearest-neighbour sorting, this is defined as the distance between the closest pair of elements, one in each group. Macnaughton-Smith has pointed out that much more intense clustering can be produced by taking the most remote pair of elements (furthest-neighbour sorting). In group-average sorting1 the distance is defined as the mean of all between-group inter-element distances; in centroid sorting it is the distance between group centroids, defined by a conventional Euclidean model. In median2 sorting the distance of a third group from two which have just fused depends on the previous three inter-group distances in the manner of Apollonius's theorem. Although the earlier of these strategies have received some comparative assessment1,3–5 no attempt seems to have been made to generalize them into a single system. As a result, quite different computer strategies have commonly been used, necessitating a separate computer program for each.

Less May Be More

Physics, like all of science, is not merely analytic. Its power of analysis is best demonstrated in its synthetic successes, the design of new questions, new experiments, even new devices. For the students whom we do not now really reach, including a large fraction of all elementary teachers, a very large proportion of women, indeed, the bulk of the college population, we need a new emphasis. It is valuable to vary the style and mood, to break with the Euclidean model at least sometimes, to employ humor, playfulness, invention; above all, to go beyond mere verbal and formula-learning. It is to be expected that the visual and graphic arts, broadly viewed, might make a merger with laboratory science at an introductory level which would genuinely change the audience we can hope to involve. The free use of simple construction materials, accompanied by some analysis of a mechanical and geometrical kind, might begin a course of rich variety which would present circuits and probability, group theory and astronomy, optics, mechanics, radioactivity, even biochemistry, as lab experience and initial analysis. Aesthetic participation and wide context appear more valuable than the necessarily didactic route to the full modern world view of the physicist. Less may be more. Some topics are more fully sketched.

THE PERCEPTION OF ATTITUDE RELATIONSHIPS: A MULTIDIMENSIONAL SCALING APPROACH TO THE STRUCTURING OF SOCIAL ATTITUDES

ABSTRACTThe primary purpose of this study was to see whether a set of perceived attitude relationships could be adequately represented by a dimensional model. The particular attitude area selected for this purpose was composed of attitudes toward war, capital punishment, and the treatment of criminals. A multidimensional method of successive intervals based upon the Euclidean model of multidimensional psychophysics was developed. If this model satisfactorily accounted for perceived relationships among statements of the three attitudes, it would also be possible to discover the number and the nature of the relevant dimensions involved. A secondary purpose of the experiment was to investigate judgments made by two diverse groups in order to discover whether they perceive attitudes as being structured in different ways.However, since the multidimensional method of successive intervals had not yet been applied even in the area of psychophysics, it was felt that an empirical evaluation of the method in an area of known dimensionality would be appropriate before any conclusions could be legitimately drawn concerning areas of unknown dimensionality. Accordingly, both multidimensional successive intervals and the previously validated method of complete triads were applied to judgments of color similarity obtained from 42 subjects. The stimuli were a set of Munsell colors of the same red hue, but differing in brightness and saturation. Both the multidimensional successive intervals and the complete triads structures, which were essentially identical, correlated highly with the Munsell color structure as an approximation to psychological color space. There were also indications of high reliability for the multidimensional successive intervals technique.In view of its validation in an area of known dimensionality, multidimensional successive intervals was applied to judgments of attitude relationships. Seven statements were selected from each of the three Thurstone scales on attitude toward war, capital punishment, and the treatment of criminals, and these 21 statements were set up in booklet form for group presentation. The attitude booklets were administered to 40 third‐year, male students at the Princeton Theological Seminary and to 82 male Air Force officer candidates at Lackland Air Force Base.The multidimensional attitude structures perceived by the two diverse groups were essentially identical, the relationships among statements of the three attitudes being adequately represented in terms of two oblique dimensions, a war dimension and a punishment dimension. Apparently individuals do perceive attitudes, at least in this particular attitude area, in terms of a definite structure, and when called upon to make judgments concerning attitude relationships, they respond in terms of this dimensional frame of reference.

Multidimensional scaling: I. Theory and method

Multidimensional scaling can be considered as involving three basic steps. In the first step, a scale of comparative distances between all pairs of stimuli is obtained. This scale is analogous to the scale of stimuli obtained in the traditional paired comparisons methods. In this scale, however, instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum. As in paired comparisons, the procedures for obtaining a scale of comparative distances leave the true zero point undetermined. Hence, a comparative distance is not a distance in the usual sense of the term, but is a distance minus an unknown constant. The second step involves estimating this unknown constant. When the unknown constant is obtained, the comparative distances can be converted into absolute distances. In the third step, the dimensionality of the psychological space necessary to account for these absolute distances is determined, and the projections of stimuli on axes of this space are obtained. A set of analytical procedures was developed for each of the three steps given above, including a least-squares solution for obtaining comparative distances by the complete method of triads, two practical methods for estimating the additive constant, and an extension of Young and Householder's Euclidean model to include procedures for obtaining the projections of stimuli on axes from fallible absolute distances.