Congruent Parts Research Articles

Dissecting the square into seven or nine congruent parts

We give a computer-based proof of the following fact: If a square is divided into seven or nine convex polygons, congruent among themselves, then the tiles are rectangles. This confirms a new case of a conjecture posed first by Yuan, Zamfirescu and Zamfirescu and later by Rao, Ren and Wang. Our method allows us to explore other variants of this question, for example, we also prove that no rectangle can be tiled by five or seven congruent non-rectangular polygons.

How fifth-grade pupils reason about fractions: a reliance on part-whole subconstructs

Fractions are an important part of the primary school mathematics curriculum, being introduced to children in the Slovenian educational system during the second grade (at age seven). For clarification of terms it is worth mentioning that the term “fraction” is only introduced in the sixth grade. Before that, the term “equal parts of a given whole” (In Slovene language, there are two different terms which are used in teaching about fractions: one is “equal parts of a given whole” as deli celote and another is “fraction” as ulomek. Fraction in our elementary school has the meaning of all other subconstructs proposed by Kieren (1976) except part-whole subconstruct. This subconstruct is covered by another term, namely “deli celote.” For easier understanding, we use in this paper only the term “fraction.”) is used. When we speak about “fractions” in primary school, we actually speak about parts of a given whole where the whole could be a set of objects or a single object that can be divided into equal parts. We used the fractional scheme theory for the theory framework of our research, focusing mainly on “parts within the whole,” the “part-whole” fraction scheme, and, to some extent, the “partitive unit” fraction scheme. Ninety fifth-grade pupils took a test we developed with the aim of achieving a better understanding of the field of pupils’ understanding of the part-whole aspect of fractions. More precisely, we were interested in how pupils understood the idea of dividing a whole into equal parts: is the idea of dividing mainly connected to dividing into congruent parts, are pupils able to reason about equal parts as parts of equal area, and do they adapt their reasoning according to the demands of the problems. The test consisted of four problems, each of them divided into parts (a total of eight problems to solve altogether). We used descriptive statistics and qualitative research based on the grounded theory method to process the data gathered from how participants solved the problems. We used the inductive process to determine how to categorize the responses and find relationship between the different categories. The findings showed that at the end of the fifth grade, pupils did not yet master the part-whole subconstruct of fractions and had problems related to linking congruency and parts of the whole. The mistakes pupils made clearly showed that a number of types of misunderstandings were prevalent: for example, not being able to connect the idea of fractions to equal parts of the whole or connecting the idea of fractions only to congruent parts of the whole as well as mistakes related to determining what part of the whole is presented.

Dissecting the square into five congruent parts

We give an affirmative answer to an old conjecture proposed by Ludwig Danzer: there is a unique dissection of the square into five congruent convex tiles.

Discontinuous additive functions: Regular behavior vs. pathological features

This paper deals with properties of discontinuous additive functions (a function f is said to be additive if f(x+y)=f(x)+f(y) for all x and y). Section 1 has an introductory character: we consider two particular functions, whose domain is Q[2] and that can be explicitly defined in an easy constructive way. We show that the considered functions are both periodic and quasiperiodic (in the sense of Definition 1); and we also find that the identity function can be obtained as the sum of two periodic functions (see also Mortola and Peirone (1999)). Then, applying the Axiom of Choice, we consider discontinuous additive functions having the whole set R as domain. We establish several properties and characterizations of additive functions: among these, the fact that the graph of an additive function is homogeneous, and that every straight line passing through a point of the graph divides it into two congruent parts. We introduce two metaphors to describe such properties informally. Finally, we examine the consequences of changing the topology on R.

Tre fiumi nello stesso mare: la relazione che cura. Bateson nella psicoterapia whitakeriana ed ericksoniana

- This paper puts into evidence how the works of Gregory Bateson (epistemologis,) Carl Whitaker (experiential-symbolical family therapist) and Milton Erickson (hypnotherapist) can be considered as main streams going in the same direction, thus completing themselves and enriching one another. The paper makes then a comparison of the ideas of the three Masters, utilizing mostly their original scripts, to demonstrate that their ideas con be considered congruent parts of the same story. Under this perspective Erickson and Whitaker in the microcosm of therapy room and in the process of therapeutic relationship are experientially utilizing the ecology of connecting structure typical of batesonian theory. The paper, divided into sections puts into evidence the connecting structure , the joining of patient and therapist; the unconscious;, the resources and resistances during psychotherapy, to finally land to the land of mystery, of sacred, and of change. The paper put into evidence also the similarity the three Masters have about teaching to students. Key words: change, circularity, experiential, hypnosis, mirror neurons, observation, resistance, therapeutic relationship.

Determinants Of The Utilization Of The Tetanus Toxoid (TT) Vaccination Coverage In Bangladesh: Evidence From A Bangladesh Demographic Health Survey 2004

Objectives: This paper employs statistical methods to determine the complete vaccination rate and to identify the factors that influence vaccination of mothers during pregnancy. The principal objective of this paper is to suggest various policy options on the basis of study findings in order to provide guidelines to fulfill the complete vaccination coverage in Bangladesh. Methods: This study analyzes data extracted from Bangladesh Demographic and Health Survey 2004 (BDHS) considering only the case of the mothers of last five years preceding the survey. To meets the objectives this study considers bivariate and multivariate analysis Results: The analysis showed that, although maximum respondents received two or more doses of TT injection a significant portion of women did not receive the TT injection yet (18.6 percent). The immunization coverage of two/more doses was higher among mothers whose husbands are highly educated and non-manual workers. This study elucidates that the rate of immunizing increases with the increase of mother's education. Receiving two/more doses of TT injection shows highest prevalence (58.3 percent) among mothers whose mobility status is unrestricted. The logistic analysis showed that receiving two or more doses of TT injection is almost 3 times higher among respondents using modern toilet facility. The model also shows that higher proportion of respondents belonging to upper category regarding household asset and quality index received two/more doses TT injection than their congruent parts. The other contributing factors for tetanus toxoid vaccination coverage were found to be mother's age at last birth, told about pregnancy complications, place of residences, mother's earning status, sources of drinking water. Conclusion: The results indicate several policy options: (1) improve the monitoring and supervision of vaccination activities especially in the rural area (2) mass media campaign to create awareness among both urban and rural women, their husbands and families about the importance of TT vaccination and the consequences of not being vaccinated (3) education for husband's and wife needs to be given very high priority (4)enable women to exercise their rights to control their concerning freedom of movement, own health acre and access to economic resources

Electron structure and phonon spectra anomalies in oxide HTS

The new results on the HTS phonon spectra with anomalous softening of LO phonon frequency ω LO were analyzed. These data could be understood as a consequence of charge superstructure in the oxygen sublattice in addition to metal sublattice ordering. The charges and spin ordering produce the doubling of lattice periods and new Brillouin zone. Doping leads to the Fermi surface with sides parallel to [100] directions. These results agree with experimental data. The phonon spectra anomalies might be associated with the nesting conditions between the congruent parts of the Fermi surface.

Dissecting orthoschemes into orthoschemes

We exhibit a dissection, with one degree of freedom, of an arbitrary orthoscheme in Euclidean, spherical or hyperbolic d-space into d+1 orthoschemes (Section 2); this can be interpreted as a set of relations in the scissors congruence group or, weaker, as a set of functional equations for the volume. Besides special cases where the dissection is into mutually congruent parts, we obtain, in the spherical case and for a special value of the parameter, scissors congruence formulae similar to Schlafli's period formulae for the spherical orthoscheme volume (see Section 5). In Section 6 we use the dissection to explain the structure of the volume formula for asymptotic hyperbolic 3-orthoschemes due to Lobachevsky. Finally, in Section 7, by exploiting symmetries, we show that two systems of special volume relations of Schlafli (in spherical d-space) and Coxeter (for all three geometries in dimension 3) hold even on the level of dissection. In particular, it seems that all the presently known exact values for the volume of special spherical 3-simplexes hold, independently of Schlafli's differential formula, as consequences of scissors congruence relations.

The Golden Mean and an Intriguing Congruence Problem

Try finding two tnanglts that art not congruent but which have five pairs of congruent parts.

One woman's work is another woman's daughter: A contribution to the sociology of childbirth

This paper considers the criticisms that women, both individually and as members of consumer groups, are making about obstetric care in Australia today, and elaborates a theoretical perspective from which depersonalization and the woman's lack of control over her own delivery can be seen as intrinsic and congruent parts of obstetric care considered as a social process. The interaction of the labour and delivery ward is seen as being negotiated between two definitions of the situation, medical work and personal experience, both of which must be sustained if a satisfactory occasion of childbirth is to take place. Structural conditions are seen to be such that ‘medical work’ is likely to be predominant, and any changes aimed at enhancing the ‘personal experience’ must take these into account.

Partitioning an Interval Into Finitely Many Congruent Parts

A subset of dense additive subgroup of real numbers will be called an interval if it is nonnull and bounded and is convex with respect to the given subgroup. By a partition of an interval we shall mean a proper decomposition of it into finitely many subsets, called parts, which are disjoint and congruent by pairs. We allow congruence by reflection as well as by translation in superposing the parts of a partition. With regard to partitioning two types of interval are to be distinguished: pivotal and nonpivotal. An interval will be called pivotal if it is symmetric but does not contain its midpoint; otherwise it will be called nonpivotal. The purpose of this paper is to prove the following two propositions concerning partitions of a nonpivotal interval: (1) Every part of a partition of a nonpivotal interval is merely a finite union of subintervals. (2) If a nonpivotal interval can be partitioned into n parts, it can also be partitioned into n subintervals. Let us see how these results apply to partitioning an interval in the group of all real numbers. Such an interval is clearly nonpivotal; it may, however, be either symmetric or asymmetric. If the interval is asymmetric, it admits many partitions, but according to (1) the parts of any partition of it consist of finite unions of subintervals hence are measurable. On the other hand, if the interval is symmetric, it evidently cannot be partitioned into subintervals and so by (2) cannot be partitioned at all. We mention in contrast to this that, as von Neumann2 has shown, every nondegenerate real interval, whether symmetric or asymmetric, can be decomposed by translation into denumerably many congruent necessarily nonmeasurable sets.

Non-Euclidean Geometry

MR. FRANKLAND (NATURE, September 7) has raised the old problem of Bertrand's proof of the parallel-axiom by a consideration of infinite areas. This is perhaps the most subtle and the most specious of all the attempted proofs, and this character it owes to the fact that a process of reasoning which is sound for finite magnitudes is extended to a field which is beyond our powers of comprehension—the field of infinity. The fallacy which underlies Bertrand's proof becomes more apparent in Legendre's simpler device (“Elements de Geometrie, ” 12 ed., Note ii.). A straight line divides a plane in which it lies into two congruent parts—;this, of course, has no real meaning, since we are dealing with infinite areas, but such is the argument—;and two rays from a point enclose an (infinite) area which is less than half the whole plane. Hence, if two intersecting lines are both parallel to the same straight line, the area of half the plane can be enclosed within an area which is less than half the plane.

Grundlagen der Geometrie

THIS fascinating work has long since attained the rank of a classic, but attention may be directed to this new edition, which has various additions, mainly bibliographical, and seven supplements, which are reprints of papers by the author on topics related to that of his famous essay. Two of these can be enjoyed by readers with no exceptional mathematical knowledge. In the one on the equality of the base angles of an isosceles triangle, Dr. Hilbert proves, inter alia, the remarkable fact that, even if we assume Euclid's theory of proportion, we cannot prove his propositions on equalities of area, unless we assume the truth of prop. 4, bk. i., of the “Elements” in the wider sense—that is, when one triangle has to be turned over to make it fit the other. It is also pointed out (p. 68) that two tetrahedra can be constructed with equal heights, and bases of equal area, which cannot be cut up into congruent polyhedra, and to which congruent polyhedra cannot be added in such a way that the solids thus produced can be sliced up into congruent parts. Consequently it is impossible to build up a theory of equality of volumes strictly analogous to Euclid's theory of equality of areas. Grundlagen der Geometrie. By D. Hilbert. Third edition. Pp. vi+280. (Leipzig and Berlin: B. G. Teubner, 1909.) Price 6 marks.